analysis.convex.gauge ⟷ Mathlib.Analysis.Convex.Gauge

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -222,7 +222,7 @@ theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
     {x | gauge s x < a} = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s :=
   by
   ext
-  simp_rw [mem_set_of_eq, mem_Union, exists_prop, mem_Ioo, and_assoc']
+  simp_rw [mem_set_of_eq, mem_Union, exists_prop, mem_Ioo, and_assoc]
   exact
     ⟨exists_lt_of_gauge_lt Absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -6,7 +6,7 @@ Authors: Yaël Dillies, Bhavik Mehta
 import Analysis.Convex.Basic
 import Analysis.NormedSpace.Pointwise
 import Analysis.Seminorm
-import Data.IsROrC.Basic
+import Analysis.RCLike.Basic
 import Tactic.Congrm
 
 #align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
@@ -368,18 +368,18 @@ theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower
 
 end LinearOrderedField
 
-section IsROrC
+section RCLike
 
-variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
+variable [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
 #print gauge_norm_smul /-
 theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖r‖ • x) = gauge s (r • x) :=
   by
   unfold gauge
   congr with θ
-  rw [@IsROrC.real_smul_eq_coe_smul 𝕜]
+  rw [@RCLike.real_smul_eq_coe_smul 𝕜]
   refine' and_congr_right fun hθ => (hs.smul _).smul_mem_iff _
-  rw [IsROrC.norm_ofReal, abs_norm]
+  rw [RCLike.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 -/
 
@@ -390,7 +390,7 @@ theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x)
 #align gauge_smul gauge_smul
 -/
 
-end IsROrC
+end RCLike
 
 section TopologicalSpace
 
@@ -467,9 +467,9 @@ theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
 #align gauge_add_le gauge_add_le
 -/
 
-section IsROrC
+section RCLike
 
-variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
+variable [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
 #print gaugeSeminorm /-
 /-- `gauge s` as a seminorm when `s` is  balanced, convex and absorbent. -/
@@ -497,7 +497,7 @@ theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs
 #align gauge_seminorm_ball_one gaugeSeminorm_ball_one
 -/
 
-end IsROrC
+end RCLike
 
 #print Seminorm.gauge_ball /-
 /-- Any seminorm arises as the gauge of its unit ball. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -194,8 +194,8 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
   · have hr' := ha.trans_lt hr
     rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
     obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
-    suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this 
-    rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ 
+    suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
+    rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ
     refine' hs₁.smul_mem_of_zero_mem hs₀ hδ ⟨by positivity, _⟩
     rw [inv_mul_le_iff hr', mul_one]
     exact hδr.le
@@ -271,7 +271,7 @@ theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s :=
 #print le_gauge_of_not_mem /-
 theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
     a ≤ gauge s x := by
-  rw [starConvex_zero_iff] at hs₀ 
+  rw [starConvex_zero_iff] at hs₀
   obtain ⟨r, hr, h⟩ := hs₂
   refine' le_csInf ⟨r, hr, singleton_subset_iff.1 <| h _ (Real.norm_of_nonneg hr.le).ge⟩ _
   rintro b ⟨hb, x, hx', rfl⟩
@@ -308,14 +308,14 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
   constructor
   · rintro ⟨hr, hx⟩
     simp_rw [mem_Ioi] at hr ⊢
-    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx 
+    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
     have := smul_pos (inv_pos.2 ha') hr
     refine' ⟨a⁻¹ • r, ⟨this, _⟩, smul_inv_smul₀ ha'.ne' _⟩
     rwa [← mem_smul_set_iff_inv_smul_mem₀ this.ne', smul_assoc,
       mem_smul_set_iff_inv_smul_mem₀ (inv_ne_zero ha'.ne'), inv_inv]
   · rintro ⟨r, ⟨hr, hx⟩, rfl⟩
     rw [mem_Ioi] at hr ⊢
-    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx 
+    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
     have := smul_pos ha' hr
     refine' ⟨this, _⟩
     rw [← mem_smul_set_iff_inv_smul_mem₀ this.ne', smul_assoc]
@@ -406,7 +406,7 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ {x | gauge s x
   have hs' : IsOpen s' := hf.is_open_preimage _ isOpen_interior
   have one_mem : (1 : ℝ) ∈ s' := by simpa only [s', f, Set.mem_preimage, one_smul]
   obtain ⟨ε, hε₀, hε⟩ := (Metric.nhds_basis_closedBall.1 _).1 (isOpen_iff_mem_nhds.1 hs' 1 one_mem)
-  rw [Real.closedBall_eq_Icc] at hε 
+  rw [Real.closedBall_eq_Icc] at hε
   have hε₁ : 0 < 1 + ε := hε₀.trans (lt_one_add ε)
   have : (1 + ε)⁻¹ < 1 := by
     rw [inv_lt_one_iff]
@@ -432,7 +432,7 @@ theorem gauge_lt_one_eq_self_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) 
 
 theorem gauge_lt_one_of_mem_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s)
     {x : E} (hx : x ∈ s) : gauge s x < 1 := by
-  rwa [← gauge_lt_one_eq_self_of_isOpen hs₁ hs₀ hs₂] at hx 
+  rwa [← gauge_lt_one_eq_self_of_isOpen hs₁ hs₀ hs₂] at hx
 #align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_isOpenₓ
 
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
@@ -441,7 +441,7 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E)
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
   have h_gauge_lt := gauge_lt_one_of_mem_of_isOpen hs₁ hs₀ hs₂ this
   rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one] at
-    h_gauge_lt 
+    h_gauge_lt
   infer_instance
 #align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ
 
@@ -456,14 +456,14 @@ theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
     exists_lt_of_gauge_lt Absorbs (lt_add_of_pos_right (gauge s x) (half_pos hε))
   obtain ⟨b, hb, hb', hy⟩ :=
     exists_lt_of_gauge_lt Absorbs (lt_add_of_pos_right (gauge s y) (half_pos hε))
-  rw [mem_smul_set_iff_inv_smul_mem₀ ha.ne'] at hx 
-  rw [mem_smul_set_iff_inv_smul_mem₀ hb.ne'] at hy 
+  rw [mem_smul_set_iff_inv_smul_mem₀ ha.ne'] at hx
+  rw [mem_smul_set_iff_inv_smul_mem₀ hb.ne'] at hy
   suffices gauge s (x + y) ≤ a + b by linarith
   have hab : 0 < a + b := add_pos ha hb
   apply gauge_le_of_mem hab.le
   have := convex_iff_div.1 hs hx hy ha.le hb.le hab
   rwa [smul_smul, smul_smul, ← mul_div_right_comm, ← mul_div_right_comm, mul_inv_cancel ha.ne',
-    mul_inv_cancel hb.ne', ← smul_add, one_div, ← mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this 
+    mul_inv_cancel hb.ne', ← smul_add, one_div, ← mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this
 #align gauge_add_le gauge_add_le
 -/
 
@@ -516,7 +516,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     exact lt_mul_of_one_lt_left hpx one_lt_two
   refine' IsGLB.csInf_eq ⟨fun r => _, fun r hr => le_of_forall_pos_le_add fun ε hε => _⟩ hp
   · rintro ⟨hr, y, hy, rfl⟩
-    rw [p.mem_ball_zero] at hy 
+    rw [p.mem_ball_zero] at hy
     rw [map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos hr]
     exact mul_le_of_le_one_right hr.le hy.le
   · have hpε : 0 < p x + ε := by positivity

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -378,7 +378,7 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖
   unfold gauge
   congr with θ
   rw [@IsROrC.real_smul_eq_coe_smul 𝕜]
-  refine' and_congr_right fun hθ => (hs.smul _).mem_smul_iff _
+  refine' and_congr_right fun hθ => (hs.smul _).smul_mem_iff _
   rw [IsROrC.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -531,7 +531,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
 theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
     gaugeSeminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) =
       p :=
-  FunLike.coe_injective p.gauge_ball
+  DFunLike.coe_injective p.gauge_ball
 #align seminorm.gauge_seminorm_ball Seminorm.gaugeSeminorm_ball
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -420,25 +420,26 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ {x | gauge s x
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
 -/
 
-#print gauge_lt_one_eq_self_of_open /-
-theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
+#print gauge_lt_one_eq_self_of_isOpen /-
+theorem gauge_lt_one_eq_self_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
     {x | gauge s x < 1} = s :=
   by
   refine' (gauge_lt_one_subset_self hs₁ ‹_› <| absorbent_nhds_zero <| hs₂.mem_nhds hs₀).antisymm _
   convert interior_subset_gauge_lt_one s
   exact hs₂.interior_eq.symm
-#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
+#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_isOpen
 -/
 
-theorem gauge_lt_one_of_mem_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) {x : E}
-    (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx 
-#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_openₓ
+theorem gauge_lt_one_of_mem_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s)
+    {x : E} (hx : x ∈ s) : gauge s x < 1 := by
+  rwa [← gauge_lt_one_eq_self_of_isOpen hs₁ hs₀ hs₂] at hx 
+#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_isOpenₓ
 
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
     (hs₂ : IsOpen s) (hx : x ∈ ε • s) : gauge s x < ε :=
   by
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
-  have h_gauge_lt := gauge_lt_one_of_mem_of_open hs₁ hs₀ hs₂ this
+  have h_gauge_lt := gauge_lt_one_of_mem_of_isOpen hs₁ hs₀ hs₂ this
   rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one] at
     h_gauge_lt 
   infer_instance
@@ -481,18 +482,18 @@ def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Abso
 variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E]
   [ContinuousSMul ℝ E]
 
-#print gaugeSeminorm_lt_one_of_open /-
-theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
+#print gaugeSeminorm_lt_one_of_isOpen /-
+theorem gaugeSeminorm_lt_one_of_isOpen (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
-  gauge_lt_one_of_mem_of_open hs₁ hs₂.zero_mem hs hx
-#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
+  gauge_lt_one_of_mem_of_isOpen hs₁ hs₂.zero_mem hs hx
+#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_isOpen
 -/
 
 #print gaugeSeminorm_ball_one /-
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s :=
   by
   rw [Seminorm.ball_zero_eq]
-  exact gauge_lt_one_eq_self_of_open hs₁ hs₂.zero_mem hs
+  exact gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs
 #align gauge_seminorm_ball_one gaugeSeminorm_ball_one
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,11 +3,11 @@ Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 -/
-import Mathbin.Analysis.Convex.Basic
-import Mathbin.Analysis.NormedSpace.Pointwise
-import Mathbin.Analysis.Seminorm
-import Mathbin.Data.IsROrC.Basic
-import Mathbin.Tactic.Congrm
+import Analysis.Convex.Basic
+import Analysis.NormedSpace.Pointwise
+import Analysis.Seminorm
+import Data.IsROrC.Basic
+import Tactic.Congrm
 
 #align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/c0c52abb75074ed8b73a948341f50521fbf43b4c

@@ -546,14 +546,14 @@ theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ := b
 #align gauge_unit_ball gauge_unit_ball
 -/
 
-#print gauge_ball /-
-theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r :=
+#print gauge_ball' /-
+theorem gauge_ball' (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r :=
   by
   rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
     abs_of_nonneg hr.le, div_eq_inv_mul]
   simp_rw [mem_ball_zero_iff, norm_neg]
   exact fun _ => id
-#align gauge_ball gauge_ball
+#align gauge_ball gauge_ball'
 -/
 
 #print mul_gauge_le_norm /-
@@ -561,7 +561,7 @@ theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤
   by
   obtain hr | hr := le_or_lt r 0
   · exact (mul_nonpos_of_nonpos_of_nonneg hr <| gauge_nonneg _).trans (norm_nonneg _)
-  rw [mul_comm, ← le_div_iff hr, ← gauge_ball hr]
+  rw [mul_comm, ← le_div_iff hr, ← gauge_ball' hr]
   exact gauge_mono (absorbent_ball_zero hr) hs x
 #align mul_gauge_le_norm mul_gauge_le_norm
 -/
@@ -575,7 +575,7 @@ theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r
       gauge s x = gauge s (y + (x - y)) := by simp
       _ ≤ gauge s y + gauge s (x - y) := (gauge_add_le hc (this.Subset hs) _ _)
       _ ≤ gauge s y + ‖x - y‖ / r :=
-        (add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr _)) _)
+        (add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball' hr _)) _)
       _ = gauge s y + r⁻¹ * dist x y := by rw [dist_eq_norm, div_eq_inv_mul]
 #align convex.lipschitz_with_gauge Convex.lipschitzWith_gauge
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
-
-! This file was ported from Lean 3 source module analysis.convex.gauge
-! leanprover-community/mathlib commit 373b03b5b9d0486534edbe94747f23cb3712f93d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Convex.Basic
 import Mathbin.Analysis.NormedSpace.Pointwise
@@ -14,6 +9,8 @@ import Mathbin.Analysis.Seminorm
 import Mathbin.Data.IsROrC.Basic
 import Mathbin.Tactic.Congrm
 
+#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
+
 /-!
 # The Minkowksi functional
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9240e8be927a0955b9a82c6c85ef499ee3a626b8

@@ -354,7 +354,7 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
 #print gauge_smul_left /-
 theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ]
     [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
-    gauge (a • s) = (|a|)⁻¹ • gauge s :=
+    gauge (a • s) = |a|⁻¹ • gauge s :=
   by
   rw [← gauge_smul_left_of_nonneg (abs_nonneg a)]
   obtain h | h := abs_choice a

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -433,13 +433,10 @@ theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈
 #align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
 -/
 
-#print gauge_lt_one_of_mem_of_open /-
 theorem gauge_lt_one_of_mem_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) {x : E}
     (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx 
-#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_open
--/
+#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_openₓ
 
-#print gauge_lt_of_mem_smul /-
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
     (hs₂ : IsOpen s) (hx : x ∈ ε • s) : gauge s x < ε :=
   by
@@ -448,8 +445,7 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E)
   rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one] at
     h_gauge_lt 
   infer_instance
-#align gauge_lt_of_mem_smul gauge_lt_of_mem_smul
--/
+#align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ
 
 end TopologicalSpace
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -67,11 +67,14 @@ def gauge (s : Set E) (x : E) : ℝ :=
 
 variable {s t : Set E} {a : ℝ} {x : E}
 
+#print gauge_def /-
 theorem gauge_def : gauge s x = sInf ({r ∈ Set.Ioi 0 | x ∈ r • s}) :=
   rfl
 #align gauge_def gauge_def
+-/
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]] -/
+#print gauge_def' /-
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
 theorem gauge_def' : gauge s x = sInf ({r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s}) :=
@@ -80,10 +83,12 @@ theorem gauge_def' : gauge s x = sInf ({r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s}) :
     "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]]"
   exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
 #align gauge_def' gauge_def'
+-/
 
 private theorem gauge_set_bdd_below : BddBelow {r : ℝ | 0 < r ∧ x ∈ r • s} :=
   ⟨0, fun r hr => hr.1.le⟩
 
+#print Absorbent.gauge_set_nonempty /-
 /-- If the given subset is `absorbent` then the set we take an infimum over in `gauge` is nonempty,
 which is useful for proving many properties about the gauge.  -/
 theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
@@ -91,18 +96,24 @@ theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
   let ⟨r, hr₁, hr₂⟩ := Absorbs x
   ⟨r, hr₁, hr₂ r (Real.norm_of_nonneg hr₁.le).ge⟩
 #align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonempty
+-/
 
+#print gauge_mono /-
 theorem gauge_mono (hs : Absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s := fun x =>
   csInf_le_csInf gauge_set_bddBelow hs.gauge_set_nonempty fun r hr => ⟨hr.1, smul_set_mono h hr.2⟩
 #align gauge_mono gauge_mono
+-/
 
+#print exists_lt_of_gauge_lt /-
 theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
     ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s :=
   by
   obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h
   exact ⟨b, hb, hba, hx⟩
 #align exists_lt_of_gauge_lt exists_lt_of_gauge_lt
+-/
 
+#print gauge_zero /-
 /-- The gauge evaluated at `0` is always zero (mathematically this requires `0` to be in the set `s`
 but, the real infimum of the empty set in Lean being defined as `0`, it holds unconditionally). -/
 @[simp]
@@ -112,7 +123,9 @@ theorem gauge_zero : gauge s 0 = 0 := by
   · simp only [smul_zero, sep_true, h, csInf_Ioi]
   · simp only [smul_zero, sep_false, h, Real.sInf_empty]
 #align gauge_zero gauge_zero
+-/
 
+#print gauge_zero' /-
 @[simp]
 theorem gauge_zero' : gauge (0 : Set E) = 0 := by
   ext
@@ -123,42 +136,58 @@ theorem gauge_zero' : gauge (0 : Set E) = 0 := by
     convert Real.sInf_empty
     exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr.1) hx
 #align gauge_zero' gauge_zero'
+-/
 
+#print gauge_empty /-
 @[simp]
 theorem gauge_empty : gauge (∅ : Set E) = 0 := by ext;
   simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
 #align gauge_empty gauge_empty
+-/
 
+#print gauge_of_subset_zero /-
 theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by
   obtain rfl | rfl := subset_singleton_iff_eq.1 h; exacts [gauge_empty, gauge_zero']
 #align gauge_of_subset_zero gauge_of_subset_zero
+-/
 
+#print gauge_nonneg /-
 /-- The gauge is always nonnegative. -/
 theorem gauge_nonneg (x : E) : 0 ≤ gauge s x :=
   Real.sInf_nonneg _ fun x hx => hx.1.le
 #align gauge_nonneg gauge_nonneg
+-/
 
+#print gauge_neg /-
 theorem gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x :=
   by
   have : ∀ x, -x ∈ s ↔ x ∈ s := fun x => ⟨fun h => by simpa using Symmetric _ h, Symmetric x⟩
   simp_rw [gauge_def', smul_neg, this]
 #align gauge_neg gauge_neg
+-/
 
+#print gauge_neg_set_neg /-
 theorem gauge_neg_set_neg (x : E) : gauge (-s) (-x) = gauge s x := by
   simp_rw [gauge_def', smul_neg, neg_mem_neg]
 #align gauge_neg_set_neg gauge_neg_set_neg
+-/
 
+#print gauge_neg_set_eq_gauge_neg /-
 theorem gauge_neg_set_eq_gauge_neg (x : E) : gauge (-s) x = gauge s (-x) := by
   rw [← gauge_neg_set_neg, neg_neg]
 #align gauge_neg_set_eq_gauge_neg gauge_neg_set_eq_gauge_neg
+-/
 
+#print gauge_le_of_mem /-
 theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
   by
   obtain rfl | ha' := ha.eq_or_lt
   · rw [mem_singleton_iff.1 (zero_smul_set_subset _ hx), gauge_zero]
   · exact csInf_le gauge_set_bdd_below ⟨ha', hx⟩
 #align gauge_le_of_mem gauge_le_of_mem
+-/
 
+#print gauge_le_eq /-
 theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
     {x | gauge s x ≤ a} = ⋂ (r : ℝ) (H : a < r), r • s :=
   by
@@ -177,7 +206,9 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
     exact
       (gauge_le_of_mem (ha.trans hε'.le) <| h _ hε').trans_lt (add_lt_add_left (half_lt_self hε) _)
 #align gauge_le_eq gauge_le_eq
+-/
 
+#print gauge_lt_eq' /-
 theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
     {x | gauge s x < a} = ⋃ (r : ℝ) (H : 0 < r) (H : r < a), r • s :=
   by
@@ -187,7 +218,9 @@ theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
     ⟨exists_lt_of_gauge_lt Absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
 #align gauge_lt_eq' gauge_lt_eq'
+-/
 
+#print gauge_lt_eq /-
 theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
     {x | gauge s x < a} = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s :=
   by
@@ -197,7 +230,9 @@ theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
     ⟨exists_lt_of_gauge_lt Absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
 #align gauge_lt_eq gauge_lt_eq
+-/
 
+#print gauge_lt_one_subset_self /-
 theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
     {x | gauge s x < 1} ⊆ s := by
   rw [gauge_lt_eq Absorbs]
@@ -205,14 +240,20 @@ theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (abs
   rintro ⟨y, hy, rfl⟩
   exact hs.smul_mem_of_zero_mem h₀ hy (Ioo_subset_Icc_self hr)
 #align gauge_lt_one_subset_self gauge_lt_one_subset_self
+-/
 
+#print gauge_le_one_of_mem /-
 theorem gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 :=
   gauge_le_of_mem zero_le_one <| by rwa [one_smul]
 #align gauge_le_one_of_mem gauge_le_one_of_mem
+-/
 
+#print self_subset_gauge_le_one /-
 theorem self_subset_gauge_le_one : s ⊆ {x | gauge s x ≤ 1} := fun x => gauge_le_one_of_mem
 #align self_subset_gauge_le_one self_subset_gauge_le_one
+-/
 
+#print Convex.gauge_le /-
 theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) (a : ℝ) :
     Convex ℝ {x | gauge s x ≤ a} := by
   by_cases ha : 0 ≤ a
@@ -221,12 +262,16 @@ theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Ab
   · convert convex_empty
     exact eq_empty_iff_forall_not_mem.2 fun x hx => ha <| (gauge_nonneg _).trans hx
 #align convex.gauge_le Convex.gauge_le
+-/
 
+#print Balanced.starConvex /-
 theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s :=
   starConvex_zero_iff.2 fun x hx a ha₀ ha₁ =>
     hs _ (by rwa [Real.norm_of_nonneg ha₀]) (smul_mem_smul_set hx)
 #align balanced.star_convex Balanced.starConvex
+-/
 
+#print le_gauge_of_not_mem /-
 theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
     a ≤ gauge s x := by
   rw [starConvex_zero_iff] at hs₀ 
@@ -240,16 +285,20 @@ theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s
     exact div_le_one_of_le hba.le ha.le
   · rw [← mul_smul, mul_inv_cancel_left₀ ha.ne']
 #align le_gauge_of_not_mem le_gauge_of_not_mem
+-/
 
+#print one_le_gauge_of_not_mem /-
 theorem one_le_gauge_of_not_mem (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ s) :
     1 ≤ gauge s x :=
   le_gauge_of_not_mem hs₁ hs₂ <| by rwa [one_smul]
 #align one_le_gauge_of_not_mem one_le_gauge_of_not_mem
+-/
 
 section LinearOrderedField
 
 variable {α : Type _} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ]
 
+#print gauge_smul_of_nonneg /-
 theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
     (ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x :=
   by
@@ -275,7 +324,9 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
     rw [← mem_smul_set_iff_inv_smul_mem₀ this.ne', smul_assoc]
     exact smul_mem_smul_set hx
 #align gauge_smul_of_nonneg gauge_smul_of_nonneg
+-/
 
+#print gauge_smul_left_of_nonneg /-
 theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ ℝ]
     [IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a) :
     gauge (a • s) = a⁻¹ • gauge s :=
@@ -298,7 +349,9 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
     refine' ⟨smul_pos (inv_pos.2 ha') hr, r⁻¹ • x, hx, _⟩
     rw [smul_inv₀, smul_assoc, inv_inv]
 #align gauge_smul_left_of_nonneg gauge_smul_left_of_nonneg
+-/
 
+#print gauge_smul_left /-
 theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ]
     [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
     gauge (a • s) = (|a|)⁻¹ • gauge s :=
@@ -314,6 +367,7 @@ theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower
     exact Symmetric _ hy
   · infer_instance
 #align gauge_smul_left gauge_smul_left
+-/
 
 end LinearOrderedField
 
@@ -321,6 +375,7 @@ section IsROrC
 
 variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
+#print gauge_norm_smul /-
 theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖r‖ • x) = gauge s (r • x) :=
   by
   unfold gauge
@@ -329,11 +384,14 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖
   refine' and_congr_right fun hθ => (hs.smul _).mem_smul_iff _
   rw [IsROrC.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
+-/
 
+#print gauge_smul /-
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/
 theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x := by
   rw [← smul_eq_mul, ← gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs]; infer_instance
 #align gauge_smul gauge_smul
+-/
 
 end IsROrC
 
@@ -341,6 +399,7 @@ section TopologicalSpace
 
 variable [TopologicalSpace E] [ContinuousSMul ℝ E]
 
+#print interior_subset_gauge_lt_one /-
 theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ {x | gauge s x < 1} :=
   by
   intro x hx
@@ -362,7 +421,9 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ {x | gauge s x
     interior_subset
       (hε ⟨(sub_le_self _ hε₀.le).trans ((le_add_iff_nonneg_right _).2 hε₀.le), le_rfl⟩)
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
+-/
 
+#print gauge_lt_one_eq_self_of_open /-
 theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
     {x | gauge s x < 1} = s :=
   by
@@ -370,11 +431,15 @@ theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈
   convert interior_subset_gauge_lt_one s
   exact hs₂.interior_eq.symm
 #align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
+-/
 
+#print gauge_lt_one_of_mem_of_open /-
 theorem gauge_lt_one_of_mem_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) {x : E}
     (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx 
 #align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_open
+-/
 
+#print gauge_lt_of_mem_smul /-
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
     (hs₂ : IsOpen s) (hx : x ∈ ε • s) : gauge s x < ε :=
   by
@@ -384,9 +449,11 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E)
     h_gauge_lt 
   infer_instance
 #align gauge_lt_of_mem_smul gauge_lt_of_mem_smul
+-/
 
 end TopologicalSpace
 
+#print gauge_add_le /-
 theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
     gauge s (x + y) ≤ gauge s x + gauge s y :=
   by
@@ -404,6 +471,7 @@ theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
   rwa [smul_smul, smul_smul, ← mul_div_right_comm, ← mul_div_right_comm, mul_inv_cancel ha.ne',
     mul_inv_cancel hb.ne', ← smul_add, one_div, ← mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this 
 #align gauge_add_le gauge_add_le
+-/
 
 section IsROrC
 
@@ -420,19 +488,24 @@ def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Abso
 variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E]
   [ContinuousSMul ℝ E]
 
+#print gaugeSeminorm_lt_one_of_open /-
 theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
   gauge_lt_one_of_mem_of_open hs₁ hs₂.zero_mem hs hx
 #align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
+-/
 
+#print gaugeSeminorm_ball_one /-
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s :=
   by
   rw [Seminorm.ball_zero_eq]
   exact gauge_lt_one_eq_self_of_open hs₁ hs₂.zero_mem hs
 #align gauge_seminorm_ball_one gaugeSeminorm_ball_one
+-/
 
 end IsROrC
 
+#print Seminorm.gauge_ball /-
 /-- Any seminorm arises as the gauge of its unit ball. -/
 @[simp]
 protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1) = p :=
@@ -458,12 +531,15 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
       inv_mul_lt_iff hpε, mul_one]
     exact lt_add_of_pos_right _ hε
 #align seminorm.gauge_ball Seminorm.gauge_ball
+-/
 
+#print Seminorm.gaugeSeminorm_ball /-
 theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
     gaugeSeminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) =
       p :=
   FunLike.coe_injective p.gauge_ball
 #align seminorm.gauge_seminorm_ball Seminorm.gaugeSeminorm_ball
+-/
 
 end AddCommGroup
 
@@ -471,10 +547,13 @@ section Norm
 
 variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E}
 
+#print gauge_unit_ball /-
 theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ := by
   rw [← ball_normSeminorm ℝ, Seminorm.gauge_ball, coe_normSeminorm]
 #align gauge_unit_ball gauge_unit_ball
+-/
 
+#print gauge_ball /-
 theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r :=
   by
   rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
@@ -482,7 +561,9 @@ theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x
   simp_rw [mem_ball_zero_iff, norm_neg]
   exact fun _ => id
 #align gauge_ball gauge_ball
+-/
 
