topology.metric_space.pi_nat ⟷ Mathlib.Topology.MetricSpace.PiNat

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

@@ -89,8 +89,8 @@ theorem apply_firstDiff_ne {x y : ∀ n, E n} (h : x ≠ y) : x (firstDiff x y)
 #print PiNat.apply_eq_of_lt_firstDiff /-
 theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDiff x y) : x n = y n :=
   by
-  rw [first_diff] at hn 
-  split_ifs at hn 
+  rw [first_diff] at hn
+  split_ifs at hn
   · convert Nat.find_min (ne_iff.1 h) hn
     simp
   · exact (not_lt_zero' hn).elim
@@ -285,8 +285,8 @@ theorem res_eq_res {x y : ℕ → α} {n : ℕ} : res x n = res y n ↔ ∀ ⦃m
   by
   constructor <;> intro h <;> induction' n with n ih; · simp
   · intro m hm
-    rw [Nat.lt_succ_iff_lt_or_eq] at hm 
-    simp only [res_succ] at h 
+    rw [Nat.lt_succ_iff_lt_or_eq] at hm
+    simp only [res_succ] at h
     cases' hm with hm hm
     · exact ih h.2 hm
     rw [hm]
@@ -394,7 +394,7 @@ protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + d
 protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y :=
   by
   rcases eq_or_ne x y with (rfl | h); · rfl
-  simp [dist_eq_of_ne h] at hxy 
+  simp [dist_eq_of_ne h] at hxy
   exact (two_ne_zero (pow_eq_zero hxy)).elim
 #align pi_nat.eq_of_dist_eq_zero PiNat.eq_of_dist_eq_zero
 -/
@@ -482,7 +482,7 @@ theorem isTopologicalBasis_cylinders :
     intro i hi
     have : y i = x i := mem_cylinder_iff.1 hy i ((hn hi).trans_lt (lt_add_one n))
     rw [this]
-    simp only [Set.mem_pi, Finset.mem_coe] at xU 
+    simp only [Set.mem_pi, Finset.mem_coe] at xU
     exact xU i hi
 #align pi_nat.is_topological_basis_cylinders PiNat.isTopologicalBasis_cylinders
 -/
@@ -499,7 +499,7 @@ theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
       ∃ (v : Set (∀ n : ℕ, E n)) (H : v ∈ {s | ∃ (x : ∀ n : ℕ, E n) (n : ℕ), s = cylinder x n}),
         x ∈ v ∧ v ⊆ s :=
       (is_topological_basis_cylinders E).exists_subset_of_mem_open hx hs
-    rw [← mem_cylinder_iff_eq.1 h'x] at h's 
+    rw [← mem_cylinder_iff_eq.1 h'x] at h's
     exact
       ⟨(1 / 2 : ℝ) ^ n, by simp, fun y hy => h's fun i hi => (apply_eq_of_dist_lt hy hi.le).symm⟩
   · intro h
@@ -552,7 +552,7 @@ protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, Unifo
         · simp only [mem_principal, set_of_subset_set_of, imp_self, imp_true_iff]
         · rintro ⟨x, y⟩ hxy
           simp only [Finset.mem_coe, Finset.mem_range, Inter_coe_set, mem_Inter, mem_set_of_eq] at
-            hxy 
+            hxy
           apply lt_of_le_of_lt _ hn
           rw [← mem_cylinder_iff_dist_le, mem_cylinder_iff]
           exact hxy
@@ -613,7 +613,7 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
   apply lt_irrefl (inf_dist x s)
   calc
     inf_dist x s ≤ dist x y := inf_dist_le_dist_of_mem ys
-    _ ≤ (1 / 2) ^ n := by rw [mem_cylinder_comm] at hy ; exact mem_cylinder_iff_dist_le.1 hy
+    _ ≤ (1 / 2) ^ n := by rw [mem_cylinder_comm] at hy; exact mem_cylinder_iff_dist_le.1 hy
     _ < inf_dist x s := hn
 #align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinder
 -/
@@ -682,8 +682,8 @@ theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClo
   obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) :=
     by
     have := Nat.find_min A B
-    push_neg at this 
-    simp_rw [not_disjoint_iff, mem_cylinder_comm] at this 
+    push_neg at this
+    simp_rw [not_disjoint_iff, mem_cylinder_comm] at this
     exact this
   refine' ⟨y, ys, _⟩
   rw [mem_cylinder_iff_eq] at hy ⊢
@@ -719,8 +719,8 @@ theorem cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff {x y : ∀ n, E
     · have Ax : (s ∩ cylinder x (longest_prefix x s)).Nonempty :=
         inter_cylinder_longest_prefix_nonempty hs hne x
       have Z := disjoint_cylinder_of_longest_prefix_lt hs ys L
-      rw [first_diff_comm] at H 
-      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H.le] at Z 
+      rw [first_diff_comm] at H
+      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H.le] at Z
       exact (Ax.not_disjoint Z).elim
     · exact L
     · have Ay : (s ∩ cylinder y (longest_prefix y s)).Nonempty :=
@@ -728,7 +728,7 @@ theorem cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff {x y : ∀ n, E
       have A'y : (s ∩ cylinder y (longest_prefix x s).succ).Nonempty :=
         Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))
       have Z := disjoint_cylinder_of_longest_prefix_lt hs xs (Nat.lt_succ_self _)
-      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H] at Z 
+      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H] at Z
       exact (A'y.not_disjoint Z).elim
   rw [l_eq, ← mem_cylinder_iff_eq]
   exact cylinder_anti y H.le (mem_cylinder_first_diff x y)
@@ -795,7 +795,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
         apply cylinder_anti y _ A.some_spec.2
         exact first_diff_le_longest_prefix hs ys xs
       rwa [← fy, ← I2, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy.symm,
-        first_diff_comm _ x] at I 
+        first_diff_comm _ x] at I
     -- case where `x ∉ s`
     · by_cases ys : y ∈ s
       -- case where `y ∈ s` (similar to the above)
@@ -808,7 +808,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
           apply cylinder_anti x _ A.some_spec.2
           apply first_diff_le_longest_prefix hs xs ys
         rw [fs y ys] at hfxfy ⊢
-        rwa [← fx, I2, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at I 
+        rwa [← fx, I2, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at I
       -- case where `y ∉ s`
       · have Ax : (s ∩ cylinder x (longest_prefix x s)).Nonempty :=
           inter_cylinder_longest_prefix_nonempty hs hne x
@@ -824,14 +824,14 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
             cases H
             · exact cylinder_longest_prefix_eq_of_longest_prefix_lt_first_diff hs hne H xs ys
             · symm
-              rw [first_diff_comm] at H 
+              rw [first_diff_comm] at H
               exact cylinder_longest_prefix_eq_of_longest_prefix_lt_first_diff hs hne H ys xs
-          rw [fx, fy] at hfxfy 
+          rw [fx, fy] at hfxfy
           apply (hfxfy _).elim
           congr
         -- case where the common prefix to `x` and `s` is long, as well as the common prefix to
         -- `y` and `s`. Then all points remain in the same cylinders.
-        · push_neg at H 
+        · push_neg at H
           have I1 : cylinder Ax.some (first_diff x y) = cylinder x (first_diff x y) :=
             by
             rw [← mem_cylinder_iff_eq]
@@ -842,7 +842,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
             exact cylinder_anti y H.2 Ay.some_spec.2
           have : cylinder Ax.some (first_diff x y) = cylinder Ay.some (first_diff x y) := by
             rw [I1, I2, I3]
-          rw [← fx, ← fy, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at this 
+          rw [← fx, ← fy, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at this
           exact this
 #align pi_nat.exists_lipschitz_retraction_of_is_closed PiNat.exists_lipschitz_retraction_of_isClosed
 -/
@@ -945,7 +945,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
     have L : tendsto (fun n : ℕ => (1 / 2 : ℝ) ^ n + (1 / 2) ^ n) at_top (𝓝 (0 + 0)) :=
       (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).add
         (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1)
-    rw [add_zero] at L 
+    rw [add_zero] at L
     exact ge_of_tendsto' L J
   have s_closed : IsClosed s :=
     by
@@ -955,7 +955,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
       by
       have : tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) at_top (𝓝 (2 * 0)) :=
         (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).const_mul _
-      rw [MulZeroClass.mul_zero] at this 
+      rw [MulZeroClass.mul_zero] at this
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closed_ball (pow_nonneg I0.le _)) this
     refine'
@@ -1126,7 +1126,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
         refine' mem_infi_of_mem δ (mem_infi_of_mem δpos _)
         simp only [Prod.forall, imp_self, mem_principal]
       · rintro ⟨x, y⟩ hxy
-        simp only [mem_Inter, mem_set_of_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy 
+        simp only [mem_Inter, mem_set_of_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy
         calc
           dist x y = ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i)) := rfl
           _ =

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

@@ -943,7 +943,8 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
           dist_triangle_right _ _ _
         _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A ⟨x, I⟩ n) (hx n)
     have L : tendsto (fun n : ℕ => (1 / 2 : ℝ) ^ n + (1 / 2) ^ n) at_top (𝓝 (0 + 0)) :=
-      (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).add (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1)
+      (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).add
+        (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1)
     rw [add_zero] at L 
     exact ge_of_tendsto' L J
   have s_closed : IsClosed s :=
@@ -953,7 +954,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
     have L : tendsto (fun n : ℕ => diam (closed_ball (u (x n)) ((1 / 2) ^ n))) at_top (𝓝 0) :=
       by
       have : tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) at_top (𝓝 (2 * 0)) :=
-        (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _
+        (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).const_mul _
       rw [MulZeroClass.mul_zero] at this 
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closed_ball (pow_nonneg I0.le _)) this

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

@@ -60,7 +60,7 @@ open scoped Classical Topology Filter
 
 open TopologicalSpace Set Metric Filter Function
 
-attribute [local simp] pow_le_pow_iff one_lt_two inv_le_inv
+attribute [local simp] pow_le_pow_iff_right one_lt_two inv_le_inv
 
 variable {E : ℕ → Type _}
 
@@ -377,7 +377,7 @@ theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y)
   rcases eq_or_ne y z with (rfl | hyz)
   · simp
   simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv, one_div, inv_pow, zero_lt_bit0, Ne.def,
-    not_false_iff, le_max_iff, zero_lt_one, pow_le_pow_iff, one_lt_two, pow_pos,
+    not_false_iff, le_max_iff, zero_lt_one, pow_le_pow_iff_right, one_lt_two, pow_pos,
     min_le_iff.1 (min_first_diff_le x y z hxz)]
 #align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarch
 -/

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

@@ -112,7 +112,7 @@ theorem firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x :=
 theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
     min (firstDiff x y) (firstDiff y z) ≤ firstDiff x z :=
   by
-  by_contra' H
+  by_contra! H
   have : x (first_diff x z) = z (first_diff x z) :=
     calc
       x (first_diff x z) = y (first_diff x z) :=
@@ -195,7 +195,7 @@ theorem mem_cylinder_iff_le_firstDiff {x y : ∀ n, E n} (hne : x ≠ y) (i : 
     x ∈ cylinder y i ↔ i ≤ firstDiff x y := by
   constructor
   · intro h
-    by_contra'
+    by_contra!
     exact apply_first_diff_ne hne (h _ this)
   · intro hi j hj
     exact apply_eq_of_lt_first_diff (hj.trans_le hi)
@@ -407,7 +407,7 @@ theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
   suffices (∀ i : ℕ, i < n → y i = x i) ↔ n ≤ first_diff y x by simpa [dist_eq_of_ne hne]
   constructor
   · intro hy
-    by_contra' H
+    by_contra! H
     exact apply_first_diff_ne hne (hy _ H)
   · intro h i hi
     exact apply_eq_of_lt_first_diff (hi.trans_le h)
@@ -911,7 +911,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
     let n := first_diff x.1 y.1 - 1
     have diff_pos : 0 < first_diff x.1 y.1 :=
       by
-      by_contra' h
+      by_contra! h
       apply apply_first_diff_ne hne'
       rw [le_zero_iff.1 h]
       apply apply_eq_of_dist_lt _ le_rfl

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

@@ -1018,7 +1018,7 @@ theorem dist_summable (x y : ∀ i, F i) :
     Summable fun i : ι => min ((1 / 2) ^ encode i) (dist (x i) (y i)) :=
   by
   refine'
-    summable_of_nonneg_of_le (fun i => _) (fun i => min_le_left _ _) summable_geometric_two_encode
+    Summable.of_nonneg_of_le (fun i => _) (fun i => min_le_left _ _) summable_geometric_two_encode
   exact le_min (pow_nonneg (by norm_num) _) dist_nonneg
 #align pi_countable.dist_summable PiCountable.dist_summable
 -/

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

@@ -3,8 +3,8 @@ Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 -/
-import Mathbin.Tactic.RingExp
-import Mathbin.Topology.MetricSpace.HausdorffDistance
+import Tactic.RingExp
+import Topology.MetricSpace.HausdorffDistance
 
 #align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
 

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

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.pi_nat
-! leanprover-community/mathlib commit 49b7f94aab3a3bdca1f9f34c5d818afb253b3993
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Tactic.RingExp
 import Mathbin.Topology.MetricSpace.HausdorffDistance
 
+#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
+
 /-!
 # Topological study of spaces `Π (n : ℕ), E n`
 

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

@@ -111,6 +111,7 @@ theorem firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x :=
 #align pi_nat.first_diff_comm PiNat.firstDiff_comm
 -/
 
+#print PiNat.min_firstDiff_le /-
 theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
     min (firstDiff x y) (firstDiff y z) ≤ firstDiff x z :=
   by
@@ -122,6 +123,7 @@ theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
       _ = z (first_diff x z) := apply_eq_of_lt_first_diff (H.trans_le (min_le_right _ _))
   exact (apply_first_diff_ne h this).elim
 #align pi_nat.min_first_diff_le PiNat.min_firstDiff_le
+-/
 
 /-! ### Cylinders -/
 
@@ -218,6 +220,7 @@ theorem cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn :
 #align pi_nat.cylinder_eq_cylinder_of_le_first_diff PiNat.cylinder_eq_cylinder_of_le_firstDiff
 -/
 
+#print PiNat.iUnion_cylinder_update /-
 theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     (⋃ k, cylinder (update x n k) (n + 1)) = cylinder x n :=
   by
@@ -232,10 +235,13 @@ theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     · simp [H i h'i, h'i.ne]
     · simp
 #align pi_nat.Union_cylinder_update PiNat.iUnion_cylinder_update
+-/
 
+#print PiNat.update_mem_cylinder /-
 theorem update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y ∈ cylinder x n :=
   mem_cylinder_iff.2 fun i hi => by simp [hi.ne]
 #align pi_nat.update_mem_cylinder PiNat.update_mem_cylinder
+-/
 
 section Res
 
@@ -338,12 +344,16 @@ protected def dist : Dist (∀ n, E n) :=
 
 attribute [local instance] PiNat.dist
 
+#print PiNat.dist_eq_of_ne /-
 theorem dist_eq_of_ne {x y : ∀ n, E n} (h : x ≠ y) : dist x y = (1 / 2 : ℝ) ^ firstDiff x y := by
   simp [dist, h]
 #align pi_nat.dist_eq_of_ne PiNat.dist_eq_of_ne
+-/
 
+#print PiNat.dist_self /-
 protected theorem dist_self (x : ∀ n, E n) : dist x x = 0 := by simp [dist]
 #align pi_nat.dist_self PiNat.dist_self
+-/
 
 #print PiNat.dist_comm /-
 protected theorem dist_comm (x y : ∀ n, E n) : dist x y = dist y x := by
@@ -351,13 +361,16 @@ protected theorem dist_comm (x y : ∀ n, E n) : dist x y = dist y x := by
 #align pi_nat.dist_comm PiNat.dist_comm
 -/
 
+#print PiNat.dist_nonneg /-
 protected theorem dist_nonneg (x y : ∀ n, E n) : 0 ≤ dist x y :=
   by
   rcases eq_or_ne x y with (rfl | h)
   · simp [dist]
   · simp [dist, h]
 #align pi_nat.dist_nonneg PiNat.dist_nonneg
+-/
 
+#print PiNat.dist_triangle_nonarch /-
 theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y) (dist y z) :=
   by
   rcases eq_or_ne x z with (rfl | hxz)
@@ -370,20 +383,26 @@ theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y)
     not_false_iff, le_max_iff, zero_lt_one, pow_le_pow_iff, one_lt_two, pow_pos,
     min_le_iff.1 (min_first_diff_le x y z hxz)]
 #align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarch
+-/
 
+#print PiNat.dist_triangle /-
 protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + dist y z :=
   calc
     dist x z ≤ max (dist x y) (dist y z) := dist_triangle_nonarch x y z
     _ ≤ dist x y + dist y z := max_le_add_of_nonneg (PiNat.dist_nonneg _ _) (PiNat.dist_nonneg _ _)
 #align pi_nat.dist_triangle PiNat.dist_triangle
+-/
 
+#print PiNat.eq_of_dist_eq_zero /-
 protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y :=
   by
   rcases eq_or_ne x y with (rfl | h); · rfl
   simp [dist_eq_of_ne h] at hxy 
   exact (two_ne_zero (pow_eq_zero hxy)).elim
 #align pi_nat.eq_of_dist_eq_zero PiNat.eq_of_dist_eq_zero
+-/
 
+#print PiNat.mem_cylinder_iff_dist_le /-
 theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
     y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n :=
   by
@@ -396,7 +415,9 @@ theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
   · intro h i hi
     exact apply_eq_of_lt_first_diff (hi.trans_le h)
 #align pi_nat.mem_cylinder_iff_dist_le PiNat.mem_cylinder_iff_dist_le
+-/
 
+#print PiNat.apply_eq_of_dist_lt /-
 theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2) ^ n) {i : ℕ}
     (hi : i ≤ n) : x i = y i :=
   by
@@ -405,7 +426,9 @@ theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2)
     simpa [dist_eq_of_ne hne, inv_lt_inv, pow_lt_pow_iff, one_lt_two] using h
   exact apply_eq_of_lt_first_diff (hi.trans_lt this)
 #align pi_nat.apply_eq_of_dist_lt PiNat.apply_eq_of_dist_lt
+-/
 
+#print PiNat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder /-
 /-- A function to a pseudo-metric-space is `1`-Lipschitz if and only if points in the same cylinder
 of length `n` are sent to points within distance `(1/2)^n`.
 Not expressed using `lipschitz_with` as we don't have a metric space structure -/
@@ -426,6 +449,7 @@ theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type _
     rw [first_diff_comm]
     exact mem_cylinder_first_diff _ _
 #align pi_nat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder PiNat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder
+-/
 
 variable (E) [∀ n, TopologicalSpace (E n)] [∀ n, DiscreteTopology (E n)]
 
@@ -468,6 +492,7 @@ theorem isTopologicalBasis_cylinders :
 
 variable {E}
 
+#print PiNat.isOpen_iff_dist /-
 theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
     IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s :=
   by
@@ -488,6 +513,7 @@ theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
     rw [PiNat.dist_comm]
     exact (mem_cylinder_iff_dist_le.1 hy).trans_lt hn
 #align pi_nat.is_open_iff_dist PiNat.isOpen_iff_dist
+-/
 
 #print PiNat.metricSpace /-
 /-- Metric space structure on `Π (n : ℕ), E n` when the spaces `E n` have the discrete topology,
@@ -577,6 +603,7 @@ where `z_w` is an element of `s` starting with `w`.
 -/
 
 
+#print PiNat.exists_disjoint_cylinder /-
 theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
     (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) :=
   by
@@ -592,6 +619,7 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
     _ ≤ (1 / 2) ^ n := by rw [mem_cylinder_comm] at hy ; exact mem_cylinder_iff_dist_le.1 hy
     _ < inf_dist x s := hn
 #align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinder
+-/
 
 #print PiNat.shortestPrefixDiff /-
 /-- Given a point `x` in a product space `Π (n : ℕ), E n`, and `s` a subset of this space, then
@@ -645,6 +673,7 @@ theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y
 #align pi_nat.first_diff_le_longest_prefix PiNat.firstDiff_le_longestPrefix
 -/
 
+#print PiNat.inter_cylinder_longestPrefix_nonempty /-
 theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
     (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
   by
@@ -663,7 +692,9 @@ theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClo
   rw [mem_cylinder_iff_eq] at hy ⊢
   rw [hy]
 #align pi_nat.inter_cylinder_longest_prefix_nonempty PiNat.inter_cylinder_longestPrefix_nonempty
+-/
 
+#print PiNat.disjoint_cylinder_of_longestPrefix_lt /-
 theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
     (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) :=
   by
@@ -674,6 +705,7 @@ theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClo
   apply (mem_cylinder_iff_le_first_diff (ne_of_mem_of_not_mem ys hx).symm _).1
   rwa [mem_cylinder_comm]
 #align pi_nat.disjoint_cylinder_of_longest_prefix_lt PiNat.disjoint_cylinder_of_longestPrefix_lt
+-/
 
 #print PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff /-
 /-- If two points `x, y` coincide up to length `n`, and the longest common prefix of `x` with `s`
@@ -850,6 +882,7 @@ end PiNat
 
 open PiNat
 
+#print exists_nat_nat_continuous_surjective_of_completeSpace /-
 /-- Any nonempty complete second countable metric space is the continuous image of the
 fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
 `exists_nat_nat_continuous_surjective_of_polish_space`. -/
@@ -952,6 +985,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
     simpa only [nonempty_coe_sort] using g_surj.nonempty
   exact ⟨g ∘ f, g_cont.comp f_cont, g_surj.comp f_surj⟩
 #align exists_nat_nat_continuous_surjective_of_complete_space exists_nat_nat_continuous_surjective_of_completeSpace
+-/
 
 namespace PiCountable
 
@@ -975,11 +1009,14 @@ protected def dist : Dist (∀ i, F i) :=
 
 attribute [local instance] PiCountable.dist
 
+#print PiCountable.dist_eq_tsum /-
 theorem dist_eq_tsum (x y : ∀ i, F i) :
     dist x y = ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i)) :=
   rfl
 #align pi_countable.dist_eq_tsum PiCountable.dist_eq_tsum
+-/
 
