analysis.normed_space.triv_sq_zero_ext ⟷ Mathlib.Analysis.NormedSpace.TrivSqZeroExt

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

@@ -79,7 +79,7 @@ theorem hasSum_snd_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R]
     sub_zero]
   simp_rw [← Nat.succ_eq_add_one, Nat.pred_succ, Nat.factorial_succ, Nat.cast_mul, ←
     Nat.succ_eq_add_one,
-    mul_div_cancel_left _ ((@Nat.cast_ne_zero 𝕜 _ _ _).mpr <| Nat.succ_ne_zero _)]
+    mul_div_cancel_left₀ _ ((@Nat.cast_ne_zero 𝕜 _ _ _).mpr <| Nat.succ_ne_zero _)]
   exact h
 #align triv_sq_zero_ext.has_sum_snd_exp_series_of_smul_comm TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm
 -/
@@ -103,7 +103,7 @@ end Topology
 
 section NormedRing
 
-variable [IsROrC 𝕜] [NormedRing R] [AddCommGroup M]
+variable [RCLike 𝕜] [NormedRing R] [AddCommGroup M]
 
 variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
 
@@ -147,7 +147,7 @@ end NormedRing
 
 section NormedCommRing
 
-variable [IsROrC 𝕜] [NormedCommRing R] [AddCommGroup M]
+variable [RCLike 𝕜] [NormedCommRing R] [AddCommGroup M]
 
 variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
 
@@ -193,7 +193,7 @@ end NormedCommRing
 
 section NormedField
 
-variable [IsROrC 𝕜] [NormedField R] [AddCommGroup M]
+variable [RCLike 𝕜] [NormedField R] [AddCommGroup M]
 
 variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
 

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

@@ -47,7 +47,6 @@ section Topology
 
 variable [TopologicalSpace R] [TopologicalSpace M]
 
-#print TrivSqZeroExt.hasSum_fst_expSeries /-
 /-- If `exp R x.fst` converges to `e` then `(exp R x).fst` converges to `e`. -/
 theorem hasSum_fst_expSeries [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module R M]
     [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
@@ -57,7 +56,6 @@ theorem hasSum_fst_expSeries [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 
     HasSum (fun n => fst (NormedSpace.expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
   simpa [NormedSpace.expSeries_apply_eq] using h
 #align triv_sq_zero_ext.has_sum_fst_exp_series TrivSqZeroExt.hasSum_fst_expSeries
--/
 
 #print TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm /-
 /-- If `exp R x.fst` converges to `e` then `(exp R x).snd` converges to `e • x.snd`. -/

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

@@ -53,9 +53,9 @@ theorem hasSum_fst_expSeries [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 
     [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
     [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M]
     [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) {e : R}
-    (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
-    HasSum (fun n => fst (expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
-  simpa [expSeries_apply_eq] using h
+    (h : HasSum (fun n => NormedSpace.expSeries 𝕜 R n fun _ => x.fst) e) :
+    HasSum (fun n => fst (NormedSpace.expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
+  simpa [NormedSpace.expSeries_apply_eq] using h
 #align triv_sq_zero_ext.has_sum_fst_exp_series TrivSqZeroExt.hasSum_fst_expSeries
 -/
 
@@ -66,10 +66,10 @@ theorem hasSum_snd_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R]
     [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M]
     [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M)
     (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R}
-    (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
-    HasSum (fun n => snd (expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd) :=
+    (h : HasSum (fun n => NormedSpace.expSeries 𝕜 R n fun _ => x.fst) e) :
+    HasSum (fun n => snd (NormedSpace.expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd) :=
   by
-  simp_rw [expSeries_apply_eq] at *
+  simp_rw [NormedSpace.expSeries_apply_eq] at *
   conv =>
     congr
     ext
@@ -92,8 +92,9 @@ theorem hasSum_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [Add
     [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
     [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M]
     [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
-    {e : R} (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
-    HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) := by
+    {e : R} (h : HasSum (fun n => NormedSpace.expSeries 𝕜 R n fun _ => x.fst) e) :
+    HasSum (fun n => NormedSpace.expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) :=
+  by
   simpa only [inl_fst_add_inr_snd_eq] using
     (has_sum_inl _ <| has_sum_fst_exp_series 𝕜 x h).add
       (has_sum_inr _ <| has_sum_snd_exp_series_of_smul_comm 𝕜 x hx h)
@@ -118,17 +119,17 @@ variable [CompleteSpace R] [T2Space R] [T2Space M]
 
 #print TrivSqZeroExt.exp_def_of_smul_comm /-
 theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) :
-    exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) :=
+    NormedSpace.exp 𝕜 x = inl (NormedSpace.exp 𝕜 x.fst) + inr (NormedSpace.exp 𝕜 x.fst • x.snd) :=
   by
-  simp_rw [exp, FormalMultilinearSeries.sum]
+  simp_rw [NormedSpace.exp, FormalMultilinearSeries.sum]
   refine' (has_sum_exp_series_of_smul_comm 𝕜 x hx _).tsum_eq
-  exact expSeries_hasSum_exp _
+  exact NormedSpace.expSeries_hasSum_exp _
 #align triv_sq_zero_ext.exp_def_of_smul_comm TrivSqZeroExt.exp_def_of_smul_comm
 -/
 
 #print TrivSqZeroExt.exp_inl /-
 @[simp]
-theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) :=
+theorem exp_inl (x : R) : NormedSpace.exp 𝕜 (inl x : tsze R M) = inl (NormedSpace.exp 𝕜 x) :=
   by
   rw [exp_def_of_smul_comm, snd_inl, fst_inl, smul_zero, inr_zero, add_zero]
   · rw [snd_inl, fst_inl, smul_zero, smul_zero]
@@ -137,9 +138,9 @@ theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) :=
 
 #print TrivSqZeroExt.exp_inr /-
 @[simp]
-theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m :=
+theorem exp_inr (m : M) : NormedSpace.exp 𝕜 (inr m : tsze R M) = 1 + inr m :=
   by
-  rw [exp_def_of_smul_comm, snd_inr, fst_inr, exp_zero, one_smul, inl_one]
+  rw [exp_def_of_smul_comm, snd_inr, fst_inr, NormedSpace.exp_zero, one_smul, inl_one]
   · rw [snd_inr, fst_inr, MulOpposite.op_zero, zero_smul, zero_smul]
 #align triv_sq_zero_ext.exp_inr TrivSqZeroExt.exp_inr
 -/
@@ -161,21 +162,22 @@ variable [TopologicalAddGroup M] [ContinuousSMul R M]
 variable [CompleteSpace R] [T2Space R] [T2Space M]
 
