algebra.lie.nilpotent ⟷ Mathlib.Algebra.Lie.Nilpotent

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

@@ -7,7 +7,7 @@ import Algebra.Lie.Solvable
 import Algebra.Lie.Quotient
 import Algebra.Lie.Normalizer
 import LinearAlgebra.Eigenspace.Basic
-import RingTheory.Nilpotent
+import RingTheory.Nilpotent.Defs
 
 #align_import algebra.lie.nilpotent from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
 

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

@@ -166,7 +166,7 @@ theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries
   by
   constructor <;> intro h
   · erw [eq_bot_iff, LieSubmodule.lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn, h.trivial]; simp
-  · rw [LieSubmodule.eq_bot_iff] at h ; apply is_trivial.mk; intro x m; apply h
+  · rw [LieSubmodule.eq_bot_iff] at h; apply is_trivial.mk; intro x m; apply h
     apply LieSubmodule.subset_lieSpan; use x, m; rfl
 #align lie_module.trivial_iff_lower_central_eq_bot LieModule.trivial_iff_lower_central_eq_bot
 -/
@@ -294,7 +294,7 @@ theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxT
   suffices lowerCentralSeries R L M k ≤ N
     by
     replace this := LieSubmodule.mono_lie_right _ _ ⊤ (le_trans this h₁)
-    rwa [ideal_oper_max_triv_submodule_eq_bot, le_bot_iff] at this 
+    rwa [ideal_oper_max_triv_submodule_eq_bot, le_bot_iff] at this
   rw [← LieSubmodule.Quotient.map_mk'_eq_bot_le, ← le_bot_iff, ← hk]
   exact map_lower_central_series_le k (LieSubmodule.Quotient.mk' N)
 #align lie_module.nilpotent_of_nilpotent_quotient LieModule.nilpotentOfNilpotentQuotient
@@ -364,7 +364,7 @@ theorem lowerCentralSeriesLast_le_max_triv :
   cases' h : nilpotency_length R L M with k
   · exact bot_le
   · rw [le_max_triv_iff_bracket_eq_bot]
-    rw [nilpotency_length_eq_succ_iff, lowerCentralSeries_succ] at h 
+    rw [nilpotency_length_eq_succ_iff, lowerCentralSeries_succ] at h
     exact h.1
 #align lie_module.lower_central_series_last_le_max_triv LieModule.lowerCentralSeriesLast_le_max_triv
 -/
@@ -375,9 +375,9 @@ theorem nontrivial_lowerCentralSeriesLast [Nontrivial M] [IsNilpotent R L M] :
   by
   rw [LieSubmodule.nontrivial_iff_ne_bot, lower_central_series_last]
   cases h : nilpotency_length R L M
-  · rw [nilpotency_length_eq_zero_iff, ← not_nontrivial_iff_subsingleton] at h 
+  · rw [nilpotency_length_eq_zero_iff, ← not_nontrivial_iff_subsingleton] at h
     contradiction
-  · rw [nilpotency_length_eq_succ_iff] at h 
+  · rw [nilpotency_length_eq_succ_iff] at h
     exact h.2
 #align lie_module.nontrivial_lower_central_series_last LieModule.nontrivial_lowerCentralSeriesLast
 -/
@@ -559,7 +559,7 @@ theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
       g '' {m | ∃ (x : L) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m} =
         {m | ∃ (x : L₂) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m}
       by
-      simp only [← LieSubmodule.mem_coeSubmodule] at this 
+      simp only [← LieSubmodule.mem_coeSubmodule] at this
       simp [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.lieIdeal_oper_eq_linear_span',
         Submodule.map_span, -Submodule.span_image, this]
     ext m₂
@@ -672,10 +672,10 @@ theorem coe_lowerCentralSeries_ideal_quot_eq {I : LieIdeal R L} (k : ℕ) :
     ext x
     constructor
     · rintro ⟨⟨y, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩
-      erw [← LieSubmodule.mem_coeSubmodule, ih, LieSubmodule.mem_coeSubmodule] at hz 
+      erw [← LieSubmodule.mem_coeSubmodule, ih, LieSubmodule.mem_coeSubmodule] at hz
       exact ⟨⟨LieSubmodule.Quotient.mk y, LieSubmodule.mem_top _⟩, ⟨z, hz⟩, rfl⟩
     · rintro ⟨⟨⟨y⟩, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩
-      erw [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.mem_coeSubmodule] at hz 
+      erw [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.mem_coeSubmodule] at hz
       exact ⟨⟨y, LieSubmodule.mem_top _⟩, ⟨z, hz⟩, rfl⟩
 #align coe_lower_central_series_ideal_quot_eq coe_lowerCentralSeries_ideal_quot_eq
 -/