+#print mul_gauge_le_norm /-
 theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤ ‖x‖ :=
   by
   obtain hr | hr := le_or_lt r 0
@@ -490,7 +571,9 @@ theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤
   rw [mul_comm, ← le_div_iff hr, ← gauge_ball hr]
   exact gauge_mono (absorbent_ball_zero hr) hs x
 #align mul_gauge_le_norm mul_gauge_le_norm
+-/
 
+#print Convex.lipschitzWith_gauge /-
 theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r)
     (hs : Metric.ball (0 : E) r ⊆ s) : LipschitzWith r⁻¹ (gauge s) :=
   have : Absorbent ℝ (Metric.ball (0 : E) r) := absorbent_ball_zero hr
@@ -502,7 +585,9 @@ theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r
         (add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr _)) _)
       _ = gauge s y + r⁻¹ * dist x y := by rw [dist_eq_norm, div_eq_inv_mul]
 #align convex.lipschitz_with_gauge Convex.lipschitzWith_gauge
+-/
 
+#print Convex.uniformContinuous_gauge /-
 theorem Convex.uniformContinuous_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
     UniformContinuous (gauge s) :=
   by
@@ -510,6 +595,7 @@ theorem Convex.uniformContinuous_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0
   lift r to ℝ≥0 using le_of_lt hr₀
   exact (hc.lipschitz_with_gauge hr₀ hr).UniformContinuous
 #align convex.uniform_continuous_gauge Convex.uniformContinuous_gauge
+-/
 
 end Norm
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -501,7 +501,6 @@ theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r
       _ ≤ gauge s y + ‖x - y‖ / r :=
         (add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr _)) _)
       _ = gauge s y + r⁻¹ * dist x y := by rw [dist_eq_norm, div_eq_inv_mul]
-      
 #align convex.lipschitz_with_gauge Convex.lipschitzWith_gauge
 
 theorem Convex.uniformContinuous_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -61,33 +61,33 @@ variable [AddCommGroup E] [Module ℝ E]
 /-- The Minkowski functional. Given a set `s` in a real vector space, `gauge s` is the functional
 which sends `x : E` to the smallest `r : ℝ` such that `x` is in `s` scaled by `r`. -/
 def gauge (s : Set E) (x : E) : ℝ :=
-  sInf { r : ℝ | 0 < r ∧ x ∈ r • s }
+  sInf {r : ℝ | 0 < r ∧ x ∈ r • s}
 #align gauge gauge
 -/
 
 variable {s t : Set E} {a : ℝ} {x : E}
 
-theorem gauge_def : gauge s x = sInf ({ r ∈ Set.Ioi 0 | x ∈ r • s }) :=
+theorem gauge_def : gauge s x = sInf ({r ∈ Set.Ioi 0 | x ∈ r • s}) :=
   rfl
 #align gauge_def gauge_def
 
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]] -/
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
-theorem gauge_def' : gauge s x = sInf ({ r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s }) :=
+theorem gauge_def' : gauge s x = sInf ({r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s}) :=
   by
   trace
     "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]]"
   exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
 #align gauge_def' gauge_def'
 
-private theorem gauge_set_bdd_below : BddBelow { r : ℝ | 0 < r ∧ x ∈ r • s } :=
+private theorem gauge_set_bdd_below : BddBelow {r : ℝ | 0 < r ∧ x ∈ r • s} :=
   ⟨0, fun r hr => hr.1.le⟩
 
 /-- If the given subset is `absorbent` then the set we take an infimum over in `gauge` is nonempty,
 which is useful for proving many properties about the gauge.  -/
 theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
-    { r : ℝ | 0 < r ∧ x ∈ r • s }.Nonempty :=
+    {r : ℝ | 0 < r ∧ x ∈ r • s}.Nonempty :=
   let ⟨r, hr₁, hr₂⟩ := Absorbs x
   ⟨r, hr₁, hr₂ r (Real.norm_of_nonneg hr₁.le).ge⟩
 #align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonempty
@@ -160,7 +160,7 @@ theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
 #align gauge_le_of_mem gauge_le_of_mem
 
 theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
-    { x | gauge s x ≤ a } = ⋂ (r : ℝ) (H : a < r), r • s :=
+    {x | gauge s x ≤ a} = ⋂ (r : ℝ) (H : a < r), r • s :=
   by
   ext
   simp_rw [Set.mem_iInter, Set.mem_setOf_eq]
@@ -179,7 +179,7 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
 #align gauge_le_eq gauge_le_eq
 
 theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
-    { x | gauge s x < a } = ⋃ (r : ℝ) (H : 0 < r) (H : r < a), r • s :=
+    {x | gauge s x < a} = ⋃ (r : ℝ) (H : 0 < r) (H : r < a), r • s :=
   by
   ext
   simp_rw [mem_set_of_eq, mem_Union, exists_prop]
@@ -189,7 +189,7 @@ theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
 #align gauge_lt_eq' gauge_lt_eq'
 
 theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
-    { x | gauge s x < a } = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s :=
+    {x | gauge s x < a} = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s :=
   by
   ext
   simp_rw [mem_set_of_eq, mem_Union, exists_prop, mem_Ioo, and_assoc']
@@ -199,7 +199,7 @@ theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
 #align gauge_lt_eq gauge_lt_eq
 
 theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
-    { x | gauge s x < 1 } ⊆ s := by
+    {x | gauge s x < 1} ⊆ s := by
   rw [gauge_lt_eq Absorbs]
   refine' Set.iUnion₂_subset fun r hr _ => _
   rintro ⟨y, hy, rfl⟩
@@ -210,11 +210,11 @@ theorem gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 :=
   gauge_le_of_mem zero_le_one <| by rwa [one_smul]
 #align gauge_le_one_of_mem gauge_le_one_of_mem
 
-theorem self_subset_gauge_le_one : s ⊆ { x | gauge s x ≤ 1 } := fun x => gauge_le_one_of_mem
+theorem self_subset_gauge_le_one : s ⊆ {x | gauge s x ≤ 1} := fun x => gauge_le_one_of_mem
 #align self_subset_gauge_le_one self_subset_gauge_le_one
 
 theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) (a : ℝ) :
-    Convex ℝ { x | gauge s x ≤ a } := by
+    Convex ℝ {x | gauge s x ≤ a} := by
   by_cases ha : 0 ≤ a
   · rw [gauge_le_eq hs h₀ Absorbs ha]
     exact convex_iInter fun i => convex_iInter fun hi => hs.smul _
@@ -341,7 +341,7 @@ section TopologicalSpace
 
 variable [TopologicalSpace E] [ContinuousSMul ℝ E]
 
-theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s x < 1 } :=
+theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ {x | gauge s x < 1} :=
   by
   intro x hx
   let f : ℝ → E := fun t => t • x
@@ -364,7 +364,7 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
 
 theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
-    { x | gauge s x < 1 } = s :=
+    {x | gauge s x < 1} = s :=
   by
   refine' (gauge_lt_one_subset_self hs₁ ‹_› <| absorbent_nhds_zero <| hs₂.mem_nhds hs₀).antisymm _
   convert interior_subset_gauge_lt_one s
@@ -438,7 +438,7 @@ end IsROrC
 protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1) = p :=
   by
   ext
-  obtain hp | hp := { r : ℝ | 0 < r ∧ x ∈ r • p.ball 0 1 }.eq_empty_or_nonempty
+  obtain hp | hp := {r : ℝ | 0 < r ∧ x ∈ r • p.ball 0 1}.eq_empty_or_nonempty
   · rw [gauge, hp, Real.sInf_empty]
     by_contra
     have hpx : 0 < p x := (map_nonneg _ _).lt_of_ne h

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -130,7 +130,7 @@ theorem gauge_empty : gauge (∅ : Set E) = 0 := by ext;
 #align gauge_empty gauge_empty
 
 theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by
-  obtain rfl | rfl := subset_singleton_iff_eq.1 h; exacts[gauge_empty, gauge_zero']
+  obtain rfl | rfl := subset_singleton_iff_eq.1 h; exacts [gauge_empty, gauge_zero']
 #align gauge_of_subset_zero gauge_of_subset_zero
 
 /-- The gauge is always nonnegative. -/
@@ -168,8 +168,8 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
   · have hr' := ha.trans_lt hr
     rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
     obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
-    suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
-    rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ
+    suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this 
+    rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ 
     refine' hs₁.smul_mem_of_zero_mem hs₀ hδ ⟨by positivity, _⟩
     rw [inv_mul_le_iff hr', mul_one]
     exact hδr.le
@@ -229,7 +229,7 @@ theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s :=
 
 theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
     a ≤ gauge s x := by
-  rw [starConvex_zero_iff] at hs₀
+  rw [starConvex_zero_iff] at hs₀ 
   obtain ⟨r, hr, h⟩ := hs₂
   refine' le_csInf ⟨r, hr, singleton_subset_iff.1 <| h _ (Real.norm_of_nonneg hr.le).ge⟩ _
   rintro b ⟨hb, x, hx', rfl⟩
@@ -261,15 +261,15 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
   constructor
   · rintro ⟨hr, hx⟩
-    simp_rw [mem_Ioi] at hr⊢
-    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
+    simp_rw [mem_Ioi] at hr ⊢
+    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx 
     have := smul_pos (inv_pos.2 ha') hr
     refine' ⟨a⁻¹ • r, ⟨this, _⟩, smul_inv_smul₀ ha'.ne' _⟩
     rwa [← mem_smul_set_iff_inv_smul_mem₀ this.ne', smul_assoc,
       mem_smul_set_iff_inv_smul_mem₀ (inv_ne_zero ha'.ne'), inv_inv]
   · rintro ⟨r, ⟨hr, hx⟩, rfl⟩
-    rw [mem_Ioi] at hr⊢
-    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
+    rw [mem_Ioi] at hr ⊢
+    rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx 
     have := smul_pos ha' hr
     refine' ⟨this, _⟩
     rw [← mem_smul_set_iff_inv_smul_mem₀ this.ne', smul_assoc]
@@ -289,11 +289,11 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
   constructor
   · rintro ⟨hr, y, hy, h⟩
-    simp_rw [mem_Ioi] at hr⊢
+    simp_rw [mem_Ioi] at hr ⊢
     refine' ⟨a • r, ⟨smul_pos ha' hr, _⟩, inv_smul_smul₀ ha'.ne' _⟩
     rwa [smul_inv₀, smul_assoc, ← h, inv_smul_smul₀ ha'.ne']
   · rintro ⟨r, ⟨hr, hx⟩, rfl⟩
-    rw [mem_Ioi] at hr⊢
+    rw [mem_Ioi] at hr ⊢
     have := smul_pos ha' hr
     refine' ⟨smul_pos (inv_pos.2 ha') hr, r⁻¹ • x, hx, _⟩
     rw [smul_inv₀, smul_assoc, inv_inv]
@@ -350,7 +350,7 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s
   have hs' : IsOpen s' := hf.is_open_preimage _ isOpen_interior
   have one_mem : (1 : ℝ) ∈ s' := by simpa only [s', f, Set.mem_preimage, one_smul]
   obtain ⟨ε, hε₀, hε⟩ := (Metric.nhds_basis_closedBall.1 _).1 (isOpen_iff_mem_nhds.1 hs' 1 one_mem)
-  rw [Real.closedBall_eq_Icc] at hε
+  rw [Real.closedBall_eq_Icc] at hε 
   have hε₁ : 0 < 1 + ε := hε₀.trans (lt_one_add ε)
   have : (1 + ε)⁻¹ < 1 := by
     rw [inv_lt_one_iff]
@@ -372,7 +372,7 @@ theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈
 #align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
 
 theorem gauge_lt_one_of_mem_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) {x : E}
-    (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx
+    (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx 
 #align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_open
 
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
@@ -381,7 +381,7 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E)
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
   have h_gauge_lt := gauge_lt_one_of_mem_of_open hs₁ hs₀ hs₂ this
   rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one] at
-    h_gauge_lt
+    h_gauge_lt 
   infer_instance
 #align gauge_lt_of_mem_smul gauge_lt_of_mem_smul
 
@@ -395,14 +395,14 @@ theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
     exists_lt_of_gauge_lt Absorbs (lt_add_of_pos_right (gauge s x) (half_pos hε))
   obtain ⟨b, hb, hb', hy⟩ :=
     exists_lt_of_gauge_lt Absorbs (lt_add_of_pos_right (gauge s y) (half_pos hε))
-  rw [mem_smul_set_iff_inv_smul_mem₀ ha.ne'] at hx
-  rw [mem_smul_set_iff_inv_smul_mem₀ hb.ne'] at hy
+  rw [mem_smul_set_iff_inv_smul_mem₀ ha.ne'] at hx 
+  rw [mem_smul_set_iff_inv_smul_mem₀ hb.ne'] at hy 
   suffices gauge s (x + y) ≤ a + b by linarith
   have hab : 0 < a + b := add_pos ha hb
   apply gauge_le_of_mem hab.le
   have := convex_iff_div.1 hs hx hy ha.le hb.le hab
   rwa [smul_smul, smul_smul, ← mul_div_right_comm, ← mul_div_right_comm, mul_inv_cancel ha.ne',
-    mul_inv_cancel hb.ne', ← smul_add, one_div, ← mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this
+    mul_inv_cancel hb.ne', ← smul_add, one_div, ← mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this 
 #align gauge_add_le gauge_add_le
 
 section IsROrC
@@ -449,7 +449,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     exact lt_mul_of_one_lt_left hpx one_lt_two
   refine' IsGLB.csInf_eq ⟨fun r => _, fun r hr => le_of_forall_pos_le_add fun ε hε => _⟩ hp
   · rintro ⟨hr, y, hy, rfl⟩
-    rw [p.mem_ball_zero] at hy
+    rw [p.mem_ball_zero] at hy 
     rw [map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos hr]
     exact mul_le_of_le_one_right hr.le hy.le
   · have hpε : 0 < p x + ε := by positivity

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -47,7 +47,7 @@ Minkowski functional, gauge
 
 open NormedField Set
 
-open Pointwise Topology NNReal
+open scoped Pointwise Topology NNReal
 
 noncomputable section
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -67,22 +67,10 @@ def gauge (s : Set E) (x : E) : ℝ :=
 
 variable {s t : Set E} {a : ℝ} {x : E}
 
-/- warning: gauge_def -> gauge_def is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align gauge_def gauge_defₓ'. -/
 theorem gauge_def : gauge s x = sInf ({ r ∈ Set.Ioi 0 | x ∈ r • s }) :=
   rfl
 #align gauge_def gauge_def
 
-/- warning: gauge_def' -> gauge_def' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align gauge_def' gauge_def'ₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]] -/
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
@@ -96,12 +84,6 @@ theorem gauge_def' : gauge s x = sInf ({ r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s })
 private theorem gauge_set_bdd_below : BddBelow { r : ℝ | 0 < r ∧ x ∈ r • s } :=
   ⟨0, fun r hr => hr.1.le⟩
 
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-Case conversion may be inaccurate. Consider using '#align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonemptyₓ'. -/
 /-- If the given subset is `absorbent` then the set we take an infimum over in `gauge` is nonempty,
 which is useful for proving many properties about the gauge.  -/
 theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
@@ -110,22 +92,10 @@ theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
   ⟨r, hr₁, hr₂ r (Real.norm_of_nonneg hr₁.le).ge⟩
 #align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonempty
 
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-Case conversion may be inaccurate. Consider using '#align gauge_mono gauge_monoₓ'. -/
 theorem gauge_mono (hs : Absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s := fun x =>
   csInf_le_csInf gauge_set_bddBelow hs.gauge_set_nonempty fun r hr => ⟨hr.1, smul_set_mono h hr.2⟩
 #align gauge_mono gauge_mono
 
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-Case conversion may be inaccurate. Consider using '#align exists_lt_of_gauge_lt exists_lt_of_gauge_ltₓ'. -/
 theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
     ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s :=
   by
@@ -133,12 +103,6 @@ theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
   exact ⟨b, hb, hba, hx⟩
 #align exists_lt_of_gauge_lt exists_lt_of_gauge_lt
 
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 /-- The gauge evaluated at `0` is always zero (mathematically this requires `0` to be in the set `s`
 but, the real infimum of the empty set in Lean being defined as `0`, it holds unconditionally). -/
 @[simp]
@@ -149,12 +113,6 @@ theorem gauge_zero : gauge s 0 = 0 := by
   · simp only [smul_zero, sep_false, h, Real.sInf_empty]
 #align gauge_zero gauge_zero
 
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 @[simp]
 theorem gauge_zero' : gauge (0 : Set E) = 0 := by
   ext
@@ -166,76 +124,34 @@ theorem gauge_zero' : gauge (0 : Set E) = 0 := by
     exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr.1) hx
 #align gauge_zero' gauge_zero'
 
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 @[simp]
 theorem gauge_empty : gauge (∅ : Set E) = 0 := by ext;
   simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
 #align gauge_empty gauge_empty
 
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 theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by
   obtain rfl | rfl := subset_singleton_iff_eq.1 h; exacts[gauge_empty, gauge_zero']
 #align gauge_of_subset_zero gauge_of_subset_zero
 
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 /-- The gauge is always nonnegative. -/
 theorem gauge_nonneg (x : E) : 0 ≤ gauge s x :=
   Real.sInf_nonneg _ fun x hx => hx.1.le
 #align gauge_nonneg gauge_nonneg
 
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 theorem gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x :=
   by
   have : ∀ x, -x ∈ s ↔ x ∈ s := fun x => ⟨fun h => by simpa using Symmetric _ h, Symmetric x⟩
   simp_rw [gauge_def', smul_neg, this]
 #align gauge_neg gauge_neg
 
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 theorem gauge_neg_set_neg (x : E) : gauge (-s) (-x) = gauge s x := by
   simp_rw [gauge_def', smul_neg, neg_mem_neg]
 #align gauge_neg_set_neg gauge_neg_set_neg
 
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-Case conversion may be inaccurate. Consider using '#align gauge_neg_set_eq_gauge_neg gauge_neg_set_eq_gauge_negₓ'. -/
 theorem gauge_neg_set_eq_gauge_neg (x : E) : gauge (-s) x = gauge s (-x) := by
   rw [← gauge_neg_set_neg, neg_neg]
 #align gauge_neg_set_eq_gauge_neg gauge_neg_set_eq_gauge_neg
 
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-Case conversion may be inaccurate. Consider using '#align gauge_le_of_mem gauge_le_of_memₓ'. -/
 theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
   by
   obtain rfl | ha' := ha.eq_or_lt
@@ -243,12 +159,6 @@ theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
   · exact csInf_le gauge_set_bdd_below ⟨ha', hx⟩
 #align gauge_le_of_mem gauge_le_of_mem
 
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-Case conversion may be inaccurate. Consider using '#align gauge_le_eq gauge_le_eqₓ'. -/
 theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
     { x | gauge s x ≤ a } = ⋂ (r : ℝ) (H : a < r), r • s :=
   by
@@ -268,12 +178,6 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
       (gauge_le_of_mem (ha.trans hε'.le) <| h _ hε').trans_lt (add_lt_add_left (half_lt_self hε) _)
 #align gauge_le_eq gauge_le_eq
 
-/- warning: gauge_lt_eq' -> gauge_lt_eq' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align gauge_lt_eq' gauge_lt_eq'ₓ'. -/
 theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ (r : ℝ) (H : 0 < r) (H : r < a), r • s :=
   by
@@ -284,12 +188,6 @@ theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
 #align gauge_lt_eq' gauge_lt_eq'
 
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-Case conversion may be inaccurate. Consider using '#align gauge_lt_eq gauge_lt_eqₓ'. -/
 theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s :=
   by
@@ -300,12 +198,6 @@ theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
 #align gauge_lt_eq gauge_lt_eq
 
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-Case conversion may be inaccurate. Consider using '#align gauge_lt_one_subset_self gauge_lt_one_subset_selfₓ'. -/
 theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
     { x | gauge s x < 1 } ⊆ s := by
   rw [gauge_lt_eq Absorbs]
@@ -314,31 +206,13 @@ theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (abs
   exact hs.smul_mem_of_zero_mem h₀ hy (Ioo_subset_Icc_self hr)
 #align gauge_lt_one_subset_self gauge_lt_one_subset_self
 
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 theorem gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 :=
   gauge_le_of_mem zero_le_one <| by rwa [one_smul]
 #align gauge_le_one_of_mem gauge_le_one_of_mem
 
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-Case conversion may be inaccurate. Consider using '#align self_subset_gauge_le_one self_subset_gauge_le_oneₓ'. -/
 theorem self_subset_gauge_le_one : s ⊆ { x | gauge s x ≤ 1 } := fun x => gauge_le_one_of_mem
 #align self_subset_gauge_le_one self_subset_gauge_le_one
 
-/- warning: convex.gauge_le -> Convex.gauge_le is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align convex.gauge_le Convex.gauge_leₓ'. -/
 theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) (a : ℝ) :
     Convex ℝ { x | gauge s x ≤ a } := by
   by_cases ha : 0 ≤ a
@@ -348,23 +222,11 @@ theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Ab
     exact eq_empty_iff_forall_not_mem.2 fun x hx => ha <| (gauge_nonneg _).trans hx
 #align convex.gauge_le Convex.gauge_le
 
-/- warning: balanced.star_convex -> Balanced.starConvex is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align balanced.star_convex Balanced.starConvexₓ'. -/
 theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s :=
   starConvex_zero_iff.2 fun x hx a ha₀ ha₁ =>
     hs _ (by rwa [Real.norm_of_nonneg ha₀]) (smul_mem_smul_set hx)
 #align balanced.star_convex Balanced.starConvex
 
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-Case conversion may be inaccurate. Consider using '#align le_gauge_of_not_mem le_gauge_of_not_memₓ'. -/
 theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
     a ≤ gauge s x := by
   rw [starConvex_zero_iff] at hs₀
@@ -379,12 +241,6 @@ theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s
   · rw [← mul_smul, mul_inv_cancel_left₀ ha.ne']
 #align le_gauge_of_not_mem le_gauge_of_not_mem
 
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-Case conversion may be inaccurate. Consider using '#align one_le_gauge_of_not_mem one_le_gauge_of_not_memₓ'. -/
 theorem one_le_gauge_of_not_mem (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ s) :
     1 ≤ gauge s x :=
   le_gauge_of_not_mem hs₁ hs₂ <| by rwa [one_smul]
@@ -394,9 +250,6 @@ section LinearOrderedField
 
 variable {α : Type _} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ]
 
-/- warning: gauge_smul_of_nonneg -> gauge_smul_of_nonneg is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align gauge_smul_of_nonneg gauge_smul_of_nonnegₓ'. -/
 theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
     (ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x :=
   by
@@ -423,9 +276,6 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
     exact smul_mem_smul_set hx
 #align gauge_smul_of_nonneg gauge_smul_of_nonneg
 
-/- warning: gauge_smul_left_of_nonneg -> gauge_smul_left_of_nonneg is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align gauge_smul_left_of_nonneg gauge_smul_left_of_nonnegₓ'. -/
 theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ ℝ]
     [IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a) :
     gauge (a • s) = a⁻¹ • gauge s :=
@@ -449,9 +299,6 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
     rw [smul_inv₀, smul_assoc, inv_inv]
 #align gauge_smul_left_of_nonneg gauge_smul_left_of_nonneg
 
-/- warning: gauge_smul_left -> gauge_smul_left is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align gauge_smul_left gauge_smul_leftₓ'. -/
 theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ]
     [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
     gauge (a • s) = (|a|)⁻¹ • gauge s :=
@@ -474,9 +321,6 @@ section IsROrC
 
 variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
-/- warning: gauge_norm_smul -> gauge_norm_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align gauge_norm_smul gauge_norm_smulₓ'. -/
 theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖r‖ • x) = gauge s (r • x) :=
   by
   unfold gauge
@@ -486,9 +330,6 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖
   rw [IsROrC.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 
-/- warning: gauge_smul -> gauge_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align gauge_smul gauge_smulₓ'. -/
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/
 theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x := by
   rw [← smul_eq_mul, ← gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs]; infer_instance
@@ -500,12 +341,6 @@ section TopologicalSpace
 
 variable [TopologicalSpace E] [ContinuousSMul ℝ E]
 
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-Case conversion may be inaccurate. Consider using '#align interior_subset_gauge_lt_one interior_subset_gauge_lt_oneₓ'. -/
 theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s x < 1 } :=
   by
   intro x hx
@@ -528,12 +363,6 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s
       (hε ⟨(sub_le_self _ hε₀.le).trans ((le_add_iff_nonneg_right _).2 hε₀.le), le_rfl⟩)
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
 
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-Case conversion may be inaccurate. Consider using '#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_openₓ'. -/
 theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
     { x | gauge s x < 1 } = s :=
   by
@@ -542,22 +371,10 @@ theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈
   exact hs₂.interior_eq.symm
 #align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
 
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-Case conversion may be inaccurate. Consider using '#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_openₓ'. -/
 theorem gauge_lt_one_of_mem_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) {x : E}
     (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx
 #align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_open
 
-/- warning: gauge_lt_of_mem_smul -> gauge_lt_of_mem_smul is a dubious translation:
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-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3] (x : E) (ε : Real), (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (IsOpen.{u1} E _inst_3 s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) ε s)) -> (LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) ε)
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-Case conversion may be inaccurate. Consider using '#align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ'. -/
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
     (hs₂ : IsOpen s) (hx : x ∈ ε • s) : gauge s x < ε :=
   by
@@ -570,12 +387,6 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E)
 
 end TopologicalSpace
 
-/- warning: gauge_add_le -> gauge_add_le is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align gauge_add_le gauge_add_leₓ'. -/
 theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
     gauge s (x + y) ≤ gauge s x + gauge s y :=
   by
@@ -609,17 +420,11 @@ def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Abso
 variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E]
   [ContinuousSMul ℝ E]
 
-/- warning: gauge_seminorm_lt_one_of_open -> gaugeSeminorm_lt_one_of_open is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_openₓ'. -/
 theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
   gauge_lt_one_of_mem_of_open hs₁ hs₂.zero_mem hs hx
 #align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
 
-/- warning: gauge_seminorm_ball_one -> gaugeSeminorm_ball_one is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align gauge_seminorm_ball_one gaugeSeminorm_ball_oneₓ'. -/
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s :=
   by
   rw [Seminorm.ball_zero_eq]
@@ -628,9 +433,6 @@ theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs
 
 end IsROrC
 
-/- warning: seminorm.gauge_ball -> Seminorm.gauge_ball is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align seminorm.gauge_ball Seminorm.gauge_ballₓ'. -/
 /-- Any seminorm arises as the gauge of its unit ball. -/
 @[simp]
 protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1) = p :=
@@ -657,9 +459,6 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     exact lt_add_of_pos_right _ hε
 #align seminorm.gauge_ball Seminorm.gauge_ball
 
-/- warning: seminorm.gauge_seminorm_ball -> Seminorm.gaugeSeminorm_ball is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align seminorm.gauge_seminorm_ball Seminorm.gaugeSeminorm_ballₓ'. -/
 theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
     gaugeSeminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) =
       p :=
@@ -672,22 +471,10 @@ section Norm
 
 variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E}
 
-/- warning: gauge_unit_ball -> gauge_unit_ball is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) x) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_1) x)
-Case conversion may be inaccurate. Consider using '#align gauge_unit_ball gauge_unit_ballₓ'. -/
 theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ := by
   rw [← ball_normSeminorm ℝ, Seminorm.gauge_ball, coe_normSeminorm]
 #align gauge_unit_ball gauge_unit_ball
 