 #print TrivSqZeroExt.exp_def /-
-theorem exp_def (x : tsze R M) : exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) :=
+theorem exp_def (x : tsze R M) :
+    NormedSpace.exp 𝕜 x = inl (NormedSpace.exp 𝕜 x.fst) + inr (NormedSpace.exp 𝕜 x.fst • x.snd) :=
   exp_def_of_smul_comm 𝕜 x (op_smul_eq_smul _ _)
 #align triv_sq_zero_ext.exp_def TrivSqZeroExt.exp_def
 -/
 
 #print TrivSqZeroExt.fst_exp /-
 @[simp]
-theorem fst_exp (x : tsze R M) : fst (exp 𝕜 x) = exp 𝕜 x.fst := by
+theorem fst_exp (x : tsze R M) : fst (NormedSpace.exp 𝕜 x) = NormedSpace.exp 𝕜 x.fst := by
   rw [exp_def, fst_add, fst_inl, fst_inr, add_zero]
 #align triv_sq_zero_ext.fst_exp TrivSqZeroExt.fst_exp
 -/
 
 #print TrivSqZeroExt.snd_exp /-
 @[simp]
-theorem snd_exp (x : tsze R M) : snd (exp 𝕜 x) = exp 𝕜 x.fst • x.snd := by
+theorem snd_exp (x : tsze R M) : snd (NormedSpace.exp 𝕜 x) = NormedSpace.exp 𝕜 x.fst • x.snd := by
   rw [exp_def, snd_add, snd_inl, snd_inr, zero_add]
 #align triv_sq_zero_ext.snd_exp TrivSqZeroExt.snd_exp
 -/
@@ -183,7 +185,7 @@ theorem snd_exp (x : tsze R M) : snd (exp 𝕜 x) = exp 𝕜 x.fst • x.snd :=
 #print TrivSqZeroExt.eq_smul_exp_of_invertible /-
 /-- Polar form of trivial-square-zero extension. -/
 theorem eq_smul_exp_of_invertible (x : tsze R M) [Invertible x.fst] :
-    x = x.fst • exp 𝕜 (⅟ x.fst • inr x.snd) := by
+    x = x.fst • NormedSpace.exp 𝕜 (⅟ x.fst • inr x.snd) := by
   rw [← inr_smul, exp_inr, smul_add, ← inl_one, ← inl_smul, ← inr_smul, smul_eq_mul, mul_one,
     smul_smul, mul_invOf_self, one_smul, inl_fst_add_inr_snd_eq]
 #align triv_sq_zero_ext.eq_smul_exp_of_invertible TrivSqZeroExt.eq_smul_exp_of_invertible
@@ -209,7 +211,7 @@ variable [CompleteSpace R] [T2Space R] [T2Space M]
 /-- More convenient version of `triv_sq_zero_ext.eq_smul_exp_of_invertible` for when `R` is a
 field. -/
 theorem eq_smul_exp_of_ne_zero (x : tsze R M) (hx : x.fst ≠ 0) :
-    x = x.fst • exp 𝕜 (x.fst⁻¹ • inr x.snd) :=
+    x = x.fst • NormedSpace.exp 𝕜 (x.fst⁻¹ • inr x.snd) :=
   letI : Invertible x.fst := invertibleOfNonzero hx
   eq_smul_exp_of_invertible _ _
 #align triv_sq_zero_ext.eq_smul_exp_of_ne_zero TrivSqZeroExt.eq_smul_exp_of_ne_zero

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

@@ -3,9 +3,9 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathbin.Analysis.NormedSpace.Basic
-import Mathbin.Analysis.NormedSpace.Exponential
-import Mathbin.Topology.Instances.TrivSqZeroExt
+import Analysis.NormedSpace.Basic
+import Analysis.NormedSpace.Exponential
+import Topology.Instances.TrivSqZeroExt
 
 #align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"
 

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

@@ -2,16 +2,13 @@
 Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
-
-! This file was ported from Lean 3 source module analysis.normed_space.triv_sq_zero_ext
-! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.NormedSpace.Basic
 import Mathbin.Analysis.NormedSpace.Exponential
 import Mathbin.Topology.Instances.TrivSqZeroExt
 
+#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"5c1efce12ba86d4901463f61019832f6a4b1a0d0"
+
 /-!
 # Results on `triv_sq_zero_ext R M` related to the norm
 

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

@@ -42,7 +42,6 @@ For now, this file contains results about `exp` for this type.
 
 variable (𝕜 : Type _) {R M : Type _}
 
--- mathport name: exprtsze
 local notation "tsze" => TrivSqZeroExt
 
 namespace TrivSqZeroExt
@@ -51,6 +50,7 @@ section Topology
 
 variable [TopologicalSpace R] [TopologicalSpace M]
 
+#print TrivSqZeroExt.hasSum_fst_expSeries /-
 /-- If `exp R x.fst` converges to `e` then `(exp R x).fst` converges to `e`. -/
 theorem hasSum_fst_expSeries [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module R M]
     [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
@@ -60,7 +60,9 @@ theorem hasSum_fst_expSeries [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 
     HasSum (fun n => fst (expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
   simpa [expSeries_apply_eq] using h
 #align triv_sq_zero_ext.has_sum_fst_exp_series TrivSqZeroExt.hasSum_fst_expSeries
+-/
 
+#print TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm /-
 /-- If `exp R x.fst` converges to `e` then `(exp R x).snd` converges to `e • x.snd`. -/
 theorem hasSum_snd_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M]
     [Algebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M]
@@ -85,7 +87,9 @@ theorem hasSum_snd_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R]
     mul_div_cancel_left _ ((@Nat.cast_ne_zero 𝕜 _ _ _).mpr <| Nat.succ_ne_zero _)]
   exact h
 #align triv_sq_zero_ext.has_sum_snd_exp_series_of_smul_comm TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm
+-/
 