+#print PiCountable.dist_summable /-
 theorem dist_summable (x y : ∀ i, F i) :
     Summable fun i : ι => min ((1 / 2) ^ encode i) (dist (x i) (y i)) :=
   by
@@ -987,16 +1024,21 @@ theorem dist_summable (x y : ∀ i, F i) :
     summable_of_nonneg_of_le (fun i => _) (fun i => min_le_left _ _) summable_geometric_two_encode
   exact le_min (pow_nonneg (by norm_num) _) dist_nonneg
 #align pi_countable.dist_summable PiCountable.dist_summable
+-/
 
+#print PiCountable.min_dist_le_dist_pi /-
 theorem min_dist_le_dist_pi (x y : ∀ i, F i) (i : ι) :
     min ((1 / 2) ^ encode i) (dist (x i) (y i)) ≤ dist x y :=
   le_tsum (dist_summable x y) i fun j hj => le_min (by simp) dist_nonneg
 #align pi_countable.min_dist_le_dist_pi PiCountable.min_dist_le_dist_pi
+-/
 
+#print PiCountable.dist_le_dist_pi_of_dist_lt /-
 theorem dist_le_dist_pi_of_dist_lt {x y : ∀ i, F i} {i : ι} (h : dist x y < (1 / 2) ^ encode i) :
     dist (x i) (y i) ≤ dist x y := by
   simpa only [not_le.2 h, false_or_iff] using min_le_iff.1 (min_dist_le_dist_pi x y i)
 #align pi_countable.dist_le_dist_pi_of_dist_lt PiCountable.dist_le_dist_pi_of_dist_lt
+-/
 
 open scoped BigOperators Topology
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -1044,8 +1044,8 @@ protected def metricSpace : MetricSpace (∀ i, F i)
     calc
       dist x z ≤
           ∑' i,
-            min ((1 / 2) ^ encode i) (dist (x i) (y i)) +
-              min ((1 / 2) ^ encode i) (dist (y i) (z i)) :=
+            (min ((1 / 2) ^ encode i) (dist (x i) (y i)) +
+              min ((1 / 2) ^ encode i) (dist (y i) (z i))) :=
         tsum_le_tsum I (dist_summable x z) ((dist_summable x y).add (dist_summable y z))
       _ = dist x y + dist y z := tsum_add (dist_summable x y) (dist_summable y z)
   eq_of_dist_eq_zero := by
@@ -1065,7 +1065,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
     · simp only [le_iInf_iff, le_principal_iff]
       intro ε εpos
       obtain ⟨K, hK⟩ :
-        ∃ K : Finset ι, (∑' i : { j // j ∉ K }, (1 / 2 : ℝ) ^ encode (i : ι)) < ε / 2 :=
+        ∃ K : Finset ι, ∑' i : { j // j ∉ K }, (1 / 2 : ℝ) ^ encode (i : ι) < ε / 2 :=
         ((tendsto_order.1 (tendsto_tsum_compl_atTop_zero fun i : ι => (1 / 2 : ℝ) ^ encode i)).2 _
             (half_pos εpos)).exists
       obtain ⟨δ, δpos, hδ⟩ : ∃ (δ : ℝ) (δpos : 0 < δ), (K.card : ℝ) * δ ≤ ε / 2 :=
@@ -1090,17 +1090,16 @@ protected def metricSpace : MetricSpace (∀ i, F i)
         calc
           dist x y = ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i)) := rfl
           _ =
-              (∑ i in K, min ((1 / 2) ^ encode i) (dist (x i) (y i))) +
+              ∑ i in K, min ((1 / 2) ^ encode i) (dist (x i) (y i)) +
                 ∑' i : (↑K : Set ι)ᶜ, min ((1 / 2) ^ encode (i : ι)) (dist (x i) (y i)) :=
             (sum_add_tsum_compl (dist_summable _ _)).symm
-          _ ≤ (∑ i in K, dist (x i) (y i)) + ∑' i : (↑K : Set ι)ᶜ, (1 / 2) ^ encode (i : ι) :=
+          _ ≤ ∑ i in K, dist (x i) (y i) + ∑' i : (↑K : Set ι)ᶜ, (1 / 2) ^ encode (i : ι) :=
             by
             refine' add_le_add (Finset.sum_le_sum fun i hi => min_le_right _ _) _
             refine' tsum_le_tsum (fun i => min_le_left _ _) _ _
             · apply Summable.subtype (dist_summable x y) ((↑K : Set ι)ᶜ)
             · apply Summable.subtype summable_geometric_two_encode ((↑K : Set ι)ᶜ)
-          _ < (∑ i in K, δ) + ε / 2 :=
-            by
+          _ < ∑ i in K, δ + ε / 2 := by
             apply add_lt_add_of_le_of_lt _ hK
             apply Finset.sum_le_sum fun i hi => _
             apply (hxy i _).le

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

@@ -120,7 +120,6 @@ theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
       x (first_diff x z) = y (first_diff x z) :=
         apply_eq_of_lt_first_diff (H.trans_le (min_le_left _ _))
       _ = z (first_diff x z) := apply_eq_of_lt_first_diff (H.trans_le (min_le_right _ _))
-      
   exact (apply_first_diff_ne h this).elim
 #align pi_nat.min_first_diff_le PiNat.min_firstDiff_le
 
@@ -376,7 +375,6 @@ protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + d
   calc
     dist x z ≤ max (dist x y) (dist y z) := dist_triangle_nonarch x y z
     _ ≤ dist x y + dist y z := max_le_add_of_nonneg (PiNat.dist_nonneg _ _) (PiNat.dist_nonneg _ _)
-    
 #align pi_nat.dist_triangle PiNat.dist_triangle
 
 protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y :=
@@ -593,7 +591,6 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
     inf_dist x s ≤ dist x y := inf_dist_le_dist_of_mem ys
     _ ≤ (1 / 2) ^ n := by rw [mem_cylinder_comm] at hy ; exact mem_cylinder_iff_dist_le.1 hy
     _ < inf_dist x s := hn
-    
 #align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinder
 
 #print PiNat.shortestPrefixDiff /-
@@ -899,7 +896,6 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
       _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [← B]
       _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := (add_le_add (A x n) (A y n))
       _ = 4 * (1 / 2) ^ (n + 1) := by ring
-      
   have g_surj : surjective g := by
     intro y
     have : ∀ n : ℕ, ∃ j, y ∈ closed_ball (u j) ((1 / 2) ^ n) :=
@@ -916,7 +912,6 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
         dist (g ⟨x, I⟩) y ≤ dist (g ⟨x, I⟩) (u (x n)) + dist y (u (x n)) :=
           dist_triangle_right _ _ _
         _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A ⟨x, I⟩ n) (hx n)
-        
     have L : tendsto (fun n : ℕ => (1 / 2 : ℝ) ^ n + (1 / 2) ^ n) at_top (𝓝 (0 + 0)) :=
       (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).add (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1)
     rw [add_zero] at L 
@@ -1046,7 +1041,6 @@ protected def metricSpace : MetricSpace (∀ i, F i)
             min ((1 / 2) ^ encode i) (dist (x i) (y i)) +
               min ((1 / 2) ^ encode i) (dist (y i) (z i)) :=
           min_le_right _ _
-        
     calc
       dist x z ≤
           ∑' i,
@@ -1054,7 +1048,6 @@ protected def metricSpace : MetricSpace (∀ i, F i)
               min ((1 / 2) ^ encode i) (dist (y i) (z i)) :=
         tsum_le_tsum I (dist_summable x z) ((dist_summable x y).add (dist_summable y z))
       _ = dist x y + dist y z := tsum_add (dist_summable x y) (dist_summable y z)
-      
   eq_of_dist_eq_zero := by
     intro x y hxy
     ext1 n
@@ -1115,7 +1108,6 @@ protected def metricSpace : MetricSpace (∀ i, F i)
           _ ≤ ε / 2 + ε / 2 :=
             (add_le_add_right (by simpa only [Finset.sum_const, nsmul_eq_mul] using hδ) _)
           _ = ε := add_halves _
-          
     · simp only [le_iInf_iff, le_principal_iff]
       intro i ε εpos
       refine' mem_infi_of_mem (min ((1 / 2) ^ encode i) ε) _
@@ -1126,7 +1118,6 @@ protected def metricSpace : MetricSpace (∀ i, F i)
       calc
         dist (x i) (y i) ≤ dist x y := dist_le_dist_pi_of_dist_lt hn
         _ < ε := hε
-        
 #align pi_countable.metric_space PiCountable.metricSpace
 -/
 

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

@@ -93,7 +93,7 @@ theorem apply_firstDiff_ne {x y : ∀ n, E n} (h : x ≠ y) : x (firstDiff x y)
 theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDiff x y) : x n = y n :=
   by
   rw [first_diff] at hn 
-  split_ifs  at hn 
+  split_ifs at hn 
   · convert Nat.find_min (ne_iff.1 h) hn
     simp
   · exact (not_lt_zero' hn).elim
@@ -133,7 +133,7 @@ theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
 such that `y i = x i` for all `i < n`.
 -/
 def cylinder (x : ∀ n, E n) (n : ℕ) : Set (∀ n, E n) :=
-  { y | ∀ i, i < n → y i = x i }
+  {y | ∀ i, i < n → y i = x i}
 #align pi_nat.cylinder PiNat.cylinder
 -/
 
@@ -308,7 +308,7 @@ theorem res_injective : Injective (@res α) :=
 
 #print PiNat.cylinder_eq_res /-
 /-- `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`.-/
-theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) : cylinder x n = { y | res y n = res x n } :=
+theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) : cylinder x n = {y | res y n = res x n} :=
   by
   ext y
   dsimp [cylinder]
@@ -441,7 +441,7 @@ theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) :=
 
 #print PiNat.isTopologicalBasis_cylinders /-
 theorem isTopologicalBasis_cylinders :
-    IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n) (n : ℕ), s = cylinder x n } :=
+    IsTopologicalBasis {s : Set (∀ n, E n) | ∃ (x : ∀ n, E n) (n : ℕ), s = cylinder x n} :=
   by
   apply is_topological_basis_of_open_of_nhds
   · rintro u ⟨x, n, rfl⟩
@@ -450,9 +450,9 @@ theorem isTopologicalBasis_cylinders :
     obtain ⟨v, ⟨U, F, hUF, rfl⟩, xU, Uu⟩ :
       ∃ (v : Set (∀ i : ℕ, E i)) (H :
         v ∈
-          { S : Set (∀ i : ℕ, E i) |
+          {S : Set (∀ i : ℕ, E i) |
             ∃ (U : ∀ i : ℕ, Set (E i)) (F : Finset ℕ),
-              (∀ i : ℕ, i ∈ F → U i ∈ { s : Set (E i) | IsOpen s }) ∧ S = (F : Set ℕ).pi U }),
+              (∀ i : ℕ, i ∈ F → U i ∈ {s : Set (E i) | IsOpen s}) ∧ S = (F : Set ℕ).pi U}),
         x ∈ v ∧ v ⊆ u :=
       (isTopologicalBasis_pi fun n : ℕ => is_topological_basis_opens).exists_subset_of_mem_open hx
         u_open
@@ -476,7 +476,7 @@ theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
   constructor
   · intro hs x hx
     obtain ⟨v, ⟨y, n, rfl⟩, h'x, h's⟩ :
-      ∃ (v : Set (∀ n : ℕ, E n)) (H : v ∈ { s | ∃ (x : ∀ n : ℕ, E n) (n : ℕ), s = cylinder x n }),
+      ∃ (v : Set (∀ n : ℕ, E n)) (H : v ∈ {s | ∃ (x : ∀ n : ℕ, E n) (n : ℕ), s = cylinder x n}),
         x ∈ v ∧ v ⊆ s :=
       (is_topological_basis_cylinders E).exists_subset_of_mem_open hx hs
     rw [← mem_cylinder_iff_eq.1 h'x] at h's 
@@ -527,7 +527,7 @@ protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, Unifo
         obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < ε := exists_pow_lt_of_lt_one εpos (by norm_num)
         apply
           @mem_infi_of_Inter _ _ _ _ _ (Finset.range n).finite_toSet fun i =>
-            { p : (∀ n : ℕ, E n) × ∀ n : ℕ, E n | p.fst i = p.snd i }
+            {p : (∀ n : ℕ, E n) × ∀ n : ℕ, E n | p.fst i = p.snd i}
         · simp only [mem_principal, set_of_subset_set_of, imp_self, imp_true_iff]
         · rintro ⟨x, y⟩ hxy
           simp only [Finset.mem_coe, Finset.mem_range, Inter_coe_set, mem_Inter, mem_set_of_eq] at
@@ -563,7 +563,7 @@ protected theorem completeSpace : CompleteSpace (∀ n, E n) :=
   intro u hu
   refine' ⟨fun n => u n n, tendsto_pi_nhds.2 fun i => _⟩
   refine' tendsto_const_nhds.congr' _
-  filter_upwards [Filter.Ici_mem_atTop i]with n hn
+  filter_upwards [Filter.Ici_mem_atTop i] with n hn
   exact apply_eq_of_dist_lt (hu i i n le_rfl hn) le_rfl
 #align pi_nat.complete_space PiNat.completeSpace
 -/
@@ -659,7 +659,7 @@ theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClo
   obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) :=
     by
     have := Nat.find_min A B
-    push_neg  at this 
+    push_neg at this 
     simp_rw [not_disjoint_iff, mem_cylinder_comm] at this 
     exact this
   refine' ⟨y, ys, _⟩
@@ -805,7 +805,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
           congr
         -- case where the common prefix to `x` and `s` is long, as well as the common prefix to
         -- `y` and `s`. Then all points remain in the same cylinders.
-        · push_neg  at H 
+        · push_neg at H 
           have I1 : cylinder Ax.some (first_diff x y) = cylinder x (first_diff x y) :=
             by
             rw [← mem_cylinder_iff_eq]
@@ -870,7 +870,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
   have I0 : (0 : ℝ) < 1 / 2 := by norm_num
   have I1 : (1 / 2 : ℝ) < 1 := by norm_num
   rcases exists_dense_seq α with ⟨u, hu⟩
-  let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closed_ball (u (x n)) ((1 / 2) ^ n)).Nonempty }
+  let s : Set (ℕ → ℕ) := {x | (⋂ n : ℕ, closed_ball (u (x n)) ((1 / 2) ^ n)).Nonempty}
   let g : s → α := fun x => x.2.some
   have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
     (mem_Inter.1 x.2.some_mem n : _)
@@ -1037,7 +1037,8 @@ protected def metricSpace : MetricSpace (∀ i, F i)
               (min ((1 / 2) ^ encode i) (dist (x i) (y i)) +
                 min ((1 / 2) ^ encode i) (dist (y i) (z i))) :=
           by
-          convert congr_arg (coe : ℝ≥0 → ℝ)
+          convert
+              congr_arg (coe : ℝ≥0 → ℝ)
                 (min_add_distrib ((1 / 2 : ℝ≥0) ^ encode i) (nndist (x i) (y i))
                   (nndist (y i) (z i))) <;>
             simp
@@ -1087,7 +1088,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
           ring
       apply
         @mem_infi_of_Inter _ _ _ _ _ K.finite_to_set fun i =>
-          { p : (∀ i : ι, F i) × ∀ i : ι, F i | dist (p.fst i) (p.snd i) < δ }
+          {p : (∀ i : ι, F i) × ∀ i : ι, F i | dist (p.fst i) (p.snd i) < δ}
       · rintro ⟨i, hi⟩
         refine' mem_infi_of_mem δ (mem_infi_of_mem δpos _)
         simp only [Prod.forall, imp_self, mem_principal]

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

@@ -92,8 +92,8 @@ theorem apply_firstDiff_ne {x y : ∀ n, E n} (h : x ≠ y) : x (firstDiff x y)
 #print PiNat.apply_eq_of_lt_firstDiff /-
 theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDiff x y) : x n = y n :=
   by
-  rw [first_diff] at hn
-  split_ifs  at hn
+  rw [first_diff] at hn 
+  split_ifs  at hn 
   · convert Nat.find_min (ne_iff.1 h) hn
     simp
   · exact (not_lt_zero' hn).elim
@@ -283,8 +283,8 @@ theorem res_eq_res {x y : ℕ → α} {n : ℕ} : res x n = res y n ↔ ∀ ⦃m
   by
   constructor <;> intro h <;> induction' n with n ih; · simp
   · intro m hm
-    rw [Nat.lt_succ_iff_lt_or_eq] at hm
-    simp only [res_succ] at h
+    rw [Nat.lt_succ_iff_lt_or_eq] at hm 
+    simp only [res_succ] at h 
     cases' hm with hm hm
     · exact ih h.2 hm
     rw [hm]
@@ -382,7 +382,7 @@ protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + d
 protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y :=
   by
   rcases eq_or_ne x y with (rfl | h); · rfl
-  simp [dist_eq_of_ne h] at hxy
+  simp [dist_eq_of_ne h] at hxy 
   exact (two_ne_zero (pow_eq_zero hxy)).elim
 #align pi_nat.eq_of_dist_eq_zero PiNat.eq_of_dist_eq_zero
 
@@ -441,17 +441,17 @@ theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) :=
 
 #print PiNat.isTopologicalBasis_cylinders /-
 theorem isTopologicalBasis_cylinders :
-    IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n)(n : ℕ), s = cylinder x n } :=
+    IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n) (n : ℕ), s = cylinder x n } :=
   by
   apply is_topological_basis_of_open_of_nhds
   · rintro u ⟨x, n, rfl⟩
     apply is_open_cylinder
   · intro x u hx u_open
     obtain ⟨v, ⟨U, F, hUF, rfl⟩, xU, Uu⟩ :
-      ∃ (v : Set (∀ i : ℕ, E i))(H :
+      ∃ (v : Set (∀ i : ℕ, E i)) (H :
         v ∈
           { S : Set (∀ i : ℕ, E i) |
-            ∃ (U : ∀ i : ℕ, Set (E i))(F : Finset ℕ),
+            ∃ (U : ∀ i : ℕ, Set (E i)) (F : Finset ℕ),
               (∀ i : ℕ, i ∈ F → U i ∈ { s : Set (E i) | IsOpen s }) ∧ S = (F : Set ℕ).pi U }),
         x ∈ v ∧ v ⊆ u :=
       (isTopologicalBasis_pi fun n : ℕ => is_topological_basis_opens).exists_subset_of_mem_open hx
@@ -463,7 +463,7 @@ theorem isTopologicalBasis_cylinders :
     intro i hi
     have : y i = x i := mem_cylinder_iff.1 hy i ((hn hi).trans_lt (lt_add_one n))
     rw [this]
-    simp only [Set.mem_pi, Finset.mem_coe] at xU
+    simp only [Set.mem_pi, Finset.mem_coe] at xU 
     exact xU i hi
 #align pi_nat.is_topological_basis_cylinders PiNat.isTopologicalBasis_cylinders
 -/
@@ -476,10 +476,10 @@ theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
   constructor
   · intro hs x hx
     obtain ⟨v, ⟨y, n, rfl⟩, h'x, h's⟩ :
-      ∃ (v : Set (∀ n : ℕ, E n))(H : v ∈ { s | ∃ (x : ∀ n : ℕ, E n)(n : ℕ), s = cylinder x n }),
+      ∃ (v : Set (∀ n : ℕ, E n)) (H : v ∈ { s | ∃ (x : ∀ n : ℕ, E n) (n : ℕ), s = cylinder x n }),
         x ∈ v ∧ v ⊆ s :=
       (is_topological_basis_cylinders E).exists_subset_of_mem_open hx hs
-    rw [← mem_cylinder_iff_eq.1 h'x] at h's
+    rw [← mem_cylinder_iff_eq.1 h'x] at h's 
     exact
       ⟨(1 / 2 : ℝ) ^ n, by simp, fun y hy => h's fun i hi => (apply_eq_of_dist_lt hy hi.le).symm⟩
   · intro h
@@ -531,7 +531,7 @@ protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, Unifo
         · simp only [mem_principal, set_of_subset_set_of, imp_self, imp_true_iff]
         · rintro ⟨x, y⟩ hxy
           simp only [Finset.mem_coe, Finset.mem_range, Inter_coe_set, mem_Inter, mem_set_of_eq] at
-            hxy
+            hxy 
           apply lt_of_le_of_lt _ hn
           rw [← mem_cylinder_iff_dist_le, mem_cylinder_iff]
           exact hxy
@@ -591,7 +591,7 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
   apply lt_irrefl (inf_dist x s)
   calc
     inf_dist x s ≤ dist x y := inf_dist_le_dist_of_mem ys
-    _ ≤ (1 / 2) ^ n := by rw [mem_cylinder_comm] at hy; exact mem_cylinder_iff_dist_le.1 hy
+    _ ≤ (1 / 2) ^ n := by rw [mem_cylinder_comm] at hy ; exact mem_cylinder_iff_dist_le.1 hy
     _ < inf_dist x s := hn
     
 #align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinder
@@ -655,15 +655,15 @@ theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClo
   have A := exists_disjoint_cylinder hs hx
   have B : longest_prefix x s < shortest_prefix_diff x s :=
     Nat.pred_lt (shortest_prefix_diff_pos hs hne hx).ne'
-  rw [longest_prefix, shortest_prefix_diff, dif_pos A] at B⊢
+  rw [longest_prefix, shortest_prefix_diff, dif_pos A] at B ⊢
   obtain ⟨y, ys, hy⟩ : ∃ y : ∀ n : ℕ, E n, y ∈ s ∧ x ∈ cylinder y (Nat.find A - 1) :=
     by
     have := Nat.find_min A B
-    push_neg  at this
-    simp_rw [not_disjoint_iff, mem_cylinder_comm] at this
+    push_neg  at this 
+    simp_rw [not_disjoint_iff, mem_cylinder_comm] at this 
     exact this
   refine' ⟨y, ys, _⟩
-  rw [mem_cylinder_iff_eq] at hy⊢
+  rw [mem_cylinder_iff_eq] at hy ⊢
   rw [hy]
 #align pi_nat.inter_cylinder_longest_prefix_nonempty PiNat.inter_cylinder_longestPrefix_nonempty
 