-/- warning: gauge_ball -> gauge_ball is a dubious translation:
-lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (SeminormedAddGroup.toAddGroup.{u1} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u1} E _inst_1))))))))) r) x) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (Norm.norm.{u1} E (SeminormedAddCommGroup.toHasNorm.{u1} E _inst_1) x) r))
-but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1)))))))) r) x) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_1) x) r))
-Case conversion may be inaccurate. Consider using '#align gauge_ball gauge_ballₓ'. -/
 theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r :=
   by
   rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
@@ -696,12 +483,6 @@ theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x
   exact fun _ => id
 #align gauge_ball gauge_ball
 
-/- warning: mul_gauge_le_norm -> mul_gauge_le_norm is a dubious translation:
-lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {s : Set.{u1} E} {r : Real} {x : E}, (HasSubset.Subset.{u1} (Set.{u1} E) (Set.hasSubset.{u1} E) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (SeminormedAddGroup.toAddGroup.{u1} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u1} E _inst_1))))))))) r) s) -> (LE.le.{0} Real Real.hasLe (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) r (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) s x)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toHasNorm.{u1} E _inst_1) x))
-but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {s : Set.{u1} E} {r : Real} {x : E}, (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1)))))))) r) s) -> (LE.le.{0} Real Real.instLEReal (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) r (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) s x)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_1) x))
-Case conversion may be inaccurate. Consider using '#align mul_gauge_le_norm mul_gauge_le_normₓ'. -/
 theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤ ‖x‖ :=
   by
   obtain hr | hr := le_or_lt r 0

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -173,8 +173,7 @@ but is expected to have type
   forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)], Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (EmptyCollection.emptyCollection.{u1} (Set.{u1} E) (Set.instEmptyCollectionSet.{u1} E))) (OfNat.ofNat.{u1} (E -> Real) 0 (Zero.toOfNat0.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.instZeroReal))))
 Case conversion may be inaccurate. Consider using '#align gauge_empty gauge_emptyₓ'. -/
 @[simp]
-theorem gauge_empty : gauge (∅ : Set E) = 0 := by
-  ext
+theorem gauge_empty : gauge (∅ : Set E) = 0 := by ext;
   simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
 #align gauge_empty gauge_empty
 
@@ -184,10 +183,8 @@ lean 3 declaration is
 but is expected to have type
   forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) s (OfNat.ofNat.{u1} (Set.{u1} E) 0 (Zero.toOfNat0.{u1} (Set.{u1} E) (Set.zero.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))))) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 s) (OfNat.ofNat.{u1} (E -> Real) 0 (Zero.toOfNat0.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.instZeroReal)))))
 Case conversion may be inaccurate. Consider using '#align gauge_of_subset_zero gauge_of_subset_zeroₓ'. -/
-theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 :=
-  by
-  obtain rfl | rfl := subset_singleton_iff_eq.1 h
-  exacts[gauge_empty, gauge_zero']
+theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by
+  obtain rfl | rfl := subset_singleton_iff_eq.1 h; exacts[gauge_empty, gauge_zero']
 #align gauge_of_subset_zero gauge_of_subset_zero
 
 /- warning: gauge_nonneg -> gauge_nonneg is a dubious translation:
@@ -493,10 +490,8 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖
 <too large>
 Case conversion may be inaccurate. Consider using '#align gauge_smul gauge_smulₓ'. -/
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/
-theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x :=
-  by
-  rw [← smul_eq_mul, ← gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs]
-  infer_instance
+theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x := by
+  rw [← smul_eq_mul, ← gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs]; infer_instance
 #align gauge_smul gauge_smul
 
 end IsROrC

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module analysis.convex.gauge
-! leanprover-community/mathlib commit dbdf71cee7bb20367cb7e37279c08b0c218cf967
+! leanprover-community/mathlib commit 373b03b5b9d0486534edbe94747f23cb3712f93d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -47,7 +47,7 @@ Minkowski functional, gauge
 
 open NormedField Set
 
-open Pointwise
+open Pointwise Topology NNReal
 
 noncomputable section
 
@@ -95,7 +95,6 @@ theorem gauge_def' : gauge s x = sInf ({ r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s })
 
 private theorem gauge_set_bdd_below : BddBelow { r : ℝ | 0 < r ∧ x ∈ r • s } :=
   ⟨0, fun r hr => hr.1.le⟩
-#align gauge_set_bdd_below gauge_set_bdd_below
 
 /- warning: absorbent.gauge_set_nonempty -> Absorbent.gauge_set_nonempty is a dubious translation:
 lean 3 declaration is
@@ -399,10 +398,7 @@ section LinearOrderedField
 variable {α : Type _} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ]
 
 /- warning: gauge_smul_of_nonneg -> gauge_smul_of_nonneg is a dubious translation:
-lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : IsScalarTower.{u2, 0, u1} α Real (Set.{u1} E) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (SMul.smul.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))) a x)) (SMul.smul.{u2, 0} α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) a (gauge.{u1} E _inst_1 _inst_2 s x)))
-but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))] [_inst_7 : IsScalarTower.{u2, 0, u1} α Real (Set.{u1} E) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6))))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedRing.toPartialOrder.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))) (OfNat.ofNat.{u2} α 0 (Zero.toOfNat0.{u2} α (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))))) a) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{u2, u1, u1} α E E (instHSMul.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6)))) a x)) (HSMul.hSMul.{u2, 0, 0} α Real Real (instHSMul.{u2, 0} α Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))) a (gauge.{u1} E _inst_1 _inst_2 s x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align gauge_smul_of_nonneg gauge_smul_of_nonnegₓ'. -/
 theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
     (ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x :=
@@ -431,10 +427,7 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
 #align gauge_smul_of_nonneg gauge_smul_of_nonneg
 
 /- warning: gauge_smul_left_of_nonneg -> gauge_smul_left_of_nonneg is a dubious translation:
-lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul)] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (SMul.smul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))) a s)) (SMul.smul.{u2, u1} α (E -> Real) (Function.hasSMul.{u1, u2, 0} E α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))) (Inv.inv.{u2} α (DivInvMonoid.toHasInv.{u2} α (DivisionRing.toDivInvMonoid.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) a) (gauge.{u1} E _inst_1 _inst_2 s)))
-but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC)))] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC))) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6)))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedRing.toPartialOrder.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))) (OfNat.ofNat.{u2} α 0 (Zero.toOfNat0.{u2} α (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))))) a) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (HSMul.hSMul.{u2, u1, u1} α (Set.{u1} E) (Set.{u1} E) (instHSMul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6))))) a s)) (HSMul.hSMul.{u2, u1, u1} α (E -> Real) (E -> Real) (instHSMul.{u2, u1} α (E -> Real) (Pi.instSMul.{u1, 0, u2} E α (fun (x : E) => Real) (fun (i : E) => SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))))) (Inv.inv.{u2} α (LinearOrderedField.toInv.{u2} α _inst_3) a) (gauge.{u1} E _inst_1 _inst_2 s)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align gauge_smul_left_of_nonneg gauge_smul_left_of_nonnegₓ'. -/
 theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ ℝ]
     [IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a) :
@@ -460,10 +453,7 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
 #align gauge_smul_left_of_nonneg gauge_smul_left_of_nonneg
 
 /- warning: gauge_smul_left -> gauge_smul_left is a dubious translation:
-lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : Module.{u2, u1} α E (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul)] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{u2, u1} α E (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6))))] {s : Set.{u1} E}, (forall (x : E), (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (Neg.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))) x) s)) -> (forall (a : α), Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (SMul.smul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{u2, u1} α E (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6))))) a s)) (SMul.smul.{u2, u1} α (E -> Real) (Function.hasSMul.{u1, u2, 0} E α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))) (Inv.inv.{u2} α (DivInvMonoid.toHasInv.{u2} α (DivisionRing.toDivInvMonoid.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (Abs.abs.{u2} α (Neg.toHasAbs.{u2} α (SubNegMonoid.toHasNeg.{u2} α (AddGroup.toSubNegMonoid.{u2} α (AddGroupWithOne.toAddGroup.{u2} α (AddCommGroupWithOne.toAddGroupWithOne.{u2} α (Ring.toAddCommGroupWithOne.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (SemilatticeSup.toHasSup.{u2} α (Lattice.toSemilatticeSup.{u2} α (LinearOrder.toLattice.{u2} α (LinearOrderedRing.toLinearOrder.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) a)) (gauge.{u1} E _inst_1 _inst_2 s)))
-but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : Module.{u2, u1} α E (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC)))] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC))) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} α E (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6))))] {s : Set.{u1} E}, (forall (x : E), (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (Neg.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) x) s)) -> (forall (a : α), Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (HSMul.hSMul.{u2, u1, u1} α (Set.{u1} E) (Set.{u1} E) (instHSMul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} α E (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6)))))) a s)) (HSMul.hSMul.{u2, u1, u1} α (E -> Real) (E -> Real) (instHSMul.{u2, u1} α (E -> Real) (Pi.instSMul.{u1, 0, u2} E α (fun (x : E) => Real) (fun (i : E) => SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))))) (Inv.inv.{u2} α (LinearOrderedField.toInv.{u2} α _inst_3) (Abs.abs.{u2} α (Neg.toHasAbs.{u2} α (Ring.toNeg.{u2} α (StrictOrderedRing.toRing.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) (SemilatticeSup.toSup.{u2} α (Lattice.toSemilatticeSup.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α (LinearOrderedRing.toLinearOrder.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))))) a)) (gauge.{u1} E _inst_1 _inst_2 s)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align gauge_smul_left gauge_smul_leftₓ'. -/
 theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ]
     [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
@@ -488,10 +478,7 @@ section IsROrC
 variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
 /- warning: gauge_norm_smul -> gauge_norm_smul is a dubious translation:
-lean 3 declaration is
-  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))], (Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u2} E _inst_1 _inst_2 s (SMul.smul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))) r) x)) (gauge.{u2} E _inst_1 _inst_2 s (SMul.smul.{u1, u2} 𝕜 E (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) r x)))
-but is expected to have type
-  forall {𝕜 : Type.{u2}} {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : IsROrC.{u2} 𝕜] [_inst_4 : Module.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_5 : IsScalarTower.{0, u2, u1} Real 𝕜 E (Algebra.toSMul.{0, u2} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u2} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u2} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))], (Balanced.{u2, u1} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{0, u1, u1} Real E E (instHSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Norm.norm.{u2} 𝕜 (NormedField.toNorm.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))) r) x)) (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{u2, u1, u1} 𝕜 E E (instHSMul.{u2, u1} 𝕜 E (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4))))) r x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align gauge_norm_smul gauge_norm_smulₓ'. -/
 theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖r‖ • x) = gauge s (r • x) :=
   by
@@ -503,10 +490,7 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖
 #align gauge_norm_smul gauge_norm_smul
 
 /- warning: gauge_smul -> gauge_smul is a dubious translation:
-lean 3 declaration is
-  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))], (Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u2} E _inst_1 _inst_2 s (SMul.smul.{u1, u2} 𝕜 E (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) r x)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))) r) (gauge.{u2} E _inst_1 _inst_2 s x)))
-but is expected to have type
-  forall {𝕜 : Type.{u2}} {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : IsROrC.{u2} 𝕜] [_inst_4 : Module.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_5 : IsScalarTower.{0, u2, u1} Real 𝕜 E (Algebra.toSMul.{0, u2} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u2} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u2} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))], (Balanced.{u2, u1} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{u2, u1, u1} 𝕜 E E (instHSMul.{u2, u1} 𝕜 E (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4))))) r x)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) (Norm.norm.{u2} 𝕜 (NormedField.toNorm.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))) r) (gauge.{u1} E _inst_1 _inst_2 s x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align gauge_smul gauge_smulₓ'. -/
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/
 theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x :=
@@ -631,10 +615,7 @@ variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ
   [ContinuousSMul ℝ E]
 
 /- warning: gauge_seminorm_lt_one_of_open -> gaugeSeminorm_lt_one_of_open is a dubious translation:
-lean 3 declaration is
-  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (forall {x : E}, (Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) x s) -> (LT.lt.{0} Real Real.hasLt (coeFn.{succ u2, succ u2} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) (fun (_x : Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) => E -> Real) (Seminorm.hasCoeToFun.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))
-but is expected to have type
-  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (Algebra.toSMul.{0, u1} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u1} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (forall {x : E}, (Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) x s) -> (LT.lt.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) x) Real.instLTReal (FunLike.coe.{succ u2, succ u2, 1} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) _x) (SubadditiveHomClass.toFunLike.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_1))))) (AddZeroClass.toAdd.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (OrderedAddCommMonoid.toPartialOrder.{0} Real Real.orderedAddCommMonoid))) (AddGroupSeminormClass.toAddLEAddHomClass.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddCommGroup.toAddGroup.{u2} E _inst_1) Real.orderedAddCommMonoid (SeminormClass.toAddGroupSeminormClass.{u2, u1, u2} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (Seminorm.instSeminormClass.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))))))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) x) (OfNat.ofNat.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) x) 1 (One.toOfNat1.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) x) Real.instOneReal))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_openₓ'. -/
 theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
@@ -642,10 +623,7 @@ theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
 #align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
 
 /- warning: gauge_seminorm_ball_one -> gaugeSeminorm_ball_one is a dubious translation:
-lean 3 declaration is
-  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (Eq.{succ u2} (Set.{u2} E) (Seminorm.ball.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) _inst_1 (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) s)
-but is expected to have type
-  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (Algebra.toSMul.{0, u1} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u1} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (Eq.{succ u2} (Set.{u2} E) (Seminorm.ball.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) _inst_1 (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) (OfNat.ofNat.{u2} E 0 (Zero.toOfNat0.{u2} E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) s)
+<too large>
 Case conversion may be inaccurate. Consider using '#align gauge_seminorm_ball_one gaugeSeminorm_ball_oneₓ'. -/
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s :=
   by
@@ -656,10 +634,7 @@ theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs
 end IsROrC
 
 /- warning: seminorm.gauge_ball -> Seminorm.gauge_ball is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align seminorm.gauge_ball Seminorm.gauge_ballₓ'. -/
 /-- Any seminorm arises as the gauge of its unit ball. -/
 @[simp]
@@ -688,10 +663,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
 #align seminorm.gauge_ball Seminorm.gauge_ball
 
 /- warning: seminorm.gauge_seminorm_ball -> Seminorm.gaugeSeminorm_ball is a dubious translation:
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(AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) p (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) Real.isROrC _inst_2 (IsScalarTower.left.{0, u1} Real E Real.monoid (MulActionWithZero.toMulAction.{0, u1} Real E Real.monoidWithZero (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) (Seminorm.balanced_ball_zero.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) (Seminorm.convex_ball.{0, u1} Real E Real.normedField _inst_1 (NormedField.toNormedSpace.{0} Real Real.normedField) _inst_2 _inst_2 (IsScalarTower.left.{0, u1} Real E Real.monoid (MulActionWithZero.toMulAction.{0, u1} Real E Real.monoidWithZero (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) p (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) (Seminorm.absorbent_ball_zero.{0, u1} Real E Real.normedField _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (zero_lt_one.{0} Real Real.hasZero Real.hasOne Real.partialOrder (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring) (NeZero.one.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) Real.nontrivial)))) p
-but is expected to have type
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Real (Semifield.toCommGroupWithZero.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (gaugeSeminorm.{0, u1} Real E _inst_1 _inst_2 (Seminorm.ball.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E (MonoidWithZero.toZero.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real 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(AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) p (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) Real.isROrC _inst_2 (IsScalarTower.left.{0, u1} Real E (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real Real.normedField)))))) (MulActionWithZero.toMulAction.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) (Seminorm.balanced_ball_zero.{u1, 0} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) (Seminorm.convex_ball.{0, u1} Real E Real.normedField _inst_1 (NormedField.toNormedSpace.{0} Real Real.normedField) _inst_2 _inst_2 (IsScalarTower.left.{0, u1} Real E (MonoidWithZero.toMonoid.{0} Real Real.instMonoidWithZeroReal) (MulActionWithZero.toMulAction.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) p (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) (Seminorm.absorbent_ball_zero.{u1, 0} Real E Real.normedField _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (zero_lt_one.{0} Real Real.instZeroReal Real.instOneReal Real.partialOrder (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring) (NeZero.one.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real.nontrivial)))) p
+<too large>
 Case conversion may be inaccurate. Consider using '#align seminorm.gauge_seminorm_ball Seminorm.gaugeSeminorm_ballₓ'. -/
 theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
     gaugeSeminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) =
@@ -711,24 +683,8 @@ lean 3 declaration is
 but is expected to have type
   forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) x) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_1) x)
 Case conversion may be inaccurate. Consider using '#align gauge_unit_ball gauge_unit_ballₓ'. -/
-theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ :=
-  by
-  obtain rfl | hx := eq_or_ne x 0
-  · rw [norm_zero, gauge_zero]
-  refine' (le_of_forall_pos_le_add fun ε hε => _).antisymm _
-  · have : 0 < ‖x‖ + ε := by positivity
-    refine' gauge_le_of_mem this.le _
-    rw [smul_ball this.ne', smul_zero, Real.norm_of_nonneg this.le, mul_one, mem_ball_zero_iff]
-    exact lt_add_of_pos_right _ hε
-  refine'
-    le_gauge_of_not_mem balanced_ball_zero.star_convex (absorbent_ball_zero zero_lt_one).Absorbs
-      fun h => _
-  obtain hx' | hx' := eq_or_ne ‖x‖ 0
-  · rw [hx'] at h
-    exact hx (zero_smul_set_subset _ h)
-  · rw [mem_smul_set_iff_inv_smul_mem₀ hx', mem_ball_zero_iff, norm_smul, norm_inv, norm_norm,
-      inv_mul_cancel hx'] at h
-    exact lt_irrefl _ h
+theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ := by
+  rw [← ball_normSeminorm ℝ, Seminorm.gauge_ball, coe_normSeminorm]
 #align gauge_unit_ball gauge_unit_ball
 
 /- warning: gauge_ball -> gauge_ball is a dubious translation:
@@ -759,5 +715,26 @@ theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤
   exact gauge_mono (absorbent_ball_zero hr) hs x
 #align mul_gauge_le_norm mul_gauge_le_norm
 
+theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r)
+    (hs : Metric.ball (0 : E) r ⊆ s) : LipschitzWith r⁻¹ (gauge s) :=
+  have : Absorbent ℝ (Metric.ball (0 : E) r) := absorbent_ball_zero hr
+  LipschitzWith.of_le_add_mul _ fun x y =>
+    calc
+      gauge s x = gauge s (y + (x - y)) := by simp
+      _ ≤ gauge s y + gauge s (x - y) := (gauge_add_le hc (this.Subset hs) _ _)
+      _ ≤ gauge s y + ‖x - y‖ / r :=
+        (add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr _)) _)
+      _ = gauge s y + r⁻¹ * dist x y := by rw [dist_eq_norm, div_eq_inv_mul]
+      
+#align convex.lipschitz_with_gauge Convex.lipschitzWith_gauge
+
+theorem Convex.uniformContinuous_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
+    UniformContinuous (gauge s) :=
+  by
+  obtain ⟨r, hr₀, hr⟩ := Metric.mem_nhds_iff.1 h₀
+  lift r to ℝ≥0 using le_of_lt hr₀
+  exact (hc.lipschitz_with_gauge hr₀ hr).UniformContinuous
+#align convex.uniform_continuous_gauge Convex.uniformContinuous_gauge
+
 end Norm
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -400,7 +400,7 @@ variable {α : Type _} [LinearOrderedField α] [MulActionWithZero α ℝ] [Order
 
 /- warning: gauge_smul_of_nonneg -> gauge_smul_of_nonneg is a dubious translation:
 lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : IsScalarTower.{u2, 0, u1} α Real (Set.{u1} E) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (SMul.smul.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))) a x)) (SMul.smul.{u2, 0} α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) a (gauge.{u1} E _inst_1 _inst_2 s x)))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : IsScalarTower.{u2, 0, u1} α Real (Set.{u1} E) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (SMul.smul.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))) a x)) (SMul.smul.{u2, 0} α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) a (gauge.{u1} E _inst_1 _inst_2 s x)))
 but is expected to have type
   forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))] [_inst_7 : IsScalarTower.{u2, 0, u1} α Real (Set.{u1} E) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6))))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedRing.toPartialOrder.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))) (OfNat.ofNat.{u2} α 0 (Zero.toOfNat0.{u2} α (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))))) a) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{u2, u1, u1} α E E (instHSMul.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6)))) a x)) (HSMul.hSMul.{u2, 0, 0} α Real Real (instHSMul.{u2, 0} α Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))) a (gauge.{u1} E _inst_1 _inst_2 s x)))
 Case conversion may be inaccurate. Consider using '#align gauge_smul_of_nonneg gauge_smul_of_nonnegₓ'. -/
@@ -432,7 +432,7 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
 
 /- warning: gauge_smul_left_of_nonneg -> gauge_smul_left_of_nonneg is a dubious translation:
 lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul)] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (SMul.smul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))) a s)) (SMul.smul.{u2, u1} α (E -> Real) (Function.hasSMul.{u1, u2, 0} E α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))) (Inv.inv.{u2} α (DivInvMonoid.toHasInv.{u2} α (DivisionRing.toDivInvMonoid.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) a) (gauge.{u1} E _inst_1 _inst_2 s)))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul)] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toHasLe.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (SMul.smul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))) a s)) (SMul.smul.{u2, u1} α (E -> Real) (Function.hasSMul.{u1, u2, 0} E α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))) (Inv.inv.{u2} α (DivInvMonoid.toHasInv.{u2} α (DivisionRing.toDivInvMonoid.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) a) (gauge.{u1} E _inst_1 _inst_2 s)))
 but is expected to have type
   forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC)))] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC))) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6)))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedRing.toPartialOrder.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))) (OfNat.ofNat.{u2} α 0 (Zero.toOfNat0.{u2} α (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))))) a) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (HSMul.hSMul.{u2, u1, u1} α (Set.{u1} E) (Set.{u1} E) (instHSMul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6))))) a s)) (HSMul.hSMul.{u2, u1, u1} α (E -> Real) (E -> Real) (instHSMul.{u2, u1} α (E -> Real) (Pi.instSMul.{u1, 0, u2} E α (fun (x : E) => Real) (fun (i : E) => SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))))) (Inv.inv.{u2} α (LinearOrderedField.toInv.{u2} α _inst_3) a) (gauge.{u1} E _inst_1 _inst_2 s)))
 Case conversion may be inaccurate. Consider using '#align gauge_smul_left_of_nonneg gauge_smul_left_of_nonnegₓ'. -/
@@ -634,7 +634,7 @@ variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ
 lean 3 declaration is
   forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (forall {x : E}, (Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) x s) -> (LT.lt.{0} Real Real.hasLt (coeFn.{succ u2, succ u2} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) (fun (_x : Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) => E -> Real) (Seminorm.hasCoeToFun.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))
 but is expected to have type
-  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (Algebra.toSMul.{0, u1} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u1} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (forall {x : E}, (Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) x s) -> (LT.lt.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) x) Real.instLTReal (FunLike.coe.{succ u2, succ u2, 1} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) _x) (SubadditiveHomClass.toFunLike.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_1))))) (AddZeroClass.toAdd.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (OrderedAddCommMonoid.toPartialOrder.{0} Real Real.orderedAddCommMonoid))) (AddGroupSeminormClass.toAddLEAddHomClass.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddCommGroup.toAddGroup.{u2} E _inst_1) Real.orderedAddCommMonoid (SeminormClass.toAddGroupSeminormClass.{u2, u1, u2} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (Seminorm.instSeminormClass.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))))))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) x) (OfNat.ofNat.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) x) 1 (One.toOfNat1.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) x) Real.instOneReal))))
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (Algebra.toSMul.{0, u1} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u1} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (forall {x : E}, (Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) x s) -> (LT.lt.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) x) Real.instLTReal (FunLike.coe.{succ u2, succ u2, 1} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) _x) (SubadditiveHomClass.toFunLike.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_1))))) (AddZeroClass.toAdd.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (OrderedAddCommMonoid.toPartialOrder.{0} Real Real.orderedAddCommMonoid))) (AddGroupSeminormClass.toAddLEAddHomClass.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddCommGroup.toAddGroup.{u2} E _inst_1) Real.orderedAddCommMonoid (SeminormClass.toAddGroupSeminormClass.{u2, u1, u2} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (Seminorm.instSeminormClass.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))))))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) x) (OfNat.ofNat.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) x) 1 (One.toOfNat1.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) x) Real.instOneReal))))
 Case conversion may be inaccurate. Consider using '#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_openₓ'. -/
 theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
@@ -659,7 +659,7 @@ end IsROrC
 lean 3 declaration is
   forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] (p : Seminorm.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))), Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (Seminorm.ball.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) p (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) (coeFn.{succ u1, succ u1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (fun (_x : Seminorm.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) => E -> Real) (Seminorm.hasCoeToFun.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) p)
 but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] (p : Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))), Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (Seminorm.ball.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) p (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) (FunLike.coe.{succ u1, succ u1, 1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) _x) (SubadditiveHomClass.toFunLike.{u1, u1, 0} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) E Real (AddZeroClass.toAdd.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (AddZeroClass.toAdd.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (OrderedAddCommMonoid.toPartialOrder.{0} Real Real.orderedAddCommMonoid))) (AddGroupSeminormClass.toAddLEAddHomClass.{u1, u1, 0} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) E Real (AddCommGroup.toAddGroup.{u1} E _inst_1) Real.orderedAddCommMonoid (SeminormClass.toAddGroupSeminormClass.{u1, 0, u1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (Seminorm.instSeminormClass.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))))) p)
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] (p : Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))), Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (Seminorm.ball.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) p (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) (FunLike.coe.{succ u1, succ u1, 1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.Seminorm._hyg.838 : E) => Real) _x) (SubadditiveHomClass.toFunLike.{u1, u1, 0} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) E Real (AddZeroClass.toAdd.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (AddZeroClass.toAdd.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (OrderedAddCommMonoid.toPartialOrder.{0} Real Real.orderedAddCommMonoid))) (AddGroupSeminormClass.toAddLEAddHomClass.{u1, u1, 0} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) E Real (AddCommGroup.toAddGroup.{u1} E _inst_1) Real.orderedAddCommMonoid (SeminormClass.toAddGroupSeminormClass.{u1, 0, u1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (Seminorm.instSeminormClass.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))))) p)
 Case conversion may be inaccurate. Consider using '#align seminorm.gauge_ball Seminorm.gauge_ballₓ'. -/
 /-- Any seminorm arises as the gauge of its unit ball. -/
 @[simp]

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module analysis.convex.gauge
-! leanprover-community/mathlib commit 3f655f5297b030a87d641ad4e825af8d9679eb0b
+! leanprover-community/mathlib commit dbdf71cee7bb20367cb7e37279c08b0c218cf967
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.Tactic.Congrm
 /-!
 # The Minkowksi functional
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file defines the Minkowski functional, aka gauge.
 