+#print TrivSqZeroExt.hasSum_expSeries_of_smul_comm /-
 /-- If `exp R x.fst` converges to `e` then `exp R x` converges to `inl e + inr (e • x.snd)`. -/
 theorem hasSum_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R]
     [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
@@ -97,6 +101,7 @@ theorem hasSum_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [Add
     (has_sum_inl _ <| has_sum_fst_exp_series 𝕜 x h).add
       (has_sum_inr _ <| has_sum_snd_exp_series_of_smul_comm 𝕜 x hx h)
 #align triv_sq_zero_ext.has_sum_exp_series_of_smul_comm TrivSqZeroExt.hasSum_expSeries_of_smul_comm
+-/
 
 end Topology
 
@@ -114,6 +119,7 @@ variable [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ
 
 variable [CompleteSpace R] [T2Space R] [T2Space M]
 
+#print TrivSqZeroExt.exp_def_of_smul_comm /-
 theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) :
     exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) :=
   by
@@ -121,20 +127,25 @@ theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd
   refine' (has_sum_exp_series_of_smul_comm 𝕜 x hx _).tsum_eq
   exact expSeries_hasSum_exp _
 #align triv_sq_zero_ext.exp_def_of_smul_comm TrivSqZeroExt.exp_def_of_smul_comm
+-/
 
+#print TrivSqZeroExt.exp_inl /-
 @[simp]
 theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) :=
   by
   rw [exp_def_of_smul_comm, snd_inl, fst_inl, smul_zero, inr_zero, add_zero]
   · rw [snd_inl, fst_inl, smul_zero, smul_zero]
 #align triv_sq_zero_ext.exp_inl TrivSqZeroExt.exp_inl
+-/
 
+#print TrivSqZeroExt.exp_inr /-
 @[simp]
 theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m :=
   by
   rw [exp_def_of_smul_comm, snd_inr, fst_inr, exp_zero, one_smul, inl_one]
   · rw [snd_inr, fst_inr, MulOpposite.op_zero, zero_smul, zero_smul]
 #align triv_sq_zero_ext.exp_inr TrivSqZeroExt.exp_inr
+-/
 
 end NormedRing
 
@@ -152,26 +163,34 @@ variable [TopologicalAddGroup M] [ContinuousSMul R M]
 
 variable [CompleteSpace R] [T2Space R] [T2Space M]
 
+#print TrivSqZeroExt.exp_def /-
 theorem exp_def (x : tsze R M) : exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) :=
   exp_def_of_smul_comm 𝕜 x (op_smul_eq_smul _ _)
 #align triv_sq_zero_ext.exp_def TrivSqZeroExt.exp_def
+-/
 
+#print TrivSqZeroExt.fst_exp /-
 @[simp]
 theorem fst_exp (x : tsze R M) : fst (exp 𝕜 x) = exp 𝕜 x.fst := by
   rw [exp_def, fst_add, fst_inl, fst_inr, add_zero]
 #align triv_sq_zero_ext.fst_exp TrivSqZeroExt.fst_exp
+-/
 
+#print TrivSqZeroExt.snd_exp /-
 @[simp]
 theorem snd_exp (x : tsze R M) : snd (exp 𝕜 x) = exp 𝕜 x.fst • x.snd := by
   rw [exp_def, snd_add, snd_inl, snd_inr, zero_add]
 #align triv_sq_zero_ext.snd_exp TrivSqZeroExt.snd_exp
+-/
 
+#print TrivSqZeroExt.eq_smul_exp_of_invertible /-
 /-- Polar form of trivial-square-zero extension. -/
 theorem eq_smul_exp_of_invertible (x : tsze R M) [Invertible x.fst] :
     x = x.fst • exp 𝕜 (⅟ x.fst • inr x.snd) := by
   rw [← inr_smul, exp_inr, smul_add, ← inl_one, ← inl_smul, ← inr_smul, smul_eq_mul, mul_one,
     smul_smul, mul_invOf_self, one_smul, inl_fst_add_inr_snd_eq]
 #align triv_sq_zero_ext.eq_smul_exp_of_invertible TrivSqZeroExt.eq_smul_exp_of_invertible
+-/
 
 end NormedCommRing
 
@@ -189,6 +208,7 @@ variable [TopologicalAddGroup M] [ContinuousSMul R M]
 
 variable [CompleteSpace R] [T2Space R] [T2Space M]
 
+#print TrivSqZeroExt.eq_smul_exp_of_ne_zero /-
 /-- More convenient version of `triv_sq_zero_ext.eq_smul_exp_of_invertible` for when `R` is a
 field. -/
 theorem eq_smul_exp_of_ne_zero (x : tsze R M) (hx : x.fst ≠ 0) :
@@ -196,6 +216,7 @@ theorem eq_smul_exp_of_ne_zero (x : tsze R M) (hx : x.fst ≠ 0) :
   letI : Invertible x.fst := invertibleOfNonzero hx
   eq_smul_exp_of_invertible _ _
 #align triv_sq_zero_ext.eq_smul_exp_of_ne_zero TrivSqZeroExt.eq_smul_exp_of_ne_zero
+-/
 
 end NormedField
 

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 
 ! This file was ported from Lean 3 source module analysis.normed_space.triv_sq_zero_ext
-! leanprover-community/mathlib commit 88a563b158f59f2983cfad685664da95502e8cdd
+! leanprover-community/mathlib commit 5c1efce12ba86d4901463f61019832f6a4b1a0d0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Topology.Instances.TrivSqZeroExt
 /-!
 # Results on `triv_sq_zero_ext R M` related to the norm
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 For now, this file contains results about `exp` for this type.
 
 ## Main results

mathlib commit https://github.com/leanprover-community/mathlib/commit/88a563b158f59f2983cfad685664da95502e8cdd

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 
 ! This file was ported from Lean 3 source module analysis.normed_space.triv_sq_zero_ext
-! leanprover-community/mathlib commit b8d2eaa69d69ce8f03179a5cda774fc0cde984e4
+! leanprover-community/mathlib commit 88a563b158f59f2983cfad685664da95502e8cdd
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -25,9 +25,14 @@ For now, this file contains results about `exp` for this type.
 * `triv_sq_zero_ext.exp_inr`
 
 ## TODO
+
 * Actually define a sensible norm on `triv_sq_zero_ext R M`, so that we have access to lemmas
   like `exp_add`.
-* Generalize some of these results to non-commutative `R`.
+* Generalize more of these results to non-commutative `R`. In principle, under sufficient conditions
+  we should expect
+ `(exp 𝕜 x).snd = ∫ t in 0..1, exp 𝕜 (t • x.fst) • op (exp 𝕜 ((1 - t) • x.fst)) • x.snd`
+  ([Physics.SE](https://physics.stackexchange.com/a/41671/185147), and
+  https://link.springer.com/chapter/10.1007/978-3-540-44953-9_2).
 