@@ -693,8 +693,8 @@ theorem cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff {x y : ∀ n, E
     · have Ax : (s ∩ cylinder x (longest_prefix x s)).Nonempty :=
         inter_cylinder_longest_prefix_nonempty hs hne x
       have Z := disjoint_cylinder_of_longest_prefix_lt hs ys L
-      rw [first_diff_comm] at H
-      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H.le] at Z
+      rw [first_diff_comm] at H 
+      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H.le] at Z 
       exact (Ax.not_disjoint Z).elim
     · exact L
     · have Ay : (s ∩ cylinder y (longest_prefix y s)).Nonempty :=
@@ -702,7 +702,7 @@ theorem cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff {x y : ∀ n, E
       have A'y : (s ∩ cylinder y (longest_prefix x s).succ).Nonempty :=
         Ay.mono (inter_subset_inter_right s (cylinder_anti _ L))
       have Z := disjoint_cylinder_of_longest_prefix_lt hs xs (Nat.lt_succ_self _)
-      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H] at Z
+      rw [cylinder_eq_cylinder_of_le_first_diff _ _ H] at Z 
       exact (A'y.not_disjoint Z).elim
   rw [l_eq, ← mem_cylinder_iff_eq]
   exact cylinder_anti y H.le (mem_cylinder_first_diff x y)
@@ -755,7 +755,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
       simpa [dist_eq_of_ne hxy, dist_eq_of_ne hfxfy]
     -- case where `x ∈ s`
     by_cases xs : x ∈ s
-    · rw [fs x xs] at hfxfy⊢
+    · rw [fs x xs] at hfxfy ⊢
       -- case where `y ∈ s`, trivial
       by_cases ys : y ∈ s;
       · rw [fs y ys]
@@ -769,7 +769,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
         apply cylinder_anti y _ A.some_spec.2
         exact first_diff_le_longest_prefix hs ys xs
       rwa [← fy, ← I2, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy.symm,
-        first_diff_comm _ x] at I
+        first_diff_comm _ x] at I 
     -- case where `x ∉ s`
     · by_cases ys : y ∈ s
       -- case where `y ∈ s` (similar to the above)
@@ -781,8 +781,8 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
           rw [← mem_cylinder_iff_eq]
           apply cylinder_anti x _ A.some_spec.2
           apply first_diff_le_longest_prefix hs xs ys
-        rw [fs y ys] at hfxfy⊢
-        rwa [← fx, I2, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at I
+        rw [fs y ys] at hfxfy ⊢
+        rwa [← fx, I2, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at I 
       -- case where `y ∉ s`
       · have Ax : (s ∩ cylinder x (longest_prefix x s)).Nonempty :=
           inter_cylinder_longest_prefix_nonempty hs hne x
@@ -798,14 +798,14 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
             cases H
             · exact cylinder_longest_prefix_eq_of_longest_prefix_lt_first_diff hs hne H xs ys
             · symm
-              rw [first_diff_comm] at H
+              rw [first_diff_comm] at H 
               exact cylinder_longest_prefix_eq_of_longest_prefix_lt_first_diff hs hne H ys xs
-          rw [fx, fy] at hfxfy
+          rw [fx, fy] at hfxfy 
           apply (hfxfy _).elim
           congr
         -- case where the common prefix to `x` and `s` is long, as well as the common prefix to
         -- `y` and `s`. Then all points remain in the same cylinders.
-        · push_neg  at H
+        · push_neg  at H 
           have I1 : cylinder Ax.some (first_diff x y) = cylinder x (first_diff x y) :=
             by
             rw [← mem_cylinder_iff_eq]
@@ -816,7 +816,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
             exact cylinder_anti y H.2 Ay.some_spec.2
           have : cylinder Ax.some (first_diff x y) = cylinder Ay.some (first_diff x y) := by
             rw [I1, I2, I3]
-          rw [← fx, ← fy, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at this
+          rw [← fx, ← fy, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at this 
           exact this
 #align pi_nat.exists_lipschitz_retraction_of_is_closed PiNat.exists_lipschitz_retraction_of_isClosed
 -/
@@ -919,7 +919,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
         
     have L : tendsto (fun n : ℕ => (1 / 2 : ℝ) ^ n + (1 / 2) ^ n) at_top (𝓝 (0 + 0)) :=
       (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).add (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1)
-    rw [add_zero] at L
+    rw [add_zero] at L 
     exact ge_of_tendsto' L J
   have s_closed : IsClosed s :=
     by
@@ -929,7 +929,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
       by
       have : tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) at_top (𝓝 (2 * 0)) :=
         (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _
-      rw [MulZeroClass.mul_zero] at this
+      rw [MulZeroClass.mul_zero] at this 
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closed_ball (pow_nonneg I0.le _)) this
     refine'
@@ -1074,7 +1074,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
         ∃ K : Finset ι, (∑' i : { j // j ∉ K }, (1 / 2 : ℝ) ^ encode (i : ι)) < ε / 2 :=
         ((tendsto_order.1 (tendsto_tsum_compl_atTop_zero fun i : ι => (1 / 2 : ℝ) ^ encode i)).2 _
             (half_pos εpos)).exists
-      obtain ⟨δ, δpos, hδ⟩ : ∃ (δ : ℝ)(δpos : 0 < δ), (K.card : ℝ) * δ ≤ ε / 2 :=
+      obtain ⟨δ, δpos, hδ⟩ : ∃ (δ : ℝ) (δpos : 0 < δ), (K.card : ℝ) * δ ≤ ε / 2 :=
         by
         rcases Nat.eq_zero_or_pos K.card with (hK | hK)
         ·
@@ -1092,7 +1092,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
         refine' mem_infi_of_mem δ (mem_infi_of_mem δpos _)
         simp only [Prod.forall, imp_self, mem_principal]
       · rintro ⟨x, y⟩ hxy
-        simp only [mem_Inter, mem_set_of_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy
+        simp only [mem_Inter, mem_set_of_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy 
         calc
           dist x y = ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i)) := rfl
           _ =

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

@@ -59,7 +59,7 @@ in general), and `ι` is countable.
 
 noncomputable section
 
-open Classical Topology Filter
+open scoped Classical Topology Filter
 
 open TopologicalSpace Set Metric Filter Function
 
@@ -1003,11 +1003,11 @@ theorem dist_le_dist_pi_of_dist_lt {x y : ∀ i, F i} {i : ι} (h : dist x y < (
   simpa only [not_le.2 h, false_or_iff] using min_le_iff.1 (min_dist_le_dist_pi x y i)
 #align pi_countable.dist_le_dist_pi_of_dist_lt PiCountable.dist_le_dist_pi_of_dist_lt
 
-open BigOperators Topology
+open scoped BigOperators Topology
 
 open Filter
 
-open NNReal
+open scoped NNReal
 
 variable (E)
 

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

@@ -111,12 +111,6 @@ theorem firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x :=
 #align pi_nat.first_diff_comm PiNat.firstDiff_comm
 -/
 
-/- warning: pi_nat.min_first_diff_le -> PiNat.min_firstDiff_le is a dubious translation:
-lean 3 declaration is
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), (Ne.{succ u1} (forall (n : Nat), E n) x z) -> (LE.le.{0} Nat Nat.hasLe (LinearOrder.min.{0} Nat Nat.linearOrder (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) y z)) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x z))
-but is expected to have type
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), (Ne.{succ u1} (forall (n : Nat), E n) x z) -> (LE.le.{0} Nat instLENat (Min.min.{0} Nat instMinNat (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) y z)) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x z))
-Case conversion may be inaccurate. Consider using '#align pi_nat.min_first_diff_le PiNat.min_firstDiff_leₓ'. -/
 theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
     min (firstDiff x y) (firstDiff y z) ≤ firstDiff x z :=
   by
@@ -225,12 +219,6 @@ theorem cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn :
 #align pi_nat.cylinder_eq_cylinder_of_le_first_diff PiNat.cylinder_eq_cylinder_of_le_firstDiff
 -/
 
-/- warning: pi_nat.Union_cylinder_update -> PiNat.iUnion_cylinder_update is a dubious translation:
-lean 3 declaration is
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.iUnion.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
-but is expected to have type
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.iUnion.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
-Case conversion may be inaccurate. Consider using '#align pi_nat.Union_cylinder_update PiNat.iUnion_cylinder_updateₓ'. -/
 theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     (⋃ k, cylinder (update x n k) (n + 1)) = cylinder x n :=
   by
@@ -246,12 +234,6 @@ theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     · simp
 #align pi_nat.Union_cylinder_update PiNat.iUnion_cylinder_update
 
-/- warning: pi_nat.update_mem_cylinder -> PiNat.update_mem_cylinder is a dubious translation:
-lean 3 declaration is
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat) (y : E n), Membership.Mem.{u1, u1} (forall (a : Nat), E a) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) x n y) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
-but is expected to have type
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat) (y : E n), Membership.mem.{u1, u1} (forall (a : Nat), E a) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) x n y) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
-Case conversion may be inaccurate. Consider using '#align pi_nat.update_mem_cylinder PiNat.update_mem_cylinderₓ'. -/
 theorem update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y ∈ cylinder x n :=
   mem_cylinder_iff.2 fun i hi => by simp [hi.ne]
 #align pi_nat.update_mem_cylinder PiNat.update_mem_cylinder
@@ -357,22 +339,10 @@ protected def dist : Dist (∀ n, E n) :=
 
 attribute [local instance] PiNat.dist
 
-/- warning: pi_nat.dist_eq_of_ne -> PiNat.dist_eq_of_ne is a dubious translation:
-lean 3 declaration is
-  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n}, (Ne.{succ u1} (forall (n : Nat), E n) x y) -> (Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y)))
-but is expected to have type
-  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n}, (Ne.{succ u1} (forall (n : Nat), E n) x y) -> (Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y)))
-Case conversion may be inaccurate. Consider using '#align pi_nat.dist_eq_of_ne PiNat.dist_eq_of_neₓ'. -/
 theorem dist_eq_of_ne {x y : ∀ n, E n} (h : x ≠ y) : dist x y = (1 / 2 : ℝ) ^ firstDiff x y := by
   simp [dist, h]
 #align pi_nat.dist_eq_of_ne PiNat.dist_eq_of_ne
 
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-Case conversion may be inaccurate. Consider using '#align pi_nat.dist_self PiNat.dist_selfₓ'. -/
 protected theorem dist_self (x : ∀ n, E n) : dist x x = 0 := by simp [dist]
 #align pi_nat.dist_self PiNat.dist_self
 
@@ -382,12 +352,6 @@ protected theorem dist_comm (x y : ∀ n, E n) : dist x y = dist y x := by
 #align pi_nat.dist_comm PiNat.dist_comm
 -/
 
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-Case conversion may be inaccurate. Consider using '#align pi_nat.dist_nonneg PiNat.dist_nonnegₓ'. -/
 protected theorem dist_nonneg (x y : ∀ n, E n) : 0 ≤ dist x y :=
   by
   rcases eq_or_ne x y with (rfl | h)
@@ -395,12 +359,6 @@ protected theorem dist_nonneg (x y : ∀ n, E n) : 0 ≤ dist x y :=
   · simp [dist, h]
 #align pi_nat.dist_nonneg PiNat.dist_nonneg
 
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-Case conversion may be inaccurate. Consider using '#align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarchₓ'. -/
 theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y) (dist y z) :=
   by
   rcases eq_or_ne x z with (rfl | hxz)
@@ -414,12 +372,6 @@ theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y)
     min_le_iff.1 (min_first_diff_le x y z hxz)]
 #align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarch
 
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-Case conversion may be inaccurate. Consider using '#align pi_nat.dist_triangle PiNat.dist_triangleₓ'. -/
 protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + dist y z :=
   calc
     dist x z ≤ max (dist x y) (dist y z) := dist_triangle_nonarch x y z
@@ -427,12 +379,6 @@ protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + d
     
 #align pi_nat.dist_triangle PiNat.dist_triangle
 
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 protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y :=
   by
   rcases eq_or_ne x y with (rfl | h); · rfl
@@ -440,12 +386,6 @@ protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x
   exact (two_ne_zero (pow_eq_zero hxy)).elim
 #align pi_nat.eq_of_dist_eq_zero PiNat.eq_of_dist_eq_zero
 
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 theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
     y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n :=
   by
@@ -459,12 +399,6 @@ theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
     exact apply_eq_of_lt_first_diff (hi.trans_le h)
 #align pi_nat.mem_cylinder_iff_dist_le PiNat.mem_cylinder_iff_dist_le
 
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 theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2) ^ n) {i : ℕ}
     (hi : i ≤ n) : x i = y i :=
   by
@@ -474,12 +408,6 @@ theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2)
   exact apply_eq_of_lt_first_diff (hi.trans_lt this)
 #align pi_nat.apply_eq_of_dist_lt PiNat.apply_eq_of_dist_lt
 
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 /-- A function to a pseudo-metric-space is `1`-Lipschitz if and only if points in the same cylinder
 of length `n` are sent to points within distance `(1/2)^n`.
 Not expressed using `lipschitz_with` as we don't have a metric space structure -/
@@ -542,12 +470,6 @@ theorem isTopologicalBasis_cylinders :
 
 variable {E}
 
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 theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
     IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s :=
   by
@@ -657,12 +579,6 @@ where `z_w` is an element of `s` starting with `w`.
 -/
 
 
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-Case conversion may be inaccurate. Consider using '#align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinderₓ'. -/
 theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
     (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) :=
   by
@@ -732,12 +648,6 @@ theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y
 #align pi_nat.first_diff_le_longest_prefix PiNat.firstDiff_le_longestPrefix
 -/
 
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 theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
     (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
   by
@@ -757,12 +667,6 @@ theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClo
   rw [hy]
 #align pi_nat.inter_cylinder_longest_prefix_nonempty PiNat.inter_cylinder_longestPrefix_nonempty
 
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 theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
     (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) :=
   by
@@ -949,12 +853,6 @@ end PiNat
 
 open PiNat
 
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 /-- Any nonempty complete second countable metric space is the continuous image of the
 fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
 `exists_nat_nat_continuous_surjective_of_polish_space`. -/
@@ -1082,23 +980,11 @@ protected def dist : Dist (∀ i, F i) :=
 
 attribute [local instance] PiCountable.dist
 
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 theorem dist_eq_tsum (x y : ∀ i, F i) :
     dist x y = ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i)) :=
   rfl
 #align pi_countable.dist_eq_tsum PiCountable.dist_eq_tsum
 
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 theorem dist_summable (x y : ∀ i, F i) :
     Summable fun i : ι => min ((1 / 2) ^ encode i) (dist (x i) (y i)) :=
   by
@@ -1107,23 +993,11 @@ theorem dist_summable (x y : ∀ i, F i) :
   exact le_min (pow_nonneg (by norm_num) _) dist_nonneg
 #align pi_countable.dist_summable PiCountable.dist_summable
 
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-  forall {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {F : ι -> Type.{u1}} [_inst_2 : forall (i : ι), MetricSpace.{u1} (F i)] (x : forall (i : ι), F i) (y : forall (i : ι), F i) (i : ι), LE.le.{0} Real Real.instLEReal (Min.min.{0} Real (LinearOrderedRing.toMin.{0} Real Real.instLinearOrderedRingReal) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Encodable.encode.{u2} ι _inst_1 i)) (Dist.dist.{u1} (F i) (PseudoMetricSpace.toDist.{u1} (F i) (MetricSpace.toPseudoMetricSpace.{u1} (F i) (_inst_2 i))) (x i) (y i))) (Dist.dist.{max u2 u1} (forall (i : ι), F i) (PiCountable.dist.{u2, u1} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y)
-Case conversion may be inaccurate. Consider using '#align pi_countable.min_dist_le_dist_pi PiCountable.min_dist_le_dist_piₓ'. -/
 theorem min_dist_le_dist_pi (x y : ∀ i, F i) (i : ι) :
     min ((1 / 2) ^ encode i) (dist (x i) (y i)) ≤ dist x y :=
   le_tsum (dist_summable x y) i fun j hj => le_min (by simp) dist_nonneg
 #align pi_countable.min_dist_le_dist_pi PiCountable.min_dist_le_dist_pi
 
-/- warning: pi_countable.dist_le_dist_pi_of_dist_lt -> PiCountable.dist_le_dist_pi_of_dist_lt is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} [_inst_1 : Encodable.{u1} ι] {F : ι -> Type.{u2}} [_inst_2 : forall (i : ι), MetricSpace.{u2} (F i)] {x : forall (i : ι), F i} {y : forall (i : ι), F i} {i : ι}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{max u1 u2} (forall (i : ι), F i) (PiCountable.dist.{u1, u2} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (Encodable.encode.{u1} ι _inst_1 i))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u2} (F i) (PseudoMetricSpace.toHasDist.{u2} (F i) (MetricSpace.toPseudoMetricSpace.{u2} (F i) (_inst_2 i))) (x i) (y i)) (Dist.dist.{max u1 u2} (forall (i : ι), F i) (PiCountable.dist.{u1, u2} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y))
-but is expected to have type
-  forall {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {F : ι -> Type.{u1}} [_inst_2 : forall (i : ι), MetricSpace.{u1} (F i)] {x : forall (i : ι), F i} {y : forall (i : ι), F i} {i : ι}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{max u2 u1} (forall (i : ι), F i) (PiCountable.dist.{u2, u1} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Encodable.encode.{u2} ι _inst_1 i))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (F i) (PseudoMetricSpace.toDist.{u1} (F i) (MetricSpace.toPseudoMetricSpace.{u1} (F i) (_inst_2 i))) (x i) (y i)) (Dist.dist.{max u2 u1} (forall (i : ι), F i) (PiCountable.dist.{u2, u1} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y))
-Case conversion may be inaccurate. Consider using '#align pi_countable.dist_le_dist_pi_of_dist_lt PiCountable.dist_le_dist_pi_of_dist_ltₓ'. -/
 theorem dist_le_dist_pi_of_dist_lt {x y : ∀ i, F i} {i : ι} (h : dist x y < (1 / 2) ^ encode i) :
     dist (x i) (y i) ≤ dist x y := by
   simpa only [not_le.2 h, false_or_iff] using min_le_iff.1 (min_dist_le_dist_pi x y i)

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

@@ -145,10 +145,7 @@ def cylinder (x : ∀ n, E n) (n : ℕ) : Set (∀ n, E n) :=
 
 #print PiNat.cylinder_eq_pi /-
 theorem cylinder_eq_pi (x : ∀ n, E n) (n : ℕ) :
-    cylinder x n = Set.pi (Finset.range n : Set ℕ) fun i : ℕ => {x i} :=
-  by
-  ext y
-  simp [cylinder]
+    cylinder x n = Set.pi (Finset.range n : Set ℕ) fun i : ℕ => {x i} := by ext y; simp [cylinder]
 #align pi_nat.cylinder_eq_pi PiNat.cylinder_eq_pi
 -/
 
@@ -452,8 +449,7 @@ Case conversion may be inaccurate. Consider using '#align pi_nat.mem_cylinder_if
 theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
     y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n :=
   by
-  rcases eq_or_ne y x with (rfl | hne)
-  · simp [PiNat.dist_self]
+  rcases eq_or_ne y x with (rfl | hne); · simp [PiNat.dist_self]
   suffices (∀ i : ℕ, i < n → y i = x i) ↔ n ≤ first_diff y x by simpa [dist_eq_of_ne hne]
   constructor
   · intro hy
@@ -472,8 +468,7 @@ Case conversion may be inaccurate. Consider using '#align pi_nat.apply_eq_of_dis
 theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2) ^ n) {i : ℕ}
     (hi : i ≤ n) : x i = y i :=
   by
-  rcases eq_or_ne x y with (rfl | hne)
-  · rfl
+  rcases eq_or_ne x y with (rfl | hne); · rfl
   have : n < first_diff x y := by
     simpa [dist_eq_of_ne hne, inv_lt_inv, pow_lt_pow_iff, one_lt_two] using h
   exact apply_eq_of_lt_first_diff (hi.trans_lt this)
@@ -499,8 +494,7 @@ theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type _
     rw [PiNat.dist_comm]
     exact mem_cylinder_iff_dist_le.1 hxy
   · intro H x y
-    rcases eq_or_ne x y with (rfl | hne)
-    · simp [PiNat.dist_nonneg]
+    rcases eq_or_ne x y with (rfl | hne); · simp [PiNat.dist_nonneg]
     rw [dist_eq_of_ne hne]
     apply H x y (first_diff x y)
     rw [first_diff_comm]
@@ -681,9 +675,7 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
   apply lt_irrefl (inf_dist x s)
   calc
     inf_dist x s ≤ dist x y := inf_dist_le_dist_of_mem ys
-    _ ≤ (1 / 2) ^ n := by
-      rw [mem_cylinder_comm] at hy
-      exact mem_cylinder_iff_dist_le.1 hy
+    _ ≤ (1 / 2) ^ n := by rw [mem_cylinder_comm] at hy; exact mem_cylinder_iff_dist_le.1 hy
     _ < inf_dist x s := hn
     