 The Minkowski functional of a set `s` is the function which associates each point to how much you

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -58,7 +58,7 @@ variable [AddCommGroup E] [Module ℝ E]
 /-- The Minkowski functional. Given a set `s` in a real vector space, `gauge s` is the functional
 which sends `x : E` to the smallest `r : ℝ` such that `x` is in `s` scaled by `r`. -/
 def gauge (s : Set E) (x : E) : ℝ :=
-  infₛ { r : ℝ | 0 < r ∧ x ∈ r • s }
+  sInf { r : ℝ | 0 < r ∧ x ∈ r • s }
 #align gauge gauge
 -/
 
@@ -66,24 +66,24 @@ variable {s t : Set E} {a : ℝ} {x : E}
 
 /- warning: gauge_def -> gauge_def is a dubious translation:
 lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.hasInf (Sep.sep.{0, 0} Real (Set.{0} Real) (Set.hasSep.{0} Real) (fun (r : Real) => Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s)) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.sInf.{0} Real Real.hasInf (Sep.sep.{0, 0} Real (Set.{0} Real) (Set.hasSep.{0} Real) (fun (r : Real) => Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s)) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))
 but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.instInfSetReal (setOf.{0} Real (fun (r : Real) => And (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s)))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.sInf.{0} Real Real.instInfSetReal (setOf.{0} Real (fun (r : Real) => And (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s)))))
 Case conversion may be inaccurate. Consider using '#align gauge_def gauge_defₓ'. -/
-theorem gauge_def : gauge s x = infₛ ({ r ∈ Set.Ioi 0 | x ∈ r • s }) :=
+theorem gauge_def : gauge s x = sInf ({ r ∈ Set.Ioi 0 | x ∈ r • s }) :=
   rfl
 #align gauge_def gauge_def
 
 /- warning: gauge_def' -> gauge_def' is a dubious translation:
 lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.hasInf (Sep.sep.{0, 0} Real (Set.{0} Real) (Set.hasSep.{0} Real) (fun (r : Real) => Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (SMul.smul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (Inv.inv.{0} Real Real.hasInv r) x) s) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.sInf.{0} Real Real.hasInf (Sep.sep.{0, 0} Real (Set.{0} Real) (Set.hasSep.{0} Real) (fun (r : Real) => Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (SMul.smul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (Inv.inv.{0} Real Real.hasInv r) x) s) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))
 but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.instInfSetReal (setOf.{0} Real (fun (r : Real) => And (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (HSMul.hSMul.{0, u1, u1} Real E E (instHSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Inv.inv.{0} Real Real.instInvReal r) x) s))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.sInf.{0} Real Real.instInfSetReal (setOf.{0} Real (fun (r : Real) => And (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (HSMul.hSMul.{0, u1, u1} Real E E (instHSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Inv.inv.{0} Real Real.instInvReal r) x) s))))
 Case conversion may be inaccurate. Consider using '#align gauge_def' gauge_def'ₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]] -/
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
-theorem gauge_def' : gauge s x = infₛ ({ r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s }) :=
+theorem gauge_def' : gauge s x = sInf ({ r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s }) :=
   by
   trace
     "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]]"
@@ -115,7 +115,7 @@ but is expected to have type
   forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {t : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) s t) -> (LE.le.{u1} (E -> Real) (Pi.hasLe.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.instLEReal)) (gauge.{u1} E _inst_1 _inst_2 t) (gauge.{u1} E _inst_1 _inst_2 s))
 Case conversion may be inaccurate. Consider using '#align gauge_mono gauge_monoₓ'. -/
 theorem gauge_mono (hs : Absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s := fun x =>
-  cinfₛ_le_cinfₛ gauge_set_bddBelow hs.gauge_set_nonempty fun r hr => ⟨hr.1, smul_set_mono h hr.2⟩
+  csInf_le_csInf gauge_set_bddBelow hs.gauge_set_nonempty fun r hr => ⟨hr.1, smul_set_mono h hr.2⟩
 #align gauge_mono gauge_mono
 
 /- warning: exists_lt_of_gauge_lt -> exists_lt_of_gauge_lt is a dubious translation:
@@ -127,7 +127,7 @@ Case conversion may be inaccurate. Consider using '#align exists_lt_of_gauge_lt
 theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
     ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s :=
   by
-  obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_cinfₛ_lt absorbs.gauge_set_nonempty h
+  obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h
   exact ⟨b, hb, hba, hx⟩
 #align exists_lt_of_gauge_lt exists_lt_of_gauge_lt
 
@@ -143,8 +143,8 @@ but, the real infimum of the empty set in Lean being defined as `0`, it holds un
 theorem gauge_zero : gauge s 0 = 0 := by
   rw [gauge_def']
   by_cases (0 : E) ∈ s
-  · simp only [smul_zero, sep_true, h, cinfₛ_Ioi]
-  · simp only [smul_zero, sep_false, h, Real.infₛ_empty]
+  · simp only [smul_zero, sep_true, h, csInf_Ioi]
+  · simp only [smul_zero, sep_false, h, Real.sInf_empty]
 #align gauge_zero gauge_zero
 
 /- warning: gauge_zero' -> gauge_zero' is a dubious translation:
@@ -158,9 +158,9 @@ theorem gauge_zero' : gauge (0 : Set E) = 0 := by
   ext
   rw [gauge_def']
   obtain rfl | hx := eq_or_ne x 0
-  · simp only [cinfₛ_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]
+  · simp only [csInf_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]
   · simp only [mem_zero, Pi.zero_apply, inv_eq_zero, smul_eq_zero]
-    convert Real.infₛ_empty
+    convert Real.sInf_empty
     exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr.1) hx
 #align gauge_zero' gauge_zero'
 
@@ -173,7 +173,7 @@ Case conversion may be inaccurate. Consider using '#align gauge_empty gauge_empt
 @[simp]
 theorem gauge_empty : gauge (∅ : Set E) = 0 := by
   ext
-  simp only [gauge_def', Real.infₛ_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
+  simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
 #align gauge_empty gauge_empty
 
 /- warning: gauge_of_subset_zero -> gauge_of_subset_zero is a dubious translation:
@@ -196,7 +196,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align gauge_nonneg gauge_nonnegₓ'. -/
 /-- The gauge is always nonnegative. -/
 theorem gauge_nonneg (x : E) : 0 ≤ gauge s x :=
-  Real.infₛ_nonneg _ fun x hx => hx.1.le
+  Real.sInf_nonneg _ fun x hx => hx.1.le
 #align gauge_nonneg gauge_nonneg
 
 /- warning: gauge_neg -> gauge_neg is a dubious translation:
@@ -241,20 +241,20 @@ theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
   by
   obtain rfl | ha' := ha.eq_or_lt
   · rw [mem_singleton_iff.1 (zero_smul_set_subset _ hx), gauge_zero]
-  · exact cinfₛ_le gauge_set_bdd_below ⟨ha', hx⟩
+  · exact csInf_le gauge_set_bdd_below ⟨ha', hx⟩
 #align gauge_le_of_mem gauge_le_of_mem
 
 /- warning: gauge_le_eq -> gauge_le_eq is a dubious translation:
 lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.interᵢ.{u1, 1} E Real (fun (r : Real) => Set.interᵢ.{u1, 0} E (LT.lt.{0} Real Real.hasLt a r) (fun (H : LT.lt.{0} Real Real.hasLt a r) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.iInter.{u1, 1} E Real (fun (r : Real) => Set.iInter.{u1, 0} E (LT.lt.{0} Real Real.hasLt a r) (fun (H : LT.lt.{0} Real Real.hasLt a r) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s))))
 but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.interᵢ.{u1, 1} E Real (fun (r : Real) => Set.interᵢ.{u1, 0} E (LT.lt.{0} Real Real.instLTReal a r) (fun (H : LT.lt.{0} Real Real.instLTReal a r) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.iInter.{u1, 1} E Real (fun (r : Real) => Set.iInter.{u1, 0} E (LT.lt.{0} Real Real.instLTReal a r) (fun (H : LT.lt.{0} Real Real.instLTReal a r) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s))))
 Case conversion may be inaccurate. Consider using '#align gauge_le_eq gauge_le_eqₓ'. -/
 theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
     { x | gauge s x ≤ a } = ⋂ (r : ℝ) (H : a < r), r • s :=
   by
   ext
-  simp_rw [Set.mem_interᵢ, Set.mem_setOf_eq]
+  simp_rw [Set.mem_iInter, Set.mem_setOf_eq]
   refine' ⟨fun h r hr => _, fun h => le_of_forall_pos_lt_add fun ε hε => _⟩
   · have hr' := ha.trans_lt hr
     rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
@@ -271,9 +271,9 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
 
 /- warning: gauge_lt_eq' -> gauge_lt_eq' is a dubious translation:
 lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) (fun (H : LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.hasLt r a) (fun (H : LT.lt.{0} Real Real.hasLt r a) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s)))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.iUnion.{u1, 1} E Real (fun (r : Real) => Set.iUnion.{u1, 0} E (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) (fun (H : LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) => Set.iUnion.{u1, 0} E (LT.lt.{0} Real Real.hasLt r a) (fun (H : LT.lt.{0} Real Real.hasLt r a) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s)))))
 but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) (fun (H : LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.instLTReal r a) (fun (H : LT.lt.{0} Real Real.instLTReal r a) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s)))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.iUnion.{u1, 1} E Real (fun (r : Real) => Set.iUnion.{u1, 0} E (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) (fun (H : LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) => Set.iUnion.{u1, 0} E (LT.lt.{0} Real Real.instLTReal r a) (fun (H : LT.lt.{0} Real Real.instLTReal r a) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s)))))
 Case conversion may be inaccurate. Consider using '#align gauge_lt_eq' gauge_lt_eq'ₓ'. -/
 theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ (r : ℝ) (H : 0 < r) (H : r < a), r • s :=
@@ -287,9 +287,9 @@ theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
 
 /- warning: gauge_lt_eq -> gauge_lt_eq is a dubious translation:
 lean 3 declaration is
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a)) (fun (H : Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a)) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.iUnion.{u1, 1} E Real (fun (r : Real) => Set.iUnion.{u1, 0} E (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a)) (fun (H : Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a)) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s))))
 but is expected to have type
-  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioo.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a)) (fun (H : Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioo.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a)) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s))))
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.iUnion.{u1, 1} E Real (fun (r : Real) => Set.iUnion.{u1, 0} E (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioo.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a)) (fun (H : Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioo.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a)) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s))))
 Case conversion may be inaccurate. Consider using '#align gauge_lt_eq gauge_lt_eqₓ'. -/
 theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s :=
@@ -310,7 +310,7 @@ Case conversion may be inaccurate. Consider using '#align gauge_lt_one_subset_se
 theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
     { x | gauge s x < 1 } ⊆ s := by
   rw [gauge_lt_eq Absorbs]
-  refine' Set.unionᵢ₂_subset fun r hr _ => _
+  refine' Set.iUnion₂_subset fun r hr _ => _
   rintro ⟨y, hy, rfl⟩
   exact hs.smul_mem_of_zero_mem h₀ hy (Ioo_subset_Icc_self hr)
 #align gauge_lt_one_subset_self gauge_lt_one_subset_self
@@ -344,7 +344,7 @@ theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Ab
     Convex ℝ { x | gauge s x ≤ a } := by
   by_cases ha : 0 ≤ a
   · rw [gauge_le_eq hs h₀ Absorbs ha]
-    exact convex_interᵢ fun i => convex_interᵢ fun hi => hs.smul _
+    exact convex_iInter fun i => convex_iInter fun hi => hs.smul _
   · convert convex_empty
     exact eq_empty_iff_forall_not_mem.2 fun x hx => ha <| (gauge_nonneg _).trans hx
 #align convex.gauge_le Convex.gauge_le
@@ -370,7 +370,7 @@ theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s
     a ≤ gauge s x := by
   rw [starConvex_zero_iff] at hs₀
   obtain ⟨r, hr, h⟩ := hs₂
-  refine' le_cinfₛ ⟨r, hr, singleton_subset_iff.1 <| h _ (Real.norm_of_nonneg hr.le).ge⟩ _
+  refine' le_csInf ⟨r, hr, singleton_subset_iff.1 <| h _ (Real.norm_of_nonneg hr.le).ge⟩ _
   rintro b ⟨hb, x, hx', rfl⟩
   refine' not_lt.1 fun hba => hx _
   have ha := hb.trans hba
@@ -406,7 +406,7 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
   by
   obtain rfl | ha' := ha.eq_or_lt
   · rw [zero_smul, gauge_zero, zero_smul]
-  rw [gauge_def', gauge_def', ← Real.infₛ_smul_of_nonneg ha]
+  rw [gauge_def', gauge_def', ← Real.sInf_smul_of_nonneg ha]
   congr 1
   ext r
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
@@ -440,7 +440,7 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
   obtain rfl | ha' := ha.eq_or_lt
   · rw [inv_zero, zero_smul, gauge_of_subset_zero (zero_smul_set_subset _)]
   ext
-  rw [gauge_def', Pi.smul_apply, gauge_def', ← Real.infₛ_smul_of_nonneg (inv_nonneg.2 ha)]
+  rw [gauge_def', Pi.smul_apply, gauge_def', ← Real.sInf_smul_of_nonneg (inv_nonneg.2 ha)]
   congr 1
   ext r
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
@@ -664,7 +664,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
   by
   ext
   obtain hp | hp := { r : ℝ | 0 < r ∧ x ∈ r • p.ball 0 1 }.eq_empty_or_nonempty
-  · rw [gauge, hp, Real.infₛ_empty]
+  · rw [gauge, hp, Real.sInf_empty]
     by_contra
     have hpx : 0 < p x := (map_nonneg _ _).lt_of_ne h
     have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx
@@ -672,7 +672,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpx₂),
       inv_mul_lt_iff hpx₂, mul_one]
     exact lt_mul_of_one_lt_left hpx one_lt_two
-  refine' IsGLB.cinfₛ_eq ⟨fun r => _, fun r hr => le_of_forall_pos_le_add fun ε hε => _⟩ hp
+  refine' IsGLB.csInf_eq ⟨fun r => _, fun r hr => le_of_forall_pos_le_add fun ε hε => _⟩ hp
   · rintro ⟨hr, y, hy, rfl⟩
     rw [p.mem_ball_zero] at hy
     rw [map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos hr]

mathlib commit https://github.com/leanprover-community/mathlib/commit/403190b5419b3f03f1a2893ad9352ca7f7d8bff6

@@ -54,18 +54,32 @@ section AddCommGroup
 
 variable [AddCommGroup E] [Module ℝ E]
 
+#print gauge /-
 /-- The Minkowski functional. Given a set `s` in a real vector space, `gauge s` is the functional
 which sends `x : E` to the smallest `r : ℝ` such that `x` is in `s` scaled by `r`. -/
 def gauge (s : Set E) (x : E) : ℝ :=
   infₛ { r : ℝ | 0 < r ∧ x ∈ r • s }
 #align gauge gauge
+-/
 
 variable {s t : Set E} {a : ℝ} {x : E}
 
+/- warning: gauge_def -> gauge_def is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.hasInf (Sep.sep.{0, 0} Real (Set.{0} Real) (Set.hasSep.{0} Real) (fun (r : Real) => Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s)) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.instInfSetReal (setOf.{0} Real (fun (r : Real) => And (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s)))))
+Case conversion may be inaccurate. Consider using '#align gauge_def gauge_defₓ'. -/
 theorem gauge_def : gauge s x = infₛ ({ r ∈ Set.Ioi 0 | x ∈ r • s }) :=
   rfl
 #align gauge_def gauge_def
 
+/- warning: gauge_def' -> gauge_def' is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.hasInf (Sep.sep.{0, 0} Real (Set.{0} Real) (Set.hasSep.{0} Real) (fun (r : Real) => Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (SMul.smul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (Inv.inv.{0} Real Real.hasInv r) x) s) (Set.Ioi.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s x) (InfSet.infₛ.{0} Real Real.instInfSetReal (setOf.{0} Real (fun (r : Real) => And (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioi.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)))) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (HSMul.hSMul.{0, u1, u1} Real E E (instHSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Inv.inv.{0} Real Real.instInvReal r) x) s))))
+Case conversion may be inaccurate. Consider using '#align gauge_def' gauge_def'ₓ'. -/
 /- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]] -/
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
@@ -80,6 +94,12 @@ private theorem gauge_set_bdd_below : BddBelow { r : ℝ | 0 < r ∧ x ∈ r •
   ⟨0, fun r hr => hr.1.le⟩
 #align gauge_set_bdd_below gauge_set_bdd_below
 
+/- warning: absorbent.gauge_set_nonempty -> Absorbent.gauge_set_nonempty is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Set.Nonempty.{0} Real (setOf.{0} Real (fun (r : Real) => And (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Set.Nonempty.{0} Real (setOf.{0} Real (fun (r : Real) => And (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s)))))
+Case conversion may be inaccurate. Consider using '#align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonemptyₓ'. -/
 /-- If the given subset is `absorbent` then the set we take an infimum over in `gauge` is nonempty,
 which is useful for proving many properties about the gauge.  -/
 theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
@@ -88,10 +108,22 @@ theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
   ⟨r, hr₁, hr₂ r (Real.norm_of_nonneg hr₁.le).ge⟩
 #align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonempty
 
+/- warning: gauge_mono -> gauge_mono is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {t : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (HasSubset.Subset.{u1} (Set.{u1} E) (Set.hasSubset.{u1} E) s t) -> (LE.le.{u1} (E -> Real) (Pi.hasLe.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.hasLe)) (gauge.{u1} E _inst_1 _inst_2 t) (gauge.{u1} E _inst_1 _inst_2 s))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {t : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) s t) -> (LE.le.{u1} (E -> Real) (Pi.hasLe.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.instLEReal)) (gauge.{u1} E _inst_1 _inst_2 t) (gauge.{u1} E _inst_1 _inst_2 s))
+Case conversion may be inaccurate. Consider using '#align gauge_mono gauge_monoₓ'. -/
 theorem gauge_mono (hs : Absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s := fun x =>
   cinfₛ_le_cinfₛ gauge_set_bddBelow hs.gauge_set_nonempty fun r hr => ⟨hr.1, smul_set_mono h hr.2⟩
 #align gauge_mono gauge_mono
 
+/- warning: exists_lt_of_gauge_lt -> exists_lt_of_gauge_lt is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real} {x : E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) a) -> (Exists.{1} Real (fun (b : Real) => And (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) b) (And (LT.lt.{0} Real Real.hasLt b a) (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) b s)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real} {x : E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) a) -> (Exists.{1} Real (fun (b : Real) => And (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) b) (And (LT.lt.{0} Real Real.instLTReal b a) (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) b s)))))
+Case conversion may be inaccurate. Consider using '#align exists_lt_of_gauge_lt exists_lt_of_gauge_ltₓ'. -/
 theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
     ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s :=
   by
@@ -99,6 +131,12 @@ theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
   exact ⟨b, hb, hba, hx⟩
 #align exists_lt_of_gauge_lt exists_lt_of_gauge_lt
 
+/- warning: gauge_zero -> gauge_zero is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))))))) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))
+Case conversion may be inaccurate. Consider using '#align gauge_zero gauge_zeroₓ'. -/
 /-- The gauge evaluated at `0` is always zero (mathematically this requires `0` to be in the set `s`
 but, the real infimum of the empty set in Lean being defined as `0`, it holds unconditionally). -/
 @[simp]
@@ -109,6 +147,12 @@ theorem gauge_zero : gauge s 0 = 0 := by
   · simp only [smul_zero, sep_false, h, Real.infₛ_empty]
 #align gauge_zero gauge_zero
 
+/- warning: gauge_zero' -> gauge_zero' is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)], Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (OfNat.ofNat.{u1} (Set.{u1} E) 0 (OfNat.mk.{u1} (Set.{u1} E) 0 (Zero.zero.{u1} (Set.{u1} E) (Set.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))))) (OfNat.ofNat.{u1} (E -> Real) 0 (OfNat.mk.{u1} (E -> Real) 0 (Zero.zero.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.hasZero)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)], Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (OfNat.ofNat.{u1} (Set.{u1} E) 0 (Zero.toOfNat0.{u1} (Set.{u1} E) (Set.zero.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))))) (OfNat.ofNat.{u1} (E -> Real) 0 (Zero.toOfNat0.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align gauge_zero' gauge_zero'ₓ'. -/
 @[simp]
 theorem gauge_zero' : gauge (0 : Set E) = 0 := by
   ext
@@ -120,37 +164,79 @@ theorem gauge_zero' : gauge (0 : Set E) = 0 := by
     exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr.1) hx
 #align gauge_zero' gauge_zero'
 
+/- warning: gauge_empty -> gauge_empty is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)], Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (EmptyCollection.emptyCollection.{u1} (Set.{u1} E) (Set.hasEmptyc.{u1} E))) (OfNat.ofNat.{u1} (E -> Real) 0 (OfNat.mk.{u1} (E -> Real) 0 (Zero.zero.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.hasZero)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)], Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (EmptyCollection.emptyCollection.{u1} (Set.{u1} E) (Set.instEmptyCollectionSet.{u1} E))) (OfNat.ofNat.{u1} (E -> Real) 0 (Zero.toOfNat0.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.instZeroReal))))
+Case conversion may be inaccurate. Consider using '#align gauge_empty gauge_emptyₓ'. -/
 @[simp]
 theorem gauge_empty : gauge (∅ : Set E) = 0 := by
   ext
   simp only [gauge_def', Real.infₛ_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
 #align gauge_empty gauge_empty
 
+/- warning: gauge_of_subset_zero -> gauge_of_subset_zero is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (HasSubset.Subset.{u1} (Set.{u1} E) (Set.hasSubset.{u1} E) s (OfNat.ofNat.{u1} (Set.{u1} E) 0 (OfNat.mk.{u1} (Set.{u1} E) 0 (Zero.zero.{u1} (Set.{u1} E) (Set.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))))) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 s) (OfNat.ofNat.{u1} (E -> Real) 0 (OfNat.mk.{u1} (E -> Real) 0 (Zero.zero.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.hasZero))))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) s (OfNat.ofNat.{u1} (Set.{u1} E) 0 (Zero.toOfNat0.{u1} (Set.{u1} E) (Set.zero.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))))) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 s) (OfNat.ofNat.{u1} (E -> Real) 0 (Zero.toOfNat0.{u1} (E -> Real) (Pi.instZero.{u1, 0} E (fun (x : E) => Real) (fun (i : E) => Real.instZeroReal)))))
+Case conversion may be inaccurate. Consider using '#align gauge_of_subset_zero gauge_of_subset_zeroₓ'. -/
 theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 :=
   by
   obtain rfl | rfl := subset_singleton_iff_eq.1 h
   exacts[gauge_empty, gauge_zero']
 #align gauge_of_subset_zero gauge_of_subset_zero
 
+/- warning: gauge_nonneg -> gauge_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} (x : E), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (gauge.{u1} E _inst_1 _inst_2 s x)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} (x : E), LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (gauge.{u1} E _inst_1 _inst_2 s x)
+Case conversion may be inaccurate. Consider using '#align gauge_nonneg gauge_nonnegₓ'. -/
 /-- The gauge is always nonnegative. -/
 theorem gauge_nonneg (x : E) : 0 ≤ gauge s x :=
   Real.infₛ_nonneg _ fun x hx => hx.1.le
 #align gauge_nonneg gauge_nonneg
 
+/- warning: gauge_neg -> gauge_neg is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (forall (x : E), (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (Neg.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))) x) s)) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (Neg.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))) x)) (gauge.{u1} E _inst_1 _inst_2 s x))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (forall (x : E), (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (Neg.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) x) s)) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (Neg.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) x)) (gauge.{u1} E _inst_1 _inst_2 s x))
+Case conversion may be inaccurate. Consider using '#align gauge_neg gauge_negₓ'. -/
 theorem gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x :=
   by
   have : ∀ x, -x ∈ s ↔ x ∈ s := fun x => ⟨fun h => by simpa using Symmetric _ h, Symmetric x⟩
   simp_rw [gauge_def', smul_neg, this]
 #align gauge_neg gauge_neg
 
+/- warning: gauge_neg_set_neg -> gauge_neg_set_neg is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 (Neg.neg.{u1} (Set.{u1} E) (Set.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))) s) (Neg.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))) x)) (gauge.{u1} E _inst_1 _inst_2 s x)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 (Neg.neg.{u1} (Set.{u1} E) (Set.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))) s) (Neg.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) x)) (gauge.{u1} E _inst_1 _inst_2 s x)
+Case conversion may be inaccurate. Consider using '#align gauge_neg_set_neg gauge_neg_set_negₓ'. -/
 theorem gauge_neg_set_neg (x : E) : gauge (-s) (-x) = gauge s x := by
   simp_rw [gauge_def', smul_neg, neg_mem_neg]
 #align gauge_neg_set_neg gauge_neg_set_neg
 
+/- warning: gauge_neg_set_eq_gauge_neg -> gauge_neg_set_eq_gauge_neg is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 (Neg.neg.{u1} (Set.{u1} E) (Set.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))) s) x) (gauge.{u1} E _inst_1 _inst_2 s (Neg.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))) x))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 (Neg.neg.{u1} (Set.{u1} E) (Set.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))) s) x) (gauge.{u1} E _inst_1 _inst_2 s (Neg.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) x))
+Case conversion may be inaccurate. Consider using '#align gauge_neg_set_eq_gauge_neg gauge_neg_set_eq_gauge_negₓ'. -/
 theorem gauge_neg_set_eq_gauge_neg (x : E) : gauge (-s) x = gauge s (-x) := by
   rw [← gauge_neg_set_neg, neg_neg]
 #align gauge_neg_set_eq_gauge_neg gauge_neg_set_eq_gauge_neg
 
+/- warning: gauge_le_of_mem -> gauge_le_of_mem is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real} {x : E}, (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) a s)) -> (LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s x) a)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real} {x : E}, (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) a s)) -> (LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s x) a)
+Case conversion may be inaccurate. Consider using '#align gauge_le_of_mem gauge_le_of_memₓ'. -/
 theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
   by
   obtain rfl | ha' := ha.eq_or_lt
@@ -158,6 +244,12 @@ theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
   · exact cinfₛ_le gauge_set_bdd_below ⟨ha', hx⟩
 #align gauge_le_of_mem gauge_le_of_mem
 
+/- warning: gauge_le_eq -> gauge_le_eq is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.interᵢ.{u1, 1} E Real (fun (r : Real) => Set.interᵢ.{u1, 0} E (LT.lt.{0} Real Real.hasLt a r) (fun (H : LT.lt.{0} Real Real.hasLt a r) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.interᵢ.{u1, 1} E Real (fun (r : Real) => Set.interᵢ.{u1, 0} E (LT.lt.{0} Real Real.instLTReal a r) (fun (H : LT.lt.{0} Real Real.instLTReal a r) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s))))
+Case conversion may be inaccurate. Consider using '#align gauge_le_eq gauge_le_eqₓ'. -/
 theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
     { x | gauge s x ≤ a } = ⋂ (r : ℝ) (H : a < r), r • s :=
   by
@@ -177,6 +269,12 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
       (gauge_le_of_mem (ha.trans hε'.le) <| h _ hε').trans_lt (add_lt_add_left (half_lt_self hε) _)
 #align gauge_le_eq gauge_le_eq
 
+/- warning: gauge_lt_eq' -> gauge_lt_eq' is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) (fun (H : LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.hasLt r a) (fun (H : LT.lt.{0} Real Real.hasLt r a) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) (fun (H : LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) => Set.unionᵢ.{u1, 0} E (LT.lt.{0} Real Real.instLTReal r a) (fun (H : LT.lt.{0} Real Real.instLTReal r a) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s)))))
+Case conversion may be inaccurate. Consider using '#align gauge_lt_eq' gauge_lt_eq'ₓ'. -/
 theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ (r : ℝ) (H : 0 < r) (H : r < a), r • s :=
   by
@@ -187,6 +285,12 @@ theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
 #align gauge_lt_eq' gauge_lt_eq'
 