 -/
 
@@ -43,20 +48,31 @@ section Topology
 
 variable [TopologicalSpace R] [TopologicalSpace M]
 
-/-- If `exp R x.fst` converges to `e` then `exp R x` converges to `inl e + inr (e • x.snd)`. -/
-theorem hasSum_expSeries [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup M] [Algebra 𝕜 R]
-    [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
-    [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] (x : tsze R M) {e : R}
+/-- If `exp R x.fst` converges to `e` then `(exp R x).fst` converges to `e`. -/
+theorem hasSum_fst_expSeries [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module R M]
+    [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
+    [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M]
+    [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) {e : R}
+    (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
+    HasSum (fun n => fst (expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
+  simpa [expSeries_apply_eq] using h
+#align triv_sq_zero_ext.has_sum_fst_exp_series TrivSqZeroExt.hasSum_fst_expSeries
+
+/-- If `exp R x.fst` converges to `e` then `(exp R x).snd` converges to `e • x.snd`. -/
+theorem hasSum_snd_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M]
+    [Algebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M]
+    [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M]
+    [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M)
+    (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R}
     (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
-    HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) :=
+    HasSum (fun n => snd (expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd) :=
   by
   simp_rw [expSeries_apply_eq] at *
   conv =>
     congr
     ext
-    rw [← inl_fst_add_inr_snd_eq (x ^ _), fst_pow, snd_pow, smul_add, ← inr_smul, ← inl_smul,
-      nsmul_eq_smul_cast 𝕜 n, smul_smul, inv_mul_eq_div, ← inv_div, ← smul_assoc]
-  refine' (has_sum_inl M h).add (has_sum_inr M _)
+    rw [snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast 𝕜 n, smul_smul, inv_mul_eq_div, ←
+      inv_div, ← smul_assoc]
   apply HasSum.smul_const
   rw [← hasSum_nat_add_iff' 1]; swap; infer_instance
   rw [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, div_zero, inv_zero, zero_smul,
@@ -65,12 +81,62 @@ theorem hasSum_expSeries [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup
     Nat.succ_eq_add_one,
     mul_div_cancel_left _ ((@Nat.cast_ne_zero 𝕜 _ _ _).mpr <| Nat.succ_ne_zero _)]
   exact h
-#align triv_sq_zero_ext.has_sum_exp_series TrivSqZeroExt.hasSum_expSeries
+#align triv_sq_zero_ext.has_sum_snd_exp_series_of_smul_comm TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm
+
+/-- If `exp R x.fst` converges to `e` then `exp R x` converges to `inl e + inr (e • x.snd)`. -/
+theorem hasSum_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R]
+    [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
+    [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M]
+    [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
+    {e : R} (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
+    HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) := by
+  simpa only [inl_fst_add_inr_snd_eq] using
+    (has_sum_inl _ <| has_sum_fst_exp_series 𝕜 x h).add
+      (has_sum_inr _ <| has_sum_snd_exp_series_of_smul_comm 𝕜 x hx h)
+#align triv_sq_zero_ext.has_sum_exp_series_of_smul_comm TrivSqZeroExt.hasSum_expSeries_of_smul_comm
 
 end Topology
 
 section NormedRing
 
+variable [IsROrC 𝕜] [NormedRing R] [AddCommGroup M]
+
+variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
+
+variable [Module 𝕜 M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M]
+
+variable [TopologicalSpace M] [TopologicalRing R]
+
+variable [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M]
+
+variable [CompleteSpace R] [T2Space R] [T2Space M]
+
+theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) :
+    exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) :=
+  by
+  simp_rw [exp, FormalMultilinearSeries.sum]
+  refine' (has_sum_exp_series_of_smul_comm 𝕜 x hx _).tsum_eq
+  exact expSeries_hasSum_exp _
+#align triv_sq_zero_ext.exp_def_of_smul_comm TrivSqZeroExt.exp_def_of_smul_comm
+
+@[simp]
+theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) :=
+  by
+  rw [exp_def_of_smul_comm, snd_inl, fst_inl, smul_zero, inr_zero, add_zero]
+  · rw [snd_inl, fst_inl, smul_zero, smul_zero]
+#align triv_sq_zero_ext.exp_inl TrivSqZeroExt.exp_inl
+
+@[simp]
+theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m :=
+  by
+  rw [exp_def_of_smul_comm, snd_inr, fst_inr, exp_zero, one_smul, inl_one]
+  · rw [snd_inr, fst_inr, MulOpposite.op_zero, zero_smul, zero_smul]
+#align triv_sq_zero_ext.exp_inr TrivSqZeroExt.exp_inr
+
+end NormedRing
+
+section NormedCommRing
+
 variable [IsROrC 𝕜] [NormedCommRing R] [AddCommGroup M]
 
 variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
@@ -84,10 +150,7 @@ variable [TopologicalAddGroup M] [ContinuousSMul R M]
 variable [CompleteSpace R] [T2Space R] [T2Space M]
 
 theorem exp_def (x : tsze R M) : exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) :=
-  by
-  simp_rw [exp, FormalMultilinearSeries.sum]
-  refine' (has_sum_exp_series 𝕜 x _).tsum_eq
-  exact expSeries_hasSum_exp _
+  exp_def_of_smul_comm 𝕜 x (op_smul_eq_smul _ _)
 #align triv_sq_zero_ext.exp_def TrivSqZeroExt.exp_def
 
 @[simp]
@@ -100,16 +163,6 @@ theorem snd_exp (x : tsze R M) : snd (exp 𝕜 x) = exp 𝕜 x.fst • x.snd :=
   rw [exp_def, snd_add, snd_inl, snd_inr, zero_add]
 #align triv_sq_zero_ext.snd_exp TrivSqZeroExt.snd_exp
 
-@[simp]
-theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) := by
-  rw [exp_def, fst_inl, snd_inl, smul_zero, inr_zero, add_zero]
-#align triv_sq_zero_ext.exp_inl TrivSqZeroExt.exp_inl
-
-@[simp]
-theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m := by
-  rw [exp_def, fst_inr, exp_zero, snd_inr, one_smul, inl_one]
-#align triv_sq_zero_ext.exp_inr TrivSqZeroExt.exp_inr
-
 /-- Polar form of trivial-square-zero extension. -/
 theorem eq_smul_exp_of_invertible (x : tsze R M) [Invertible x.fst] :
     x = x.fst • exp 𝕜 (⅟ x.fst • inr x.snd) := by
@@ -117,7 +170,7 @@ theorem eq_smul_exp_of_invertible (x : tsze R M) [Invertible x.fst] :
     smul_smul, mul_invOf_self, one_smul, inl_fst_add_inr_snd_eq]
 #align triv_sq_zero_ext.eq_smul_exp_of_invertible TrivSqZeroExt.eq_smul_exp_of_invertible
 