 #align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinder
@@ -749,8 +741,7 @@ Case conversion may be inaccurate. Consider using '#align pi_nat.inter_cylinder_
 theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
     (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
   by
-  by_cases hx : x ∈ s
-  · exact ⟨x, hx, self_mem_cylinder _ _⟩
+  by_cases hx : x ∈ s; · exact ⟨x, hx, self_mem_cylinder _ _⟩
   have A := exists_disjoint_cylinder hs hx
   have B : longest_prefix x s < shortest_prefix_diff x s :=
     Nat.pred_lt (shortest_prefix_diff_pos hs hne hx).ne'
@@ -841,8 +832,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
   -- check that the range of `f` is `s`.
   · apply subset.antisymm
     · rintro x ⟨y, rfl⟩
-      by_cases hy : y ∈ s
-      · rwa [fs y hy]
+      by_cases hy : y ∈ s; · rwa [fs y hy]
       simpa [hf, if_neg hy] using (inter_cylinder_longest_prefix_nonempty hs hne y).choose_spec.1
     · intro x hx
       rw [← fs x hx]
@@ -850,10 +840,9 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
   -- check that `f` is `1`-Lipschitz, by a case analysis.
   · apply LipschitzWith.mk_one fun x y => _
     -- exclude the trivial cases where `x = y`, or `f x = f y`.
-    rcases eq_or_ne x y with (rfl | hxy)
+    rcases eq_or_ne x y with (rfl | hxy);
     · simp
-    rcases eq_or_ne (f x) (f y) with (h' | hfxfy)
-    · simp [h', dist_nonneg]
+    rcases eq_or_ne (f x) (f y) with (h' | hfxfy); · simp [h', dist_nonneg]
     have I2 : cylinder x (first_diff x y) = cylinder y (first_diff x y) :=
       by
       rw [← mem_cylinder_iff_eq]
@@ -864,7 +853,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
     by_cases xs : x ∈ s
     · rw [fs x xs] at hfxfy⊢
       -- case where `y ∈ s`, trivial
-      by_cases ys : y ∈ s
+      by_cases ys : y ∈ s;
       · rw [fs y ys]
       -- case where `y ∉ s`
       have A : (s ∩ cylinder y (longest_prefix y s)).Nonempty :=
@@ -991,8 +980,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
     by
     apply continuous_iff_continuousAt.2 fun y => _
     apply continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => _
-    rcases eq_or_ne x y with (rfl | hne)
-    · simp
+    rcases eq_or_ne x y with (rfl | hne); · simp
     have hne' : x.1 ≠ y.1 := subtype.coe_injective.ne hne
     have dist' : dist x y = dist x.1 y.1 := rfl
     let n := first_diff x.1 y.1 - 1

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

@@ -228,13 +228,13 @@ theorem cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn :
 #align pi_nat.cylinder_eq_cylinder_of_le_first_diff PiNat.cylinder_eq_cylinder_of_le_firstDiff
 -/
 
-/- warning: pi_nat.Union_cylinder_update -> PiNat.unionᵢ_cylinder_update is a dubious translation:
+/- warning: pi_nat.Union_cylinder_update -> PiNat.iUnion_cylinder_update is a dubious translation:
 lean 3 declaration is
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.unionᵢ.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.iUnion.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
 but is expected to have type
-  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.unionᵢ.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
-Case conversion may be inaccurate. Consider using '#align pi_nat.Union_cylinder_update PiNat.unionᵢ_cylinder_updateₓ'. -/
-theorem unionᵢ_cylinder_update (x : ∀ n, E n) (n : ℕ) :
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.iUnion.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
+Case conversion may be inaccurate. Consider using '#align pi_nat.Union_cylinder_update PiNat.iUnion_cylinder_updateₓ'. -/
+theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     (⋃ k, cylinder (update x n k) (n + 1)) = cylinder x n :=
   by
   ext y
@@ -247,7 +247,7 @@ theorem unionᵢ_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     rcases Nat.lt_succ_iff_lt_or_eq.1 hi with (h'i | rfl)
     · simp [H i h'i, h'i.ne]
     · simp
-#align pi_nat.Union_cylinder_update PiNat.unionᵢ_cylinder_update
+#align pi_nat.Union_cylinder_update PiNat.iUnion_cylinder_update
 
 /- warning: pi_nat.update_mem_cylinder -> PiNat.update_mem_cylinder is a dubious translation:
 lean 3 declaration is
@@ -606,7 +606,7 @@ protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, Unifo
       simp [Pi.uniformity, comap_infi, gt_iff_lt, preimage_set_of_eq, comap_principal,
         PseudoMetricSpace.uniformity_dist, h, idRel]
       apply le_antisymm
-      · simp only [le_infᵢ_iff, le_principal_iff]
+      · simp only [le_iInf_iff, le_principal_iff]
         intro ε εpos
         obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < ε := exists_pow_lt_of_lt_one εpos (by norm_num)
         apply
@@ -619,7 +619,7 @@ protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, Unifo
           apply lt_of_le_of_lt _ hn
           rw [← mem_cylinder_iff_dist_le, mem_cylinder_iff]
           exact hxy
-      · simp only [le_infᵢ_iff, le_principal_iff]
+      · simp only [le_iInf_iff, le_principal_iff]
         intro n
         refine' mem_infi_of_mem ((1 / 2) ^ n) _
         refine' mem_infi_of_mem (by positivity) _
@@ -1206,7 +1206,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
     simp only [Pi.uniformity, comap_infi, gt_iff_lt, preimage_set_of_eq, comap_principal,
       PseudoMetricSpace.uniformity_dist]
     apply le_antisymm
-    · simp only [le_infᵢ_iff, le_principal_iff]
+    · simp only [le_iInf_iff, le_principal_iff]
       intro ε εpos
       obtain ⟨K, hK⟩ :
         ∃ K : Finset ι, (∑' i : { j // j ∉ K }, (1 / 2 : ℝ) ^ encode (i : ι)) < ε / 2 :=
@@ -1253,7 +1253,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
             (add_le_add_right (by simpa only [Finset.sum_const, nsmul_eq_mul] using hδ) _)
           _ = ε := add_halves _
           
-    · simp only [le_infᵢ_iff, le_principal_iff]
+    · simp only [le_iInf_iff, le_principal_iff]
       intro i ε εpos
       refine' mem_infi_of_mem (min ((1 / 2) ^ encode i) ε) _
       have : 0 < min ((1 / 2) ^ encode i) ε := lt_min (by simp) εpos

mathlib commit https://github.com/leanprover-community/mathlib/commit/cd8fafa2fac98e1a67097e8a91ad9901cfde48af

@@ -266,6 +266,7 @@ variable {α : Type _}
 open List
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print PiNat.res /-
 /-- In the case where `E` has constant value `α`,
 the cylinder `cylinder x n` can be identified with the element of `list α`
 consisting of the first `n` entries of `x`. See `cylinder_eq_res`.
@@ -274,22 +275,30 @@ def res (x : ℕ → α) : ℕ → List α
   | 0 => nil
   | Nat.succ n => x n::res n
 #align pi_nat.res PiNat.res
+-/
 
+#print PiNat.res_zero /-
 @[simp]
 theorem res_zero (x : ℕ → α) : res x 0 = @nil α :=
   rfl
 #align pi_nat.res_zero PiNat.res_zero
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print PiNat.res_succ /-
 @[simp]
 theorem res_succ (x : ℕ → α) (n : ℕ) : res x n.succ = x n::res x n :=
   rfl
 #align pi_nat.res_succ PiNat.res_succ
+-/
 
+#print PiNat.res_length /-
 @[simp]
 theorem res_length (x : ℕ → α) (n : ℕ) : (res x n).length = n := by induction n <;> simp [*]
 #align pi_nat.res_length PiNat.res_length
+-/
 
+#print PiNat.res_eq_res /-
 /-- The restrictions of `x` and `y` to `n` are equal if and only if `x m = y m` for all `m < n`.-/
 theorem res_eq_res {x y : ℕ → α} {n : ℕ} : res x n = res y n ↔ ∀ ⦃m⦄, m < n → x m = y m :=
   by
@@ -306,7 +315,9 @@ theorem res_eq_res {x y : ℕ → α} {n : ℕ} : res x n = res y n ↔ ∀ ⦃m
   refine' ⟨h (Nat.lt_succ_self _), ih fun m hm => _⟩
   exact h (hm.trans (Nat.lt_succ_self _))
 #align pi_nat.res_eq_res PiNat.res_eq_res
+-/
 
+#print PiNat.res_injective /-
 theorem res_injective : Injective (@res α) :=
   by
   intro x y h
@@ -314,7 +325,9 @@ theorem res_injective : Injective (@res α) :=
   apply res_eq_res.mp _ (Nat.lt_succ_self _)
   rw [h]
 #align pi_nat.res_injective PiNat.res_injective
+-/
 
+#print PiNat.cylinder_eq_res /-
 /-- `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`.-/
 theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) : cylinder x n = { y | res y n = res x n } :=
   by
@@ -322,6 +335,7 @@ theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) : cylinder x n = { y | res y
   dsimp [cylinder]
   rw [res_eq_res]
 #align pi_nat.cylinder_eq_res PiNat.cylinder_eq_res
+-/
 
 end Res
 
@@ -495,11 +509,13 @@ theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type _
 
 variable (E) [∀ n, TopologicalSpace (E n)] [∀ n, DiscreteTopology (E n)]
 
+#print PiNat.isOpen_cylinder /-
 theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) :=
   by
   rw [PiNat.cylinder_eq_pi]
   exact isOpen_set_pi (Finset.range n).finite_toSet fun a ha => isOpen_discrete _
 #align pi_nat.is_open_cylinder PiNat.isOpen_cylinder
+-/
 
 #print PiNat.isTopologicalBasis_cylinders /-
 theorem isTopologicalBasis_cylinders :

mathlib commit https://github.com/leanprover-community/mathlib/commit/49b7f94aab3a3bdca1f9f34c5d818afb253b3993

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.pi_nat
-! leanprover-community/mathlib commit 25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e
+! leanprover-community/mathlib commit 49b7f94aab3a3bdca1f9f34c5d818afb253b3993
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -259,6 +259,72 @@ theorem update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y
   mem_cylinder_iff.2 fun i hi => by simp [hi.ne]
 #align pi_nat.update_mem_cylinder PiNat.update_mem_cylinder
 
+section Res
+
+variable {α : Type _}
+
+open List
+
+/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+/-- In the case where `E` has constant value `α`,
+the cylinder `cylinder x n` can be identified with the element of `list α`
+consisting of the first `n` entries of `x`. See `cylinder_eq_res`.
+We call this list `res x n`, the restriction of `x` to `n`.-/
+def res (x : ℕ → α) : ℕ → List α
+  | 0 => nil
+  | Nat.succ n => x n::res n
+#align pi_nat.res PiNat.res
+
+@[simp]
+theorem res_zero (x : ℕ → α) : res x 0 = @nil α :=
+  rfl
+#align pi_nat.res_zero PiNat.res_zero
+
+/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+@[simp]
+theorem res_succ (x : ℕ → α) (n : ℕ) : res x n.succ = x n::res x n :=
+  rfl
+#align pi_nat.res_succ PiNat.res_succ
+
+@[simp]
+theorem res_length (x : ℕ → α) (n : ℕ) : (res x n).length = n := by induction n <;> simp [*]
+#align pi_nat.res_length PiNat.res_length
+
+/-- The restrictions of `x` and `y` to `n` are equal if and only if `x m = y m` for all `m < n`.-/
+theorem res_eq_res {x y : ℕ → α} {n : ℕ} : res x n = res y n ↔ ∀ ⦃m⦄, m < n → x m = y m :=
+  by
+  constructor <;> intro h <;> induction' n with n ih; · simp
+  · intro m hm
+    rw [Nat.lt_succ_iff_lt_or_eq] at hm
+    simp only [res_succ] at h
+    cases' hm with hm hm
+    · exact ih h.2 hm
+    rw [hm]
+    exact h.1
+  · simp
+  simp only [res_succ]
+  refine' ⟨h (Nat.lt_succ_self _), ih fun m hm => _⟩
+  exact h (hm.trans (Nat.lt_succ_self _))
+#align pi_nat.res_eq_res PiNat.res_eq_res
+
+theorem res_injective : Injective (@res α) :=
+  by
+  intro x y h
+  ext n
+  apply res_eq_res.mp _ (Nat.lt_succ_self _)
+  rw [h]
+#align pi_nat.res_injective PiNat.res_injective
+
+/-- `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`.-/
+theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) : cylinder x n = { y | res y n = res x n } :=
+  by
+  ext y
+  dsimp [cylinder]
+  rw [res_eq_res]
+#align pi_nat.cylinder_eq_res PiNat.cylinder_eq_res
+
+end Res
+
 /-!
 ### A distance function on `Π n, E n`
 
@@ -429,14 +495,19 @@ theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type _
 
 variable (E) [∀ n, TopologicalSpace (E n)] [∀ n, DiscreteTopology (E n)]
 
+theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) :=
+  by
+  rw [PiNat.cylinder_eq_pi]
+  exact isOpen_set_pi (Finset.range n).finite_toSet fun a ha => isOpen_discrete _
+#align pi_nat.is_open_cylinder PiNat.isOpen_cylinder
+
 #print PiNat.isTopologicalBasis_cylinders /-
 theorem isTopologicalBasis_cylinders :
     IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n)(n : ℕ), s = cylinder x n } :=
   by
   apply is_topological_basis_of_open_of_nhds
   · rintro u ⟨x, n, rfl⟩
-    rw [cylinder_eq_pi]
-    exact isOpen_set_pi (Finset.range n).finite_toSet fun a ha => isOpen_discrete _
+    apply is_open_cylinder
   · intro x u hx u_open
     obtain ⟨v, ⟨U, F, hUF, rfl⟩, xU, Uu⟩ :
       ∃ (v : Set (∀ i : ℕ, E i))(H :

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.pi_nat
-! leanprover-community/mathlib commit e1a7bdeb4fd826b7e71d130d34988f0a2d26a177
+! leanprover-community/mathlib commit 25a9423c6b2c8626e91c688bfd6c1d0a986a3e6e
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.Topology.MetricSpace.HausdorffDistance
 /-!
 # Topological study of spaces `Π (n : ℕ), E n`
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 When `E n` are topological spaces, the space `Π (n : ℕ), E n` is naturally a topological space
 (with the product topology). When `E n` are uniform spaces, it also inherits a uniform structure.
 However, it does not inherit a canonical metric space structure of the `E n`. Nevertheless, one

mathlib commit https://github.com/leanprover-community/mathlib/commit/e05ead7993520a432bec94ac504842d90707ad63

@@ -69,19 +69,24 @@ namespace PiNat
 /-! ### The first_diff function -/
 
 
+#print PiNat.firstDiff /-
 /-- In a product space `Π n, E n`, then `first_diff x y` is the first index at which `x` and `y`
 differ. If `x = y`, then by convention we set `first_diff x x = 0`. -/
 @[pp_nodot]
 irreducible_def firstDiff (x y : ∀ n, E n) : ℕ :=
   if h : x ≠ y then Nat.find (ne_iff.1 h) else 0
 #align pi_nat.first_diff PiNat.firstDiff
+-/
 
+#print PiNat.apply_firstDiff_ne /-
 theorem apply_firstDiff_ne {x y : ∀ n, E n} (h : x ≠ y) : x (firstDiff x y) ≠ y (firstDiff x y) :=
   by
   rw [first_diff, dif_pos h]
   exact Nat.find_spec (ne_iff.1 h)
 #align pi_nat.apply_first_diff_ne PiNat.apply_firstDiff_ne
+-/
 
+#print PiNat.apply_eq_of_lt_firstDiff /-
 theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDiff x y) : x n = y n :=
   by
   rw [first_diff] at hn
@@ -90,7 +95,9 @@ theorem apply_eq_of_lt_firstDiff {x y : ∀ n, E n} {n : ℕ} (hn : n < firstDif
     simp
   · exact (not_lt_zero' hn).elim
 #align pi_nat.apply_eq_of_lt_first_diff PiNat.apply_eq_of_lt_firstDiff
+-/
 
+#print PiNat.firstDiff_comm /-
 theorem firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x :=
   by
   rcases eq_or_ne x y with (rfl | hxy); · rfl
@@ -99,7 +106,14 @@ theorem firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x :=
   · exact h
   · exact (apply_first_diff_ne hxy.symm (apply_eq_of_lt_first_diff h).symm).elim
 #align pi_nat.first_diff_comm PiNat.firstDiff_comm
+-/
 
+/- warning: pi_nat.min_first_diff_le -> PiNat.min_firstDiff_le is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), (Ne.{succ u1} (forall (n : Nat), E n) x z) -> (LE.le.{0} Nat Nat.hasLe (LinearOrder.min.{0} Nat Nat.linearOrder (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) y z)) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x z))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), (Ne.{succ u1} (forall (n : Nat), E n) x z) -> (LE.le.{0} Nat instLENat (Min.min.{0} Nat instMinNat (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) y z)) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x z))
+Case conversion may be inaccurate. Consider using '#align pi_nat.min_first_diff_le PiNat.min_firstDiff_leₓ'. -/
 theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
     min (firstDiff x y) (firstDiff y z) ≤ firstDiff x z :=
   by
@@ -116,6 +130,7 @@ theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
 /-! ### Cylinders -/
 
 
+#print PiNat.cylinder /-
 /-- In a product space `Π n, E n`, the cylinder set of length `n` around `x`, denoted
 `cylinder x n`, is the set of sequences `y` that coincide with `x` on the first `n` symbols, i.e.,
 such that `y i = x i` for all `i < n`.
@@ -123,30 +138,42 @@ such that `y i = x i` for all `i < n`.
 def cylinder (x : ∀ n, E n) (n : ℕ) : Set (∀ n, E n) :=
   { y | ∀ i, i < n → y i = x i }
 #align pi_nat.cylinder PiNat.cylinder
+-/
 
+#print PiNat.cylinder_eq_pi /-
 theorem cylinder_eq_pi (x : ∀ n, E n) (n : ℕ) :
     cylinder x n = Set.pi (Finset.range n : Set ℕ) fun i : ℕ => {x i} :=
   by
   ext y
   simp [cylinder]
 #align pi_nat.cylinder_eq_pi PiNat.cylinder_eq_pi
+-/
 
+#print PiNat.cylinder_zero /-
 @[simp]
 theorem cylinder_zero (x : ∀ n, E n) : cylinder x 0 = univ := by simp [cylinder_eq_pi]
 #align pi_nat.cylinder_zero PiNat.cylinder_zero
+-/
 
+#print PiNat.cylinder_anti /-
 theorem cylinder_anti (x : ∀ n, E n) {m n : ℕ} (h : m ≤ n) : cylinder x n ⊆ cylinder x m :=
   fun y hy i hi => hy i (hi.trans_le h)
 #align pi_nat.cylinder_anti PiNat.cylinder_anti
+-/
 
+#print PiNat.mem_cylinder_iff /-
 @[simp]
 theorem mem_cylinder_iff {x y : ∀ n, E n} {n : ℕ} : y ∈ cylinder x n ↔ ∀ i, i < n → y i = x i :=
   Iff.rfl
 #align pi_nat.mem_cylinder_iff PiNat.mem_cylinder_iff
+-/
 
+#print PiNat.self_mem_cylinder /-
 theorem self_mem_cylinder (x : ∀ n, E n) (n : ℕ) : x ∈ cylinder x n := by simp
 #align pi_nat.self_mem_cylinder PiNat.self_mem_cylinder
+-/
 
+#print PiNat.mem_cylinder_iff_eq /-
 theorem mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
     y ∈ cylinder x n ↔ cylinder y n = cylinder x n :=
   by
@@ -163,11 +190,15 @@ theorem mem_cylinder_iff_eq {x y : ∀ n, E n} {n : ℕ} :
     rw [← h]
     exact self_mem_cylinder _ _
 #align pi_nat.mem_cylinder_iff_eq PiNat.mem_cylinder_iff_eq
+-/
 
+#print PiNat.mem_cylinder_comm /-
 theorem mem_cylinder_comm (x y : ∀ n, E n) (n : ℕ) : y ∈ cylinder x n ↔ x ∈ cylinder y n := by
   simp [mem_cylinder_iff_eq, eq_comm]
 #align pi_nat.mem_cylinder_comm PiNat.mem_cylinder_comm
+-/
 
+#print PiNat.mem_cylinder_iff_le_firstDiff /-
 theorem mem_cylinder_iff_le_firstDiff {x y : ∀ n, E n} (hne : x ≠ y) (i : ℕ) :
     x ∈ cylinder y i ↔ i ≤ firstDiff x y := by
   constructor
@@ -177,18 +208,29 @@ theorem mem_cylinder_iff_le_firstDiff {x y : ∀ n, E n} (hne : x ≠ y) (i : 
   · intro hi j hj
     exact apply_eq_of_lt_first_diff (hj.trans_le hi)
 #align pi_nat.mem_cylinder_iff_le_first_diff PiNat.mem_cylinder_iff_le_firstDiff
+-/
 
+#print PiNat.mem_cylinder_firstDiff /-
 theorem mem_cylinder_firstDiff (x y : ∀ n, E n) : x ∈ cylinder y (firstDiff x y) := fun i hi =>
   apply_eq_of_lt_firstDiff hi
 #align pi_nat.mem_cylinder_first_diff PiNat.mem_cylinder_firstDiff
+-/
 
+#print PiNat.cylinder_eq_cylinder_of_le_firstDiff /-
 theorem cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn : n ≤ firstDiff x y) :
     cylinder x n = cylinder y n := by
   rw [← mem_cylinder_iff_eq]
   intro i hi
   exact apply_eq_of_lt_first_diff (hi.trans_le hn)
 #align pi_nat.cylinder_eq_cylinder_of_le_first_diff PiNat.cylinder_eq_cylinder_of_le_firstDiff
+-/
 
+/- warning: pi_nat.Union_cylinder_update -> PiNat.unionᵢ_cylinder_update is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.unionᵢ.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat), Eq.{succ u1} (Set.{u1} (forall (n : Nat), E n)) (Set.unionᵢ.{u1, succ u1} (forall (n : Nat), E n) (E n) (fun (k : E n) => PiNat.cylinder.{u1} (fun (n : Nat) => E n) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) x n k) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
+Case conversion may be inaccurate. Consider using '#align pi_nat.Union_cylinder_update PiNat.unionᵢ_cylinder_updateₓ'. -/
 theorem unionᵢ_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     (⋃ k, cylinder (update x n k) (n + 1)) = cylinder x n :=
   by
@@ -204,6 +246,12 @@ theorem unionᵢ_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     · simp
 #align pi_nat.Union_cylinder_update PiNat.unionᵢ_cylinder_update
 
+/- warning: pi_nat.update_mem_cylinder -> PiNat.update_mem_cylinder is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat) (y : E n), Membership.Mem.{u1, u1} (forall (a : Nat), E a) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => Nat.decidableEq a b) x n y) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (n : Nat) (y : E n), Membership.mem.{u1, u1} (forall (a : Nat), E a) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) (Function.update.{1, succ u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) (b : Nat) => instDecidableEqNat a b) x n y) (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)
+Case conversion may be inaccurate. Consider using '#align pi_nat.update_mem_cylinder PiNat.update_mem_cylinderₓ'. -/
 theorem update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y ∈ cylinder x n :=
   mem_cylinder_iff.2 fun i hi => by simp [hi.ne]
 #align pi_nat.update_mem_cylinder PiNat.update_mem_cylinder
@@ -219,25 +267,47 @@ local instances in this section.
 -/
 