+/- warning: gauge_lt_eq -> gauge_lt_eq is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a)) (fun (H : Membership.Mem.{0, 0} Real (Set.{0} Real) (Set.hasMem.{0} Real) r (Set.Ioo.{0} Real Real.preorder (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) a)) => SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) r s))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) a)) (Set.unionᵢ.{u1, 1} E Real (fun (r : Real) => Set.unionᵢ.{u1, 0} E (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioo.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a)) (fun (H : Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembershipSet.{0} Real) r (Set.Ioo.{0} Real Real.instPreorderReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) a)) => HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) r s))))
+Case conversion may be inaccurate. Consider using '#align gauge_lt_eq gauge_lt_eqₓ'. -/
 theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s :=
   by
@@ -197,6 +301,12 @@ theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
 #align gauge_lt_eq gauge_lt_eq
 
+/- warning: gauge_lt_one_subset_self -> gauge_lt_one_subset_self is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (HasSubset.Subset.{u1} (Set.{u1} E) (Set.hasSubset.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) s)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) s)
+Case conversion may be inaccurate. Consider using '#align gauge_lt_one_subset_self gauge_lt_one_subset_selfₓ'. -/
 theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
     { x | gauge s x < 1 } ⊆ s := by
   rw [gauge_lt_eq Absorbs]
@@ -205,13 +315,31 @@ theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (abs
   exact hs.smul_mem_of_zero_mem h₀ hy (Ioo_subset_Icc_self hr)
 #align gauge_lt_one_subset_self gauge_lt_one_subset_self
 
+/- warning: gauge_le_one_of_mem -> gauge_le_one_of_mem is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x s) -> (LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x s) -> (LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))
+Case conversion may be inaccurate. Consider using '#align gauge_le_one_of_mem gauge_le_one_of_memₓ'. -/
 theorem gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 :=
   gauge_le_of_mem zero_le_one <| by rwa [one_smul]
 #align gauge_le_one_of_mem gauge_le_one_of_mem
 
+/- warning: self_subset_gauge_le_one -> self_subset_gauge_le_one is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, HasSubset.Subset.{u1} (Set.{u1} E) (Set.hasSubset.{u1} E) s (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) s (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))
+Case conversion may be inaccurate. Consider using '#align self_subset_gauge_le_one self_subset_gauge_le_oneₓ'. -/
 theorem self_subset_gauge_le_one : s ⊆ { x | gauge s x ≤ 1 } := fun x => gauge_le_one_of_mem
 #align self_subset_gauge_le_one self_subset_gauge_le_one
 
+/- warning: convex.gauge_le -> Convex.gauge_le is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s x) a)))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (a : Real), Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (setOf.{u1} E (fun (x : E) => LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s x) a)))
+Case conversion may be inaccurate. Consider using '#align convex.gauge_le Convex.gauge_leₓ'. -/
 theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) (a : ℝ) :
     Convex ℝ { x | gauge s x ≤ a } := by
   by_cases ha : 0 ≤ a
@@ -221,11 +349,23 @@ theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Ab
     exact eq_empty_iff_forall_not_mem.2 fun x hx => ha <| (gauge_nonneg _).trans hx
 #align convex.gauge_le Convex.gauge_le
 
+/- warning: balanced.star_convex -> Balanced.starConvex is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Balanced.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (StarConvex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Balanced.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (StarConvex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s)
+Case conversion may be inaccurate. Consider using '#align balanced.star_convex Balanced.starConvexₓ'. -/
 theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s :=
   starConvex_zero_iff.2 fun x hx a ha₀ ha₁ =>
     hs _ (by rwa [Real.norm_of_nonneg ha₀]) (smul_mem_smul_set hx)
 #align balanced.star_convex Balanced.starConvex
 
+/- warning: le_gauge_of_not_mem -> le_gauge_of_not_mem is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real} {x : E}, (StarConvex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Absorbs.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s (Singleton.singleton.{u1, u1} E (Set.{u1} E) (Set.hasSingleton.{u1} E) x)) -> (Not (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) a s))) -> (LE.le.{0} Real Real.hasLe a (gauge.{u1} E _inst_1 _inst_2 s x))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {a : Real} {x : E}, (StarConvex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Absorbs.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s (Singleton.singleton.{u1, u1} E (Set.{u1} E) (Set.instSingletonSet.{u1} E) x)) -> (Not (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) a s))) -> (LE.le.{0} Real Real.instLEReal a (gauge.{u1} E _inst_1 _inst_2 s x))
+Case conversion may be inaccurate. Consider using '#align le_gauge_of_not_mem le_gauge_of_not_memₓ'. -/
 theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
     a ≤ gauge s x := by
   rw [starConvex_zero_iff] at hs₀
@@ -240,6 +380,12 @@ theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s
   · rw [← mul_smul, mul_inv_cancel_left₀ ha.ne']
 #align le_gauge_of_not_mem le_gauge_of_not_mem
 
+/- warning: one_le_gauge_of_not_mem -> one_le_gauge_of_not_mem is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, (StarConvex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Absorbs.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s (Singleton.singleton.{u1, u1} E (Set.{u1} E) (Set.hasSingleton.{u1} E) x)) -> (Not (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x s)) -> (LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (gauge.{u1} E _inst_1 _inst_2 s x))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} {x : E}, (StarConvex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Absorbs.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s (Singleton.singleton.{u1, u1} E (Set.{u1} E) (Set.instSingletonSet.{u1} E) x)) -> (Not (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x s)) -> (LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (gauge.{u1} E _inst_1 _inst_2 s x))
+Case conversion may be inaccurate. Consider using '#align one_le_gauge_of_not_mem one_le_gauge_of_not_memₓ'. -/
 theorem one_le_gauge_of_not_mem (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ s) :
     1 ≤ gauge s x :=
   le_gauge_of_not_mem hs₁ hs₂ <| by rwa [one_smul]
@@ -249,6 +395,12 @@ section LinearOrderedField
 
 variable {α : Type _} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ]
 
+/- warning: gauge_smul_of_nonneg -> gauge_smul_of_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : IsScalarTower.{u2, 0, u1} α Real (Set.{u1} E) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (SMul.smul.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6))) a x)) (SMul.smul.{u2, 0} α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) a (gauge.{u1} E _inst_1 _inst_2 s x)))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))] [_inst_7 : IsScalarTower.{u2, 0, u1} α Real (Set.{u1} E) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6))))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedRing.toPartialOrder.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))) (OfNat.ofNat.{u2} α 0 (Zero.toOfNat0.{u2} α (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))))) a) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{u2, u1, u1} α E E (instHSMul.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6)))) a x)) (HSMul.hSMul.{u2, 0, 0} α Real Real (instHSMul.{u2, 0} α Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))) a (gauge.{u1} E _inst_1 _inst_2 s x)))
+Case conversion may be inaccurate. Consider using '#align gauge_smul_of_nonneg gauge_smul_of_nonnegₓ'. -/
 theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
     (ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x :=
   by
@@ -275,6 +427,12 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
     exact smul_mem_smul_set hx
 #align gauge_smul_of_nonneg gauge_smul_of_nonneg
 
+/- warning: gauge_smul_left_of_nonneg -> gauge_smul_left_of_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul)] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (OrderedAddCommGroup.toPartialOrder.{u2} α (StrictOrderedRing.toOrderedAddCommGroup.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) (OfNat.ofNat.{u2} α 0 (OfNat.mk.{u2} α 0 (Zero.zero.{u2} α (MulZeroClass.toHasZero.{u2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} α (NonAssocRing.toNonUnitalNonAssocRing.{u2} α (Ring.toNonAssocRing.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))))))))) a) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (SMul.smul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))))) _inst_6)))) a s)) (SMul.smul.{u2, u1} α (E -> Real) (Function.hasSMul.{u1, u2, 0} E α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))) (Inv.inv.{u2} α (DivInvMonoid.toHasInv.{u2} α (DivisionRing.toDivInvMonoid.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) a) (gauge.{u1} E _inst_1 _inst_2 s)))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : MulActionWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1)))))] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC)))] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC))) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6)))] {s : Set.{u1} E} {a : α}, (LE.le.{u2} α (Preorder.toLE.{u2} α (PartialOrder.toPreorder.{u2} α (StrictOrderedRing.toPartialOrder.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))) (OfNat.ofNat.{u2} α 0 (Zero.toOfNat0.{u2} α (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))))) a) -> (Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (HSMul.hSMul.{u2, u1, u1} α (Set.{u1} E) (Set.{u1} E) (instHSMul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) _inst_6))))) a s)) (HSMul.hSMul.{u2, u1, u1} α (E -> Real) (E -> Real) (instHSMul.{u2, u1} α (E -> Real) (Pi.instSMul.{u1, 0, u2} E α (fun (x : E) => Real) (fun (i : E) => SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))))) (Inv.inv.{u2} α (LinearOrderedField.toInv.{u2} α _inst_3) a) (gauge.{u1} E _inst_1 _inst_2 s)))
+Case conversion may be inaccurate. Consider using '#align gauge_smul_left_of_nonneg gauge_smul_left_of_nonnegₓ'. -/
 theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ ℝ]
     [IsScalarTower α ℝ ℝ] [IsScalarTower α ℝ E] {s : Set E} {a : α} (ha : 0 ≤ a) :
     gauge (a • s) = a⁻¹ • gauge s :=
@@ -298,6 +456,12 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
     rw [smul_inv₀, smul_assoc, inv_inv]
 #align gauge_smul_left_of_nonneg gauge_smul_left_of_nonneg
 
+/- warning: gauge_smul_left -> gauge_smul_left is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero] [_inst_5 : OrderedSMul.{u2, 0} α Real (StrictOrderedSemiring.toOrderedSemiring.{u2} α (StrictOrderedRing.toStrictOrderedSemiring.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) _inst_4)] [_inst_6 : Module.{u2, u1} α E (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul)] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (Mul.toSMul.{0} Real Real.hasMul) (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4))) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{u2, u1} α E (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6))))] {s : Set.{u1} E}, (forall (x : E), (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (Neg.neg.{u1} E (SubNegMonoid.toHasNeg.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1))) x) s)) -> (forall (a : α), Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (SMul.smul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toHasSmul.{u2, u1} α E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u2, u1} α E (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{u2, u1} α E (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6))))) a s)) (SMul.smul.{u2, u1} α (E -> Real) (Function.hasSMul.{u1, u2, 0} E α Real (SMulZeroClass.toHasSmul.{u2, 0} α Real Real.hasZero (SMulWithZero.toSmulZeroClass.{u2, 0} α Real (MulZeroClass.toHasZero.{u2} α (MulZeroOneClass.toMulZeroClass.{u2} α (MonoidWithZero.toMulZeroOneClass.{u2} α (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) Real.hasZero (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (Ring.toSemiring.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3))))) Real.hasZero _inst_4)))) (Inv.inv.{u2} α (DivInvMonoid.toHasInv.{u2} α (DivisionRing.toDivInvMonoid.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))) (Abs.abs.{u2} α (Neg.toHasAbs.{u2} α (SubNegMonoid.toHasNeg.{u2} α (AddGroup.toSubNegMonoid.{u2} α (AddGroupWithOne.toAddGroup.{u2} α (AddCommGroupWithOne.toAddGroupWithOne.{u2} α (Ring.toAddCommGroupWithOne.{u2} α (DivisionRing.toRing.{u2} α (Field.toDivisionRing.{u2} α (LinearOrderedField.toField.{u2} α _inst_3)))))))) (SemilatticeSup.toHasSup.{u2} α (Lattice.toSemilatticeSup.{u2} α (LinearOrder.toLattice.{u2} α (LinearOrderedRing.toLinearOrder.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))))) a)) (gauge.{u1} E _inst_1 _inst_2 s)))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {α : Type.{u2}} [_inst_3 : LinearOrderedField.{u2} α] [_inst_4 : MulActionWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal] [_inst_5 : OrderedSMul.{u2, 0} α Real (OrderedCommSemiring.toOrderedSemiring.{u2} α (StrictOrderedCommSemiring.toOrderedCommSemiring.{u2} α (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.orderedAddCommMonoid (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (AddMonoid.toZero.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid))) _inst_4)] [_inst_6 : Module.{u2, u1} α E (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_7 : SMulCommClass.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC)))] [_inst_8 : IsScalarTower.{u2, 0, 0} α Real Real (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (Algebra.toSMul.{0, 0} Real Real Real.instCommSemiringReal Real.semiring (NormedAlgebra.toAlgebra.{0, 0} Real Real Real.normedField (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (IsROrC.toNormedAlgebra.{0} Real Real.isROrC))) (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4)))] [_inst_9 : IsScalarTower.{u2, 0, u1} α Real E (SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} α E (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6))))] {s : Set.{u1} E}, (forall (x : E), (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (Neg.neg.{u1} E (NegZeroClass.toNeg.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) x) s)) -> (forall (a : α), Eq.{succ u1} (E -> Real) (gauge.{u1} E _inst_1 _inst_2 (HSMul.hSMul.{u2, u1, u1} α (Set.{u1} E) (Set.{u1} E) (instHSMul.{u2, u1} α (Set.{u1} E) (Set.smulSet.{u2, u1} α E (SMulZeroClass.toSMul.{u2, u1} α E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} α E (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} α E (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} α E (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_6)))))) a s)) (HSMul.hSMul.{u2, u1, u1} α (E -> Real) (E -> Real) (instHSMul.{u2, u1} α (E -> Real) (Pi.instSMul.{u1, 0, u2} E α (fun (x : E) => Real) (fun (i : E) => SMulZeroClass.toSMul.{u2, 0} α Real Real.instZeroReal (SMulWithZero.toSMulZeroClass.{u2, 0} α Real (CommMonoidWithZero.toZero.{u2} α (CommGroupWithZero.toCommMonoidWithZero.{u2} α (Semifield.toCommGroupWithZero.{u2} α (LinearOrderedSemifield.toSemifield.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3))))) Real.instZeroReal (MulActionWithZero.toSMulWithZero.{u2, 0} α Real (Semiring.toMonoidWithZero.{u2} α (StrictOrderedSemiring.toSemiring.{u2} α (LinearOrderedSemiring.toStrictOrderedSemiring.{u2} α (LinearOrderedCommSemiring.toLinearOrderedSemiring.{u2} α (LinearOrderedSemifield.toLinearOrderedCommSemiring.{u2} α (LinearOrderedField.toLinearOrderedSemifield.{u2} α _inst_3)))))) Real.instZeroReal _inst_4))))) (Inv.inv.{u2} α (LinearOrderedField.toInv.{u2} α _inst_3) (Abs.abs.{u2} α (Neg.toHasAbs.{u2} α (Ring.toNeg.{u2} α (StrictOrderedRing.toRing.{u2} α (LinearOrderedRing.toStrictOrderedRing.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3))))) (SemilatticeSup.toSup.{u2} α (Lattice.toSemilatticeSup.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α (LinearOrderedRing.toLinearOrder.{u2} α (LinearOrderedCommRing.toLinearOrderedRing.{u2} α (LinearOrderedField.toLinearOrderedCommRing.{u2} α _inst_3)))))))) a)) (gauge.{u1} E _inst_1 _inst_2 s)))
+Case conversion may be inaccurate. Consider using '#align gauge_smul_left gauge_smul_leftₓ'. -/
 theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ]
     [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
     gauge (a • s) = (|a|)⁻¹ • gauge s :=
@@ -320,6 +484,12 @@ section IsROrC
 
 variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
+/- warning: gauge_norm_smul -> gauge_norm_smul is a dubious translation:
+lean 3 declaration is
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))], (Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u2} E _inst_1 _inst_2 s (SMul.smul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))) r) x)) (gauge.{u2} E _inst_1 _inst_2 s (SMul.smul.{u1, u2} 𝕜 E (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) r x)))
+but is expected to have type
+  forall {𝕜 : Type.{u2}} {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : IsROrC.{u2} 𝕜] [_inst_4 : Module.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_5 : IsScalarTower.{0, u2, u1} Real 𝕜 E (Algebra.toSMul.{0, u2} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u2} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u2} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))], (Balanced.{u2, u1} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{0, u1, u1} Real E E (instHSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (Norm.norm.{u2} 𝕜 (NormedField.toNorm.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))) r) x)) (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{u2, u1, u1} 𝕜 E E (instHSMul.{u2, u1} 𝕜 E (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4))))) r x)))
+Case conversion may be inaccurate. Consider using '#align gauge_norm_smul gauge_norm_smulₓ'. -/
 theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖r‖ • x) = gauge s (r • x) :=
   by
   unfold gauge
@@ -329,6 +499,12 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖
   rw [IsROrC.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 
+/- warning: gauge_smul -> gauge_smul is a dubious translation:
+lean 3 declaration is
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))], (Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u2} E _inst_1 _inst_2 s (SMul.smul.{u1, u2} 𝕜 E (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) r x)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) (Norm.norm.{u1} 𝕜 (NormedField.toHasNorm.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))) r) (gauge.{u2} E _inst_1 _inst_2 s x)))
+but is expected to have type
+  forall {𝕜 : Type.{u2}} {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : IsROrC.{u2} 𝕜] [_inst_4 : Module.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_5 : IsScalarTower.{0, u2, u1} Real 𝕜 E (Algebra.toSMul.{0, u2} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u2} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u2} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))], (Balanced.{u2, u1} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u2} 𝕜 (NormedCommRing.toSeminormedCommRing.{u2} 𝕜 (NormedField.toNormedCommRing.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4)))) s) -> (forall (r : 𝕜) (x : E), Eq.{1} Real (gauge.{u1} E _inst_1 _inst_2 s (HSMul.hSMul.{u2, u1, u1} 𝕜 E E (instHSMul.{u2, u1} 𝕜 E (SMulZeroClass.toSMul.{u2, u1} 𝕜 E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u2, u1} 𝕜 E (CommMonoidWithZero.toZero.{u2} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u2} 𝕜 (Semifield.toCommGroupWithZero.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u2, u1} 𝕜 E (Semiring.toMonoidWithZero.{u2} 𝕜 (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{u2, u1} 𝕜 E (DivisionSemiring.toSemiring.{u2} 𝕜 (Semifield.toDivisionSemiring.{u2} 𝕜 (Field.toSemifield.{u2} 𝕜 (NormedField.toField.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_4))))) r x)) (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) (Norm.norm.{u2} 𝕜 (NormedField.toNorm.{u2} 𝕜 (DenselyNormedField.toNormedField.{u2} 𝕜 (IsROrC.toDenselyNormedField.{u2} 𝕜 _inst_3))) r) (gauge.{u1} E _inst_1 _inst_2 s x)))
+Case conversion may be inaccurate. Consider using '#align gauge_smul gauge_smulₓ'. -/
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/
 theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x) = ‖r‖ * gauge s x :=
   by
@@ -342,6 +518,12 @@ section TopologicalSpace
 
 variable [TopologicalSpace E] [ContinuousSMul ℝ E]
 
+/- warning: interior_subset_gauge_lt_one -> interior_subset_gauge_lt_one is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3] (s : Set.{u1} E), HasSubset.Subset.{u1} (Set.{u1} E) (Set.hasSubset.{u1} E) (interior.{u1} E _inst_3 s) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3] (s : Set.{u1} E), HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) (interior.{u1} E _inst_3 s) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))
+Case conversion may be inaccurate. Consider using '#align interior_subset_gauge_lt_one interior_subset_gauge_lt_oneₓ'. -/
 theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s x < 1 } :=
   by
   intro x hx
@@ -364,6 +546,12 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s
       (hε ⟨(sub_le_self _ hε₀.le).trans ((le_add_iff_nonneg_right _).2 hε₀.le), le_rfl⟩)
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
 
+/- warning: gauge_lt_one_eq_self_of_open -> gauge_lt_one_eq_self_of_open is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3], (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (IsOpen.{u1} E _inst_3 s) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))))) s)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3], (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (IsOpen.{u1} E _inst_3 s) -> (Eq.{succ u1} (Set.{u1} E) (setOf.{u1} E (fun (x : E) => LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)))) s)
+Case conversion may be inaccurate. Consider using '#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_openₓ'. -/
 theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
     { x | gauge s x < 1 } = s :=
   by
@@ -372,10 +560,22 @@ theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈
   exact hs₂.interior_eq.symm
 #align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
 
+/- warning: gauge_lt_one_of_mem_of_open -> gauge_lt_one_of_mem_of_open is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3], (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (IsOpen.{u1} E _inst_3 s) -> (forall {x : E}, (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x s) -> (LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3], (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (IsOpen.{u1} E _inst_3 s) -> (forall {x : E}, (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x s) -> (LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))))
+Case conversion may be inaccurate. Consider using '#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_openₓ'. -/
 theorem gauge_lt_one_of_mem_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) {x : E}
     (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx
 #align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_open
 
+/- warning: gauge_lt_of_mem_smul -> gauge_lt_of_mem_smul is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3] (x : E) (ε : Real), (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) ε) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) s) -> (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (IsOpen.{u1} E _inst_3 s) -> (Membership.Mem.{u1, u1} E (Set.{u1} E) (Set.hasMem.{u1} E) x (SMul.smul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) ε s)) -> (LT.lt.{0} Real Real.hasLt (gauge.{u1} E _inst_1 _inst_2 s x) ε)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E} [_inst_3 : TopologicalSpace.{u1} E] [_inst_4 : ContinuousSMul.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_3] (x : E) (ε : Real), (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) ε) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) s) -> (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (IsOpen.{u1} E _inst_3 s) -> (Membership.mem.{u1, u1} E (Set.{u1} E) (Set.instMembershipSet.{u1} E) x (HSMul.hSMul.{0, u1, u1} Real (Set.{u1} E) (Set.{u1} E) (instHSMul.{0, u1} Real (Set.{u1} E) (Set.smulSet.{0, u1} Real E (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))) ε s)) -> (LT.lt.{0} Real Real.instLTReal (gauge.{u1} E _inst_1 _inst_2 s x) ε)
+Case conversion may be inaccurate. Consider using '#align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ'. -/
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
     (hs₂ : IsOpen s) (hx : x ∈ ε • s) : gauge s x < ε :=
   by
@@ -388,6 +588,12 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E)
 
 end TopologicalSpace
 
+/- warning: gauge_add_le -> gauge_add_le is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (x : E) (y : E), LE.le.{0} Real Real.hasLe (gauge.{u1} E _inst_1 _inst_2 s (HAdd.hAdd.{u1, u1, u1} E E E (instHAdd.{u1} E (AddZeroClass.toHasAdd.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))) x y)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (gauge.{u1} E _inst_1 _inst_2 s x) (gauge.{u1} E _inst_1 _inst_2 s y)))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] {s : Set.{u1} E}, (Convex.{0, u1} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (Absorbent.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) s) -> (forall (x : E) (y : E), LE.le.{0} Real Real.instLEReal (gauge.{u1} E _inst_1 _inst_2 s (HAdd.hAdd.{u1, u1, u1} E E E (instHAdd.{u1} E (AddZeroClass.toAdd.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))) x y)) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (gauge.{u1} E _inst_1 _inst_2 s x) (gauge.{u1} E _inst_1 _inst_2 s y)))
+Case conversion may be inaccurate. Consider using '#align gauge_add_le gauge_add_leₓ'. -/
 theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
     gauge s (x + y) ≤ gauge s x + gauge s y :=
   by
@@ -410,20 +616,34 @@ section IsROrC
 
 variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
+#print gaugeSeminorm /-
 /-- `gauge s` as a seminorm when `s` is  balanced, convex and absorbent. -/
 @[simps]
 def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Absorbent ℝ s) : Seminorm 𝕜 E :=
   Seminorm.of (gauge s) (gauge_add_le hs₁ hs₂) (gauge_smul hs₀)
 #align gauge_seminorm gaugeSeminorm
+-/
 
 variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E]
   [ContinuousSMul ℝ E]
 
+/- warning: gauge_seminorm_lt_one_of_open -> gaugeSeminorm_lt_one_of_open is a dubious translation:
+lean 3 declaration is
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (forall {x : E}, (Membership.Mem.{u2, u2} E (Set.{u2} E) (Set.hasMem.{u2} E) x s) -> (LT.lt.{0} Real Real.hasLt (coeFn.{succ u2, succ u2} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) (fun (_x : Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) => E -> Real) (Seminorm.hasCoeToFun.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) x) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))))
+but is expected to have type
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (Algebra.toSMul.{0, u1} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u1} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (forall {x : E}, (Membership.mem.{u2, u2} E (Set.{u2} E) (Set.instMembershipSet.{u2} E) x s) -> (LT.lt.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) x) Real.instLTReal (FunLike.coe.{succ u2, succ u2, 1} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E (fun (_x : E) => (fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) _x) (SubadditiveHomClass.toFunLike.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddZeroClass.toAdd.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_1))))) (AddZeroClass.toAdd.{0} Real (AddMonoid.toAddZeroClass.{0} Real (AddCommMonoid.toAddMonoid.{0} Real (OrderedAddCommMonoid.toAddCommMonoid.{0} Real Real.orderedAddCommMonoid)))) (Preorder.toLE.{0} Real (PartialOrder.toPreorder.{0} Real (OrderedAddCommMonoid.toPartialOrder.{0} Real Real.orderedAddCommMonoid))) (AddGroupSeminormClass.toAddLEAddHomClass.{u2, u2, 0} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) E Real (AddCommGroup.toAddGroup.{u2} E _inst_1) Real.orderedAddCommMonoid (SeminormClass.toAddGroupSeminormClass.{u2, u1, u2} (Seminorm.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4))))) 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (Seminorm.instSeminormClass.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (AddCommGroup.toAddGroup.{u2} E _inst_1) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))))))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) x) (OfNat.ofNat.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) x) 1 (One.toOfNat1.{0} ((fun (a._@.Mathlib.Analysis.Seminorm._hyg.840 : E) => Real) x) Real.instOneReal))))
+Case conversion may be inaccurate. Consider using '#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_openₓ'. -/
 theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
   gauge_lt_one_of_mem_of_open hs₁ hs₂.zero_mem hs hx
 #align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
 