-end NormedRing
+end NormedCommRing
 
 section NormedField
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/22131150f88a2d125713ffa0f4693e3355b1eb49

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 
 ! This file was ported from Lean 3 source module analysis.normed_space.triv_sq_zero_ext
-! leanprover-community/mathlib commit 806c0bb86f6128cfa2f702285727518eb5244390
+! leanprover-community/mathlib commit b8d2eaa69d69ce8f03179a5cda774fc0cde984e4
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,9 +17,6 @@ import Mathbin.Topology.Instances.TrivSqZeroExt
 
 For now, this file contains results about `exp` for this type.
 
-TODO: actually define a sensible norm on `triv_sq_zero_ext R M`, so that we have access to lemmas
-like `exp_add`.
-
 ## Main results
 
 * `triv_sq_zero_ext.fst_exp`
@@ -27,6 +24,11 @@ like `exp_add`.
 * `triv_sq_zero_ext.exp_inl`
 * `triv_sq_zero_ext.exp_inr`
 
+## TODO
+* Actually define a sensible norm on `triv_sq_zero_ext R M`, so that we have access to lemmas
+  like `exp_add`.
+* Generalize some of these results to non-commutative `R`.
+
 -/
 
 
@@ -43,8 +45,8 @@ variable [TopologicalSpace R] [TopologicalSpace M]
 
 /-- If `exp R x.fst` converges to `e` then `exp R x` converges to `inl e + inr (e • x.snd)`. -/
 theorem hasSum_expSeries [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup M] [Algebra 𝕜 R]
-    [Module R M] [Module 𝕜 M] [IsScalarTower 𝕜 R M] [TopologicalRing R] [TopologicalAddGroup M]
-    [ContinuousSMul R M] (x : tsze R M) {e : R}
+    [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
+    [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] (x : tsze R M) {e : R}
     (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
     HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) :=
   by
@@ -71,7 +73,9 @@ section NormedRing
 
 variable [IsROrC 𝕜] [NormedCommRing R] [AddCommGroup M]
 
-variable [NormedAlgebra 𝕜 R] [Module R M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
+variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
+
+variable [Module 𝕜 M] [IsScalarTower 𝕜 R M]
 
 variable [TopologicalSpace M] [TopologicalRing R]
 
@@ -119,7 +123,9 @@ section NormedField
 
 variable [IsROrC 𝕜] [NormedField R] [AddCommGroup M]
 
-variable [NormedAlgebra 𝕜 R] [Module R M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
+variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
+
+variable [Module 𝕜 M] [IsScalarTower 𝕜 R M]
 
 variable [TopologicalSpace M] [TopologicalRing R]
 

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

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

@@ -316,7 +316,7 @@ end Normed
 
 section
 
-variable [IsROrC 𝕜] [NormedRing R] [NormedAddCommGroup M]
+variable [RCLike 𝕜] [NormedRing R] [NormedAddCommGroup M]
 variable [NormedAlgebra 𝕜 R] [NormedSpace 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M]
 variable [BoundedSMul R M] [BoundedSMul Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
 variable [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M]

This way we don't switch between general normed spaces and real normed spaces back and forth throughout the file.

@@ -3,7 +3,6 @@ Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
 -/
-import Mathlib.Analysis.NormedSpace.Basic
 import Mathlib.Analysis.NormedSpace.Exponential
 import Mathlib.Analysis.NormedSpace.ProdLp
 import Mathlib.Topology.Instances.TrivSqZeroExt

The motivation here is to enable using exp_add_of_commute on dual numbers.

This Zulip thread has some discussion about other possible norms.

@@ -5,6 +5,7 @@ Authors: Eric Wieser
 -/
 import Mathlib.Analysis.NormedSpace.Basic
 import Mathlib.Analysis.NormedSpace.Exponential
+import Mathlib.Analysis.NormedSpace.ProdLp
 import Mathlib.Topology.Instances.TrivSqZeroExt
 
 #align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
@@ -12,7 +13,15 @@ import Mathlib.Topology.Instances.TrivSqZeroExt
 /-!
 # Results on `TrivSqZeroExt R M` related to the norm
 
-For now, this file contains results about `exp` for this type.
+This file contains results about `NormedSpace.exp` for `TrivSqZeroExt`.
+
+It also contains a definition of the $ℓ^1$ norm,
+which defines $\|r + m\| \coloneqq \|r\| + \|m\|$.
+This is not a particularly canonical choice of definition,
+but it is sufficient to provide a `NormedAlgebra` instance,
+and thus enables `NormedSpace.exp_add_of_commute` to be used on `TrivSqZeroExt`.
+If the non-canonicity becomes problematic in future,
+we could keep the collection of instances behind an `open scoped`.
 
 ## Main results
 
@@ -20,21 +29,29 @@ For now, this file contains results about `exp` for this type.
 * `TrivSqZeroExt.snd_exp`
 * `TrivSqZeroExt.exp_inl`
 * `TrivSqZeroExt.exp_inr`
+* The $ℓ^1$ norm on `TrivSqZeroExt`:
+  * `TrivSqZeroExt.instL1SeminormedAddCommGroup`
+  * `TrivSqZeroExt.instL1SeminormedRing`
+  * `TrivSqZeroExt.instL1SeminormedCommRing`
+  * `TrivSqZeroExt.instL1BoundedSMul`
+  * `TrivSqZeroExt.instL1NormedAddCommGroup`
+  * `TrivSqZeroExt.instL1NormedRing`
+  * `TrivSqZeroExt.instL1NormedCommRing`
+  * `TrivSqZeroExt.instL1NormedSpace`
+  * `TrivSqZeroExt.instL1NormedAlgebra`
 
 ## TODO
 
-* Actually define a sensible norm on `TrivSqZeroExt R M`, so that we have access to lemmas
-  like `exp_add`.
 * Generalize more of these results to non-commutative `R`. In principle, under sufficient conditions
   we should expect
- `(exp 𝕜 x).snd = ∫ t in 0..1, exp 𝕜 (t • x.fst) • op (exp 𝕜 ((1 - t) • x.fst)) • x.snd`
+  `(exp 𝕜 x).snd = ∫ t in 0..1, exp 𝕜 (t • x.fst) • op (exp 𝕜 ((1 - t) • x.fst)) • x.snd`
   ([Physics.SE](https://physics.stackexchange.com/a/41671/185147), and
   https://link.springer.com/chapter/10.1007/978-3-540-44953-9_2).
 