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

@@ -3,11 +3,11 @@ Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
-import Mathbin.Algebra.Lie.Solvable
-import Mathbin.Algebra.Lie.Quotient
-import Mathbin.Algebra.Lie.Normalizer
-import Mathbin.LinearAlgebra.Eigenspace.Basic
-import Mathbin.RingTheory.Nilpotent
+import Algebra.Lie.Solvable
+import Algebra.Lie.Quotient
+import Algebra.Lie.Normalizer
+import LinearAlgebra.Eigenspace.Basic
+import RingTheory.Nilpotent
 
 #align_import algebra.lie.nilpotent from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
 

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

@@ -262,7 +262,6 @@ theorem exists_forall_pow_toEndomorphism_eq_zero [hM : IsNilpotent R L M] :
 #align lie_module.nilpotent_endo_of_nilpotent_module LieModule.exists_forall_pow_toEndomorphism_eq_zero
 -/
 
-#print LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent /-
 /-- For a nilpotent Lie module, the weight space of the 0 weight is the whole module.
 
 This result will be used downstream to show that weight spaces are Lie submodules, at which time
@@ -278,7 +277,6 @@ theorem iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
   use k; rw [hk]
   exact LinearMap.zero_apply m
 #align lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
--/
 
 #print LieModule.nilpotentOfNilpotentQuotient /-
 /-- If the quotient of a Lie module `M` by a Lie submodule on which the Lie algebra acts trivially
@@ -645,13 +643,11 @@ theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
 #align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra
 -/
 
-#print LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent /-
 /-- See also `lie_algebra.zero_root_space_eq_top_of_nilpotent`. -/
 theorem LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L] :
     (⨅ x : L, (ad R L x).maximalGeneralizedEigenspace 0) = ⊤ :=
   LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L
 #align lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
--/
 
 -- TODO Generalise the below to Lie modules if / when we define morphisms, equivs of Lie modules
 -- covering a Lie algebra morphism of (possibly different) Lie algebras.

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -250,8 +250,8 @@ instance (priority := 100) trivialIsNilpotent [IsTrivial L M] : IsNilpotent R L
 #align lie_module.trivial_is_nilpotent LieModule.trivialIsNilpotent
 -/
 
-#print LieModule.nilpotent_endo_of_nilpotent_module /-
-theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
+#print LieModule.exists_forall_pow_toEndomorphism_eq_zero /-
+theorem exists_forall_pow_toEndomorphism_eq_zero [hM : IsNilpotent R L M] :
     ∃ k : ℕ, ∀ x : L, toEndomorphism R L M x ^ k = 0 :=
   by
   obtain ⟨k, hM⟩ := hM
@@ -259,7 +259,7 @@ theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
   intro x; ext m
   rw [LinearMap.pow_apply, LinearMap.zero_apply, ← @LieSubmodule.mem_bot R L M, ← hM]
   exact iterate_to_endomorphism_mem_lower_central_series R L M x m k
-#align lie_module.nilpotent_endo_of_nilpotent_module LieModule.nilpotent_endo_of_nilpotent_module
+#align lie_module.nilpotent_endo_of_nilpotent_module LieModule.exists_forall_pow_toEndomorphism_eq_zero
 -/
 
 #print LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent /-
@@ -641,7 +641,7 @@ open LieAlgebra
 #print LieAlgebra.nilpotent_ad_of_nilpotent_algebra /-
 theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
     ∃ k : ℕ, ∀ x : L, ad R L x ^ k = 0 :=
-  LieModule.nilpotent_endo_of_nilpotent_module R L L
+  LieModule.exists_forall_pow_toEndomorphism_eq_zero R L L
 #align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra
 -/
 

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module algebra.lie.nilpotent
-! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.Lie.Solvable
 import Mathbin.Algebra.Lie.Quotient
@@ -14,6 +9,8 @@ import Mathbin.Algebra.Lie.Normalizer
 import Mathbin.LinearAlgebra.Eigenspace.Basic
 import Mathbin.RingTheory.Nilpotent
 