 
+#print PiNat.dist /-
 /-- The distance function on a product space `Π n, E n`, given by `dist x y = (1/2)^n` where `n` is
 the first index at which `x` and `y` differ. -/
-protected def hasDist : Dist (∀ n, E n) :=
+protected def dist : Dist (∀ n, E n) :=
   ⟨fun x y => if h : x ≠ y then (1 / 2 : ℝ) ^ firstDiff x y else 0⟩
-#align pi_nat.has_dist PiNat.hasDist
+#align pi_nat.has_dist PiNat.dist
+-/
 
-attribute [local instance] PiNat.hasDist
+attribute [local instance] PiNat.dist
 
+/- warning: pi_nat.dist_eq_of_ne -> PiNat.dist_eq_of_ne is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n}, (Ne.{succ u1} (forall (n : Nat), E n) x y) -> (Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y)))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n}, (Ne.{succ u1} (forall (n : Nat), E n) x y) -> (Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (PiNat.firstDiff.{u1} (fun (n : Nat) => E n) x y)))
+Case conversion may be inaccurate. Consider using '#align pi_nat.dist_eq_of_ne PiNat.dist_eq_of_neₓ'. -/
 theorem dist_eq_of_ne {x y : ∀ n, E n} (h : x ≠ y) : dist x y = (1 / 2 : ℝ) ^ firstDiff x y := by
   simp [dist, h]
 #align pi_nat.dist_eq_of_ne PiNat.dist_eq_of_ne
 
+/- warning: pi_nat.dist_self -> PiNat.dist_self is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n), Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x x) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n), Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x x) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))
+Case conversion may be inaccurate. Consider using '#align pi_nat.dist_self PiNat.dist_selfₓ'. -/
 protected theorem dist_self (x : ∀ n, E n) : dist x x = 0 := by simp [dist]
 #align pi_nat.dist_self PiNat.dist_self
 
+#print PiNat.dist_comm /-
 protected theorem dist_comm (x y : ∀ n, E n) : dist x y = dist y x := by
   simp [dist, @eq_comm _ x y, first_diff_comm]
 #align pi_nat.dist_comm PiNat.dist_comm
+-/
 
+/- warning: pi_nat.dist_nonneg -> PiNat.dist_nonneg is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n), LE.le.{0} Real Real.hasLe (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero))) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y)
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n), LE.le.{0} Real Real.instLEReal (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal)) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y)
+Case conversion may be inaccurate. Consider using '#align pi_nat.dist_nonneg PiNat.dist_nonnegₓ'. -/
 protected theorem dist_nonneg (x y : ∀ n, E n) : 0 ≤ dist x y :=
   by
   rcases eq_or_ne x y with (rfl | h)
@@ -245,6 +315,12 @@ protected theorem dist_nonneg (x y : ∀ n, E n) : 0 ≤ dist x y :=
   · simp [dist, h]
 #align pi_nat.dist_nonneg PiNat.dist_nonneg
 
+/- warning: pi_nat.dist_triangle_nonarch -> PiNat.dist_triangle_nonarch is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x z) (LinearOrder.max.{0} Real Real.linearOrder (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) y z))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x z) (Max.max.{0} Real (LinearOrderedRing.toMax.{0} Real Real.instLinearOrderedRingReal) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) y z))
+Case conversion may be inaccurate. Consider using '#align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarchₓ'. -/
 theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y) (dist y z) :=
   by
   rcases eq_or_ne x z with (rfl | hxz)
@@ -258,6 +334,12 @@ theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y)
     min_le_iff.1 (min_first_diff_le x y z hxz)]
 #align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarch
 
+/- warning: pi_nat.dist_triangle -> PiNat.dist_triangle is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.hasAdd) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) y z))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (z : forall (n : Nat), E n), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x z) (HAdd.hAdd.{0, 0, 0} Real Real Real (instHAdd.{0} Real Real.instAddReal) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) y z))
+Case conversion may be inaccurate. Consider using '#align pi_nat.dist_triangle PiNat.dist_triangleₓ'. -/
 protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + dist y z :=
   calc
     dist x z ≤ max (dist x y) (dist y z) := dist_triangle_nonarch x y z
@@ -265,6 +347,12 @@ protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + d
     
 #align pi_nat.dist_triangle PiNat.dist_triangle
 
+/- warning: pi_nat.eq_of_dist_eq_zero -> PiNat.eq_of_dist_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n), (Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) -> (Eq.{succ u1} (forall (n : Nat), E n) x y)
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} (x : forall (n : Nat), E n) (y : forall (n : Nat), E n), (Eq.{1} Real (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) -> (Eq.{succ u1} (forall (n : Nat), E n) x y)
+Case conversion may be inaccurate. Consider using '#align pi_nat.eq_of_dist_eq_zero PiNat.eq_of_dist_eq_zeroₓ'. -/
 protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y :=
   by
   rcases eq_or_ne x y with (rfl | h); · rfl
@@ -272,6 +360,12 @@ protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x
   exact (two_ne_zero (pow_eq_zero hxy)).elim
 #align pi_nat.eq_of_dist_eq_zero PiNat.eq_of_dist_eq_zero
 
+/- warning: pi_nat.mem_cylinder_iff_dist_le -> PiNat.mem_cylinder_iff_dist_le is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n} {n : Nat}, Iff (Membership.Mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) y (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)) (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) y x) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) n))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n} {n : Nat}, Iff (Membership.mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) y (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)) (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) y x) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) n))
+Case conversion may be inaccurate. Consider using '#align pi_nat.mem_cylinder_iff_dist_le PiNat.mem_cylinder_iff_dist_leₓ'. -/
 theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
     y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n :=
   by
@@ -286,6 +380,12 @@ theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
     exact apply_eq_of_lt_first_diff (hi.trans_le h)
 #align pi_nat.mem_cylinder_iff_dist_le PiNat.mem_cylinder_iff_dist_le
 
+/- warning: pi_nat.apply_eq_of_dist_lt -> PiNat.apply_eq_of_dist_lt is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n} {n : Nat}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) n)) -> (forall {i : Nat}, (LE.le.{0} Nat Nat.hasLe i n) -> (Eq.{succ u1} (E i) (x i) (y i)))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} {x : forall (n : Nat), E n} {y : forall (n : Nat), E n} {n : Nat}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) n)) -> (forall {i : Nat}, (LE.le.{0} Nat instLENat i n) -> (Eq.{succ u1} (E i) (x i) (y i)))
+Case conversion may be inaccurate. Consider using '#align pi_nat.apply_eq_of_dist_lt PiNat.apply_eq_of_dist_ltₓ'. -/
 theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2) ^ n) {i : ℕ}
     (hi : i ≤ n) : x i = y i :=
   by
@@ -296,6 +396,12 @@ theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2)
   exact apply_eq_of_lt_first_diff (hi.trans_lt this)
 #align pi_nat.apply_eq_of_dist_lt PiNat.apply_eq_of_dist_lt
 
+/- warning: pi_nat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder -> PiNat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {f : (forall (n : Nat), E n) -> α}, Iff (forall (x : forall (n : Nat), E n) (y : forall (n : Nat), E n), LE.le.{0} Real Real.hasLe (Dist.dist.{u2} α (PseudoMetricSpace.toHasDist.{u2} α _inst_1) (f x) (f y)) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y)) (forall (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (n : Nat), (Membership.Mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) y (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u2} α (PseudoMetricSpace.toHasDist.{u2} α _inst_1) (f x) (f y)) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) n)))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} {α : Type.{u2}} [_inst_1 : PseudoMetricSpace.{u2} α] {f : (forall (n : Nat), E n) -> α}, Iff (forall (x : forall (n : Nat), E n) (y : forall (n : Nat), E n), LE.le.{0} Real Real.instLEReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f x) (f y)) (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y)) (forall (x : forall (n : Nat), E n) (y : forall (n : Nat), E n) (n : Nat), (Membership.mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) y (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u2} α (PseudoMetricSpace.toDist.{u2} α _inst_1) (f x) (f y)) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) n)))
+Case conversion may be inaccurate. Consider using '#align pi_nat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder PiNat.lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinderₓ'. -/
 /-- A function to a pseudo-metric-space is `1`-Lipschitz if and only if points in the same cylinder
 of length `n` are sent to points within distance `(1/2)^n`.
 Not expressed using `lipschitz_with` as we don't have a metric space structure -/
@@ -320,6 +426,7 @@ theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type _
 
 variable (E) [∀ n, TopologicalSpace (E n)] [∀ n, DiscreteTopology (E n)]
 
+#print PiNat.isTopologicalBasis_cylinders /-
 theorem isTopologicalBasis_cylinders :
     IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n)(n : ℕ), s = cylinder x n } :=
   by
@@ -347,9 +454,16 @@ theorem isTopologicalBasis_cylinders :
     simp only [Set.mem_pi, Finset.mem_coe] at xU
     exact xU i hi
 #align pi_nat.is_topological_basis_cylinders PiNat.isTopologicalBasis_cylinders
+-/
 
 variable {E}
 
+/- warning: pi_nat.is_open_iff_dist -> PiNat.isOpen_iff_dist is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] (s : Set.{u1} (forall (n : Nat), E n)), Iff (IsOpen.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) (forall (x : forall (n : Nat), E n), (Membership.Mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) x s) -> (Exists.{1} Real (fun (ε : Real) => Exists.{0} (GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) (fun (H : GT.gt.{0} Real Real.hasLt ε (OfNat.ofNat.{0} Real 0 (OfNat.mk.{0} Real 0 (Zero.zero.{0} Real Real.hasZero)))) => forall (y : forall (n : Nat), E n), (LT.lt.{0} Real Real.hasLt (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) ε) -> (Membership.Mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) y s)))))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] (s : Set.{u1} (forall (n : Nat), E n)), Iff (IsOpen.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) (forall (x : forall (n : Nat), E n), (Membership.mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) x s) -> (Exists.{1} Real (fun (ε : Real) => And (GT.gt.{0} Real Real.instLTReal ε (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZeroReal))) (forall (y : forall (n : Nat), E n), (LT.lt.{0} Real Real.instLTReal (Dist.dist.{u1} (forall (n : Nat), E n) (PiNat.dist.{u1} (fun (n : Nat) => E n)) x y) ε) -> (Membership.mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) y s)))))
+Case conversion may be inaccurate. Consider using '#align pi_nat.is_open_iff_dist PiNat.isOpen_iff_distₓ'. -/
 theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
     IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ y, dist x y < ε → y ∈ s :=
   by
@@ -371,6 +485,7 @@ theorem isOpen_iff_dist (s : Set (∀ n, E n)) :
     exact (mem_cylinder_iff_dist_le.1 hy).trans_lt hn
 #align pi_nat.is_open_iff_dist PiNat.isOpen_iff_dist
 
+#print PiNat.metricSpace /-
 /-- Metric space structure on `Π (n : ℕ), E n` when the spaces `E n` have the discrete topology,
 where the distance is given by `dist x y = (1/2)^n`, where `n` is the smallest index where `x` and
 `y` differ. Not registered as a global instance by default.
@@ -382,7 +497,9 @@ protected def metricSpace : MetricSpace (∀ n, E n) :=
   MetricSpace.ofDistTopology dist PiNat.dist_self PiNat.dist_comm PiNat.dist_triangle
     isOpen_iff_dist PiNat.eq_of_dist_eq_zero
 #align pi_nat.metric_space PiNat.metricSpace
+-/
 
+#print PiNat.metricSpaceOfDiscreteUniformity /-
 /-- Metric space structure on `Π (n : ℕ), E n` when the spaces `E n` have the discrete uniformity,
 where the distance is given by `dist x y = (1/2)^n`, where `n` is the smallest index where `x` and
 `y` differ. Not registered as a global instance by default. -/
@@ -420,16 +537,20 @@ protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, Unifo
         intro x y hxy
         exact apply_eq_of_dist_lt hxy le_rfl }
 #align pi_nat.metric_space_of_discrete_uniformity PiNat.metricSpaceOfDiscreteUniformity
+-/
 
+#print PiNat.metricSpaceNatNat /-
 /-- Metric space structure on `ℕ → ℕ` where the distance is given by `dist x y = (1/2)^n`,
 where `n` is the smallest index where `x` and `y` differ.
 Not registered as a global instance by default. -/
 def metricSpaceNatNat : MetricSpace (ℕ → ℕ) :=
   PiNat.metricSpaceOfDiscreteUniformity fun n => rfl
 #align pi_nat.metric_space_nat_nat PiNat.metricSpaceNatNat
+-/
 
 attribute [local instance] PiNat.metricSpace
 
+#print PiNat.completeSpace /-
 protected theorem completeSpace : CompleteSpace (∀ n, E n) :=
   by
   refine' Metric.complete_of_convergent_controlled_sequences (fun n => (1 / 2) ^ n) (by simp) _
@@ -439,6 +560,7 @@ protected theorem completeSpace : CompleteSpace (∀ n, E n) :=
   filter_upwards [Filter.Ici_mem_atTop i]with n hn
   exact apply_eq_of_dist_lt (hu i i n le_rfl hn) le_rfl
 #align pi_nat.complete_space PiNat.completeSpace
+-/
 
 /-!
 ### Retractions inside product spaces
@@ -451,6 +573,12 @@ where `z_w` is an element of `s` starting with `w`.
 -/
 
 
+/- warning: pi_nat.exists_disjoint_cylinder -> PiNat.exists_disjoint_cylinder is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] {s : Set.{u1} (forall (n : Nat), E n)}, (IsClosed.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) -> (forall {x : forall (n : Nat), E n}, (Not (Membership.Mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) x s)) -> (Exists.{1} Nat (fun (n : Nat) => Disjoint.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.completeBooleanAlgebra.{u1} (forall (n : Nat), E n))))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} (forall (n : Nat), E n)) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.booleanAlgebra.{u1} (forall (n : Nat), E n)))) s (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n))))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] {s : Set.{u1} (forall (n : Nat), E n)}, (IsClosed.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) -> (forall {x : forall (n : Nat), E n}, (Not (Membership.mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) x s)) -> (Exists.{1} Nat (fun (n : Nat) => Disjoint.{u1} (Set.{u1} (forall (n : Nat), E n)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.instCompleteBooleanAlgebraSet.{u1} (forall (n : Nat), E n))))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} (forall (n : Nat), E n)) (Preorder.toLE.{u1} (Set.{u1} (forall (n : Nat), E n)) (PartialOrder.toPreorder.{u1} (Set.{u1} (forall (n : Nat), E n)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.instCompleteBooleanAlgebraSet.{u1} (forall (n : Nat), E n))))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.instCompleteBooleanAlgebraSet.{u1} (forall (n : Nat), E n))))))) s (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n))))
+Case conversion may be inaccurate. Consider using '#align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinderₓ'. -/
 theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
     (hx : x ∉ s) : ∃ n, Disjoint s (cylinder x n) :=
   by
@@ -470,13 +598,16 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
     
 #align pi_nat.exists_disjoint_cylinder PiNat.exists_disjoint_cylinder
 
+#print PiNat.shortestPrefixDiff /-
 /-- Given a point `x` in a product space `Π (n : ℕ), E n`, and `s` a subset of this space, then
 `shortest_prefix_diff x s` if the smallest `n` for which there is no element of `s` having the same
 prefix of length `n` as `x`. If there is no such `n`, then use `0` by convention. -/
 def shortestPrefixDiff {E : ℕ → Type _} (x : ∀ n, E n) (s : Set (∀ n, E n)) : ℕ :=
   if h : ∃ n, Disjoint s (cylinder x n) then Nat.find h else 0
 #align pi_nat.shortest_prefix_diff PiNat.shortestPrefixDiff
+-/
 
+#print PiNat.firstDiff_lt_shortestPrefixDiff /-
 theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
     (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y < shortestPrefixDiff x s :=
   by
@@ -489,21 +620,27 @@ theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s)
   rw [mem_cylinder_comm]
   exact cylinder_anti y B (mem_cylinder_first_diff x y)
 #align pi_nat.first_diff_lt_shortest_prefix_diff PiNat.firstDiff_lt_shortestPrefixDiff
+-/
 
+#print PiNat.shortestPrefixDiff_pos /-
 theorem shortestPrefixDiff_pos {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty)
     {x : ∀ n, E n} (hx : x ∉ s) : 0 < shortestPrefixDiff x s :=
   by
   rcases hne with ⟨y, hy⟩
   exact (zero_le _).trans_lt (first_diff_lt_shortest_prefix_diff hs hx hy)
 #align pi_nat.shortest_prefix_diff_pos PiNat.shortestPrefixDiff_pos
+-/
 
+#print PiNat.longestPrefix /-
 /-- Given a point `x` in a product space `Π (n : ℕ), E n`, and `s` a subset of this space, then
 `longest_prefix x s` if the largest `n` for which there is an element of `s` having the same
 prefix of length `n` as `x`. If there is no such `n`, use `0` by convention. -/
 def longestPrefix {E : ℕ → Type _} (x : ∀ n, E n) (s : Set (∀ n, E n)) : ℕ :=
   shortestPrefixDiff x s - 1
 #align pi_nat.longest_prefix PiNat.longestPrefix
+-/
 
+#print PiNat.firstDiff_le_longestPrefix /-
 theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y : ∀ n, E n}
     (hx : x ∉ s) (hy : y ∈ s) : firstDiff x y ≤ longestPrefix x s :=
   by
@@ -511,7 +648,14 @@ theorem firstDiff_le_longestPrefix {s : Set (∀ n, E n)} (hs : IsClosed s) {x y
   · exact first_diff_lt_shortest_prefix_diff hs hx hy
   · exact shortest_prefix_diff_pos hs ⟨y, hy⟩ hx
 #align pi_nat.first_diff_le_longest_prefix PiNat.firstDiff_le_longestPrefix
+-/
 
+/- warning: pi_nat.inter_cylinder_longest_prefix_nonempty -> PiNat.inter_cylinder_longestPrefix_nonempty is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] {s : Set.{u1} (forall (n : Nat), E n)}, (IsClosed.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) -> (Set.Nonempty.{u1} (forall (n : Nat), E n) s) -> (forall (x : forall (n : Nat), E n), Set.Nonempty.{u1} (forall (n : Nat), E n) (Inter.inter.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.hasInter.{u1} (forall (n : Nat), E n)) s (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x (PiNat.longestPrefix.{u1} (fun (n : Nat) => E n) x s))))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] {s : Set.{u1} (forall (n : Nat), E n)}, (IsClosed.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) -> (Set.Nonempty.{u1} (forall (n : Nat), E n) s) -> (forall (x : forall (n : Nat), E n), Set.Nonempty.{u1} (forall (n : Nat), E n) (Inter.inter.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.instInterSet.{u1} (forall (n : Nat), E n)) s (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x (PiNat.longestPrefix.{u1} (fun (n : Nat) => E n) x s))))
+Case conversion may be inaccurate. Consider using '#align pi_nat.inter_cylinder_longest_prefix_nonempty PiNat.inter_cylinder_longestPrefix_nonemptyₓ'. -/
 theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClosed s)
     (hne : s.Nonempty) (x : ∀ n, E n) : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
   by
@@ -532,6 +676,12 @@ theorem inter_cylinder_longestPrefix_nonempty {s : Set (∀ n, E n)} (hs : IsClo
   rw [hy]
 #align pi_nat.inter_cylinder_longest_prefix_nonempty PiNat.inter_cylinder_longestPrefix_nonempty
 
+/- warning: pi_nat.disjoint_cylinder_of_longest_prefix_lt -> PiNat.disjoint_cylinder_of_longestPrefix_lt is a dubious translation:
+lean 3 declaration is
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] {s : Set.{u1} (forall (n : Nat), E n)}, (IsClosed.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) -> (forall {x : forall (n : Nat), E n}, (Not (Membership.Mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.hasMem.{u1} (forall (n : Nat), E n)) x s)) -> (forall {n : Nat}, (LT.lt.{0} Nat Nat.hasLt (PiNat.longestPrefix.{u1} (fun (n : Nat) => E n) x s) n) -> (Disjoint.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteSemilatticeInf.toPartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteLattice.toCompleteSemilatticeInf.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.completeBooleanAlgebra.{u1} (forall (n : Nat), E n))))))) (GeneralizedBooleanAlgebra.toOrderBot.{u1} (Set.{u1} (forall (n : Nat), E n)) (BooleanAlgebra.toGeneralizedBooleanAlgebra.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.booleanAlgebra.{u1} (forall (n : Nat), E n)))) s (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n))))
+but is expected to have type
+  forall {E : Nat -> Type.{u1}} [_inst_1 : forall (n : Nat), TopologicalSpace.{u1} (E n)] [_inst_2 : forall (n : Nat), DiscreteTopology.{u1} (E n) (_inst_1 n)] {s : Set.{u1} (forall (n : Nat), E n)}, (IsClosed.{u1} (forall (n : Nat), E n) (Pi.topologicalSpace.{0, u1} Nat (fun (n : Nat) => E n) (fun (a : Nat) => _inst_1 a)) s) -> (forall {x : forall (n : Nat), E n}, (Not (Membership.mem.{u1, u1} (forall (n : Nat), E n) (Set.{u1} (forall (n : Nat), E n)) (Set.instMembershipSet.{u1} (forall (n : Nat), E n)) x s)) -> (forall {n : Nat}, (LT.lt.{0} Nat instLTNat (PiNat.longestPrefix.{u1} (fun (n : Nat) => E n) x s) n) -> (Disjoint.{u1} (Set.{u1} (forall (n : Nat), E n)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.instCompleteBooleanAlgebraSet.{u1} (forall (n : Nat), E n))))))) (BoundedOrder.toOrderBot.{u1} (Set.{u1} (forall (n : Nat), E n)) (Preorder.toLE.{u1} (Set.{u1} (forall (n : Nat), E n)) (PartialOrder.toPreorder.{u1} (Set.{u1} (forall (n : Nat), E n)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.instCompleteBooleanAlgebraSet.{u1} (forall (n : Nat), E n))))))))) (CompleteLattice.toBoundedOrder.{u1} (Set.{u1} (forall (n : Nat), E n)) (Order.Coframe.toCompleteLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteDistribLattice.toCoframe.{u1} (Set.{u1} (forall (n : Nat), E n)) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u1} (Set.{u1} (forall (n : Nat), E n)) (Set.instCompleteBooleanAlgebraSet.{u1} (forall (n : Nat), E n))))))) s (PiNat.cylinder.{u1} (fun (n : Nat) => E n) x n))))
+Case conversion may be inaccurate. Consider using '#align pi_nat.disjoint_cylinder_of_longest_prefix_lt PiNat.disjoint_cylinder_of_longestPrefix_ltₓ'. -/
 theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClosed s) {x : ∀ n, E n}
     (hx : x ∉ s) {n : ℕ} (hn : longestPrefix x s < n) : Disjoint s (cylinder x n) :=
   by
@@ -543,6 +693,7 @@ theorem disjoint_cylinder_of_longestPrefix_lt {s : Set (∀ n, E n)} (hs : IsClo
   rwa [mem_cylinder_comm]
 #align pi_nat.disjoint_cylinder_of_longest_prefix_lt PiNat.disjoint_cylinder_of_longestPrefix_lt
 