 -/
 
 
-variable (𝕜 : Type*) {R M : Type*}
+variable (𝕜 : Type*) {S R M : Type*}
 
 local notation "tsze" => TrivSqZeroExt
 
@@ -176,4 +193,140 @@ end Field
 
 end Topology
 
+/-!
+### The $ℓ^1$ norm on the trivial square zero extension
+-/
+
+noncomputable section Seminormed
+
+section Ring
+variable [SeminormedCommRing S] [SeminormedRing R] [SeminormedAddCommGroup M]
+variable [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M]
+variable [BoundedSMul S R] [BoundedSMul S M] [BoundedSMul R M] [BoundedSMul Rᵐᵒᵖ M]
+variable [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M]
+
+instance instL1SeminormedAddCommGroup : SeminormedAddCommGroup (tsze R M) :=
+  inferInstanceAs <| SeminormedAddCommGroup (WithLp 1 <| R × M)
+
+example :
+    (TrivSqZeroExt.instUniformSpace : UniformSpace (tsze R M)) =
+    PseudoMetricSpace.toUniformSpace := rfl
+
+theorem norm_def (x : tsze R M) : ‖x‖ = ‖fst x‖ + ‖snd x‖ := by
+  rw [WithLp.prod_norm_eq_add (by norm_num)]
+  simp only [ENNReal.one_toReal, Real.rpow_one, div_one]
+  rfl
+
+theorem nnnorm_def (x : tsze R M) : ‖x‖₊ = ‖fst x‖₊ + ‖snd x‖₊ := by
+  ext; simp [norm_def]
+
+@[simp] theorem norm_inl (r : R) : ‖(inl r : tsze R M)‖ = ‖r‖ := by simp [norm_def]
+@[simp] theorem norm_inr (m : M) : ‖(inr m : tsze R M)‖ = ‖m‖ := by simp [norm_def]
+
+@[simp] theorem nnnorm_inl (r : R) : ‖(inl r : tsze R M)‖₊ = ‖r‖₊ := by simp [nnnorm_def]
+@[simp] theorem nnnorm_inr (m : M) : ‖(inr m : tsze R M)‖₊ = ‖m‖₊ := by simp [nnnorm_def]
+
+instance instL1SeminormedRing : SeminormedRing (tsze R M) where
+  norm_mul
+  | ⟨r₁, m₁⟩, ⟨r₂, m₂⟩ => by
+    dsimp
+    rw [norm_def, norm_def, norm_def, add_mul, mul_add, mul_add, snd_mul, fst_mul]
+    dsimp [fst, snd]
+    rw [add_assoc]
+    gcongr
+    · exact norm_mul_le _ _
+    refine (norm_add_le _ _).trans ?_
+    gcongr
+    · exact norm_smul_le _ _
+    refine (_root_.norm_smul_le _ _).trans ?_
+    rw [mul_comm, MulOpposite.norm_op]
+    exact le_add_of_nonneg_right <| by positivity
+  __ : SeminormedAddCommGroup (tsze R M) := inferInstance
+  __ : Ring (tsze R M) := inferInstance
+
+instance instL1BoundedSMul : BoundedSMul S (tsze R M) :=
+  inferInstanceAs <| BoundedSMul S (WithLp 1 <| R × M)
+
+instance [NormOneClass R] : NormOneClass (tsze R M) where
+  norm_one := by rw [norm_def, fst_one, snd_one, norm_zero, norm_one, add_zero]
+
+
+end Ring
+
+section CommRing
+
+variable [SeminormedCommRing R] [SeminormedAddCommGroup M]
+variable [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
+variable [BoundedSMul R M]
+
+instance instL1SeminormedCommRing : SeminormedCommRing (tsze R M) where
+  __ : CommRing (tsze R M) := inferInstance
+  __ : SeminormedRing (tsze R M) := inferInstance
+
+end CommRing
+
+end Seminormed
+
+noncomputable section Normed
+
+section Ring
+
+variable [NormedCommRing S] [NormedRing R] [NormedAddCommGroup M]
+variable [Algebra S R] [Module S M] [Module R M] [Module Rᵐᵒᵖ M]
+variable [BoundedSMul S R] [BoundedSMul S M] [BoundedSMul R M] [BoundedSMul Rᵐᵒᵖ M]
+variable [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower S R M] [IsScalarTower S Rᵐᵒᵖ M]
+
+instance instL1NormedAddCommGroup : NormedAddCommGroup (tsze R M) :=
+  inferInstanceAs <| NormedAddCommGroup (WithLp 1 <| R × M)
+
+instance instL1NormedRing : NormedRing (tsze R M) where
+  __ : NormedAddCommGroup (tsze R M) := inferInstance
+  __ : SeminormedRing (tsze R M) := inferInstance
+
+end Ring
+
+section CommRing
+
+variable [NormedCommRing R] [NormedAddCommGroup M]
+variable [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
+variable [BoundedSMul R M]
+
+instance instL1NormedCommRing : NormedCommRing (tsze R M) where
+  __ : CommRing (tsze R M) := inferInstance
+  __ : NormedRing (tsze R M) := inferInstance
+
+end CommRing
+
+section Algebra
+
+variable [NormedField 𝕜] [NormedRing R] [NormedAddCommGroup M]
+variable [NormedAlgebra 𝕜 R] [NormedSpace 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M]
+variable [BoundedSMul R M] [BoundedSMul Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
+variable [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M]
+
+instance instL1NormedSpace : NormedSpace 𝕜 (tsze R M) :=
+  inferInstanceAs <| NormedSpace 𝕜 (WithLp 1 <| R × M)
+
+instance instL1NormedAlgebra : NormedAlgebra 𝕜 (tsze R M) where
+  norm_smul_le := _root_.norm_smul_le
+
+end Algebra
+
+
+end Normed
+
+section
+
+variable [IsROrC 𝕜] [NormedRing R] [NormedAddCommGroup M]
+variable [NormedAlgebra 𝕜 R] [NormedSpace 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M]
+variable [BoundedSMul R M] [BoundedSMul Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
+variable [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M]
+variable [CompleteSpace R] [CompleteSpace M]
+
+-- Evidence that we have sufficient instances on `tsze R N` to make `exp_add_of_commute` usable
+example (a b : tsze R M) (h : Commute a b) : exp 𝕜 (a + b) = exp 𝕜 a * exp 𝕜 b :=
+  exp_add_of_commute h
+
+end
+
 end TrivSqZeroExt

These results are still true even if the exponential does not converge, because in those cases all the terms are zero.