+#align_import algebra.lie.nilpotent from "leanprover-community/mathlib"@"fd4551cfe4b7484b81c2c9ba3405edae27659676"
+
 /-!
 # Nilpotent Lie algebras
 

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

@@ -48,6 +48,7 @@ variable (k : ℕ) (N : LieSubmodule R L M)
 
 namespace LieSubmodule
 
+#print LieSubmodule.lcs /-
 /-- A generalisation of the lower central series. The zeroth term is a specified Lie submodule of
 a Lie module. In the case when we specify the top ideal `⊤` of the Lie algebra, regarded as a Lie
 module over itself, we get the usual lower central series of a Lie algebra.
@@ -62,16 +63,21 @@ See also `lie_module.lower_central_series_eq_lcs_comap` and
 def lcs : LieSubmodule R L M → LieSubmodule R L M :=
   (fun N => ⁅(⊤ : LieIdeal R L), N⁆)^[k]
 #align lie_submodule.lcs LieSubmodule.lcs
+-/
 
+#print LieSubmodule.lcs_zero /-
 @[simp]
 theorem lcs_zero (N : LieSubmodule R L M) : N.lcs 0 = N :=
   rfl
 #align lie_submodule.lcs_zero LieSubmodule.lcs_zero
+-/
 
+#print LieSubmodule.lcs_succ /-
 @[simp]
 theorem lcs_succ : N.lcs (k + 1) = ⁅(⊤ : LieIdeal R L), N.lcs k⁆ :=
   Function.iterate_succ_apply' (fun N' => ⁅⊤, N'⁆) k N
 #align lie_submodule.lcs_succ LieSubmodule.lcs_succ
+-/
 
 end LieSubmodule
 
@@ -79,21 +85,27 @@ namespace LieModule
 
 variable (R L M)
 
+#print LieModule.lowerCentralSeries /-
 /-- The lower central series of Lie submodules of a Lie module. -/
 def lowerCentralSeries : LieSubmodule R L M :=
   (⊤ : LieSubmodule R L M).lcs k
 #align lie_module.lower_central_series LieModule.lowerCentralSeries
+-/
 
+#print LieModule.lowerCentralSeries_zero /-
 @[simp]
 theorem lowerCentralSeries_zero : lowerCentralSeries R L M 0 = ⊤ :=
   rfl
 #align lie_module.lower_central_series_zero LieModule.lowerCentralSeries_zero
+-/
 
+#print LieModule.lowerCentralSeries_succ /-
 @[simp]
 theorem lowerCentralSeries_succ :
     lowerCentralSeries R L M (k + 1) = ⁅(⊤ : LieIdeal R L), lowerCentralSeries R L M k⁆ :=
   (⊤ : LieSubmodule R L M).lcs_succ k
 #align lie_module.lower_central_series_succ LieModule.lowerCentralSeries_succ
+-/
 
 end LieModule
 
@@ -103,12 +115,14 @@ open LieModule
 
 variable {R L M}
 
+#print LieSubmodule.lcs_le_self /-
 theorem lcs_le_self : N.lcs k ≤ N := by
   induction' k with k ih
   · simp
   · simp only [lcs_succ]
     exact (LieSubmodule.mono_lie_right _ _ ⊤ ih).trans (N.lie_le_right ⊤)
 #align lie_submodule.lcs_le_self LieSubmodule.lcs_le_self
+-/
 
 #print LieSubmodule.lowerCentralSeries_eq_lcs_comap /-
 theorem lowerCentralSeries_eq_lcs_comap : lowerCentralSeries R L N k = (N.lcs k).comap N.incl :=
@@ -137,6 +151,7 @@ namespace LieModule
 
 variable (R L M)
 
+#print LieModule.antitone_lowerCentralSeries /-
 theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M :=
   by
   intro l k
@@ -147,7 +162,9 @@ theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M :=
       exact (LieSubmodule.mono_lie_right _ _ ⊤ (ih hk)).trans (LieSubmodule.lie_le_right _ _)
     · exact hk.symm ▸ le_rfl
 #align lie_module.antitone_lower_central_series LieModule.antitone_lowerCentralSeries
+-/
 