+#print PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff /-
 /-- If two points `x, y` coincide up to length `n`, and the longest common prefix of `x` with `s`
 is strictly shorter than `n`, then the longest common prefix of `y` with `s` is the same, and both
 cylinders of this length based at `x` and `y` coincide. -/
@@ -571,7 +722,9 @@ theorem cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff {x y : ∀ n, E
   rw [l_eq, ← mem_cylinder_iff_eq]
   exact cylinder_anti y H.le (mem_cylinder_first_diff x y)
 #align pi_nat.cylinder_longest_prefix_eq_of_longest_prefix_lt_first_diff PiNat.cylinder_longestPrefix_eq_of_longestPrefix_lt_firstDiff
+-/
 
+#print PiNat.exists_lipschitz_retraction_of_isClosed /-
 /-- Given a closed nonempty subset `s` of `Π (n : ℕ), E n`, there exists a Lipschitz retraction
 onto this set, i.e., a Lipschitz map with range equal to `s`, equal to the identity on `s`. -/
 theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
@@ -683,7 +836,9 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
           rw [← fx, ← fy, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_first_diff hfxfy] at this
           exact this
 #align pi_nat.exists_lipschitz_retraction_of_is_closed PiNat.exists_lipschitz_retraction_of_isClosed
+-/
 
+#print PiNat.exists_retraction_of_isClosed /-
 /-- Given a closed nonempty subset `s` of `Π (n : ℕ), E n`, there exists a retraction onto this
 set, i.e., a continuous map with range equal to `s`, equal to the identity on `s`. -/
 theorem exists_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s.Nonempty) :
@@ -692,7 +847,9 @@ theorem exists_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s) (
   rcases exists_lipschitz_retraction_of_is_closed hs hne with ⟨f, fs, frange, hf⟩
   exact ⟨f, fs, frange, hf.continuous⟩
 #align pi_nat.exists_retraction_of_is_closed PiNat.exists_retraction_of_isClosed
+-/
 
+#print PiNat.exists_retraction_subtype_of_isClosed /-
 theorem exists_retraction_subtype_of_isClosed {s : Set (∀ n, E n)} (hs : IsClosed s)
     (hne : s.Nonempty) : ∃ f : (∀ n, E n) → s, (∀ x : s, f x = x) ∧ Surjective f ∧ Continuous f :=
   by
@@ -707,11 +864,18 @@ theorem exists_retraction_subtype_of_isClosed {s : Set (∀ n, E n)} (hs : IsClo
     simpa only using fs x.val x.property
   exact ⟨cod_restrict f s A, B, fun x => ⟨x, B x⟩, f_cont.subtype_mk _⟩
 #align pi_nat.exists_retraction_subtype_of_is_closed PiNat.exists_retraction_subtype_of_isClosed
+-/
 
 end PiNat
 
 open PiNat
 
+/- warning: exists_nat_nat_continuous_surjective_of_complete_space -> exists_nat_nat_continuous_surjective_of_completeSpace is a dubious translation:
+lean 3 declaration is
+  forall (α : Type.{u1}) [_inst_1 : MetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))] [_inst_3 : TopologicalSpace.SecondCountableTopology.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))] [_inst_4 : Nonempty.{succ u1} α], Exists.{succ u1} ((Nat -> Nat) -> α) (fun (f : (Nat -> Nat) -> α) => And (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => Nat.topologicalSpace)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))) f) (Function.Surjective.{1, succ u1} (Nat -> Nat) α f))
+but is expected to have type
+  forall (α : Type.{u1}) [_inst_1 : MetricSpace.{u1} α] [_inst_2 : CompleteSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))] [_inst_3 : TopologicalSpace.SecondCountableTopology.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1)))] [_inst_4 : Nonempty.{succ u1} α], Exists.{succ u1} ((Nat -> Nat) -> α) (fun (f : (Nat -> Nat) -> α) => And (Continuous.{0, u1} (Nat -> Nat) α (Pi.topologicalSpace.{0, 0} Nat (fun (ᾰ : Nat) => Nat) (fun (a : Nat) => instTopologicalSpaceNat)) (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α (MetricSpace.toPseudoMetricSpace.{u1} α _inst_1))) f) (Function.Surjective.{1, succ u1} (Nat -> Nat) α f))
+Case conversion may be inaccurate. Consider using '#align exists_nat_nat_continuous_surjective_of_complete_space exists_nat_nat_continuous_surjective_of_completeSpaceₓ'. -/
 /-- Any nonempty complete second countable metric space is the continuous image of the
 fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
 `exists_nat_nat_continuous_surjective_of_polish_space`. -/
@@ -829,20 +993,34 @@ variable {ι : Type _} [Encodable ι] {F : ι → Type _} [∀ i, MetricSpace (F
 
 open Encodable
 
+#print PiCountable.dist /-
 /-- Given a countable family of metric spaces, one may put a distance on their product `Π i, E i`.
 It is highly non-canonical, though, and therefore not registered as a global instance.
 The distance we use here is `dist x y = ∑' i, min (1/2)^(encode i) (dist (x i) (y i))`. -/
-protected def hasDist : Dist (∀ i, F i) :=
+protected def dist : Dist (∀ i, F i) :=
   ⟨fun x y => ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i))⟩
-#align pi_countable.has_dist PiCountable.hasDist
+#align pi_countable.has_dist PiCountable.dist
+-/
 
-attribute [local instance] PiCountable.hasDist
+attribute [local instance] PiCountable.dist
 
+/- warning: pi_countable.dist_eq_tsum -> PiCountable.dist_eq_tsum is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} [_inst_1 : Encodable.{u1} ι] {F : ι -> Type.{u2}} [_inst_2 : forall (i : ι), MetricSpace.{u2} (F i)] (x : forall (i : ι), F i) (y : forall (i : ι), F i), Eq.{1} Real (Dist.dist.{max u1 u2} (forall (i : ι), F i) (PiCountable.dist.{u1, u2} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y) (tsum.{0, u1} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) ι (fun (i : ι) => LinearOrder.min.{0} Real Real.linearOrder (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (Encodable.encode.{u1} ι _inst_1 i)) (Dist.dist.{u2} (F i) (PseudoMetricSpace.toHasDist.{u2} (F i) (MetricSpace.toPseudoMetricSpace.{u2} (F i) (_inst_2 i))) (x i) (y i))))
+but is expected to have type
+  forall {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {F : ι -> Type.{u1}} [_inst_2 : forall (i : ι), MetricSpace.{u1} (F i)] (x : forall (i : ι), F i) (y : forall (i : ι), F i), Eq.{1} Real (Dist.dist.{max u2 u1} (forall (i : ι), F i) (PiCountable.dist.{u2, u1} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y) (tsum.{0, u2} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) ι (fun (i : ι) => Min.min.{0} Real (LinearOrderedRing.toMin.{0} Real Real.instLinearOrderedRingReal) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Encodable.encode.{u2} ι _inst_1 i)) (Dist.dist.{u1} (F i) (PseudoMetricSpace.toDist.{u1} (F i) (MetricSpace.toPseudoMetricSpace.{u1} (F i) (_inst_2 i))) (x i) (y i))))
+Case conversion may be inaccurate. Consider using '#align pi_countable.dist_eq_tsum PiCountable.dist_eq_tsumₓ'. -/
 theorem dist_eq_tsum (x y : ∀ i, F i) :
     dist x y = ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i)) :=
   rfl
 #align pi_countable.dist_eq_tsum PiCountable.dist_eq_tsum
 
+/- warning: pi_countable.dist_summable -> PiCountable.dist_summable is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} [_inst_1 : Encodable.{u1} ι] {F : ι -> Type.{u2}} [_inst_2 : forall (i : ι), MetricSpace.{u2} (F i)] (x : forall (i : ι), F i) (y : forall (i : ι), F i), Summable.{0, u1} Real ι Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (i : ι) => LinearOrder.min.{0} Real Real.linearOrder (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (Encodable.encode.{u1} ι _inst_1 i)) (Dist.dist.{u2} (F i) (PseudoMetricSpace.toHasDist.{u2} (F i) (MetricSpace.toPseudoMetricSpace.{u2} (F i) (_inst_2 i))) (x i) (y i)))
+but is expected to have type
+  forall {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {F : ι -> Type.{u1}} [_inst_2 : forall (i : ι), MetricSpace.{u1} (F i)] (x : forall (i : ι), F i) (y : forall (i : ι), F i), Summable.{0, u2} Real ι Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (i : ι) => Min.min.{0} Real (LinearOrderedRing.toMin.{0} Real Real.instLinearOrderedRingReal) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Encodable.encode.{u2} ι _inst_1 i)) (Dist.dist.{u1} (F i) (PseudoMetricSpace.toDist.{u1} (F i) (MetricSpace.toPseudoMetricSpace.{u1} (F i) (_inst_2 i))) (x i) (y i)))
+Case conversion may be inaccurate. Consider using '#align pi_countable.dist_summable PiCountable.dist_summableₓ'. -/
 theorem dist_summable (x y : ∀ i, F i) :
     Summable fun i : ι => min ((1 / 2) ^ encode i) (dist (x i) (y i)) :=
   by
@@ -851,11 +1029,23 @@ theorem dist_summable (x y : ∀ i, F i) :
   exact le_min (pow_nonneg (by norm_num) _) dist_nonneg
 #align pi_countable.dist_summable PiCountable.dist_summable
 
+/- warning: pi_countable.min_dist_le_dist_pi -> PiCountable.min_dist_le_dist_pi is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} [_inst_1 : Encodable.{u1} ι] {F : ι -> Type.{u2}} [_inst_2 : forall (i : ι), MetricSpace.{u2} (F i)] (x : forall (i : ι), F i) (y : forall (i : ι), F i) (i : ι), LE.le.{0} Real Real.hasLe (LinearOrder.min.{0} Real Real.linearOrder (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (Encodable.encode.{u1} ι _inst_1 i)) (Dist.dist.{u2} (F i) (PseudoMetricSpace.toHasDist.{u2} (F i) (MetricSpace.toPseudoMetricSpace.{u2} (F i) (_inst_2 i))) (x i) (y i))) (Dist.dist.{max u1 u2} (forall (i : ι), F i) (PiCountable.dist.{u1, u2} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y)
+but is expected to have type
+  forall {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {F : ι -> Type.{u1}} [_inst_2 : forall (i : ι), MetricSpace.{u1} (F i)] (x : forall (i : ι), F i) (y : forall (i : ι), F i) (i : ι), LE.le.{0} Real Real.instLEReal (Min.min.{0} Real (LinearOrderedRing.toMin.{0} Real Real.instLinearOrderedRingReal) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Encodable.encode.{u2} ι _inst_1 i)) (Dist.dist.{u1} (F i) (PseudoMetricSpace.toDist.{u1} (F i) (MetricSpace.toPseudoMetricSpace.{u1} (F i) (_inst_2 i))) (x i) (y i))) (Dist.dist.{max u2 u1} (forall (i : ι), F i) (PiCountable.dist.{u2, u1} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y)
+Case conversion may be inaccurate. Consider using '#align pi_countable.min_dist_le_dist_pi PiCountable.min_dist_le_dist_piₓ'. -/
 theorem min_dist_le_dist_pi (x y : ∀ i, F i) (i : ι) :
     min ((1 / 2) ^ encode i) (dist (x i) (y i)) ≤ dist x y :=
   le_tsum (dist_summable x y) i fun j hj => le_min (by simp) dist_nonneg
 #align pi_countable.min_dist_le_dist_pi PiCountable.min_dist_le_dist_pi
 
+/- warning: pi_countable.dist_le_dist_pi_of_dist_lt -> PiCountable.dist_le_dist_pi_of_dist_lt is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} [_inst_1 : Encodable.{u1} ι] {F : ι -> Type.{u2}} [_inst_2 : forall (i : ι), MetricSpace.{u2} (F i)] {x : forall (i : ι), F i} {y : forall (i : ι), F i} {i : ι}, (LT.lt.{0} Real Real.hasLt (Dist.dist.{max u1 u2} (forall (i : ι), F i) (PiCountable.dist.{u1, u2} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.monoid)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (DivInvMonoid.toHasDiv.{0} Real (DivisionRing.toDivInvMonoid.{0} Real Real.divisionRing))) (OfNat.ofNat.{0} Real 1 (OfNat.mk.{0} Real 1 (One.one.{0} Real Real.hasOne))) (OfNat.ofNat.{0} Real 2 (OfNat.mk.{0} Real 2 (bit0.{0} Real Real.hasAdd (One.one.{0} Real Real.hasOne))))) (Encodable.encode.{u1} ι _inst_1 i))) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u2} (F i) (PseudoMetricSpace.toHasDist.{u2} (F i) (MetricSpace.toPseudoMetricSpace.{u2} (F i) (_inst_2 i))) (x i) (y i)) (Dist.dist.{max u1 u2} (forall (i : ι), F i) (PiCountable.dist.{u1, u2} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y))
+but is expected to have type
+  forall {ι : Type.{u2}} [_inst_1 : Encodable.{u2} ι] {F : ι -> Type.{u1}} [_inst_2 : forall (i : ι), MetricSpace.{u1} (F i)] {x : forall (i : ι), F i} {y : forall (i : ι), F i} {i : ι}, (LT.lt.{0} Real Real.instLTReal (Dist.dist.{max u2 u1} (forall (i : ι), F i) (PiCountable.dist.{u2, u1} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y) (HPow.hPow.{0, 0, 0} Real Nat Real (instHPow.{0, 0} Real Nat (Monoid.Pow.{0} Real Real.instMonoidReal)) (HDiv.hDiv.{0, 0, 0} Real Real Real (instHDiv.{0} Real (LinearOrderedField.toDiv.{0} Real Real.instLinearOrderedFieldReal)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOneReal)) (OfNat.ofNat.{0} Real 2 (instOfNat.{0} Real 2 Real.natCast (instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) (Encodable.encode.{u2} ι _inst_1 i))) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} (F i) (PseudoMetricSpace.toDist.{u1} (F i) (MetricSpace.toPseudoMetricSpace.{u1} (F i) (_inst_2 i))) (x i) (y i)) (Dist.dist.{max u2 u1} (forall (i : ι), F i) (PiCountable.dist.{u2, u1} ι _inst_1 (fun (i : ι) => F i) (fun (i : ι) => _inst_2 i)) x y))
+Case conversion may be inaccurate. Consider using '#align pi_countable.dist_le_dist_pi_of_dist_lt PiCountable.dist_le_dist_pi_of_dist_ltₓ'. -/
 theorem dist_le_dist_pi_of_dist_lt {x y : ∀ i, F i} {i : ι} (h : dist x y < (1 / 2) ^ encode i) :
     dist (x i) (y i) ≤ dist x y := by
   simpa only [not_le.2 h, false_or_iff] using min_le_iff.1 (min_dist_le_dist_pi x y i)
@@ -869,6 +1059,7 @@ open NNReal
 
 variable (E)
 
+#print PiCountable.metricSpace /-
 /-- Given a countable family of metric spaces, one may put a distance on their product `Π i, E i`,
 defining the right topology and uniform structure. It is highly non-canonical, though, and therefore
 not registered as a global instance.
@@ -984,6 +1175,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
         _ < ε := hε
         
 #align pi_countable.metric_space PiCountable.metricSpace
+-/
 
 end PiCountable
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -894,8 +894,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
               (min ((1 / 2) ^ encode i) (dist (x i) (y i)) +
                 min ((1 / 2) ^ encode i) (dist (y i) (z i))) :=
           by
-          convert
-              congr_arg (coe : ℝ≥0 → ℝ)
+          convert congr_arg (coe : ℝ≥0 → ℝ)
                 (min_add_distrib ((1 / 2 : ℝ≥0) ^ encode i) (nndist (x i) (y i))
                   (nndist (y i) (z i))) <;>
             simp

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -789,7 +789,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
       by
       have : tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) at_top (𝓝 (2 * 0)) :=
         (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _
-      rw [mul_zero] at this
+      rw [MulZeroClass.mul_zero] at this
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closed_ball (pow_nonneg I0.le _)) this
     refine'
@@ -937,7 +937,8 @@ protected def metricSpace : MetricSpace (∀ i, F i)
         rcases Nat.eq_zero_or_pos K.card with (hK | hK)
         ·
           exact
-            ⟨1, zero_lt_one, by simpa only [hK, Nat.cast_zero, zero_mul] using (half_pos εpos).le⟩
+            ⟨1, zero_lt_one, by
+              simpa only [hK, Nat.cast_zero, MulZeroClass.zero_mul] using (half_pos εpos).le⟩
         · have Kpos : 0 < (K.card : ℝ) := Nat.cast_pos.2 hK
           refine' ⟨ε / 2 / (K.card : ℝ), div_pos (half_pos εpos) Kpos, le_of_eq _⟩
           field_simp [Kpos.ne']

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -221,7 +221,7 @@ local instances in this section.
 
 /-- The distance function on a product space `Π n, E n`, given by `dist x y = (1/2)^n` where `n` is
 the first index at which `x` and `y` differ. -/
-protected def hasDist : HasDist (∀ n, E n) :=
+protected def hasDist : Dist (∀ n, E n) :=
   ⟨fun x y => if h : x ≠ y then (1 / 2 : ℝ) ^ firstDiff x y else 0⟩
 #align pi_nat.has_dist PiNat.hasDist
 
@@ -832,7 +832,7 @@ open Encodable
 /-- Given a countable family of metric spaces, one may put a distance on their product `Π i, E i`.
 It is highly non-canonical, though, and therefore not registered as a global instance.
 The distance we use here is `dist x y = ∑' i, min (1/2)^(encode i) (dist (x i) (y i))`. -/
-protected def hasDist : HasDist (∀ i, F i) :=
+protected def hasDist : Dist (∀ i, F i) :=
   ⟨fun x y => ∑' i : ι, min ((1 / 2) ^ encode i) (dist (x i) (y i))⟩
 #align pi_countable.has_dist PiCountable.hasDist
 

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

@@ -757,7 +757,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
       dist (g x) (g y) ≤ dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) :=
         dist_triangle_right _ _ _
       _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [← B]
-      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A x n) (A y n)
+      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := (add_le_add (A x n) (A y n))
       _ = 4 * (1 / 2) ^ (n + 1) := by ring
       
   have g_surj : surjective g := by
@@ -969,7 +969,7 @@ protected def metricSpace : MetricSpace (∀ i, F i)
             apply (hxy i _).le
             simpa using hi
           _ ≤ ε / 2 + ε / 2 :=
-            add_le_add_right (by simpa only [Finset.sum_const, nsmul_eq_mul] using hδ) _
+            (add_le_add_right (by simpa only [Finset.sum_const, nsmul_eq_mul] using hδ) _)
           _ = ε := add_halves _
           
     · simp only [le_infᵢ_iff, le_principal_iff]

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.pi_nat
-! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! leanprover-community/mathlib commit e1a7bdeb4fd826b7e71d130d34988f0a2d26a177
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -379,8 +379,8 @@ but it does not take care of a possible uniformity. If the `E n` have a uniform
 there will be two non-defeq uniform structures on `Π n, E n`, the product one and the one coming
 from the metric structure. In this case, use `metric_space_of_discrete_uniformity` instead. -/
 protected def metricSpace : MetricSpace (∀ n, E n) :=
-  MetricSpace.ofMetrizable dist PiNat.dist_self PiNat.dist_comm PiNat.dist_triangle isOpen_iff_dist
-    PiNat.eq_of_dist_eq_zero
+  MetricSpace.ofDistTopology dist PiNat.dist_self PiNat.dist_comm PiNat.dist_triangle
+    isOpen_iff_dist PiNat.eq_of_dist_eq_zero
 #align pi_nat.metric_space PiNat.metricSpace
 
 /-- Metric space structure on `Π (n : ℕ), E n` when the spaces `E n` have the discrete uniformity,

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

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -490,7 +490,7 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
   · exact ⟨0, by simp⟩
   have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx
   obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one
-  refine' ⟨n, disjoint_left.2 fun y ys hy => ?_⟩
+  refine ⟨n, disjoint_left.2 fun y ys hy => ?_⟩
   apply lt_irrefl (infDist x s)
   calc
     infDist x s ≤ dist x y := infDist_le_dist_of_mem ys

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -758,7 +758,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [Metr
       dist (g x) (g y) ≤ dist (g x) (u (x.1 n)) + dist (g y) (u (x.1 n)) :=
         dist_triangle_right _ _ _
       _ = dist (g x) (u (x.1 n)) + dist (g y) (u (y.1 n)) := by rw [← B]
-      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := (add_le_add (A x n) (A y n))
+      _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A x n) (A y n)
       _ = 4 * (1 / 2) ^ (n + 1) := by ring
   have g_surj : Surjective g := fun y ↦ by
     have : ∀ n : ℕ, ∃ j, y ∈ closedBall (u j) ((1 / 2) ^ n) := fun n ↦ by

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -199,7 +199,7 @@ open List
 /-- In the case where `E` has constant value `α`,
 the cylinder `cylinder x n` can be identified with the element of `List α`
 consisting of the first `n` entries of `x`. See `cylinder_eq_res`.
-We call this list `res x n`, the restriction of `x` to `n`.-/
+We call this list `res x n`, the restriction of `x` to `n`. -/
 def res (x : ℕ → α) : ℕ → List α
   | 0 => nil
   | Nat.succ n => x n :: res x n
@@ -219,7 +219,7 @@ theorem res_succ (x : ℕ → α) (n : ℕ) : res x n.succ = x n :: res x n :=
 theorem res_length (x : ℕ → α) (n : ℕ) : (res x n).length = n := by induction n <;> simp [*]
 #align pi_nat.res_length PiNat.res_length
 
-/-- The restrictions of `x` and `y` to `n` are equal if and only if `x m = y m` for all `m < n`.-/
+/-- The restrictions of `x` and `y` to `n` are equal if and only if `x m = y m` for all `m < n`. -/
 theorem res_eq_res {x y : ℕ → α} {n : ℕ} :
     res x n = res y n ↔ ∀ ⦃m⦄, m < n → x m = y m := by
   constructor <;> intro h <;> induction' n with n ih; · simp
@@ -243,7 +243,7 @@ theorem res_injective : Injective (@res α) := by
   rw [h]
 #align pi_nat.res_injective PiNat.res_injective
 
-/-- `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`.-/
+/-- `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`. -/
 theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) :
     cylinder x n = { y | res y n = res x n } := by
   ext y

@@ -295,7 +295,7 @@ theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y)
   · simp
   rcases eq_or_ne y z with (rfl | hyz)
   · simp
-  simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv, one_div, inv_pow, zero_lt_two, Ne.def,
+  simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv, one_div, inv_pow, zero_lt_two, Ne,
     not_false_iff, le_max_iff, pow_le_pow_iff_right, one_lt_two, pow_pos,
     min_le_iff.1 (min_firstDiff_le x y z hxz)]
 #align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarch

Reconstitute the file Algebra.Order.Monoid.WithZero from three files:

• Algebra.Order.Monoid.WithZero.Defs
• Algebra.Order.Monoid.WithZero.Basic
• Algebra.Order.WithZero

Avoid importing it in many files. Most uses were just to get le_zero_iff to work on Nat.