The hasSum_fst_expSeries lemma has been dropped because it's a trivial consequence of the new fst_expSeries; the fact that the series are elementwise equal.

@@ -44,76 +44,70 @@ namespace TrivSqZeroExt
 
 section Topology
 
-variable [TopologicalSpace R] [TopologicalSpace M]
-
-/-- If `exp R x.fst` converges to `e` then `(exp R x).fst` converges to `e`. -/
-theorem hasSum_fst_expSeries [Field 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R] [Module R M]
-    [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
-    [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M]
-    [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) {e : R}
-    (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
-    HasSum (fun n => fst (expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
-  simpa [expSeries_apply_eq] using h
-#align triv_sq_zero_ext.has_sum_fst_exp_series TrivSqZeroExt.hasSum_fst_expSeries
+section not_charZero
+variable [Field 𝕜] [Ring R] [AddCommGroup M]
+  [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M]
+  [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M]
+  [TopologicalSpace R] [TopologicalSpace M]
+  [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M]
+
+@[simp] theorem fst_expSeries (x : tsze R M) (n : ℕ) :
+    fst (expSeries 𝕜 (tsze R M) n fun _ => x) = expSeries 𝕜 R n fun _ => x.fst := by
+  simp [expSeries_apply_eq]
+
+end not_charZero
+
+section Ring
+variable [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M]
+  [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M]
+  [SMulCommClass R Rᵐᵒᵖ M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M]
+  [TopologicalSpace R] [TopologicalSpace M]
+  [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M]
+
+theorem snd_expSeries_of_smul_comm
+    (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) (n : ℕ) :
+    snd (expSeries 𝕜 (tsze R M) (n + 1) fun _ => x) = (expSeries 𝕜 R n fun _ => x.fst) • x.snd := by
+  simp_rw [expSeries_apply_eq, snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast 𝕜 (n + 1),
+    smul_smul, smul_assoc, Nat.factorial_succ, Nat.pred_succ, Nat.cast_mul, mul_inv_rev,
+    inv_mul_cancel_right₀ ((Nat.cast_ne_zero (R := 𝕜)).mpr <| Nat.succ_ne_zero n)]
 
 /-- If `exp R x.fst` converges to `e` then `(exp R x).snd` converges to `e • x.snd`. -/
-theorem hasSum_snd_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M]
-    [Algebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M]
-    [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M]
-    [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M)
+theorem hasSum_snd_expSeries_of_smul_comm (x : tsze R M)
     (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) {e : R}
     (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
     HasSum (fun n => snd (expSeries 𝕜 (tsze R M) n fun _ => x)) (e • x.snd) := by
-  simp_rw [expSeries_apply_eq] at *
-  conv =>
-    congr
-    ext n
-    rw [snd_smul, snd_pow_of_smul_comm _ _ hx, nsmul_eq_smul_cast 𝕜 n, smul_smul, inv_mul_eq_div, ←
-      inv_div, ← smul_assoc]
-  apply HasSum.smul_const
   rw [← hasSum_nat_add_iff' 1]
-  rw [Finset.range_one, Finset.sum_singleton, Nat.cast_zero, div_zero, inv_zero, zero_smul,
-    sub_zero]
-  simp_rw [← Nat.succ_eq_add_one, Nat.pred_succ, Nat.factorial_succ, Nat.cast_mul, ←
-    Nat.succ_eq_add_one,
-    mul_div_cancel_left _ ((@Nat.cast_ne_zero 𝕜 _ _ _).mpr <| Nat.succ_ne_zero _)]
-  exact h
+  simp_rw [snd_expSeries_of_smul_comm _ _ hx]
+  simp_rw [expSeries_apply_eq] at *
+  rw [Finset.range_one, Finset.sum_singleton, Nat.factorial_zero, Nat.cast_one, pow_zero,
+    inv_one, one_smul, snd_one, sub_zero]
+  exact h.smul_const _
 #align triv_sq_zero_ext.has_sum_snd_exp_series_of_smul_comm TrivSqZeroExt.hasSum_snd_expSeries_of_smul_comm
 
 /-- If `exp R x.fst` converges to `e` then `exp R x` converges to `inl e + inr (e • x.snd)`. -/
-theorem hasSum_expSeries_of_smul_comm [Field 𝕜] [CharZero 𝕜] [Ring R] [AddCommGroup M] [Algebra 𝕜 R]
-    [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M] [Module 𝕜 M] [IsScalarTower 𝕜 R M]
-    [IsScalarTower 𝕜 Rᵐᵒᵖ M] [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M]
-    [ContinuousSMul Rᵐᵒᵖ M] (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
+theorem hasSum_expSeries_of_smul_comm
+    (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd)
     {e : R} (h : HasSum (fun n => expSeries 𝕜 R n fun _ => x.fst) e) :
     HasSum (fun n => expSeries 𝕜 (tsze R M) n fun _ => x) (inl e + inr (e • x.snd)) := by
+  have : HasSum (fun n => fst (expSeries 𝕜 (tsze R M) n fun _ => x)) e := by
+    simpa [fst_expSeries] using h
   simpa only [inl_fst_add_inr_snd_eq] using
-    (hasSum_inl _ <| hasSum_fst_expSeries 𝕜 x h).add
-      (hasSum_inr _ <| hasSum_snd_expSeries_of_smul_comm 𝕜 x hx h)
+    (hasSum_inl _ <| this).add (hasSum_inr _ <| hasSum_snd_expSeries_of_smul_comm 𝕜 x hx h)
 #align triv_sq_zero_ext.has_sum_exp_series_of_smul_comm TrivSqZeroExt.hasSum_expSeries_of_smul_comm
 