+/- warning: gauge_seminorm_ball_one -> gaugeSeminorm_ball_one is a dubious translation:
+lean 3 declaration is
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (SMulZeroClass.toHasSmul.{0, u1} Real 𝕜 (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (SMulWithZero.toSmulZeroClass.{0, u1} Real 𝕜 (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real 𝕜 (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField))))) (AddZeroClass.toHasZero.{u1} 𝕜 (AddMonoid.toAddZeroClass.{u1} 𝕜 (AddCommMonoid.toAddMonoid.{u1} 𝕜 (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))))) (Module.toMulActionWithZero.{0, u1} Real 𝕜 (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real Real.normedField)))) (AddCommGroup.toAddCommMonoid.{u1} 𝕜 (SeminormedAddCommGroup.toAddCommGroup.{u1} 𝕜 (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))))) (NormedSpace.toModule.{0, u1} Real 𝕜 Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{u1} 𝕜 (NonUnitalNormedRing.toNonUnitalSeminormedRing.{u1} 𝕜 (NormedRing.toNonUnitalNormedRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NormedAlgebra.toNormedSpace'.{0, u1} Real Real.normedField 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toHasSmul.{0, u2} Real E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u2} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (Eq.{succ u2} (Set.{u2} E) (Seminorm.ball.{u1, u2} 𝕜 E (SeminormedCommRing.toSemiNormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) _inst_1 (SMulZeroClass.toHasSmul.{u1, u2} 𝕜 E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{u1, u2} 𝕜 E (MulZeroClass.toHasZero.{u1} 𝕜 (MulZeroOneClass.toMulZeroClass.{u1} 𝕜 (MonoidWithZero.toMulZeroOneClass.{u1} 𝕜 (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (AddCommMonoid.toAddMonoid.{u2} E (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (Ring.toSemiring.{u1} 𝕜 (NormedRing.toRing.{u1} 𝕜 (NormedCommRing.toNormedRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) (OfNat.ofNat.{u2} E 0 (OfNat.mk.{u2} E 0 (Zero.zero.{u2} E (AddZeroClass.toHasZero.{u2} E (AddMonoid.toAddZeroClass.{u2} E (SubNegMonoid.toAddMonoid.{u2} E (AddGroup.toSubNegMonoid.{u2} E (AddCommGroup.toAddGroup.{u2} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) s)
+but is expected to have type
+  forall {𝕜 : Type.{u1}} {E : Type.{u2}} [_inst_1 : AddCommGroup.{u2} E] [_inst_2 : Module.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] {s : Set.{u2} E} [_inst_3 : IsROrC.{u1} 𝕜] [_inst_4 : Module.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1)] [_inst_5 : IsScalarTower.{0, u1, u2} Real 𝕜 E (Algebra.toSMul.{0, u1} Real 𝕜 Real.instCommSemiringReal (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (NormedAlgebra.toAlgebra.{0, u1} Real 𝕜 Real.normedField (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (IsROrC.toNormedAlgebra.{u1} 𝕜 _inst_3))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2))))] {hs₀ : Balanced.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) s} {hs₁ : Convex.{0, u2} Real E Real.orderedSemiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} {hs₂ : Absorbent.{0, u2} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) s} [_inst_6 : TopologicalSpace.{u2} E] [_inst_7 : ContinuousSMul.{0, u2} Real E (SMulZeroClass.toSMul.{0, u2} Real E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u2} Real E Real.instZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u2} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{0, u2} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_2)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) _inst_6], (IsOpen.{u2} E _inst_6 s) -> (Eq.{succ u2} (Set.{u2} E) (Seminorm.ball.{u1, u2} 𝕜 E (SeminormedCommRing.toSeminormedRing.{u1} 𝕜 (NormedCommRing.toSeminormedCommRing.{u1} 𝕜 (NormedField.toNormedCommRing.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))) _inst_1 (SMulZeroClass.toSMul.{u1, u2} 𝕜 E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{u1, u2} 𝕜 E (CommMonoidWithZero.toZero.{u1} 𝕜 (CommGroupWithZero.toCommMonoidWithZero.{u1} 𝕜 (Semifield.toCommGroupWithZero.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{u1, u2} 𝕜 E (Semiring.toMonoidWithZero.{u1} 𝕜 (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3))))))) (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))) (Module.toMulActionWithZero.{u1, u2} 𝕜 E (DivisionSemiring.toSemiring.{u1} 𝕜 (Semifield.toDivisionSemiring.{u1} 𝕜 (Field.toSemifield.{u1} 𝕜 (NormedField.toField.{u1} 𝕜 (DenselyNormedField.toNormedField.{u1} 𝕜 (IsROrC.toDenselyNormedField.{u1} 𝕜 _inst_3)))))) (AddCommGroup.toAddCommMonoid.{u2} E _inst_1) _inst_4)))) (gaugeSeminorm.{u1, u2} 𝕜 E _inst_1 _inst_2 s _inst_3 _inst_4 _inst_5 hs₀ hs₁ hs₂) (OfNat.ofNat.{u2} E 0 (Zero.toOfNat0.{u2} E (NegZeroClass.toZero.{u2} E (SubNegZeroMonoid.toNegZeroClass.{u2} E (SubtractionMonoid.toSubNegZeroMonoid.{u2} E (SubtractionCommMonoid.toSubtractionMonoid.{u2} E (AddCommGroup.toDivisionAddCommMonoid.{u2} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) s)
+Case conversion may be inaccurate. Consider using '#align gauge_seminorm_ball_one gaugeSeminorm_ball_oneₓ'. -/
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s :=
   by
   rw [Seminorm.ball_zero_eq]
@@ -432,6 +652,12 @@ theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs
 
 end IsROrC
 
+/- warning: seminorm.gauge_ball -> Seminorm.gauge_ball is a dubious translation:
+lean 3 declaration is
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(Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) p)
+but is expected to have type
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_inst_1) Real.orderedAddCommMonoid (SeminormClass.toAddGroupSeminormClass.{u1, 0, u1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) (Seminorm.instSeminormClass.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))))))) p)
+Case conversion may be inaccurate. Consider using '#align seminorm.gauge_ball Seminorm.gauge_ballₓ'. -/
 /-- Any seminorm arises as the gauge of its unit ball. -/
 @[simp]
 protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1) = p :=
@@ -458,6 +684,12 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     exact lt_add_of_pos_right _ hε
 #align seminorm.gauge_ball Seminorm.gauge_ball
 
+/- warning: seminorm.gauge_seminorm_ball -> Seminorm.gaugeSeminorm_ball is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] (p : Seminorm.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real Real.semiring)))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real Real.semiring) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))), Eq.{succ u1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real (NormedField.toNormedCommRing.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC))))) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC)))))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E (Ring.toSemiring.{0} Real (NormedRing.toRing.{0} Real (NormedCommRing.toNormedRing.{0} Real (NormedField.toNormedCommRing.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (gaugeSeminorm.{0, u1} Real E _inst_1 _inst_2 (Seminorm.ball.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 (SMulZeroClass.toHasSmul.{0, u1} Real E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (SMulWithZero.toSmulZeroClass.{0, u1} Real E (MulZeroClass.toHasZero.{0} Real (MulZeroOneClass.toMulZeroClass.{0} Real (MonoidWithZero.toMulZeroOneClass.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing))))) (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) p (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) Real.isROrC _inst_2 (IsScalarTower.left.{0, u1} Real E Real.monoid (MulActionWithZero.toMulAction.{0, u1} Real E Real.monoidWithZero (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) (Seminorm.balanced_ball_zero.{0, u1} Real E (SeminormedCommRing.toSemiNormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) (Seminorm.convex_ball.{0, u1} Real E Real.normedField _inst_1 (NormedField.toNormedSpace.{0} Real Real.normedField) _inst_2 _inst_2 (IsScalarTower.left.{0, u1} Real E Real.monoid (MulActionWithZero.toMulAction.{0, u1} Real E Real.monoidWithZero (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (AddCommMonoid.toAddMonoid.{u1} E (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) p (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (AddCommGroup.toAddGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) (Seminorm.absorbent_ball_zero.{0, u1} Real E Real.normedField _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (zero_lt_one.{0} Real Real.hasZero Real.hasOne Real.partialOrder (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring) (NeZero.one.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (NonAssocRing.toNonAssocSemiring.{0} Real (Ring.toNonAssocRing.{0} Real Real.ring))) Real.nontrivial)))) p
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : AddCommGroup.{u1} E] [_inst_2 : Module.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1)] (p : Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E Real.instZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))), Eq.{succ u1} (Seminorm.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real (NormedField.toNormedCommRing.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC))))) (AddCommGroup.toAddGroup.{u1} E _inst_1) (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E (CommMonoidWithZero.toZero.{0} Real (CommGroupWithZero.toCommMonoidWithZero.{0} Real (Semifield.toCommGroupWithZero.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC))))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (DenselyNormedField.toNormedField.{0} Real (IsROrC.toDenselyNormedField.{0} Real Real.isROrC)))))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))))) (gaugeSeminorm.{0, u1} Real E _inst_1 _inst_2 (Seminorm.ball.{0, u1} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 (SMulZeroClass.toSMul.{0, u1} Real E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (SMulWithZero.toSMulZeroClass.{0, u1} Real E (MonoidWithZero.toZero.{0} Real (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (MulActionWithZero.toSMulWithZero.{0, u1} Real E (Semiring.toMonoidWithZero.{0} Real (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing))))) (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E (Ring.toSemiring.{0} Real (SeminormedRing.toRing.{0} Real (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2)))) p (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) Real.isROrC _inst_2 (IsScalarTower.left.{0, u1} Real E (MonoidWithZero.toMonoid.{0} Real (Semiring.toMonoidWithZero.{0} Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real Real.normedField)))))) (MulActionWithZero.toMulAction.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) (Seminorm.balanced_ball_zero.{u1, 0} Real E (SeminormedCommRing.toSeminormedRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)) _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) (Seminorm.convex_ball.{0, u1} Real E Real.normedField _inst_1 (NormedField.toNormedSpace.{0} Real Real.normedField) _inst_2 _inst_2 (IsScalarTower.left.{0, u1} Real E (MonoidWithZero.toMonoid.{0} Real Real.instMonoidWithZeroReal) (MulActionWithZero.toMulAction.{0, u1} Real E Real.instMonoidWithZeroReal (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))) (Module.toMulActionWithZero.{0, u1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u1} E _inst_1) _inst_2))) p (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E _inst_1))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) (Seminorm.absorbent_ball_zero.{u1, 0} Real E Real.normedField _inst_1 _inst_2 p (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (zero_lt_one.{0} Real Real.instZeroReal Real.instOneReal Real.partialOrder (OrderedSemiring.zeroLEOneClass.{0} Real Real.orderedSemiring) (NeZero.one.{0} Real (NonAssocSemiring.toMulZeroOneClass.{0} Real (Semiring.toNonAssocSemiring.{0} Real Real.semiring)) Real.nontrivial)))) p
+Case conversion may be inaccurate. Consider using '#align seminorm.gauge_seminorm_ball Seminorm.gaugeSeminorm_ballₓ'. -/
 theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
     gaugeSeminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) =
       p :=
@@ -470,6 +702,12 @@ section Norm
 
 variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E}
 
+/- warning: gauge_unit_ball -> gauge_unit_ball is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (SeminormedAddGroup.toAddGroup.{u1} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u1} E _inst_1))))))))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne)))) x) (Norm.norm.{u1} E (SeminormedAddCommGroup.toHasNorm.{u1} E _inst_1) x)
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1)))))))) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal))) x) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_1) x)
+Case conversion may be inaccurate. Consider using '#align gauge_unit_ball gauge_unit_ballₓ'. -/
 theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ :=
   by
   obtain rfl | hx := eq_or_ne x 0
@@ -490,6 +728,12 @@ theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ :=
     exact lt_irrefl _ h
 #align gauge_unit_ball gauge_unit_ball
 
+/- warning: gauge_ball -> gauge_ball is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {r : Real}, (LT.lt.{0} Real Real.hasLt (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) r) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (SeminormedAddGroup.toAddGroup.{u1} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u1} E _inst_1))))))))) r) x) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (Norm.norm.{u1} E (SeminormedAddCommGroup.toHasNorm.{u1} E _inst_1) x) r))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {r : Real}, (LT.lt.{0} Real Real.instLTReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) r) -> (forall (x : E), Eq.{1} Real (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1)))))))) r) x) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_1) x) r))
+Case conversion may be inaccurate. Consider using '#align gauge_ball gauge_ballₓ'. -/
 theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r :=
   by
   rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
@@ -498,6 +742,12 @@ theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x
   exact fun _ => id
 #align gauge_ball gauge_ball
 
+/- warning: mul_gauge_le_norm -> mul_gauge_le_norm is a dubious translation:
+lean 3 declaration is
+  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {s : Set.{u1} E} {r : Real} {x : E}, (HasSubset.Subset.{u1} (Set.{u1} E) (Set.hasSubset.{u1} E) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (OfNat.mk.{u1} E 0 (Zero.zero.{u1} E (AddZeroClass.toHasZero.{u1} E (AddMonoid.toAddZeroClass.{u1} E (SubNegMonoid.toAddMonoid.{u1} E (AddGroup.toSubNegMonoid.{u1} E (SeminormedAddGroup.toAddGroup.{u1} E (SeminormedAddCommGroup.toSeminormedAddGroup.{u1} E _inst_1))))))))) r) s) -> (LE.le.{0} Real Real.hasLe (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.hasMul) r (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) s x)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toHasNorm.{u1} E _inst_1) x))
+but is expected to have type
+  forall {E : Type.{u1}} [_inst_1 : SeminormedAddCommGroup.{u1} E] [_inst_2 : NormedSpace.{0, u1} Real E Real.normedField _inst_1] {s : Set.{u1} E} {r : Real} {x : E}, (HasSubset.Subset.{u1} (Set.{u1} E) (Set.instHasSubsetSet.{u1} E) (Metric.ball.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E _inst_1) (OfNat.ofNat.{u1} E 0 (Zero.toOfNat0.{u1} E (NegZeroClass.toZero.{u1} E (SubNegZeroMonoid.toNegZeroClass.{u1} E (SubtractionMonoid.toSubNegZeroMonoid.{u1} E (SubtractionCommMonoid.toSubtractionMonoid.{u1} E (AddCommGroup.toDivisionAddCommMonoid.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1)))))))) r) s) -> (LE.le.{0} Real Real.instLEReal (HMul.hMul.{0, 0, 0} Real Real Real (instHMul.{0} Real Real.instMulReal) r (gauge.{u1} E (SeminormedAddCommGroup.toAddCommGroup.{u1} E _inst_1) (NormedSpace.toModule.{0, u1} Real E Real.normedField _inst_1 _inst_2) s x)) (Norm.norm.{u1} E (SeminormedAddCommGroup.toNorm.{u1} E _inst_1) x))
+Case conversion may be inaccurate. Consider using '#align mul_gauge_le_norm mul_gauge_le_normₓ'. -/
 theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤ ‖x‖ :=
   by
   obtain hr | hr := le_or_lt r 0

mathlib commit https://github.com/leanprover-community/mathlib/commit/2f8347015b12b0864dfaf366ec4909eb70c78740

@@ -326,7 +326,7 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖
   congr with θ
   rw [@IsROrC.real_smul_eq_coe_smul 𝕜]
   refine' and_congr_right fun hθ => (hs.smul _).mem_smul_iff _
-  rw [IsROrC.norm_of_real, abs_norm]
+  rw [IsROrC.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module analysis.convex.gauge
-! leanprover-community/mathlib commit 468b141b14016d54b479eb7a0fff1e360b7e3cf6
+! leanprover-community/mathlib commit 3f655f5297b030a87d641ad4e825af8d9679eb0b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -322,13 +322,11 @@ variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
 theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (‖r‖ • x) = gauge s (r • x) :=
   by
-  rw [@IsROrC.real_smul_eq_coe_smul 𝕜]
-  obtain rfl | hr := eq_or_ne r 0
-  · simp only [norm_zero, IsROrC.of_real_zero]
   unfold gauge
   congr with θ
+  rw [@IsROrC.real_smul_eq_coe_smul 𝕜]
   refine' and_congr_right fun hθ => (hs.smul _).mem_smul_iff _
-  rw [IsROrC.norm_of_real, norm_norm]
+  rw [IsROrC.norm_of_real, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/17ad94b4953419f3e3ce3e77da3239c62d1d09f0

@@ -494,7 +494,7 @@ theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ :=
 
 theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r :=
   by
-  rw [← smul_unit_ball_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
+  rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
     abs_of_nonneg hr.le, div_eq_inv_mul]
   simp_rw [mem_ball_zero_iff, norm_neg]
   exact fun _ => id

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -66,13 +66,13 @@ theorem gauge_def : gauge s x = infₛ ({ r ∈ Set.Ioi 0 | x ∈ r • s }) :=
   rfl
 #align gauge_def gauge_def
 
-/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:76:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]] -/
+/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]] -/
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
 theorem gauge_def' : gauge s x = infₛ ({ r ∈ Set.Ioi 0 | r⁻¹ • x ∈ s }) :=
   by
   trace
-    "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:76:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]]"
+    "./././Mathport/Syntax/Translate/Tactic/Builtin.lean:73:14: unsupported tactic `congrm #[[expr Inf (λ r, _)]]"
   exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
 #align gauge_def' gauge_def'
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

@@ -581,7 +581,7 @@ theorem gauge_ball (hr : 0 ≤ r) (x : E) : gauge (ball (0 : E) r) x = ‖x‖ /
     simp_rw [mem_ball_zero_iff, norm_neg]
     exact fun _ => id
 
-@[deprecated gauge_ball]
+@[deprecated gauge_ball] -- 2023-07-24
 theorem gauge_ball' (hr : 0 < r) (x : E) : gauge (ball (0 : E) r) x = ‖x‖ / r :=
   gauge_ball hr.le x
 #align gauge_ball gauge_ball'

@@ -365,7 +365,7 @@ theorem gauge_eq_zero (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s
 
 theorem gauge_pos (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
     0 < gauge s x ↔ x ≠ 0 := by
-  simp only [(gauge_nonneg _).gt_iff_ne, Ne.def, gauge_eq_zero hs hb]
+  simp only [(gauge_nonneg _).gt_iff_ne, Ne, gauge_eq_zero hs hb]
 
 end TopologicalSpace
 

RCLike is an analytic typeclass, hence should be under Analysis

@@ -7,7 +7,7 @@ import Mathlib.Analysis.Convex.Topology
 import Mathlib.Analysis.NormedSpace.Pointwise
 import Mathlib.Analysis.Seminorm
 import Mathlib.Analysis.LocallyConvex.Bounded
-import Mathlib.Data.RCLike.Basic
+import Mathlib.Analysis.RCLike.Basic
 
 #align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
 

IsROrC contains data, which goes against the expectation that classes prefixed with Is are prop-valued. People have been complaining about this on and off, so this PR renames IsROrC to RCLike.

@@ -7,7 +7,7 @@ import Mathlib.Analysis.Convex.Topology
 import Mathlib.Analysis.NormedSpace.Pointwise
 import Mathlib.Analysis.Seminorm
 import Mathlib.Analysis.LocallyConvex.Bounded
-import Mathlib.Data.IsROrC.Basic
+import Mathlib.Data.RCLike.Basic
 
 #align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
 
@@ -318,17 +318,17 @@ theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower
 
 end LinearOrderedField
 
-section IsROrC
+section RCLike
 
-variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
+variable [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
 theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) :
     gauge s (‖r‖ • x) = gauge s (r • x) := by
   unfold gauge
   congr with θ
-  rw [@IsROrC.real_smul_eq_coe_smul 𝕜]
+  rw [@RCLike.real_smul_eq_coe_smul 𝕜]
   refine' and_congr_right fun hθ => (hs.smul _).smul_mem_iff _
-  rw [IsROrC.norm_ofReal, abs_norm]
+  rw [RCLike.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 
 /-- If `s` is balanced, then the Minkowski functional is ℂ-homogeneous. -/
@@ -336,7 +336,7 @@ theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x)
   rw [← smul_eq_mul, ← gauge_smul_of_nonneg (norm_nonneg r), gauge_norm_smul hs]
 #align gauge_smul gauge_smul
 
-end IsROrC
+end RCLike
 
 open Filter
 
@@ -504,9 +504,9 @@ theorem gauge_eq_one_iff_mem_frontier (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
 
 end TopologicalVectorSpace
 
-section IsROrC
+section RCLike
 
-variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
+variable [RCLike 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
 /-- `gauge s` as a seminorm when `s` is balanced, convex and absorbent. -/
 @[simps!]
@@ -527,7 +527,7 @@ theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs
   exact gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs
 #align gauge_seminorm_ball_one gaugeSeminorm_ball_one
 
-end IsROrC
+end RCLike
 
 /-- Any seminorm arises as the gauge of its unit ball. -/
 @[simp]

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -394,13 +394,13 @@ theorem gauge_lt_one_eq_self_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) 
   exact hs₂.interior_eq.symm
 #align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_isOpen
 
--- porting note: droped unneeded assumptions
+-- Porting note: droped unneeded assumptions
 theorem gauge_lt_one_of_mem_of_isOpen (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
     gauge s x < 1 :=
   interior_subset_gauge_lt_one s <| by rwa [hs₂.interior_eq]
 #align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_isOpenₓ
 
--- porting note: droped unneeded assumptions
+-- Porting note: droped unneeded assumptions
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s) (hx : x ∈ ε • s) :
     gauge s x < ε := by
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -379,8 +379,8 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s
   have H₁ : Tendsto (fun r : ℝ ↦ r⁻¹ • x) (𝓝[<] 1) (𝓝 ((1 : ℝ)⁻¹ • x)) :=
     ((tendsto_id.inv₀ one_ne_zero).smul tendsto_const_nhds).mono_left inf_le_left
   rw [inv_one, one_smul] at H₁
-  have H₂ : ∀ᶠ r in 𝓝[<] (1 : ℝ), x ∈ r • s ∧ 0 < r ∧ r < 1
-  · filter_upwards [H₁ (mem_interior_iff_mem_nhds.1 hx), Ioo_mem_nhdsWithin_Iio' one_pos]
+  have H₂ : ∀ᶠ r in 𝓝[<] (1 : ℝ), x ∈ r • s ∧ 0 < r ∧ r < 1 := by
+    filter_upwards [H₁ (mem_interior_iff_mem_nhds.1 hx), Ioo_mem_nhdsWithin_Iio' one_pos]
     intro r h₁ h₂
     exact ⟨(mem_smul_set_iff_inv_smul_mem₀ h₂.1.ne' _ _).2 h₁, h₂⟩
   rcases H₂.exists with ⟨r, hxr, hr₀, hr₁⟩
@@ -411,8 +411,8 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s
 
 theorem mem_closure_of_gauge_le_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s)
     (h : gauge s x ≤ 1) : x ∈ closure s := by
-  have : ∀ᶠ r : ℝ in 𝓝[<] 1, r • x ∈ s
-  · filter_upwards [Ico_mem_nhdsWithin_Iio' one_pos] with r ⟨hr₀, hr₁⟩
+  have : ∀ᶠ r : ℝ in 𝓝[<] 1, r • x ∈ s := by
+    filter_upwards [Ico_mem_nhdsWithin_Iio' one_pos] with r ⟨hr₀, hr₁⟩
     apply gauge_lt_one_subset_self hc hs₀ ha
     rw [mem_setOf_eq, gauge_smul_of_nonneg hr₀]
     exact mul_lt_one_of_nonneg_of_lt_one_left hr₀ hr₁ h
@@ -460,8 +460,8 @@ theorem continuousAt_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuo
   have ha : Absorbent ℝ s := absorbent_nhds_zero hs₀
   refine (nhds_basis_Icc_pos _).tendsto_right_iff.2 fun ε hε₀ ↦ ?_
   rw [← map_add_left_nhds_zero, eventually_map]
-  have : ε • s ∩ -(ε • s) ∈ 𝓝 0
-  · exact inter_mem ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)
+  have : ε • s ∩ -(ε • s) ∈ 𝓝 0 :=
+    inter_mem ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)
       (neg_mem_nhds_zero _ ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀))
   filter_upwards [this] with y hy
   constructor
@@ -603,8 +603,8 @@ theorem gauge_closedBall (hr : 0 ≤ r) (x : E) : gauge (closedBall (0 : E) r) x
   · apply le_antisymm
     · rw [← gauge_ball hr]
       exact gauge_mono (absorbent_ball_zero hr') ball_subset_closedBall x
-    · suffices : ∀ᶠ R in 𝓝[>] r, ‖x‖ / R ≤ gauge (closedBall 0 r) x
-      · refine le_of_tendsto ?_ this
+    · suffices ∀ᶠ R in 𝓝[>] r, ‖x‖ / R ≤ gauge (closedBall 0 r) x by
+        refine le_of_tendsto ?_ this
         exact tendsto_const_nhds.div inf_le_left hr'.ne'
       filter_upwards [self_mem_nhdsWithin] with R hR
       rw [← gauge_ball (hr.trans hR.out.le)]

... to avoid conflict with _root_.map_nonneg, see Zulip.

@@ -536,7 +536,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
   obtain hp | hp := { r : ℝ | 0 < r ∧ x ∈ r • p.ball 0 1 }.eq_empty_or_nonempty
   · rw [gauge, hp, Real.sInf_empty]
     by_contra h
-    have hpx : 0 < p x := (map_nonneg _ _).lt_of_ne h
+    have hpx : 0 < p x := (apply_nonneg _ _).lt_of_ne h
     have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx
     refine' hp.subset ⟨hpx₂, (2 * p x)⁻¹ • x, _, smul_inv_smul₀ hpx₂.ne' _⟩
     rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpx₂),
@@ -549,7 +549,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     exact mul_le_of_le_one_right hr.le hy.le
   · have hpε : 0 < p x + ε :=
       -- Porting note: was `by positivity`
-      add_pos_of_nonneg_of_pos (map_nonneg _ _) hε
+      add_pos_of_nonneg_of_pos (apply_nonneg _ _) hε
     refine' hr ⟨hpε, (p x + ε)⁻¹ • x, _, smul_inv_smul₀ hpε.ne' _⟩
     rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpε),
       inv_mul_lt_iff hpε, mul_one]

• add balanced_iff_closedBall_smul, balanced_neg;
• generalize Balanced.neg_mem_iff to a SeminormedRing + NormOneClass, add Balanced.neg_eq
• add Balanced.smul_mem_mono and Balanced.smul_congr;
• rename Balanced.mem_smul_iff to Balanced.smul_mem_iff;
• rename balanced_zero_union_interior to Balanced.zero_insert_interior, use insert 0 (interior A) instead of 0 ∪ interior A;
• make Balanced.interior and Balanced.closure protected;
• deprecate Absorbs.zero_mem';
• rename balanced_convexHull_of_balanced to Balanced.convexHull;
• add absorbs_iff_eventually_cobounded_mapsTo, use it to golf some proofs.

@@ -327,7 +327,7 @@ theorem gauge_norm_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) :
   unfold gauge
   congr with θ
   rw [@IsROrC.real_smul_eq_coe_smul 𝕜]
-  refine' and_congr_right fun hθ => (hs.smul _).mem_smul_iff _
+  refine' and_congr_right fun hθ => (hs.smul _).smul_mem_iff _
   rw [IsROrC.norm_ofReal, abs_norm]
 #align gauge_norm_smul gauge_norm_smul
 

Add comap_gauge_nhds_zero, comap_gauge_nhds_zero_le, tendsto_gauge_nhds_zero, tendsto_gauge_nhds_zero', and continuousAt_gauge_zero.