Before

After

@@ -747,7 +747,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [Metr
     have diff_pos : 0 < firstDiff x.1 y.1 := by
       by_contra! h
       apply apply_firstDiff_ne hne'
-      rw [le_zero_iff.1 h]
+      rw [Nat.le_zero.1 h]
       apply apply_eq_of_dist_lt _ le_rfl
       rw [pow_zero]
       exact hxy

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -52,7 +52,8 @@ in general), and `ι` is countable.
 
 noncomputable section
 
-open Classical Topology Filter
+open scoped Classical
+open Topology Filter
 
 open TopologicalSpace Set Metric Filter Function
 

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -608,14 +608,14 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
     Otherwise, `f x` remains in the same `n`-cylinder as `x`. Similarly for `y`. Finally, `f x` and
     `f y` are again in the same `n`-cylinder, as desired. -/
   set f := fun x => if x ∈ s then x else (inter_cylinder_longestPrefix_nonempty hs hne x).some
-  have fs : ∀ x ∈ s, f x = x := fun x xs => by simp [xs]
+  have fs : ∀ x ∈ s, f x = x := fun x xs => by simp [f, xs]
   refine' ⟨f, fs, _, _⟩
   -- check that the range of `f` is `s`.
   · apply Subset.antisymm
     · rintro x ⟨y, rfl⟩
       by_cases hy : y ∈ s
       · rwa [fs y hy]
-      simpa [if_neg hy] using (inter_cylinder_longestPrefix_nonempty hs hne y).choose_spec.1
+      simpa [f, if_neg hy] using (inter_cylinder_longestPrefix_nonempty hs hne y).choose_spec.1
     · intro x hx
       rw [← fs x hx]
       exact mem_range_self _
@@ -640,7 +640,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
       -- case where `y ∉ s`
       have A : (s ∩ cylinder y (longestPrefix y s)).Nonempty :=
         inter_cylinder_longestPrefix_nonempty hs hne y
-      have fy : f y = A.some := by simp_rw [if_neg ys]
+      have fy : f y = A.some := by simp_rw [f, if_neg ys]
       have I : cylinder A.some (firstDiff x y) = cylinder y (firstDiff x y) := by
         rw [← mem_cylinder_iff_eq, firstDiff_comm]
         apply cylinder_anti y _ A.some_mem.2
@@ -652,7 +652,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
       -- case where `y ∈ s` (similar to the above)
       · have A : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
           inter_cylinder_longestPrefix_nonempty hs hne x
-        have fx : f x = A.some := by simp_rw [if_neg xs]
+        have fx : f x = A.some := by simp_rw [f, if_neg xs]
         have I : cylinder A.some (firstDiff x y) = cylinder x (firstDiff x y) := by
           rw [← mem_cylinder_iff_eq]
           apply cylinder_anti x _ A.some_mem.2
@@ -662,10 +662,10 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
       -- case where `y ∉ s`
       · have Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
           inter_cylinder_longestPrefix_nonempty hs hne x
-        have fx : f x = Ax.some := by simp_rw [if_neg xs]
+        have fx : f x = Ax.some := by simp_rw [f, if_neg xs]
         have Ay : (s ∩ cylinder y (longestPrefix y s)).Nonempty :=
           inter_cylinder_longestPrefix_nonempty hs hne y
-        have fy : f y = Ay.some := by simp_rw [if_neg ys]
+        have fy : f y = Ay.some := by simp_rw [f, if_neg ys]
         -- case where the common prefix to `x` and `s`, or `y` and `s`, is shorter than the
         -- common part to `x` and `y` -- then `f x = f y`.
         by_cases H : longestPrefix x s < firstDiff x y ∨ longestPrefix y s < firstDiff x y

See here on Zulip.

This PR changes a bunch of names containing nhds_0 or/and lt_1 to nhds_zero or/and lt_one.

@@ -773,14 +773,15 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [Metr
           dist_triangle_right _ _ _
         _ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A ⟨x, I⟩ n) (hx n)
     have L : Tendsto (fun n : ℕ => (1 / 2 : ℝ) ^ n + (1 / 2) ^ n) atTop (𝓝 (0 + 0)) :=
-      (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).add (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1)
+      (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).add
+        (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1)
     rw [add_zero] at L
     exact ge_of_tendsto' L J
   have s_closed : IsClosed s := by
     refine isClosed_iff_clusterPt.mpr fun x hx ↦ ?_
     have L : Tendsto (fun n : ℕ => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0) := by
       have : Tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) atTop (𝓝 (2 * 0)) :=
-        (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _
+        (tendsto_pow_atTop_nhds_zero_of_lt_one I0.le I1).const_mul _
       rw [mul_zero] at this
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this

The names for lemmas about monotonicity of (a ^ ·) and (· ^ n) were a mess. This PR tidies up everything related by following the naming convention for (a * ·) and (· * b). Namely, (a ^ ·) is pow_right and (· ^ n) is pow_left in lemma names. All lemma renames follow the corresponding multiplication lemma names closely.

Renames

Algebra.GroupPower.Order

• pow_mono → pow_right_mono
• pow_le_pow → pow_le_pow_right
• pow_le_pow_of_le_left → pow_le_pow_left
• pow_lt_pow_of_lt_left → pow_lt_pow_left
• strictMonoOn_pow → pow_left_strictMonoOn
• pow_strictMono_right → pow_right_strictMono
• pow_lt_pow → pow_lt_pow_right
• pow_lt_pow_iff → pow_lt_pow_iff_right
• pow_le_pow_iff → pow_le_pow_iff_right
• self_lt_pow → lt_self_pow
• strictAnti_pow → pow_right_strictAnti
• pow_lt_pow_iff_of_lt_one → pow_lt_pow_iff_right_of_lt_one
• pow_lt_pow_of_lt_one → pow_lt_pow_right_of_lt_one
• lt_of_pow_lt_pow → lt_of_pow_lt_pow_left
• le_of_pow_le_pow → le_of_pow_le_pow_left
• pow_lt_pow₀ → pow_lt_pow_right₀

Algebra.GroupPower.CovariantClass

• pow_le_pow_of_le_left' → pow_le_pow_left'
• nsmul_le_nsmul_of_le_right → nsmul_le_nsmul_right
• pow_lt_pow' → pow_lt_pow_right'
• nsmul_lt_nsmul → nsmul_lt_nsmul_left
• pow_strictMono_left → pow_right_strictMono'
• nsmul_strictMono_right → nsmul_left_strictMono
• StrictMono.pow_right' → StrictMono.pow_const
• StrictMono.nsmul_left → StrictMono.const_nsmul
• pow_strictMono_right' → pow_left_strictMono
• nsmul_strictMono_left → nsmul_right_strictMono
• Monotone.pow_right → Monotone.pow_const
• Monotone.nsmul_left → Monotone.const_nsmul
• lt_of_pow_lt_pow' → lt_of_pow_lt_pow_left'
• lt_of_nsmul_lt_nsmul → lt_of_nsmul_lt_nsmul_right
• pow_le_pow' → pow_le_pow_right'
• nsmul_le_nsmul → nsmul_le_nsmul_left
• pow_le_pow_of_le_one' → pow_le_pow_right_of_le_one'
• nsmul_le_nsmul_of_nonpos → nsmul_le_nsmul_left_of_nonpos
• le_of_pow_le_pow' → le_of_pow_le_pow_left'
• le_of_nsmul_le_nsmul' → le_of_nsmul_le_nsmul_right'
• pow_le_pow_iff' → pow_le_pow_iff_right'
• nsmul_le_nsmul_iff → nsmul_le_nsmul_iff_left
• pow_lt_pow_iff' → pow_lt_pow_iff_right'
• nsmul_lt_nsmul_iff → nsmul_lt_nsmul_iff_left

Data.Nat.Pow

• Nat.pow_lt_pow_of_lt_left → Nat.pow_lt_pow_left
• Nat.pow_le_iff_le_left → Nat.pow_le_pow_iff_left
• Nat.pow_lt_iff_lt_left → Nat.pow_lt_pow_iff_left

Lemmas added

• pow_le_pow_iff_left
• pow_lt_pow_iff_left
• pow_right_injective
• pow_right_inj
• Nat.pow_le_pow_left to have the correct name since Nat.pow_le_pow_of_le_left is in Std.
• Nat.pow_le_pow_right to have the correct name since Nat.pow_le_pow_of_le_right is in Std.

Lemmas removed

• self_le_pow was a duplicate of le_self_pow.
• Nat.pow_lt_pow_of_lt_right is defeq to pow_lt_pow_right.
• Nat.pow_right_strictMono is defeq to pow_right_strictMono.
• Nat.pow_le_iff_le_right is defeq to pow_le_pow_iff_right.
• Nat.pow_lt_iff_lt_right is defeq to pow_lt_pow_iff_right.

Other changes

• A bunch of proofs have been golfed.
• Some lemma assumptions have been turned from 0 < n or 1 ≤ n to n ≠ 0.
• A few Nat lemmas have been protected.
• One docstring has been fixed.

@@ -56,7 +56,7 @@ open Classical Topology Filter
 
 open TopologicalSpace Set Metric Filter Function
 
-attribute [local simp] pow_le_pow_iff one_lt_two inv_le_inv zero_le_two zero_lt_two
+attribute [local simp] pow_le_pow_iff_right one_lt_two inv_le_inv zero_le_two zero_lt_two
 
 variable {E : ℕ → Type*}
 
@@ -295,7 +295,7 @@ theorem dist_triangle_nonarch (x y z : ∀ n, E n) : dist x z ≤ max (dist x y)
   rcases eq_or_ne y z with (rfl | hyz)
   · simp
   simp only [dist_eq_of_ne, hxz, hxy, hyz, inv_le_inv, one_div, inv_pow, zero_lt_two, Ne.def,
-    not_false_iff, le_max_iff, pow_le_pow_iff, one_lt_two, pow_pos,
+    not_false_iff, le_max_iff, pow_le_pow_iff_right, one_lt_two, pow_pos,
     min_le_iff.1 (min_firstDiff_le x y z hxz)]
 #align pi_nat.dist_triangle_nonarch PiNat.dist_triangle_nonarch
 
@@ -328,7 +328,7 @@ theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2)
   rcases eq_or_ne x y with (rfl | hne)
   · rfl
   have : n < firstDiff x y := by
-    simpa [dist_eq_of_ne hne, inv_lt_inv, pow_lt_pow_iff, one_lt_two] using h
+    simpa [dist_eq_of_ne hne, inv_lt_inv, pow_lt_pow_iff_right, one_lt_two] using h
   exact apply_eq_of_lt_firstDiff (hi.trans_lt this)
 #align pi_nat.apply_eq_of_dist_lt PiNat.apply_eq_of_dist_lt
 

To fit with the "please try harder" convention of ! tactics.

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

@@ -90,7 +90,7 @@ theorem firstDiff_comm (x y : ∀ n, E n) : firstDiff x y = firstDiff y x := by
 
 theorem min_firstDiff_le (x y z : ∀ n, E n) (h : x ≠ z) :
     min (firstDiff x y) (firstDiff y z) ≤ firstDiff x z := by
-  by_contra' H
+  by_contra! H
   rw [lt_min_iff] at H
   refine apply_firstDiff_ne h ?_
   calc
@@ -154,7 +154,7 @@ theorem mem_cylinder_iff_le_firstDiff {x y : ∀ n, E n} (hne : x ≠ y) (i : 
     x ∈ cylinder y i ↔ i ≤ firstDiff x y := by
   constructor
   · intro h
-    by_contra'
+    by_contra!
     exact apply_firstDiff_ne hne (h _ this)
   · intro hi j hj
     exact apply_eq_of_lt_firstDiff (hj.trans_le hi)
@@ -317,7 +317,7 @@ theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :
   suffices (∀ i : ℕ, i < n → y i = x i) ↔ n ≤ firstDiff y x by simpa [dist_eq_of_ne hne]
   constructor
   · intro hy
-    by_contra' H
+    by_contra! H
     exact apply_firstDiff_ne hne (hy _ H)
   · intro h i hi
     exact apply_eq_of_lt_firstDiff (hi.trans_le h)
@@ -744,7 +744,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [Metr
     have dist' : dist x y = dist x.1 y.1 := rfl
     let n := firstDiff x.1 y.1 - 1
     have diff_pos : 0 < firstDiff x.1 y.1 := by
-      by_contra' h
+      by_contra! h
       apply apply_firstDiff_ne hne'
       rw [le_zero_iff.1 h]
       apply apply_eq_of_dist_lt _ le_rfl

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

@@ -363,7 +363,7 @@ theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) := by
 
 theorem isTopologicalBasis_cylinders :
     IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n) (n : ℕ), s = cylinder x n } := by
-  apply isTopologicalBasis_of_open_of_nhds
+  apply isTopologicalBasis_of_isOpen_of_nhds
   · rintro u ⟨x, n, rfl⟩
     apply isOpen_cylinder
   · intro x u hx u_open

• Remove simp-lemma bitwise_of_ne_zero, since it wasn't used, and could cause loops in an inconsistent context.

@@ -308,7 +308,6 @@ protected theorem dist_triangle (x y z : ∀ n, E n) : dist x z ≤ dist x y + d
 protected theorem eq_of_dist_eq_zero (x y : ∀ n, E n) (hxy : dist x y = 0) : x = y := by
   rcases eq_or_ne x y with (rfl | h); · rfl
   simp [dist_eq_of_ne h] at hxy
-  exact (two_ne_zero (pow_eq_zero hxy)).elim
 #align pi_nat.eq_of_dist_eq_zero PiNat.eq_of_dist_eq_zero
 
 theorem mem_cylinder_iff_dist_le {x y : ∀ n, E n} {n : ℕ} :

Rename

• summable_of_norm_bounded -> Summable.of_norm_bounded;
• summable_of_norm_bounded_eventually -> Summable.of_norm_bounded_eventually;
• summable_of_nnnorm_bounded -> Summable.of_nnnorm_bounded;
• summable_of_summable_norm -> Summable.of_norm;
• summable_of_summable_nnnorm -> Summable.of_nnnorm;

New lemmas

• Summable.of_norm_bounded_eventually_nat
• Summable.norm

Misc changes

• Golf a few proofs.

@@ -832,7 +832,7 @@ theorem dist_eq_tsum (x y : ∀ i, F i) :
 
 theorem dist_summable (x y : ∀ i, F i) :
     Summable fun i : ι => min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) := by
-  refine summable_of_nonneg_of_le (fun i => ?_) (fun i => min_le_left _ _)
+  refine .of_nonneg_of_le (fun i => ?_) (fun i => min_le_left _ _)
     summable_geometric_two_encode
   exact le_min (pow_nonneg (by norm_num) _) dist_nonneg
 #align pi_countable.dist_summable PiCountable.dist_summable

mathport was forgetting a space in filter_upwards [...]with instead of filter_upwards [...] with.

@@ -470,7 +470,7 @@ protected theorem completeSpace : CompleteSpace (∀ n, E n) := by
   intro u hu
   refine' ⟨fun n => u n n, tendsto_pi_nhds.2 fun i => _⟩
   refine' tendsto_const_nhds.congr' _
-  filter_upwards [Filter.Ici_mem_atTop i]with n hn
+  filter_upwards [Filter.Ici_mem_atTop i] with n hn
   exact apply_eq_of_dist_lt (hu i i n le_rfl hn) le_rfl
 #align pi_nat.complete_space PiNat.completeSpace
 

Use Bornology.IsBounded instead.

@@ -786,7 +786,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [Metr
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this
     refine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_ball)
-      (fun n => bounded_closedBall) (fun N ↦ ?_) L
+      (fun n => isBounded_closedBall) (fun N ↦ ?_) L
     obtain ⟨y, hxy, ys⟩ : ∃ y, y ∈ ball x ((1 / 2) ^ N) ∩ s :=
       clusterPt_principal_iff.1 hx _ (ball_mem_nhds x (pow_pos I0 N))
     have E :

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -782,7 +782,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [Metr
     have L : Tendsto (fun n : ℕ => diam (closedBall (u (x n)) ((1 / 2) ^ n))) atTop (𝓝 0) := by
       have : Tendsto (fun n : ℕ => (2 : ℝ) * (1 / 2) ^ n) atTop (𝓝 (2 * 0)) :=
         (tendsto_pow_atTop_nhds_0_of_lt_1 I0.le I1).const_mul _
-      rw [MulZeroClass.mul_zero] at this
+      rw [mul_zero] at this
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this
     refine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_ball)

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

This has nice performance benefits.