-end Topology
-
-section NormedRing
-
-variable [IsROrC 𝕜] [NormedRing R] [AddCommGroup M]
-
-variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [SMulCommClass R Rᵐᵒᵖ M]
-
-variable [Module 𝕜 M] [IsScalarTower 𝕜 R M] [IsScalarTower 𝕜 Rᵐᵒᵖ M]
-
-variable [TopologicalSpace M] [TopologicalRing R]
-
-variable [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M]
-
-variable [CompleteSpace R] [T2Space R] [T2Space M]
+variable [T2Space R] [T2Space M]
 
 theorem exp_def_of_smul_comm (x : tsze R M) (hx : MulOpposite.op x.fst • x.snd = x.fst • x.snd) :
     exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) := by
   simp_rw [exp, FormalMultilinearSeries.sum]
-  refine' (hasSum_expSeries_of_smul_comm 𝕜 x hx _).tsum_eq
-  exact expSeries_hasSum_exp _
-#align triv_sq_zero_ext.exp_def_of_smul_comm TrivSqZeroExt.exp_def_of_smul_comm
+  by_cases h : Summable (fun (n : ℕ) => (expSeries 𝕜 R n) fun x_1 ↦ fst x)
+  · refine (hasSum_expSeries_of_smul_comm 𝕜 x hx ?_).tsum_eq
+    exact h.hasSum
+  · rw [tsum_eq_zero_of_not_summable h, zero_smul, inr_zero, inl_zero, zero_add,
+      tsum_eq_zero_of_not_summable]
+    simp_rw [← fst_expSeries] at h
+    refine mt ?_ h
+    exact (Summable.map · (TrivSqZeroExt.fstHom 𝕜 R M).toLinearMap continuous_fst)
 
 @[simp]
 theorem exp_inl (x : R) : exp 𝕜 (inl x : tsze R M) = inl (exp 𝕜 x) := by
@@ -127,21 +121,16 @@ theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m := by
   · rw [snd_inr, fst_inr, MulOpposite.op_zero, zero_smul, zero_smul]
 #align triv_sq_zero_ext.exp_inr TrivSqZeroExt.exp_inr
 
-end NormedRing
+end Ring
 
-section NormedCommRing
+section CommRing
+variable [Field 𝕜] [CharZero 𝕜] [CommRing R] [AddCommGroup M]
+  [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M]
+  [IsCentralScalar R M] [IsScalarTower 𝕜 R M]
+  [TopologicalSpace R] [TopologicalSpace M]
+  [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M]
 
-variable [IsROrC 𝕜] [NormedCommRing R] [AddCommGroup M]
-
-variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
-
-variable [Module 𝕜 M] [IsScalarTower 𝕜 R M]
-
-variable [TopologicalSpace M] [TopologicalRing R]
-
-variable [TopologicalAddGroup M] [ContinuousSMul R M]
-
-variable [CompleteSpace R] [T2Space R] [T2Space M]
+variable [T2Space R] [T2Space M]
 
 theorem exp_def (x : tsze R M) : exp 𝕜 x = inl (exp 𝕜 x.fst) + inr (exp 𝕜 x.fst • x.snd) :=
   exp_def_of_smul_comm 𝕜 x (op_smul_eq_smul _ _)
@@ -164,21 +153,16 @@ theorem eq_smul_exp_of_invertible (x : tsze R M) [Invertible x.fst] :
     smul_smul, mul_invOf_self, one_smul, inl_fst_add_inr_snd_eq]
 #align triv_sq_zero_ext.eq_smul_exp_of_invertible TrivSqZeroExt.eq_smul_exp_of_invertible
 
-end NormedCommRing
-
-section NormedField
+end CommRing
 
-variable [IsROrC 𝕜] [NormedField R] [AddCommGroup M]
+section Field
+variable [Field 𝕜] [CharZero 𝕜] [Field R] [AddCommGroup M]
+  [Algebra 𝕜 R] [Module 𝕜 M] [Module R M] [Module Rᵐᵒᵖ M]
+  [IsCentralScalar R M] [IsScalarTower 𝕜 R M]
+  [TopologicalSpace R] [TopologicalSpace M]
+  [TopologicalRing R] [TopologicalAddGroup M] [ContinuousSMul R M] [ContinuousSMul Rᵐᵒᵖ M]
 
-variable [NormedAlgebra 𝕜 R] [Module R M] [Module Rᵐᵒᵖ M] [IsCentralScalar R M]
-
-variable [Module 𝕜 M] [IsScalarTower 𝕜 R M]
-
-variable [TopologicalSpace M] [TopologicalRing R]
-
-variable [TopologicalAddGroup M] [ContinuousSMul R M]
-
-variable [CompleteSpace R] [T2Space R] [T2Space M]
+variable [T2Space R] [T2Space M]
 
 /-- More convenient version of `TrivSqZeroExt.eq_smul_exp_of_invertible` for when `R` is a
 field. -/
@@ -188,6 +172,8 @@ theorem eq_smul_exp_of_ne_zero (x : tsze R M) (hx : x.fst ≠ 0) :
   eq_smul_exp_of_invertible _ _
 #align triv_sq_zero_ext.eq_smul_exp_of_ne_zero TrivSqZeroExt.eq_smul_exp_of_ne_zero
 
-end NormedField
+end Field
+
+end Topology
 
 end TrivSqZeroExt

Per discussion at zulip

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

@@ -38,6 +38,8 @@ variable (𝕜 : Type*) {R M : Type*}
 
 local notation "tsze" => TrivSqZeroExt
 
+open NormedSpace -- For `exp`.
+
 namespace TrivSqZeroExt
 
 section Topology

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

This has nice performance benefits.

@@ -34,7 +34,7 @@ For now, this file contains results about `exp` for this type.
 -/
 
 
-variable (𝕜 : Type _) {R M : Type _}
+variable (𝕜 : Type*) {R M : Type*}
 
 local notation "tsze" => TrivSqZeroExt
 

• depends on: #5966

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

@@ -2,16 +2,13 @@
 Copyright (c) 2023 Eric Wieser. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eric Wieser
-
-! This file was ported from Lean 3 source module analysis.normed_space.triv_sq_zero_ext
-! leanprover-community/mathlib commit 88a563b158f59f2983cfad685664da95502e8cdd
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.NormedSpace.Basic
 import Mathlib.Analysis.NormedSpace.Exponential
 import Mathlib.Topology.Instances.TrivSqZeroExt
 
+#align_import analysis.normed_space.triv_sq_zero_ext from "leanprover-community/mathlib"@"88a563b158f59f2983cfad685664da95502e8cdd"
+
 /-!
 # Results on `TrivSqZeroExt R M` related to the norm