+#print LieModule.trivial_iff_lower_central_eq_bot /-
 theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries R L M 1 = ⊥ :=
   by
   constructor <;> intro h
@@ -155,6 +172,7 @@ theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries
   · rw [LieSubmodule.eq_bot_iff] at h ; apply is_trivial.mk; intro x m; apply h
     apply LieSubmodule.subset_lieSpan; use x, m; rfl
 #align lie_module.trivial_iff_lower_central_eq_bot LieModule.trivial_iff_lower_central_eq_bot
+-/
 
 #print LieModule.iterate_toEndomorphism_mem_lowerCentralSeries /-
 theorem iterate_toEndomorphism_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ) :
@@ -186,6 +204,7 @@ variable (R L M)
 
 open LieAlgebra
 
+#print LieModule.derivedSeries_le_lowerCentralSeries /-
 theorem derivedSeries_le_lowerCentralSeries (k : ℕ) :
     derivedSeries R L k ≤ lowerCentralSeries R L L k :=
   by
@@ -196,17 +215,22 @@ theorem derivedSeries_le_lowerCentralSeries (k : ℕ) :
     rw [derived_series_def, derived_series_of_ideal_succ, lowerCentralSeries_succ]
     exact LieSubmodule.mono_lie _ _ _ _ h' h
 #align lie_module.derived_series_le_lower_central_series LieModule.derivedSeries_le_lowerCentralSeries
+-/
 
+#print LieModule.IsNilpotent /-
 /-- A Lie module is nilpotent if its lower central series reaches 0 (in a finite number of
 steps). -/
 class IsNilpotent : Prop where
   nilpotent : ∃ k, lowerCentralSeries R L M k = ⊥
 #align lie_module.is_nilpotent LieModule.IsNilpotent
+-/
 
+#print LieModule.isNilpotent_iff /-
 /-- See also `lie_module.is_nilpotent_iff_exists_ucs_eq_top`. -/
 theorem isNilpotent_iff : IsNilpotent R L M ↔ ∃ k, lowerCentralSeries R L M k = ⊥ :=
   ⟨fun h => h.nilpotent, fun h => ⟨h⟩⟩
 #align lie_module.is_nilpotent_iff LieModule.isNilpotent_iff
+-/
 
 variable {R L M}
 
@@ -223,9 +247,11 @@ theorem LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot (N : LieSubmodule R L M)
 
 variable (R L M)
 
+#print LieModule.trivialIsNilpotent /-
 instance (priority := 100) trivialIsNilpotent [IsTrivial L M] : IsNilpotent R L M :=
   ⟨by use 1; change ⁅⊤, ⊤⁆ = ⊥; simp⟩
 #align lie_module.trivial_is_nilpotent LieModule.trivialIsNilpotent
+-/
 
 #print LieModule.nilpotent_endo_of_nilpotent_module /-
 theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
@@ -279,6 +305,7 @@ theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxT
 #align lie_module.nilpotent_of_nilpotent_quotient LieModule.nilpotentOfNilpotentQuotient
 -/
 
+#print LieModule.nilpotencyLength /-
 /-- Given a nilpotent Lie module `M` with lower central series `M = C₀ ≥ C₁ ≥ ⋯ ≥ Cₖ = ⊥`, this is
 the natural number `k` (the number of inclusions).
 
@@ -286,7 +313,9 @@ For a non-nilpotent module, we use the junk value 0. -/
 noncomputable def nilpotencyLength : ℕ :=
   sInf {k | lowerCentralSeries R L M k = ⊥}
 #align lie_module.nilpotency_length LieModule.nilpotencyLength
+-/
 
+#print LieModule.nilpotencyLength_eq_zero_iff /-
 theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
     nilpotencyLength R L M = 0 ↔ Subsingleton M :=
   by
@@ -302,7 +331,9 @@ theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
   rw [Nat.sInf_eq_zero]
   exact Or.inl h
 #align lie_module.nilpotency_length_eq_zero_iff LieModule.nilpotencyLength_eq_zero_iff
+-/
 
+#print LieModule.nilpotencyLength_eq_succ_iff /-
 theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
     nilpotencyLength R L M = k + 1 ↔
       lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥ :=
@@ -315,6 +346,7 @@ theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
     exact eq_bot_iff.mpr (h₁ ▸ antitone_lower_central_series R L M h₁₂)
   exact Nat.sInf_upward_closed_eq_succ_iff hs k
 #align lie_module.nilpotency_length_eq_succ_iff LieModule.nilpotencyLength_eq_succ_iff
+-/
 