@@ -58,7 +58,7 @@ open TopologicalSpace Set Metric Filter Function
 
 attribute [local simp] pow_le_pow_iff one_lt_two inv_le_inv zero_le_two zero_lt_two
 
-variable {E : ℕ → Type _}
+variable {E : ℕ → Type*}
 
 namespace PiNat
 
@@ -191,7 +191,7 @@ theorem update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y
 
 section Res
 
-variable {α : Type _}
+variable {α : Type*}
 
 open List
 
@@ -336,7 +336,7 @@ theorem apply_eq_of_dist_lt {x y : ∀ n, E n} {n : ℕ} (h : dist x y < (1 / 2)
 /-- A function to a pseudo-metric-space is `1`-Lipschitz if and only if points in the same cylinder
 of length `n` are sent to points within distance `(1/2)^n`.
 Not expressed using `LipschitzWith` as we don't have a metric space structure -/
-theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type _}
+theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type*}
     [PseudoMetricSpace α] {f : (∀ n, E n) → α} :
     (∀ x y : ∀ n, E n, dist (f x) (f y) ≤ dist x y) ↔
       ∀ x y n, y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n := by
@@ -421,7 +421,7 @@ protected def metricSpace : MetricSpace (∀ n, E n) :=
 /-- Metric space structure on `Π (n : ℕ), E n` when the spaces `E n` have the discrete uniformity,
 where the distance is given by `dist x y = (1/2)^n`, where `n` is the smallest index where `x` and
 `y` differ. Not registered as a global instance by default. -/
-protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, UniformSpace (E n)]
+protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type*} [∀ n, UniformSpace (E n)]
     (h : ∀ n, uniformity (E n) = 𝓟 idRel) : MetricSpace (∀ n, E n) :=
   haveI : ∀ n, DiscreteTopology (E n) := fun n => discreteTopology_of_discrete_uniformity (h n)
   { dist_triangle := PiNat.dist_triangle
@@ -503,7 +503,7 @@ theorem exists_disjoint_cylinder {s : Set (∀ n, E n)} (hs : IsClosed s) {x : 
 /-- Given a point `x` in a product space `Π (n : ℕ), E n`, and `s` a subset of this space, then
 `shortestPrefixDiff x s` if the smallest `n` for which there is no element of `s` having the same
 prefix of length `n` as `x`. If there is no such `n`, then use `0` by convention. -/
-def shortestPrefixDiff {E : ℕ → Type _} (x : ∀ n, E n) (s : Set (∀ n, E n)) : ℕ :=
+def shortestPrefixDiff {E : ℕ → Type*} (x : ∀ n, E n) (s : Set (∀ n, E n)) : ℕ :=
   if h : ∃ n, Disjoint s (cylinder x n) then Nat.find h else 0
 #align pi_nat.shortest_prefix_diff PiNat.shortestPrefixDiff
 
@@ -528,7 +528,7 @@ theorem shortestPrefixDiff_pos {s : Set (∀ n, E n)} (hs : IsClosed s) (hne : s
 /-- Given a point `x` in a product space `Π (n : ℕ), E n`, and `s` a subset of this space, then
 `longestPrefix x s` if the largest `n` for which there is an element of `s` having the same
 prefix of length `n` as `x`. If there is no such `n`, use `0` by convention. -/
-def longestPrefix {E : ℕ → Type _} (x : ∀ n, E n) (s : Set (∀ n, E n)) : ℕ :=
+def longestPrefix {E : ℕ → Type*} (x : ∀ n, E n) (s : Set (∀ n, E n)) : ℕ :=
   shortestPrefixDiff x s - 1
 #align pi_nat.longest_prefix PiNat.longestPrefix
 
@@ -719,7 +719,7 @@ open PiNat
 /-- Any nonempty complete second countable metric space is the continuous image of the
 fundamental space `ℕ → ℕ`. For a version of this theorem in the context of Polish spaces, see
 `exists_nat_nat_continuous_surjective_of_polishSpace`. -/
-theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [MetricSpace α]
+theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
     [CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
     ∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by
   /- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
@@ -812,7 +812,7 @@ namespace PiCountable
 -/
 
 
-variable {ι : Type _} [Encodable ι] {F : ι → Type _} [∀ i, MetricSpace (F i)]
+variable {ι : Type*} [Encodable ι] {F : ι → Type*} [∀ i, MetricSpace (F i)]
 
 open Encodable
 

• depends on: #5966

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

@@ -2,14 +2,11 @@
 Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module topology.metric_space.pi_nat
-! leanprover-community/mathlib commit 49b7f94aab3a3bdca1f9f34c5d818afb253b3993
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Topology.MetricSpace.HausdorffDistance
 
+#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5d818afb253b3993"
+
 /-!
 # Topological study of spaces `Π (n : ℕ), E n`
 

• Also remove most superfluous parentheses around big operators (∑, ∏ and variants).
• roughly the used regex: ([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤]) replaced by $1 $2 $3

@@ -877,8 +877,8 @@ protected def metricSpace : MetricSpace (∀ i, F i) where
         _ ≤ min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) +
               min ((1 / 2) ^ encode i : ℝ) (dist (y i) (z i)) :=
           min_le_right _ _
-    calc dist x z ≤ ∑' i, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) +
-          min ((1 / 2) ^ encode i : ℝ) (dist (y i) (z i)) :=
+    calc dist x z ≤ ∑' i, (min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) +
+          min ((1 / 2) ^ encode i : ℝ) (dist (y i) (z i))) :=
         tsum_le_tsum I (dist_summable x z) ((dist_summable x y).add (dist_summable y z))
       _ = dist x y + dist y z := tsum_add (dist_summable x y) (dist_summable y z)
   edist_dist _ _ := by exact ENNReal.coe_nnreal_eq _

@@ -175,7 +175,7 @@ theorem cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn :
 #align pi_nat.cylinder_eq_cylinder_of_le_first_diff PiNat.cylinder_eq_cylinder_of_le_firstDiff
 
 theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
-    (⋃ k, cylinder (update x n k) (n + 1)) = cylinder x n := by
+    ⋃ k, cylinder (update x n k) (n + 1) = cylinder x n := by
   ext y
   simp only [mem_cylinder_iff, mem_iUnion]
   constructor
@@ -793,7 +793,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
     obtain ⟨y, hxy, ys⟩ : ∃ y, y ∈ ball x ((1 / 2) ^ N) ∩ s :=
       clusterPt_principal_iff.1 hx _ (ball_mem_nhds x (pow_pos I0 N))
     have E :
-      (⋂ (n : ℕ) (H : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n)) =
+      ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n) =
         ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n) := by
       refine iInter_congr fun n ↦ iInter_congr fun hn ↦ ?_
       have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn

Co-authored-by: Yury G. Kudryashov <urkud@urkud.name>

@@ -911,14 +911,14 @@ protected def metricSpace : MetricSpace (∀ i, F i) where
         calc
           dist x y = ∑' i : ι, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) := rfl
           _ = (∑ i in K, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i))) +
-                ∑' i : ↑((K : Set ι)ᶜ), min ((1 / 2) ^ encode (i : ι) : ℝ) (dist (x i) (y i)) :=
+                ∑' i : ↑(K : Set ι)ᶜ, min ((1 / 2) ^ encode (i : ι) : ℝ) (dist (x i) (y i)) :=
             (sum_add_tsum_compl (dist_summable _ _)).symm
           _ ≤ (∑ i in K, dist (x i) (y i)) +
-                ∑' i : ↑((K : Set ι)ᶜ), ((1 / 2) ^ encode (i : ι) : ℝ) := by
+                ∑' i : ↑(K : Set ι)ᶜ, ((1 / 2) ^ encode (i : ι) : ℝ) := by
             refine' add_le_add (Finset.sum_le_sum fun i _ => min_le_right _ _) _
             refine' tsum_le_tsum (fun i => min_le_left _ _) _ _
-            · apply Summable.subtype (dist_summable x y) ((↑K : Set ι)ᶜ)
-            · apply Summable.subtype summable_geometric_two_encode ((↑K : Set ι)ᶜ)
+            · apply Summable.subtype (dist_summable x y) (↑K : Set ι)ᶜ
+            · apply Summable.subtype summable_geometric_two_encode (↑K : Set ι)ᶜ
           _ < (∑ _i in K, δ) + ε / 2 := by
             apply add_lt_add_of_le_of_lt _ hK
             refine Finset.sum_le_sum fun i hi => (hxy i ?_).le

Changes are of the form

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

@@ -661,7 +661,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
           rw [← mem_cylinder_iff_eq]
           apply cylinder_anti x _ A.some_mem.2
           apply firstDiff_le_longestPrefix hs xs ys
-        rw [fs y ys] at hfxfy⊢
+        rw [fs y ys] at hfxfy ⊢
         rwa [← fx, I2, ← mem_cylinder_iff_eq, mem_cylinder_iff_le_firstDiff hfxfy] at I
       -- case where `y ∉ s`
       · have Ax : (s ∩ cylinder x (longestPrefix x s)).Nonempty :=
@@ -684,7 +684,7 @@ theorem exists_lipschitz_retraction_of_isClosed {s : Set (∀ n, E n)} (hs : IsC
           congr
         -- case where the common prefix to `x` and `s` is long, as well as the common prefix to
         -- `y` and `s`. Then all points remain in the same cylinders.
-        · push_neg  at H
+        · push_neg at H
           have I1 : cylinder Ax.some (firstDiff x y) = cylinder x (firstDiff x y) := by
             rw [← mem_cylinder_iff_eq]
             exact cylinder_anti x H.1 Ax.some_mem.2

Exprs now have an mdata field. It seems that this gets in the way of push_neg, as reported on Zulip.

The above seems to fix the reported errors.

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -515,7 +515,7 @@ theorem firstDiff_lt_shortestPrefixDiff {s : Set (∀ n, E n)} (hs : IsClosed s)
   have A := exists_disjoint_cylinder hs hx
   rw [shortestPrefixDiff, dif_pos A]
   have B := Nat.find_spec A
-  contrapose! B; rw [not_lt] at B -- porting note: why this `rw` is needed?
+  contrapose! B
   rw [not_disjoint_iff_nonempty_inter]
   refine' ⟨y, hy, _⟩
   rw [mem_cylinder_comm]

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -366,7 +366,7 @@ theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) := by
 #align pi_nat.is_open_cylinder PiNat.isOpen_cylinder
 
 theorem isTopologicalBasis_cylinders :
-    IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n)(n : ℕ), s = cylinder x n } := by
+    IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n) (n : ℕ), s = cylinder x n } := by
   apply isTopologicalBasis_of_open_of_nhds
   · rintro u ⟨x, n, rfl⟩
     apply isOpen_cylinder

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>

@@ -174,10 +174,10 @@ theorem cylinder_eq_cylinder_of_le_firstDiff (x y : ∀ n, E n) {n : ℕ} (hn :
   exact apply_eq_of_lt_firstDiff (hi.trans_le hn)
 #align pi_nat.cylinder_eq_cylinder_of_le_first_diff PiNat.cylinder_eq_cylinder_of_le_firstDiff
 
-theorem unionᵢ_cylinder_update (x : ∀ n, E n) (n : ℕ) :
+theorem iUnion_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     (⋃ k, cylinder (update x n k) (n + 1)) = cylinder x n := by
   ext y
-  simp only [mem_cylinder_iff, mem_unionᵢ]
+  simp only [mem_cylinder_iff, mem_iUnion]
   constructor
   · rintro ⟨k, hk⟩ i hi
     simpa [hi.ne] using hk i (Nat.lt_succ_of_lt hi)
@@ -186,7 +186,7 @@ theorem unionᵢ_cylinder_update (x : ∀ n, E n) (n : ℕ) :
     rcases Nat.lt_succ_iff_lt_or_eq.1 hi with (h'i | rfl)
     · simp [H i h'i, h'i.ne]
     · simp
-#align pi_nat.Union_cylinder_update PiNat.unionᵢ_cylinder_update
+#align pi_nat.Union_cylinder_update PiNat.iUnion_cylinder_update
 
 theorem update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y ∈ cylinder x n :=
   mem_cylinder_iff.2 fun i hi => by simp [hi.ne]
@@ -434,26 +434,26 @@ protected def metricSpaceOfDiscreteUniformity {E : ℕ → Type _} [∀ n, Unifo
     edist_dist := fun _ _ ↦ by exact ENNReal.coe_nnreal_eq _
     toUniformSpace := Pi.uniformSpace _
     uniformity_dist := by
-      simp [Pi.uniformity, comap_infᵢ, gt_iff_lt, preimage_setOf_eq, comap_principal,
+      simp [Pi.uniformity, comap_iInf, gt_iff_lt, preimage_setOf_eq, comap_principal,
         PseudoMetricSpace.uniformity_dist, h, idRel]
       apply le_antisymm
-      · simp only [le_infᵢ_iff, le_principal_iff]
+      · simp only [le_iInf_iff, le_principal_iff]
         intro ε εpos
         obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < ε := exists_pow_lt_of_lt_one εpos (by norm_num)
         apply
-          @mem_infᵢ_of_interᵢ _ _ _ _ _ (Finset.range n).finite_toSet fun i =>
+          @mem_iInf_of_iInter _ _ _ _ _ (Finset.range n).finite_toSet fun i =>
             { p : (∀ n : ℕ, E n) × ∀ n : ℕ, E n | p.fst i = p.snd i }
         · simp only [mem_principal, setOf_subset_setOf, imp_self, imp_true_iff]
         · rintro ⟨x, y⟩ hxy
-          simp only [Finset.mem_coe, Finset.mem_range, interᵢ_coe_set, mem_interᵢ, mem_setOf_eq]
+          simp only [Finset.mem_coe, Finset.mem_range, iInter_coe_set, mem_iInter, mem_setOf_eq]
             at hxy
           apply lt_of_le_of_lt _ hn
           rw [← mem_cylinder_iff_dist_le, mem_cylinder_iff]
           exact hxy
-      · simp only [le_infᵢ_iff, le_principal_iff]
+      · simp only [le_iInf_iff, le_principal_iff]
         intro n
-        refine' mem_infᵢ_of_mem ((1 / 2) ^ n : ℝ) _
-        refine' mem_infᵢ_of_mem (by positivity) _
+        refine' mem_iInf_of_mem ((1 / 2) ^ n : ℝ) _
+        refine' mem_iInf_of_mem (by positivity) _
         simp only [mem_principal, setOf_subset_setOf, Prod.forall]
         intro x y hxy
         exact apply_eq_of_dist_lt hxy le_rfl }
@@ -738,7 +738,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
   let s : Set (ℕ → ℕ) := { x | (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty }
   let g : s → α := fun x => x.2.some
   have A : ∀ (x : s) (n : ℕ), dist (g x) (u ((x : ℕ → ℕ) n)) ≤ (1 / 2) ^ n := fun x n =>
-    (mem_interᵢ.1 x.2.some_mem n : _)
+    (mem_iInter.1 x.2.some_mem n : _)
   have g_cont : Continuous g := by
     refine continuous_iff_continuousAt.2 fun y => ?_
     refine continuousAt_of_locally_lipschitz zero_lt_one 4 fun x hxy => ?_
@@ -768,7 +768,7 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
       rcases hu.exists_dist_lt y (by simp : (0 : ℝ) < (1 / 2) ^ n) with ⟨j, hj⟩
       exact ⟨j, hj.le⟩
     choose x hx using this
-    have I : (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty := ⟨y, mem_interᵢ.2 hx⟩
+    have I : (⋂ n : ℕ, closedBall (u (x n)) ((1 / 2) ^ n)).Nonempty := ⟨y, mem_iInter.2 hx⟩
     refine' ⟨⟨x, I⟩, _⟩
     refine' dist_le_zero.1 _
     have J : ∀ n : ℕ, dist (g ⟨x, I⟩) y ≤ (1 / 2) ^ n + (1 / 2) ^ n := fun n =>
@@ -788,19 +788,19 @@ theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type _) [Met
       rw [MulZeroClass.mul_zero] at this
       exact
         squeeze_zero (fun n => diam_nonneg) (fun n => diam_closedBall (pow_nonneg I0.le _)) this
-    refine nonempty_interᵢ_of_nonempty_binterᵢ (fun n => isClosed_ball)
+    refine nonempty_iInter_of_nonempty_biInter (fun n => isClosed_ball)
       (fun n => bounded_closedBall) (fun N ↦ ?_) L
     obtain ⟨y, hxy, ys⟩ : ∃ y, y ∈ ball x ((1 / 2) ^ N) ∩ s :=
       clusterPt_principal_iff.1 hx _ (ball_mem_nhds x (pow_pos I0 N))
     have E :
       (⋂ (n : ℕ) (H : n ≤ N), closedBall (u (x n)) ((1 / 2) ^ n)) =
         ⋂ (n : ℕ) (H : n ≤ N), closedBall (u (y n)) ((1 / 2) ^ n) := by
-      refine interᵢ_congr fun n ↦ interᵢ_congr fun hn ↦ ?_
+      refine iInter_congr fun n ↦ iInter_congr fun hn ↦ ?_
       have : x n = y n := apply_eq_of_dist_lt (mem_ball'.1 hxy) hn
       rw [this]
     rw [E]
     apply Nonempty.mono _ ys
-    apply interᵢ_subset_interᵢ₂
+    apply iInter_subset_iInter₂
   obtain ⟨f, -, f_surj, f_cont⟩ :
     ∃ f : (ℕ → ℕ) → s, (∀ x : s, f x = x) ∧ Surjective f ∧ Continuous f := by
     apply exists_retraction_subtype_of_isClosed s_closed
@@ -890,10 +890,10 @@ protected def metricSpace : MetricSpace (∀ i, F i) where
     simp
   toUniformSpace := Pi.uniformSpace _
   uniformity_dist := by
-    simp only [Pi.uniformity, comap_infᵢ, gt_iff_lt, preimage_setOf_eq, comap_principal,
+    simp only [Pi.uniformity, comap_iInf, gt_iff_lt, preimage_setOf_eq, comap_principal,
       PseudoMetricSpace.uniformity_dist]
     apply le_antisymm
-    · simp only [le_infᵢ_iff, le_principal_iff]
+    · simp only [le_iInf_iff, le_principal_iff]
       intro ε εpos
       obtain ⟨K, hK⟩ :
         ∃ K : Finset ι, (∑' i : { j // j ∉ K }, (1 / 2 : ℝ) ^ encode (i : ι)) < ε / 2 :=
@@ -901,13 +901,13 @@ protected def metricSpace : MetricSpace (∀ i, F i) where
             (half_pos εpos)).exists
       obtain ⟨δ, δpos, hδ⟩ : ∃ δ : ℝ, 0 < δ ∧ (K.card : ℝ) * δ < ε / 2 :=
         exists_pos_mul_lt (half_pos εpos) _
-      apply @mem_infᵢ_of_interᵢ _ _ _ _ _ K.finite_toSet fun i =>
+      apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
           { p : (∀ i : ι, F i) × ∀ i : ι, F i | dist (p.fst i) (p.snd i) < δ }
       · rintro ⟨i, hi⟩
-        refine' mem_infᵢ_of_mem δ (mem_infᵢ_of_mem δpos _)
+        refine' mem_iInf_of_mem δ (mem_iInf_of_mem δpos _)
         simp only [Prod.forall, imp_self, mem_principal, Subset.rfl]
       · rintro ⟨x, y⟩ hxy
-        simp only [mem_interᵢ, mem_setOf_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy
+        simp only [mem_iInter, mem_setOf_eq, SetCoe.forall, Finset.mem_range, Finset.mem_coe] at hxy
         calc
           dist x y = ∑' i : ι, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i)) := rfl
           _ = (∑ i in K, min ((1 / 2) ^ encode i : ℝ) (dist (x i) (y i))) +
@@ -926,11 +926,11 @@ protected def metricSpace : MetricSpace (∀ i, F i) where
           _ ≤ ε / 2 + ε / 2 :=
             (add_le_add_right (by simpa only [Finset.sum_const, nsmul_eq_mul] using hδ.le) _)
           _ = ε := add_halves _
-    · simp only [le_infᵢ_iff, le_principal_iff]
+    · simp only [le_iInf_iff, le_principal_iff]
       intro i ε εpos
-      refine' mem_infᵢ_of_mem (min ((1 / 2) ^ encode i : ℝ) ε) _
+      refine' mem_iInf_of_mem (min ((1 / 2) ^ encode i : ℝ) ε) _
       have : 0 < min ((1 / 2) ^ encode i : ℝ) ε := lt_min (by simp) εpos
-      refine' mem_infᵢ_of_mem this _
+      refine' mem_iInf_of_mem this _
       simp only [and_imp, Prod.forall, setOf_subset_setOf, lt_min_iff, mem_principal]
       intro x y hn hε
       calc

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel
 
 ! This file was ported from Lean 3 source module topology.metric_space.pi_nat
-! leanprover-community/mathlib commit e1a7bdeb4fd826b7e71d130d34988f0a2d26a177
+! leanprover-community/mathlib commit 49b7f94aab3a3bdca1f9f34c5d818afb253b3993
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -192,6 +192,69 @@ theorem update_mem_cylinder (x : ∀ n, E n) (n : ℕ) (y : E n) : update x n y
   mem_cylinder_iff.2 fun i hi => by simp [hi.ne]
 #align pi_nat.update_mem_cylinder PiNat.update_mem_cylinder
 
+section Res
+
+variable {α : Type _}
+
+open List
+
+/-- In the case where `E` has constant value `α`,
+the cylinder `cylinder x n` can be identified with the element of `List α`
+consisting of the first `n` entries of `x`. See `cylinder_eq_res`.
+We call this list `res x n`, the restriction of `x` to `n`.-/
+def res (x : ℕ → α) : ℕ → List α
+  | 0 => nil
+  | Nat.succ n => x n :: res x n
+#align pi_nat.res PiNat.res
+
+@[simp]
+theorem res_zero (x : ℕ → α) : res x 0 = @nil α :=
+  rfl
+#align pi_nat.res_zero PiNat.res_zero
+
+@[simp]
+theorem res_succ (x : ℕ → α) (n : ℕ) : res x n.succ = x n :: res x n :=
+  rfl
+#align pi_nat.res_succ PiNat.res_succ
+
+@[simp]
+theorem res_length (x : ℕ → α) (n : ℕ) : (res x n).length = n := by induction n <;> simp [*]
+#align pi_nat.res_length PiNat.res_length
+
+/-- The restrictions of `x` and `y` to `n` are equal if and only if `x m = y m` for all `m < n`.-/
+theorem res_eq_res {x y : ℕ → α} {n : ℕ} :
+    res x n = res y n ↔ ∀ ⦃m⦄, m < n → x m = y m := by
+  constructor <;> intro h <;> induction' n with n ih; · simp
+  · intro m hm
+    rw [Nat.lt_succ_iff_lt_or_eq] at hm
+    simp only [res_succ, cons.injEq] at h
+    cases' hm with hm hm
+    · exact ih h.2 hm
+    rw [hm]
+    exact h.1
+  · simp
+  simp only [res_succ, cons.injEq]
+  refine' ⟨h (Nat.lt_succ_self _), ih fun m hm => _⟩
+  exact h (hm.trans (Nat.lt_succ_self _))
+#align pi_nat.res_eq_res PiNat.res_eq_res
+
+theorem res_injective : Injective (@res α) := by
+  intro x y h
+  ext n
+  apply res_eq_res.mp _ (Nat.lt_succ_self _)
+  rw [h]
+#align pi_nat.res_injective PiNat.res_injective
+
+/-- `cylinder x n` is equal to the set of sequences `y` with the same restriction to `n` as `x`.-/
+theorem cylinder_eq_res (x : ℕ → α) (n : ℕ) :
+    cylinder x n = { y | res y n = res x n } := by
+  ext y
+  dsimp [cylinder]
+  rw [res_eq_res]
+#align pi_nat.cylinder_eq_res PiNat.cylinder_eq_res
+
+end Res
+
 /-!
 ### A distance function on `Π n, E n`
 
@@ -297,12 +360,16 @@ theorem lipschitz_with_one_iff_forall_dist_image_le_of_mem_cylinder {α : Type _
 variable (E)
 variable [∀ n, TopologicalSpace (E n)] [∀ n, DiscreteTopology (E n)]
 
+theorem isOpen_cylinder (x : ∀ n, E n) (n : ℕ) : IsOpen (cylinder x n) := by
+  rw [PiNat.cylinder_eq_pi]
+  exact isOpen_set_pi (Finset.range n).finite_toSet fun a _ => isOpen_discrete _
+#align pi_nat.is_open_cylinder PiNat.isOpen_cylinder
+
 theorem isTopologicalBasis_cylinders :
     IsTopologicalBasis { s : Set (∀ n, E n) | ∃ (x : ∀ n, E n)(n : ℕ), s = cylinder x n } := by
   apply isTopologicalBasis_of_open_of_nhds
   · rintro u ⟨x, n, rfl⟩
-    rw [cylinder_eq_pi]
-    exact isOpen_set_pi (Finset.range n).finite_toSet fun a _ => isOpen_discrete _
+    apply isOpen_cylinder
   · intro x u hx u_open
     obtain ⟨v, ⟨U, F, -, rfl⟩, xU, Uu⟩ :
         ∃ v ∈ { S : Set (∀ i : ℕ, E i) | ∃ (U : ∀ i : ℕ, Set (E i)) (F : Finset ℕ),

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