@@ -338,8 +338,39 @@ theorem gauge_smul (hs : Balanced 𝕜 s) (r : 𝕜) (x : E) : gauge s (r • x)
 
 end IsROrC
 
+open Filter
+
 section TopologicalSpace
 
+variable [TopologicalSpace E]
+
+theorem comap_gauge_nhds_zero_le (ha : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
+    comap (gauge s) (𝓝 0) ≤ 𝓝 0 := fun u hu ↦ by
+  rcases (hb hu).exists_pos with ⟨r, hr₀, hr⟩
+  filter_upwards [preimage_mem_comap (gt_mem_nhds (inv_pos.2 hr₀))] with x (hx : gauge s x < r⁻¹)
+  rcases exists_lt_of_gauge_lt ha hx with ⟨c, hc₀, hcr, y, hy, rfl⟩
+  have hrc := (lt_inv hr₀ hc₀).2 hcr
+  rcases hr c⁻¹ (hrc.le.trans (le_abs_self _)) hy with ⟨z, hz, rfl⟩
+  simpa only [smul_inv_smul₀ hc₀.ne']
+
+variable [T1Space E]
+
+theorem gauge_eq_zero (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
+    gauge s x = 0 ↔ x = 0 := by
+  refine ⟨fun h₀ ↦ by_contra fun (hne : x ≠ 0) ↦ ?_, fun h ↦ h.symm ▸ gauge_zero⟩
+  have : {x}ᶜ ∈ comap (gauge s) (𝓝 0) :=
+    comap_gauge_nhds_zero_le hs hb (isOpen_compl_singleton.mem_nhds hne.symm)
+  rcases ((nhds_basis_zero_abs_sub_lt _).comap _).mem_iff.1 this with ⟨r, hr₀, hr⟩
+  exact hr (by simpa [h₀]) rfl
+
+theorem gauge_pos (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
+    0 < gauge s x ↔ x ≠ 0 := by
+  simp only [(gauge_nonneg _).gt_iff_ne, Ne.def, gauge_eq_zero hs hb]
+
+end TopologicalSpace
+
+section ContinuousSMul
+
 variable [TopologicalSpace E] [ContinuousSMul ℝ E]
 
 open Filter in
@@ -394,9 +425,29 @@ theorem mem_frontier_of_gauge_eq_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha :
   ⟨mem_closure_of_gauge_le_one hc hs₀ ha h.le, fun h' ↦
     (interior_subset_gauge_lt_one s h').out.ne h⟩
 
-end TopologicalSpace
+theorem tendsto_gauge_nhds_zero' (hs : s ∈ 𝓝 0) : Tendsto (gauge s) (𝓝 0) (𝓝[≥] 0) := by
+  refine nhdsWithin_Ici_basis_Icc.tendsto_right_iff.2 fun ε hε ↦ ?_
+  rw [← set_smul_mem_nhds_zero_iff hε.ne'] at hs
+  filter_upwards [hs] with x hx
+  exact ⟨gauge_nonneg _, gauge_le_of_mem hε.le hx⟩
+
+theorem tendsto_gauge_nhds_zero (hs : s ∈ 𝓝 0) : Tendsto (gauge s) (𝓝 0) (𝓝 0) :=
+  (tendsto_gauge_nhds_zero' hs).mono_right inf_le_left
+
+/-- If `s` is a neighborhood of the origin, then `gauge s` is continuous at the origin.
+See also `continuousAt_gauge`. -/
+theorem continuousAt_gauge_zero (hs : s ∈ 𝓝 0) : ContinuousAt (gauge s) 0 := by
+  rw [ContinuousAt, gauge_zero]
+  exact tendsto_gauge_nhds_zero hs
+
+theorem comap_gauge_nhds_zero (hb : Bornology.IsVonNBounded ℝ s) (h₀ : s ∈ 𝓝 0) :
+    comap (gauge s) (𝓝 0) = 𝓝 0 :=
+  (comap_gauge_nhds_zero_le (absorbent_nhds_zero h₀) hb).antisymm
+    (tendsto_gauge_nhds_zero h₀).le_comap
 
-section TopologicalAddGroup
+end ContinuousSMul
+
+section TopologicalVectorSpace
 
 open Filter
 
@@ -451,24 +502,7 @@ theorem gauge_eq_one_iff_mem_frontier (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
   rw [eq_iff_le_not_lt, gauge_le_one_iff_mem_closure hc hs₀, gauge_lt_one_iff_mem_interior hc hs₀]
   rfl
 
-theorem gauge_eq_zero [T1Space E] (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
-    gauge s x = 0 ↔ x = 0 := by
-  refine ⟨not_imp_not.1 fun (h : x ≠ 0) ↦ ne_of_gt ?_, fun h ↦ h.symm ▸ gauge_zero⟩
-  rcases (hb (isOpen_compl_singleton.mem_nhds h.symm)).exists_pos with ⟨c, hc₀, hc⟩
-  refine (inv_pos.2 hc₀).trans_le <| le_csInf hs.gauge_set_nonempty ?_
-  rintro r ⟨hr₀, x, hx, rfl⟩
-  contrapose! hc
-  refine ⟨r⁻¹, ?_, fun h ↦ ?_⟩
-  · rw [norm_inv, Real.norm_of_nonneg hr₀.le, le_inv hc₀ hr₀]
-    exact hc.le
-  · rcases h hx with ⟨y, hy, rfl⟩
-    simp [hr₀.ne'] at hy
-
-theorem gauge_pos [T1Space E] (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
-    0 < gauge s x ↔ x ≠ 0 := by
-  simp only [(gauge_nonneg _).gt_iff_ne, Ne.def, gauge_eq_zero hs hb]
-
-end TopologicalAddGroup
+end TopologicalVectorSpace
 
 section IsROrC
 

@@ -405,10 +405,9 @@ variable [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E]
 /-- If `s` is a convex neighborhood of the origin in a topological real vector space, then `gauge s`
 is continuous. If the ambient space is a normed space, then `gauge s` is Lipschitz continuous, see
 `Convex.lipschitz_gauge`. -/
-theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous (gauge s) := by
+theorem continuousAt_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : ContinuousAt (gauge s) x := by
   have ha : Absorbent ℝ s := absorbent_nhds_zero hs₀
-  simp only [continuous_iff_continuousAt, ContinuousAt, (nhds_basis_Icc_pos _).tendsto_right_iff]
-  intro x ε hε₀
+  refine (nhds_basis_Icc_pos _).tendsto_right_iff.2 fun ε hε₀ ↦ ?_
   rw [← map_add_left_nhds_zero, eventually_map]
   have : ε • s ∩ -(ε • s) ∈ 𝓝 0
   · exact inter_mem ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)
@@ -424,6 +423,13 @@ theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous
       gauge s (x + y) ≤ gauge s x + gauge s y := gauge_add_le hc ha _ _
       _ ≤ gauge s x + ε := add_le_add_left (gauge_le_of_mem hε₀.le hy.1) _
 
+/-- If `s` is a convex neighborhood of the origin in a topological real vector space, then `gauge s`
+is continuous. If the ambient space is a normed space, then `gauge s` is Lipschitz continuous, see
+`Convex.lipschitz_gauge`. -/
+@[continuity]
+theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous (gauge s) :=
+  continuous_iff_continuousAt.2 fun _ ↦ continuousAt_gauge hc hs₀
+
 theorem gauge_lt_one_eq_interior (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
     { x | gauge s x < 1 } = interior s := by
   refine Subset.antisymm (fun x hx ↦ ?_) (interior_subset_gauge_lt_one s)

This prepares for the introduction of a non-dependent synonym of FunLike, which helps a lot with keeping #8386 readable.

This is entirely search-and-replace in 680197f combined with manual fixes in 4145626, e900597 and b8428f8. The commands that generated this change:

sed -i 's/\bFunLike\b/DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoFunLike\b/toDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/import Mathlib.Data.DFunLike/import Mathlib.Data.FunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bHom_FunLike\b/Hom_DFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean     
sed -i 's/\binstFunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\bfunLike\b/instDFunLike/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean
sed -i 's/\btoo many metavariables to apply `fun_like.has_coe_to_fun`/too many metavariables to apply `DFunLike.hasCoeToFun`/g' {Archive,Counterexamples,Mathlib,test}/**/*.lean

Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

@@ -519,7 +519,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
 theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
     gaugeSeminorm (p.balanced_ball_zero 1) (p.convex_ball 0 1) (p.absorbent_ball_zero zero_lt_one) =
       p :=
-  FunLike.coe_injective p.gauge_ball
+  DFunLike.coe_injective p.gauge_ball
 #align seminorm.gauge_seminorm_ball Seminorm.gaugeSeminorm_ball
 
 end AddCommGroup

Redefine Absorbs and Absorbent in terms of the cobounded filter.

@@ -75,8 +75,8 @@ private theorem gauge_set_bddBelow : BddBelow { r : ℝ | 0 < r ∧ x ∈ r •
 which is useful for proving many properties about the gauge.  -/
 theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
     { r : ℝ | 0 < r ∧ x ∈ r • s }.Nonempty :=
-  let ⟨r, hr₁, hr₂⟩ := absorbs x
-  ⟨r, hr₁, hr₂ r (Real.norm_of_nonneg hr₁.le).ge⟩
+  let ⟨r, hr₁, hr₂⟩ := (absorbs x).exists_pos
+  ⟨r, hr₁, hr₂ r (Real.norm_of_nonneg hr₁.le).ge rfl⟩
 #align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonempty
 
 theorem gauge_mono (hs : Absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s := fun _ =>
@@ -233,7 +233,7 @@ theorem Balanced.starConvex (hs : Balanced ℝ s) : StarConvex ℝ 0 s :=
 theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s {x}) (hx : x ∉ a • s) :
     a ≤ gauge s x := by
   rw [starConvex_zero_iff] at hs₀
-  obtain ⟨r, hr, h⟩ := hs₂
+  obtain ⟨r, hr, h⟩ := hs₂.exists_pos
   refine' le_csInf ⟨r, hr, singleton_subset_iff.1 <| h _ (Real.norm_of_nonneg hr.le).ge⟩ _
   rintro b ⟨hb, x, hx', rfl⟩
   refine' not_lt.1 fun hba => hx _
@@ -448,7 +448,7 @@ theorem gauge_eq_one_iff_mem_frontier (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
 theorem gauge_eq_zero [T1Space E] (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
     gauge s x = 0 ↔ x = 0 := by
   refine ⟨not_imp_not.1 fun (h : x ≠ 0) ↦ ne_of_gt ?_, fun h ↦ h.symm ▸ gauge_zero⟩
-  rcases hb (isOpen_compl_singleton.mem_nhds h.symm) with ⟨c, hc₀, hc⟩
+  rcases (hb (isOpen_compl_singleton.mem_nhds h.symm)).exists_pos with ⟨c, hc₀, hc⟩
   refine (inv_pos.2 hc₀).trans_le <| le_csInf hs.gauge_set_nonempty ?_
   rintro r ⟨hr₀, x, hx, rfl⟩
   contrapose! hc

After this PR, no file in Geometry uses autoImplicit, and in Analysis it's scoped to six declarations.

@@ -38,9 +38,6 @@ For a real vector space,
 Minkowski functional, gauge
 -/
 
-set_option autoImplicit true
-
-
 open NormedField Set
 open scoped Pointwise Topology NNReal
 
@@ -58,7 +55,7 @@ def gauge (s : Set E) (x : E) : ℝ :=
   sInf { r : ℝ | 0 < r ∧ x ∈ r • s }
 #align gauge gauge
 
-variable {s t : Set E} {a : ℝ}
+variable {s t : Set E} {x : E} {a : ℝ}
 
 theorem gauge_def : gauge s x = sInf ({ r ∈ Set.Ioi (0 : ℝ) | x ∈ r • s }) :=
   rfl

Mostly, this means replacing "of_open" by "of_isOpen". A few lemmas names were misleading and are corrected differently. Zulip discussion.

@@ -359,24 +359,24 @@ theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s
   exact (gauge_le_of_mem hr₀.le hxr).trans_lt hr₁
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
 
-theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
+theorem gauge_lt_one_eq_self_of_isOpen (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
     { x | gauge s x < 1 } = s := by
   refine' (gauge_lt_one_subset_self hs₁ ‹_› <| absorbent_nhds_zero <| hs₂.mem_nhds hs₀).antisymm _
   convert interior_subset_gauge_lt_one s
   exact hs₂.interior_eq.symm
-#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
+#align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_isOpen
 
 -- porting note: droped unneeded assumptions
-theorem gauge_lt_one_of_mem_of_open (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
+theorem gauge_lt_one_of_mem_of_isOpen (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
     gauge s x < 1 :=
   interior_subset_gauge_lt_one s <| by rwa [hs₂.interior_eq]
-#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_openₓ
+#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_isOpenₓ
 
 -- porting note: droped unneeded assumptions
 theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s) (hx : x ∈ ε • s) :
     gauge s x < ε := by
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
-  have h_gauge_lt := gauge_lt_one_of_mem_of_open hs₂ this
+  have h_gauge_lt := gauge_lt_one_of_mem_of_isOpen hs₂ this
   rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one]
     at h_gauge_lt
 #align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ
@@ -480,14 +480,14 @@ def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Abso
 variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [TopologicalSpace E]
   [ContinuousSMul ℝ E]
 
-theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
+theorem gaugeSeminorm_lt_one_of_isOpen (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
-  gauge_lt_one_of_mem_of_open hs hx
-#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
+  gauge_lt_one_of_mem_of_isOpen hs hx
+#align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_isOpen
 
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s := by
   rw [Seminorm.ball_zero_eq]
-  exact gauge_lt_one_eq_self_of_open hs₁ hs₂.zero_mem hs
+  exact gauge_lt_one_eq_self_of_isOpen hs₁ hs₂.zero_mem hs
 #align gauge_seminorm_ball_one gaugeSeminorm_ball_one
 
 end IsROrC

Adds a term elaborator for congr(...) "congruence quotations". For example, if hf : f = f' and hx : x = x', then we have congr($hf $x) : f x = f' x'. This supports the functions having implicit arguments, and it has support for subsingleton instance arguments. So for example, if s t : Set X are sets with Fintype instances and h : s = t then congr(Fintype.card $h) : Fintype.card s = Fintype.card t works.

Ports the congrm tactic as a convenient frontend for applying a congruence quotation to the goal. Holes are turned into congruence holes. For example, congrm 1 + ?_ uses congr(1 + $(?_)). Placeholders (_) do not turn into congruence holes; that's not to say they have to be identical on the LHS and RHS, but congrm itself is responsible for finding a congruence lemma for such arguments.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Moritz Doll <moritz.doll@googlemail.com>

@@ -66,12 +66,8 @@ theorem gauge_def : gauge s x = sInf ({ r ∈ Set.Ioi (0 : ℝ) | x ∈ r • s
 
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
-theorem gauge_def' : gauge s x = sInf ({ r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s }) := by
-  -- Porting note: used `congrm`
-  rw [gauge]
-  apply congr_arg
-  ext
-  simp only [mem_setOf, mem_Ioi]
+theorem gauge_def' : gauge s x = sInf {r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s} := by
+  congrm sInf {r | ?_}
   exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
 #align gauge_def' gauge_def'
 

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -38,6 +38,8 @@ For a real vector space,
 Minkowski functional, gauge
 -/
 
+set_option autoImplicit true
+
 
 open NormedField Set
 open scoped Pointwise Topology NNReal

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -44,7 +44,7 @@ open scoped Pointwise Topology NNReal
 
 noncomputable section
 
-variable {𝕜 E F : Type _}
+variable {𝕜 E F : Type*}
 
 section AddCommGroup
 
@@ -257,7 +257,7 @@ theorem one_le_gauge_of_not_mem (hs₁ : StarConvex ℝ 0 s) (hs₂ : Absorbs 
 
 section LinearOrderedField
 
-variable {α : Type _} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ]
+variable {α : Type*} [LinearOrderedField α] [MulActionWithZero α ℝ] [OrderedSMul α ℝ]
 
 theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set E)] {s : Set E} {a : α}
     (ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x := by

@@ -3,9 +3,10 @@ Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 -/
-import Mathlib.Analysis.Convex.Basic
+import Mathlib.Analysis.Convex.Topology
 import Mathlib.Analysis.NormedSpace.Pointwise
 import Mathlib.Analysis.Seminorm
+import Mathlib.Analysis.LocallyConvex.Bounded
 import Mathlib.Data.IsROrC.Basic
 
 #align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
@@ -185,18 +186,37 @@ theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
 #align gauge_lt_eq gauge_lt_eq
 
+theorem mem_openSegment_of_gauge_lt_one (absorbs : Absorbent ℝ s) (hgauge : gauge s x < 1) :
+    ∃ y ∈ s, x ∈ openSegment ℝ 0 y := by
+  rcases exists_lt_of_gauge_lt absorbs hgauge with ⟨r, hr₀, hr₁, y, hy, rfl⟩
+  refine ⟨y, hy, 1 - r, r, ?_⟩
+  simp [*]
+
 theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
-    { x | gauge s x < 1 } ⊆ s := by
-  rw [gauge_lt_eq absorbs]
-  refine' Set.iUnion₂_subset fun r hr _ => _
-  rintro ⟨y, hy, rfl⟩
-  exact hs.smul_mem_of_zero_mem h₀ hy (Ioo_subset_Icc_self hr)
+    { x | gauge s x < 1 } ⊆ s := fun _x hx ↦
+  let ⟨_y, hys, hx⟩ := mem_openSegment_of_gauge_lt_one absorbs hx
+  hs.openSegment_subset h₀ hys hx
 #align gauge_lt_one_subset_self gauge_lt_one_subset_self
 
 theorem gauge_le_one_of_mem {x : E} (hx : x ∈ s) : gauge s x ≤ 1 :=
   gauge_le_of_mem zero_le_one <| by rwa [one_smul]
 #align gauge_le_one_of_mem gauge_le_one_of_mem
 
+/-- Gauge is subadditive. -/
+theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
+    gauge s (x + y) ≤ gauge s x + gauge s y := by
+  refine' le_of_forall_pos_lt_add fun ε hε => _
+  obtain ⟨a, ha, ha', x, hx, rfl⟩ :=
+    exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s x) (half_pos hε))
+  obtain ⟨b, hb, hb', y, hy, rfl⟩ :=
+    exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s y) (half_pos hε))
+  calc
+    gauge s (a • x + b • y) ≤ a + b := gauge_le_of_mem (by positivity) <| by
+      rw [hs.add_smul ha.le hb.le]
+      exact add_mem_add (smul_mem_smul_set hx) (smul_mem_smul_set hy)
+    _ < gauge s (a • x) + gauge s (b • y) + ε := by linarith
+#align gauge_add_le gauge_add_le
+
 theorem self_subset_gauge_le_one : s ⊆ { x | gauge s x ≤ 1 } := fun _ => gauge_le_one_of_mem
 #align self_subset_gauge_le_one self_subset_gauge_le_one
 
@@ -363,24 +383,91 @@ theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s
     at h_gauge_lt
 #align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ
 
+theorem mem_closure_of_gauge_le_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s)
+    (h : gauge s x ≤ 1) : x ∈ closure s := by
+  have : ∀ᶠ r : ℝ in 𝓝[<] 1, r • x ∈ s
+  · filter_upwards [Ico_mem_nhdsWithin_Iio' one_pos] with r ⟨hr₀, hr₁⟩
+    apply gauge_lt_one_subset_self hc hs₀ ha
+    rw [mem_setOf_eq, gauge_smul_of_nonneg hr₀]
+    exact mul_lt_one_of_nonneg_of_lt_one_left hr₀ hr₁ h
+  refine mem_closure_of_tendsto ?_ this
+  exact Filter.Tendsto.mono_left (Continuous.tendsto' (by continuity) _ _ (one_smul _ _))
+    inf_le_left
+
+theorem mem_frontier_of_gauge_eq_one (hc : Convex ℝ s) (hs₀ : 0 ∈ s) (ha : Absorbent ℝ s)
+    (h : gauge s x = 1) : x ∈ frontier s :=
+  ⟨mem_closure_of_gauge_le_one hc hs₀ ha h.le, fun h' ↦
+    (interior_subset_gauge_lt_one s h').out.ne h⟩
+
 end TopologicalSpace
 
-theorem gauge_add_le (hs : Convex ℝ s) (absorbs : Absorbent ℝ s) (x y : E) :
-    gauge s (x + y) ≤ gauge s x + gauge s y := by
-  refine' le_of_forall_pos_lt_add fun ε hε => _
-  obtain ⟨a, ha, ha', hx⟩ :=
-    exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s x) (half_pos hε))
-  obtain ⟨b, hb, hb', hy⟩ :=
-    exists_lt_of_gauge_lt absorbs (lt_add_of_pos_right (gauge s y) (half_pos hε))
-  rw [mem_smul_set_iff_inv_smul_mem₀ ha.ne'] at hx
-  rw [mem_smul_set_iff_inv_smul_mem₀ hb.ne'] at hy
-  suffices gauge s (x + y) ≤ a + b by linarith
-  have hab : 0 < a + b := add_pos ha hb
-  apply gauge_le_of_mem hab.le
-  have := convex_iff_div.1 hs hx hy ha.le hb.le hab
-  rwa [smul_smul, smul_smul, ← mul_div_right_comm, ← mul_div_right_comm, mul_inv_cancel ha.ne',
-    mul_inv_cancel hb.ne', ← smul_add, one_div, ← mem_smul_set_iff_inv_smul_mem₀ hab.ne'] at this
-#align gauge_add_le gauge_add_le
+section TopologicalAddGroup
+
+open Filter
+
+variable [TopologicalSpace E] [TopologicalAddGroup E] [ContinuousSMul ℝ E]
+
+/-- If `s` is a convex neighborhood of the origin in a topological real vector space, then `gauge s`
+is continuous. If the ambient space is a normed space, then `gauge s` is Lipschitz continuous, see
+`Convex.lipschitz_gauge`. -/
+theorem continuous_gauge (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) : Continuous (gauge s) := by
+  have ha : Absorbent ℝ s := absorbent_nhds_zero hs₀
+  simp only [continuous_iff_continuousAt, ContinuousAt, (nhds_basis_Icc_pos _).tendsto_right_iff]
+  intro x ε hε₀
+  rw [← map_add_left_nhds_zero, eventually_map]
+  have : ε • s ∩ -(ε • s) ∈ 𝓝 0
+  · exact inter_mem ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀)
+      (neg_mem_nhds_zero _ ((set_smul_mem_nhds_zero_iff hε₀.ne').2 hs₀))
+  filter_upwards [this] with y hy
+  constructor
+  · rw [sub_le_iff_le_add]
+    calc
+      gauge s x = gauge s (x + y + (-y)) := by simp
+      _ ≤ gauge s (x + y) + gauge s (-y) := gauge_add_le hc ha _ _
+      _ ≤ gauge s (x + y) + ε := add_le_add_left (gauge_le_of_mem hε₀.le (mem_neg.1 hy.2)) _
+  · calc
+      gauge s (x + y) ≤ gauge s x + gauge s y := gauge_add_le hc ha _ _
+      _ ≤ gauge s x + ε := add_le_add_left (gauge_le_of_mem hε₀.le hy.1) _
+
+theorem gauge_lt_one_eq_interior (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
+    { x | gauge s x < 1 } = interior s := by
+  refine Subset.antisymm (fun x hx ↦ ?_) (interior_subset_gauge_lt_one s)
+  rcases mem_openSegment_of_gauge_lt_one (absorbent_nhds_zero hs₀) hx with ⟨y, hys, hxy⟩
+  exact hc.openSegment_interior_self_subset_interior (mem_interior_iff_mem_nhds.2 hs₀) hys hxy
+
+theorem gauge_lt_one_iff_mem_interior (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
+    gauge s x < 1 ↔ x ∈ interior s :=
+  Set.ext_iff.1 (gauge_lt_one_eq_interior hc hs₀) _
+
+theorem gauge_le_one_iff_mem_closure (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
+    gauge s x ≤ 1 ↔ x ∈ closure s :=
+  ⟨mem_closure_of_gauge_le_one hc (mem_of_mem_nhds hs₀) (absorbent_nhds_zero hs₀), fun h ↦
+    le_on_closure (fun _ ↦ gauge_le_one_of_mem) (continuous_gauge hc hs₀).continuousOn
+      continuousOn_const h⟩
+
+theorem gauge_eq_one_iff_mem_frontier (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
+    gauge s x = 1 ↔ x ∈ frontier s := by
+  rw [eq_iff_le_not_lt, gauge_le_one_iff_mem_closure hc hs₀, gauge_lt_one_iff_mem_interior hc hs₀]
+  rfl
+
+theorem gauge_eq_zero [T1Space E] (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
+    gauge s x = 0 ↔ x = 0 := by
+  refine ⟨not_imp_not.1 fun (h : x ≠ 0) ↦ ne_of_gt ?_, fun h ↦ h.symm ▸ gauge_zero⟩
+  rcases hb (isOpen_compl_singleton.mem_nhds h.symm) with ⟨c, hc₀, hc⟩
+  refine (inv_pos.2 hc₀).trans_le <| le_csInf hs.gauge_set_nonempty ?_
+  rintro r ⟨hr₀, x, hx, rfl⟩
+  contrapose! hc
+  refine ⟨r⁻¹, ?_, fun h ↦ ?_⟩
+  · rw [norm_inv, Real.norm_of_nonneg hr₀.le, le_inv hc₀ hr₀]
+    exact hc.le
+  · rcases h hx with ⟨y, hy, rfl⟩
+    simp [hr₀.ne'] at hy
+
+theorem gauge_pos [T1Space E] (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
+    0 < gauge s x ↔ x ≠ 0 := by
+  simp only [(gauge_nonneg _).gt_iff_ne, Ne.def, gauge_eq_zero hs hb]
+
+end TopologicalAddGroup
 
 section IsROrC
 
@@ -442,25 +529,57 @@ theorem Seminorm.gaugeSeminorm_ball (p : Seminorm ℝ E) :
 
 end AddCommGroup
 
-section Norm
+section Seminormed
 
 variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E}
+open Metric
 
-theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ := by
+theorem gauge_unit_ball (x : E) : gauge (ball (0 : E) 1) x = ‖x‖ := by
   rw [← ball_normSeminorm ℝ, Seminorm.gauge_ball, coe_normSeminorm]
 #align gauge_unit_ball gauge_unit_ball
 
-theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r := by
-  rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
+theorem gauge_ball (hr : 0 ≤ r) (x : E) : gauge (ball (0 : E) r) x = ‖x‖ / r := by
+  rcases hr.eq_or_lt with rfl | hr
+  · simp
+  · rw [← smul_unitBall_of_pos hr, gauge_smul_left, Pi.smul_apply, gauge_unit_ball, smul_eq_mul,
     abs_of_nonneg hr.le, div_eq_inv_mul]
-  simp_rw [mem_ball_zero_iff, norm_neg]
-  exact fun _ => id
-#align gauge_ball gauge_ball
+    simp_rw [mem_ball_zero_iff, norm_neg]
+    exact fun _ => id
+
+@[deprecated gauge_ball]
+theorem gauge_ball' (hr : 0 < r) (x : E) : gauge (ball (0 : E) r) x = ‖x‖ / r :=
+  gauge_ball hr.le x
+#align gauge_ball gauge_ball'
+
+@[simp]
+theorem gauge_closure_zero : gauge (closure (0 : Set E)) = 0 := funext fun x ↦ by
+  simp only [← singleton_zero, gauge_def', mem_closure_zero_iff_norm, norm_smul, mul_eq_zero,
+    norm_eq_zero, inv_eq_zero]
+  rcases (norm_nonneg x).eq_or_gt with hx | hx
+  · convert csInf_Ioi (a := (0 : ℝ))
+    exact Set.ext fun r ↦ and_iff_left (.inr hx)
+  · convert Real.sInf_empty
+    exact eq_empty_of_forall_not_mem fun r ⟨hr₀, hr⟩ ↦ hx.ne' <| hr.resolve_left hr₀.out.ne'
+
+@[simp]
+theorem gauge_closedBall (hr : 0 ≤ r) (x : E) : gauge (closedBall (0 : E) r) x = ‖x‖ / r := by
+  rcases hr.eq_or_lt with rfl | hr'
+  · rw [div_zero, closedBall_zero', singleton_zero, gauge_closure_zero]; rfl
+  · apply le_antisymm
+    · rw [← gauge_ball hr]
+      exact gauge_mono (absorbent_ball_zero hr') ball_subset_closedBall x
+    · suffices : ∀ᶠ R in 𝓝[>] r, ‖x‖ / R ≤ gauge (closedBall 0 r) x
+      · refine le_of_tendsto ?_ this
+        exact tendsto_const_nhds.div inf_le_left hr'.ne'
+      filter_upwards [self_mem_nhdsWithin] with R hR
+      rw [← gauge_ball (hr.trans hR.out.le)]
+      refine gauge_mono ?_ (closedBall_subset_ball hR) _
+      exact (absorbent_ball_zero hr').subset ball_subset_closedBall
 
 theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤ ‖x‖ := by
   obtain hr | hr := le_or_lt r 0
   · exact (mul_nonpos_of_nonpos_of_nonneg hr <| gauge_nonneg _).trans (norm_nonneg _)
-  rw [mul_comm, ← le_div_iff hr, ← gauge_ball hr]
+  rw [mul_comm, ← le_div_iff hr, ← gauge_ball hr.le]
   exact gauge_mono (absorbent_ball_zero hr) hs x
 #align mul_gauge_le_norm mul_gauge_le_norm
 
@@ -472,15 +591,30 @@ theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r
       gauge s x = gauge s (y + (x - y)) := by simp
       _ ≤ gauge s y + gauge s (x - y) := gauge_add_le hc (this.subset hs) _ _
       _ ≤ gauge s y + ‖x - y‖ / r :=
-        add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr _)) _
+        add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr.le _)) _
       _ = gauge s y + r⁻¹ * dist x y := by rw [dist_eq_norm, div_eq_inv_mul, NNReal.coe_inv]
 #align convex.lipschitz_with_gauge Convex.lipschitzWith_gauge
 
+theorem Convex.lipschitz_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
+    ∃ K, LipschitzWith K (gauge s) :=
+  let ⟨r, hr₀, hr⟩ := Metric.mem_nhds_iff.1 h₀
+  ⟨(⟨r, hr₀.le⟩ : ℝ≥0)⁻¹, hc.lipschitzWith_gauge hr₀ hr⟩
+
 theorem Convex.uniformContinuous_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
-    UniformContinuous (gauge s) := by
-  obtain ⟨r, hr₀, hr⟩ := Metric.mem_nhds_iff.1 h₀
-  lift r to ℝ≥0 using le_of_lt hr₀
-  exact (hc.lipschitzWith_gauge hr₀ hr).uniformContinuous
+    UniformContinuous (gauge s) :=
+  let ⟨_K, hK⟩ := hc.lipschitz_gauge h₀; hK.uniformContinuous
 #align convex.uniform_continuous_gauge Convex.uniformContinuous_gauge
 
-end Norm
+end Seminormed
+
+section Normed
+
+variable [NormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E}
+open Metric
+
+theorem le_gauge_of_subset_closedBall (hs : Absorbent ℝ s) (hr : 0 ≤ r) (hsr : s ⊆ closedBall 0 r) :
+    ‖x‖ / r ≤ gauge s x := by
+  rw [← gauge_closedBall hr]
+  exact gauge_mono hs hsr _
+
+end Normed

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,17 +2,14 @@
 Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
-
-! This file was ported from Lean 3 source module analysis.convex.gauge
-! leanprover-community/mathlib commit 373b03b5b9d0486534edbe94747f23cb3712f93d
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Convex.Basic
 import Mathlib.Analysis.NormedSpace.Pointwise
 import Mathlib.Analysis.Seminorm
 import Mathlib.Data.IsROrC.Basic
 
+#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
+
 /-!
 # The Minkowski functional
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -389,7 +389,7 @@ section IsROrC
 
 variable [IsROrC 𝕜] [Module 𝕜 E] [IsScalarTower ℝ 𝕜 E]
 
-/-- `gauge s` as a seminorm when `s` is  balanced, convex and absorbent. -/
+/-- `gauge s` as a seminorm when `s` is balanced, convex and absorbent. -/
 @[simps!]
 def gaugeSeminorm (hs₀ : Balanced 𝕜 s) (hs₁ : Convex ℝ s) (hs₂ : Absorbent ℝ s) : Seminorm 𝕜 E :=
   Seminorm.of (gauge s) (gauge_add_le hs₁ hs₂) (gauge_smul hs₀)

@@ -290,7 +290,7 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
 
 theorem gauge_smul_left [Module α E] [SMulCommClass α ℝ ℝ] [IsScalarTower α ℝ ℝ]
     [IsScalarTower α ℝ E] {s : Set E} (symmetric : ∀ x ∈ s, -x ∈ s) (a : α) :
-    gauge (a • s) = (|a|)⁻¹ • gauge s := by
+    gauge (a • s) = |a|⁻¹ • gauge s := by
   rw [← gauge_smul_left_of_nonneg (abs_nonneg a)]
   obtain h | h := abs_choice a
   · rw [h]

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -252,14 +252,14 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
   constructor
   · rintro ⟨hr, hx⟩
-    simp_rw [mem_Ioi] at hr⊢
+    simp_rw [mem_Ioi] at hr ⊢
     rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
     have := smul_pos (inv_pos.2 ha') hr
     refine' ⟨a⁻¹ • r, ⟨this, _⟩, smul_inv_smul₀ ha'.ne' _⟩
     rwa [← mem_smul_set_iff_inv_smul_mem₀ this.ne', smul_assoc,
       mem_smul_set_iff_inv_smul_mem₀ (inv_ne_zero ha'.ne'), inv_inv]
   · rintro ⟨r, ⟨hr, hx⟩, rfl⟩
-    rw [mem_Ioi] at hr⊢
+    rw [mem_Ioi] at hr ⊢
     rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
     have := smul_pos ha' hr
     refine' ⟨this, _⟩
@@ -279,11 +279,11 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
   constructor
   · rintro ⟨hr, y, hy, h⟩
-    simp_rw [mem_Ioi] at hr⊢
+    simp_rw [mem_Ioi] at hr ⊢
     refine' ⟨a • r, ⟨smul_pos ha' hr, _⟩, inv_smul_smul₀ ha'.ne' _⟩
     rwa [smul_inv₀, smul_assoc, ← h, inv_smul_smul₀ ha'.ne']
   · rintro ⟨r, ⟨hr, hx⟩, rfl⟩
-    rw [mem_Ioi] at hr⊢
+    rw [mem_Ioi] at hr ⊢
     refine' ⟨smul_pos (inv_pos.2 ha') hr, r⁻¹ • x, hx, _⟩
     rw [smul_inv₀, smul_assoc, inv_inv]
 #align gauge_smul_left_of_nonneg gauge_smul_left_of_nonneg

Drop unneeded assumptions in gauge_lt_one_of_mem_of_open and gauge_lt_of_mem_smul

@@ -330,25 +330,18 @@ section TopologicalSpace
 
 variable [TopologicalSpace E] [ContinuousSMul ℝ E]
 
+open Filter in
 theorem interior_subset_gauge_lt_one (s : Set E) : interior s ⊆ { x | gauge s x < 1 } := by
   intro x hx
-  let f : ℝ → E := fun t => t • x
-  have hf : Continuous f := by continuity
-  let s' := f ⁻¹' interior s
-  have hs' : IsOpen s' := hf.isOpen_preimage _ isOpen_interior
-  have one_mem : (1 : ℝ) ∈ s' := by simpa only [Set.mem_preimage, one_smul]
-  obtain ⟨ε, hε₀, hε⟩ := (Metric.nhds_basis_closedBall.1 _).1 (isOpen_iff_mem_nhds.1 hs' 1 one_mem)
-  rw [Real.closedBall_eq_Icc] at hε
-  have hε₁ : 0 < 1 + ε := hε₀.trans (lt_one_add ε)
-  have : (1 + ε)⁻¹ < 1 := by
-    rw [inv_lt_one_iff]
-    right
-    linarith
-  refine' (gauge_le_of_mem (inv_nonneg.2 hε₁.le) _).trans_lt this
-  rw [mem_inv_smul_set_iff₀ hε₁.ne']
-  exact
-    interior_subset
-      (hε ⟨(sub_le_self _ hε₀.le).trans ((le_add_iff_nonneg_right _).2 hε₀.le), le_rfl⟩)
+  have H₁ : Tendsto (fun r : ℝ ↦ r⁻¹ • x) (𝓝[<] 1) (𝓝 ((1 : ℝ)⁻¹ • x)) :=
+    ((tendsto_id.inv₀ one_ne_zero).smul tendsto_const_nhds).mono_left inf_le_left
+  rw [inv_one, one_smul] at H₁
+  have H₂ : ∀ᶠ r in 𝓝[<] (1 : ℝ), x ∈ r • s ∧ 0 < r ∧ r < 1
+  · filter_upwards [H₁ (mem_interior_iff_mem_nhds.1 hx), Ioo_mem_nhdsWithin_Iio' one_pos]
+    intro r h₁ h₂
+    exact ⟨(mem_smul_set_iff_inv_smul_mem₀ h₂.1.ne' _ _).2 h₁, h₂⟩
+  rcases H₂.exists with ⟨r, hxr, hr₀, hr₁⟩
+  exact (gauge_le_of_mem hr₀.le hxr).trans_lt hr₁
 #align interior_subset_gauge_lt_one interior_subset_gauge_lt_one
 
 theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) :
@@ -358,17 +351,20 @@ theorem gauge_lt_one_eq_self_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈
   exact hs₂.interior_eq.symm
 #align gauge_lt_one_eq_self_of_open gauge_lt_one_eq_self_of_open
 
-theorem gauge_lt_one_of_mem_of_open (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : IsOpen s) {x : E}
-    (hx : x ∈ s) : gauge s x < 1 := by rwa [← gauge_lt_one_eq_self_of_open hs₁ hs₀ hs₂] at hx
-#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_open
+-- porting note: droped unneeded assumptions
+theorem gauge_lt_one_of_mem_of_open (hs₂ : IsOpen s) {x : E} (hx : x ∈ s) :
+    gauge s x < 1 :=
+  interior_subset_gauge_lt_one s <| by rwa [hs₂.interior_eq]
+#align gauge_lt_one_of_mem_of_open gauge_lt_one_of_mem_of_openₓ
 
-theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₀ : (0 : E) ∈ s) (hs₁ : Convex ℝ s)
-    (hs₂ : IsOpen s) (hx : x ∈ ε • s) : gauge s x < ε := by
+-- porting note: droped unneeded assumptions
+theorem gauge_lt_of_mem_smul (x : E) (ε : ℝ) (hε : 0 < ε) (hs₂ : IsOpen s) (hx : x ∈ ε • s) :
+    gauge s x < ε := by
   have : ε⁻¹ • x ∈ s := by rwa [← mem_smul_set_iff_inv_smul_mem₀ hε.ne']
-  have h_gauge_lt := gauge_lt_one_of_mem_of_open hs₁ hs₀ hs₂ this
+  have h_gauge_lt := gauge_lt_one_of_mem_of_open hs₂ this
   rwa [gauge_smul_of_nonneg (inv_nonneg.2 hε.le), smul_eq_mul, inv_mul_lt_iff hε, mul_one]
     at h_gauge_lt
-#align gauge_lt_of_mem_smul gauge_lt_of_mem_smul
+#align gauge_lt_of_mem_smul gauge_lt_of_mem_smulₓ
 
 end TopologicalSpace
 
@@ -404,7 +400,7 @@ variable {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ
 
 theorem gaugeSeminorm_lt_one_of_open (hs : IsOpen s) {x : E} (hx : x ∈ s) :
     gaugeSeminorm hs₀ hs₁ hs₂ x < 1 :=
-  gauge_lt_one_of_mem_of_open hs₁ hs₂.zero_mem hs hx
+  gauge_lt_one_of_mem_of_open hs hx
 #align gauge_seminorm_lt_one_of_open gaugeSeminorm_lt_one_of_open
 
 theorem gaugeSeminorm_ball_one (hs : IsOpen s) : (gaugeSeminorm hs₀ hs₁ hs₂).ball 0 1 = s := by

@@ -153,7 +153,7 @@ theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a :=
 #align gauge_le_of_mem gauge_le_of_mem
 
 theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
-    { x | gauge s x ≤ a } = ⋂ (r : ℝ) (_H : a < r), r • s := by
+    { x | gauge s x ≤ a } = ⋂ (r : ℝ) (_ : a < r), r • s := by
   ext x
   simp_rw [Set.mem_iInter, Set.mem_setOf_eq]
   refine' ⟨fun h r hr => _, fun h => le_of_forall_pos_lt_add fun ε hε => _⟩
@@ -171,7 +171,7 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
 #align gauge_le_eq gauge_le_eq
 
 theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
-    { x | gauge s x < a } = ⋃ (r : ℝ) (_H : 0 < r) (_H : r < a), r • s := by
+    { x | gauge s x < a } = ⋃ (r : ℝ) (_ : 0 < r) (_ : r < a), r • s := by
   ext
   simp_rw [mem_setOf, mem_iUnion, exists_prop]
   exact

Forward-port of leanprover-community/mathlib#19102

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yaël Dillies, Bhavik Mehta
 
 ! This file was ported from Lean 3 source module analysis.convex.gauge
-! leanprover-community/mathlib commit 3f655f5297b030a87d641ad4e825af8d9679eb0b
+! leanprover-community/mathlib commit 373b03b5b9d0486534edbe94747f23cb3712f93d
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -41,7 +41,8 @@ Minkowski functional, gauge
 -/
 
 
-open NormedField Set Pointwise
+open NormedField Set
+open scoped Pointwise Topology NNReal
 
 noncomputable section
 
@@ -453,24 +454,7 @@ section Norm
 variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] {s : Set E} {r : ℝ} {x : E}
 
 theorem gauge_unit_ball (x : E) : gauge (Metric.ball (0 : E) 1) x = ‖x‖ := by
-  obtain rfl | hx := eq_or_ne x 0
-  · rw [norm_zero, gauge_zero]
-  refine' (le_of_forall_pos_le_add fun ε hε => _).antisymm _
-  · have : 0 < ‖x‖ + ε :=
-      -- Porting note: was `by positivity`
-      add_pos_of_nonneg_of_pos (norm_nonneg _) hε
-    refine' gauge_le_of_mem this.le _
-    rw [smul_ball this.ne', smul_zero, Real.norm_of_nonneg this.le, mul_one, mem_ball_zero_iff]
-    exact lt_add_of_pos_right _ hε
-  refine'
-    le_gauge_of_not_mem balanced_ball_zero.starConvex (absorbent_ball_zero zero_lt_one).absorbs
-      fun h => _
-  obtain hx' | hx' := eq_or_ne ‖x‖ 0
-  · rw [hx'] at h
-    exact hx (zero_smul_set_subset _ h)
-  · rw [mem_smul_set_iff_inv_smul_mem₀ hx', mem_ball_zero_iff, norm_smul, norm_inv, norm_norm,
-      inv_mul_cancel hx'] at h
-    exact lt_irrefl _ h
+  rw [← ball_normSeminorm ℝ, Seminorm.gauge_ball, coe_normSeminorm]
 #align gauge_unit_ball gauge_unit_ball
 
 theorem gauge_ball (hr : 0 < r) (x : E) : gauge (Metric.ball (0 : E) r) x = ‖x‖ / r := by
@@ -487,4 +471,23 @@ theorem mul_gauge_le_norm (hs : Metric.ball (0 : E) r ⊆ s) : r * gauge s x ≤
   exact gauge_mono (absorbent_ball_zero hr) hs x
 #align mul_gauge_le_norm mul_gauge_le_norm
 
+theorem Convex.lipschitzWith_gauge {r : ℝ≥0} (hc : Convex ℝ s) (hr : 0 < r)
+    (hs : Metric.ball (0 : E) r ⊆ s) : LipschitzWith r⁻¹ (gauge s) :=
+  have : Absorbent ℝ (Metric.ball (0 : E) r) := absorbent_ball_zero hr
+  LipschitzWith.of_le_add_mul _ fun x y =>
+    calc
+      gauge s x = gauge s (y + (x - y)) := by simp
+      _ ≤ gauge s y + gauge s (x - y) := gauge_add_le hc (this.subset hs) _ _
+      _ ≤ gauge s y + ‖x - y‖ / r :=
+        add_le_add_left ((gauge_mono this hs (x - y)).trans_eq (gauge_ball hr _)) _
+      _ = gauge s y + r⁻¹ * dist x y := by rw [dist_eq_norm, div_eq_inv_mul, NNReal.coe_inv]
+#align convex.lipschitz_with_gauge Convex.lipschitzWith_gauge
+
+theorem Convex.uniformContinuous_gauge (hc : Convex ℝ s) (h₀ : s ∈ 𝓝 (0 : E)) :
+    UniformContinuous (gauge s) := by
+  obtain ⟨r, hr₀, hr⟩ := Metric.mem_nhds_iff.1 h₀
+  lift r to ℝ≥0 using le_of_lt hr₀
+  exact (hc.lipschitzWith_gauge hr₀ hr).uniformContinuous
+#align convex.uniform_continuous_gauge Convex.uniformContinuous_gauge
+
 end Norm

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -54,18 +54,18 @@ variable [AddCommGroup E] [Module ℝ E]
 /-- The Minkowski functional. Given a set `s` in a real vector space, `gauge s` is the functional
 which sends `x : E` to the smallest `r : ℝ` such that `x` is in `s` scaled by `r`. -/
 def gauge (s : Set E) (x : E) : ℝ :=
-  infₛ { r : ℝ | 0 < r ∧ x ∈ r • s }
+  sInf { r : ℝ | 0 < r ∧ x ∈ r • s }
 #align gauge gauge
 
 variable {s t : Set E} {a : ℝ}
 
-theorem gauge_def : gauge s x = infₛ ({ r ∈ Set.Ioi (0 : ℝ) | x ∈ r • s }) :=
+theorem gauge_def : gauge s x = sInf ({ r ∈ Set.Ioi (0 : ℝ) | x ∈ r • s }) :=
   rfl
 #align gauge_def gauge_def
 
 /-- An alternative definition of the gauge using scalar multiplication on the element rather than on
 the set. -/
-theorem gauge_def' : gauge s x = infₛ ({ r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s }) := by
+theorem gauge_def' : gauge s x = sInf ({ r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s }) := by
   -- Porting note: used `congrm`
   rw [gauge]
   apply congr_arg
@@ -86,12 +86,12 @@ theorem Absorbent.gauge_set_nonempty (absorbs : Absorbent ℝ s) :
 #align absorbent.gauge_set_nonempty Absorbent.gauge_set_nonempty
 
 theorem gauge_mono (hs : Absorbent ℝ s) (h : s ⊆ t) : gauge t ≤ gauge s := fun _ =>
-  cinfₛ_le_cinfₛ gauge_set_bddBelow hs.gauge_set_nonempty fun _ hr => ⟨hr.1, smul_set_mono h hr.2⟩
+  csInf_le_csInf gauge_set_bddBelow hs.gauge_set_nonempty fun _ hr => ⟨hr.1, smul_set_mono h hr.2⟩
 #align gauge_mono gauge_mono
 
 theorem exists_lt_of_gauge_lt (absorbs : Absorbent ℝ s) (h : gauge s x < a) :
     ∃ b, 0 < b ∧ b < a ∧ x ∈ b • s := by
-  obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_cinfₛ_lt absorbs.gauge_set_nonempty h
+  obtain ⟨b, ⟨hb, hx⟩, hba⟩ := exists_lt_of_csInf_lt absorbs.gauge_set_nonempty h
   exact ⟨b, hb, hba, hx⟩
 #align exists_lt_of_gauge_lt exists_lt_of_gauge_lt
 
@@ -101,8 +101,8 @@ but, the real infimum of the empty set in Lean being defined as `0`, it holds un
 theorem gauge_zero : gauge s 0 = 0 := by
   rw [gauge_def']
   by_cases h : (0 : E) ∈ s
-  · simp only [smul_zero, sep_true, h, cinfₛ_Ioi]
-  · simp only [smul_zero, sep_false, h, Real.infₛ_empty]
+  · simp only [smul_zero, sep_true, h, csInf_Ioi]
+  · simp only [smul_zero, sep_false, h, Real.sInf_empty]
 #align gauge_zero gauge_zero
 
 @[simp]
@@ -110,16 +110,16 @@ theorem gauge_zero' : gauge (0 : Set E) = 0 := by
   ext x
   rw [gauge_def']
   obtain rfl | hx := eq_or_ne x 0
-  · simp only [cinfₛ_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]
+  · simp only [csInf_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]
   · simp only [mem_zero, Pi.zero_apply, inv_eq_zero, smul_eq_zero]
-    convert Real.infₛ_empty
+    convert Real.sInf_empty
     exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr.1) hx
 #align gauge_zero' gauge_zero'
 
 @[simp]
 theorem gauge_empty : gauge (∅ : Set E) = 0 := by
   ext
-  simp only [gauge_def', Real.infₛ_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
+  simp only [gauge_def', Real.sInf_empty, mem_empty_iff_false, Pi.zero_apply, sep_false]
 #align gauge_empty gauge_empty
 
 theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by
@@ -129,7 +129,7 @@ theorem gauge_of_subset_zero (h : s ⊆ 0) : gauge s = 0 := by
 
 /-- The gauge is always nonnegative. -/
 theorem gauge_nonneg (x : E) : 0 ≤ gauge s x :=
-  Real.infₛ_nonneg _ fun _ hx => hx.1.le
+  Real.sInf_nonneg _ fun _ hx => hx.1.le
 #align gauge_nonneg gauge_nonneg
 
 theorem gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x := by
@@ -148,13 +148,13 @@ theorem gauge_neg_set_eq_gauge_neg (x : E) : gauge (-s) x = gauge s (-x) := by
 theorem gauge_le_of_mem (ha : 0 ≤ a) (hx : x ∈ a • s) : gauge s x ≤ a := by
   obtain rfl | ha' := ha.eq_or_lt
   · rw [mem_singleton_iff.1 (zero_smul_set_subset _ hx), gauge_zero]
-  · exact cinfₛ_le gauge_set_bddBelow ⟨ha', hx⟩
+  · exact csInf_le gauge_set_bddBelow ⟨ha', hx⟩
 #align gauge_le_of_mem gauge_le_of_mem
 
 theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Absorbent ℝ s) (ha : 0 ≤ a) :
     { x | gauge s x ≤ a } = ⋂ (r : ℝ) (_H : a < r), r • s := by
   ext x
-  simp_rw [Set.mem_interᵢ, Set.mem_setOf_eq]
+  simp_rw [Set.mem_iInter, Set.mem_setOf_eq]
   refine' ⟨fun h r hr => _, fun h => le_of_forall_pos_lt_add fun ε hε => _⟩
   · have hr' := ha.trans_lt hr
     rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
@@ -172,7 +172,7 @@ theorem gauge_le_eq (hs₁ : Convex ℝ s) (hs₀ : (0 : E) ∈ s) (hs₂ : Abso
 theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ (r : ℝ) (_H : 0 < r) (_H : r < a), r • s := by
   ext
-  simp_rw [mem_setOf, mem_unionᵢ, exists_prop]
+  simp_rw [mem_setOf, mem_iUnion, exists_prop]
   exact
     ⟨exists_lt_of_gauge_lt absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
@@ -181,7 +181,7 @@ theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
 theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
     { x | gauge s x < a } = ⋃ r ∈ Set.Ioo 0 (a : ℝ), r • s := by
   ext
-  simp_rw [mem_setOf, mem_unionᵢ, exists_prop, mem_Ioo, and_assoc]
+  simp_rw [mem_setOf, mem_iUnion, exists_prop, mem_Ioo, and_assoc]
   exact
     ⟨exists_lt_of_gauge_lt absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
       (gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
@@ -190,7 +190,7 @@ theorem gauge_lt_eq (absorbs : Absorbent ℝ s) (a : ℝ) :
 theorem gauge_lt_one_subset_self (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Absorbent ℝ s) :
     { x | gauge s x < 1 } ⊆ s := by
   rw [gauge_lt_eq absorbs]
-  refine' Set.unionᵢ₂_subset fun r hr _ => _
+  refine' Set.iUnion₂_subset fun r hr _ => _
   rintro ⟨y, hy, rfl⟩
   exact hs.smul_mem_of_zero_mem h₀ hy (Ioo_subset_Icc_self hr)
 #align gauge_lt_one_subset_self gauge_lt_one_subset_self
@@ -206,7 +206,7 @@ theorem Convex.gauge_le (hs : Convex ℝ s) (h₀ : (0 : E) ∈ s) (absorbs : Ab
     Convex ℝ { x | gauge s x ≤ a } := by
   by_cases ha : 0 ≤ a
   · rw [gauge_le_eq hs h₀ absorbs ha]
-    exact convex_interᵢ fun i => convex_interᵢ fun _ => hs.smul _
+    exact convex_iInter fun i => convex_iInter fun _ => hs.smul _
   · -- Porting note: `convert` needed help
     convert convex_empty (𝕜 := ℝ) (E := E)
     exact eq_empty_iff_forall_not_mem.2 fun x hx => ha <| (gauge_nonneg _).trans hx
@@ -221,7 +221,7 @@ theorem le_gauge_of_not_mem (hs₀ : StarConvex ℝ 0 s) (hs₂ : Absorbs ℝ s
     a ≤ gauge s x := by
   rw [starConvex_zero_iff] at hs₀
   obtain ⟨r, hr, h⟩ := hs₂
-  refine' le_cinfₛ ⟨r, hr, singleton_subset_iff.1 <| h _ (Real.norm_of_nonneg hr.le).ge⟩ _
+  refine' le_csInf ⟨r, hr, singleton_subset_iff.1 <| h _ (Real.norm_of_nonneg hr.le).ge⟩ _
   rintro b ⟨hb, x, hx', rfl⟩
   refine' not_lt.1 fun hba => hx _
   have ha := hb.trans hba
@@ -245,7 +245,7 @@ theorem gauge_smul_of_nonneg [MulActionWithZero α E] [IsScalarTower α ℝ (Set
     (ha : 0 ≤ a) (x : E) : gauge s (a • x) = a • gauge s x := by
   obtain rfl | ha' := ha.eq_or_lt
   · rw [zero_smul, gauge_zero, zero_smul]
-  rw [gauge_def', gauge_def', ← Real.infₛ_smul_of_nonneg ha]
+  rw [gauge_def', gauge_def', ← Real.sInf_smul_of_nonneg ha]
   congr 1
   ext r
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
@@ -272,7 +272,7 @@ theorem gauge_smul_left_of_nonneg [MulActionWithZero α E] [SMulCommClass α ℝ
   obtain rfl | ha' := ha.eq_or_lt
   · rw [inv_zero, zero_smul, gauge_of_subset_zero (zero_smul_set_subset _)]
   ext x
-  rw [gauge_def', Pi.smul_apply, gauge_def', ← Real.infₛ_smul_of_nonneg (inv_nonneg.2 ha)]
+  rw [gauge_def', Pi.smul_apply, gauge_def', ← Real.sInf_smul_of_nonneg (inv_nonneg.2 ha)]
   congr 1
   ext r
   simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
@@ -418,7 +418,7 @@ end IsROrC
 protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1) = p := by
   ext x
   obtain hp | hp := { r : ℝ | 0 < r ∧ x ∈ r • p.ball 0 1 }.eq_empty_or_nonempty
-  · rw [gauge, hp, Real.infₛ_empty]
+  · rw [gauge, hp, Real.sInf_empty]
     by_contra h
     have hpx : 0 < p x := (map_nonneg _ _).lt_of_ne h
     have hpx₂ : 0 < 2 * p x := mul_pos zero_lt_two hpx
@@ -426,7 +426,7 @@ protected theorem Seminorm.gauge_ball (p : Seminorm ℝ E) : gauge (p.ball 0 1)
     rw [p.mem_ball_zero, map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos (inv_pos.2 hpx₂),
       inv_mul_lt_iff hpx₂, mul_one]
     exact lt_mul_of_one_lt_left hpx one_lt_two
-  refine' IsGLB.cinfₛ_eq ⟨fun r => _, fun r hr => le_of_forall_pos_le_add fun ε hε => _⟩ hp
+  refine' IsGLB.csInf_eq ⟨fun r => _, fun r hr => le_of_forall_pos_le_add fun ε hε => _⟩ hp
   · rintro ⟨hr, y, hy, rfl⟩
     rw [p.mem_ball_zero] at hy
     rw [map_smul_eq_mul, Real.norm_eq_abs, abs_of_pos hr]

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>