 #print LieModule.lowerCentralSeriesLast /-
 /-- Given a non-trivial nilpotent Lie module `M` with lower central series
@@ -329,6 +361,7 @@ noncomputable def lowerCentralSeriesLast : LieSubmodule R L M :=
 #align lie_module.lower_central_series_last LieModule.lowerCentralSeriesLast
 -/
 
+#print LieModule.lowerCentralSeriesLast_le_max_triv /-
 theorem lowerCentralSeriesLast_le_max_triv :
     lowerCentralSeriesLast R L M ≤ maxTrivSubmodule R L M :=
   by
@@ -339,7 +372,9 @@ theorem lowerCentralSeriesLast_le_max_triv :
     rw [nilpotency_length_eq_succ_iff, lowerCentralSeries_succ] at h 
     exact h.1
 #align lie_module.lower_central_series_last_le_max_triv LieModule.lowerCentralSeriesLast_le_max_triv
+-/
 
+#print LieModule.nontrivial_lowerCentralSeriesLast /-
 theorem nontrivial_lowerCentralSeriesLast [Nontrivial M] [IsNilpotent R L M] :
     Nontrivial (lowerCentralSeriesLast R L M) :=
   by
@@ -350,6 +385,7 @@ theorem nontrivial_lowerCentralSeriesLast [Nontrivial M] [IsNilpotent R L M] :
   · rw [nilpotency_length_eq_succ_iff] at h 
     exact h.2
 #align lie_module.nontrivial_lower_central_series_last LieModule.nontrivial_lowerCentralSeriesLast
+-/
 
 #print LieModule.nontrivial_max_triv_of_isNilpotent /-
 theorem nontrivial_max_triv_of_isNilpotent [Nontrivial M] [IsNilpotent R L M] :
@@ -518,8 +554,7 @@ variable {f : L →ₗ⁅R⁆ L₂} {g : M →ₗ[R] M₂}
 
 variable (hf : Surjective f) (hg : Surjective g) (hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆)
 
-include hf hg hfg
-
+#print Function.Surjective.lieModule_lcs_map_eq /-
 theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
     (lowerCentralSeries R L M k : Submodule R M).map g = lowerCentralSeries R L₂ M₂ k :=
   by
@@ -540,7 +575,9 @@ theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
       obtain ⟨y, rfl⟩ := hf x
       exact ⟨⁅y, n⁆, ⟨y, n, hn, rfl⟩, (hfg y n).symm⟩
 #align function.surjective.lie_module_lcs_map_eq Function.Surjective.lieModule_lcs_map_eq
+-/
 
+#print Function.Surjective.lieModuleIsNilpotent /-
 theorem Function.Surjective.lieModuleIsNilpotent [IsNilpotent R L M] : IsNilpotent R L₂ M₂ :=
   by
   obtain ⟨k, hk⟩ := id (by infer_instance : IsNilpotent R L M)
@@ -548,9 +585,9 @@ theorem Function.Surjective.lieModuleIsNilpotent [IsNilpotent R L M] : IsNilpote
   rw [← LieSubmodule.coe_toSubmodule_eq_iff] at hk ⊢
   simp [← hf.lie_module_lcs_map_eq hg hfg k, hk]
 #align function.surjective.lie_module_is_nilpotent Function.Surjective.lieModuleIsNilpotent
+-/
 
-omit hf hg hfg
-
+#print Equiv.lieModule_isNilpotent_iff /-
 theorem Equiv.lieModule_isNilpotent_iff (f : L ≃ₗ⁅R⁆ L₂) (g : M ≃ₗ[R] M₂)
     (hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆) : IsNilpotent R L M ↔ IsNilpotent R L₂ M₂ :=
   by
@@ -562,6 +599,7 @@ theorem Equiv.lieModule_isNilpotent_iff (f : L ≃ₗ⁅R⁆ L₂) (g : M ≃ₗ
     rw [LinearEquiv.coe_coe, LieEquiv.coe_to_lieHom, ← g.symm_apply_apply ⁅f.symm x, g.symm m⁆, ←
       hfg, f.apply_symm_apply, g.apply_symm_apply]
 #align equiv.lie_module_is_nilpotent_iff Equiv.lieModule_isNilpotent_iff
+-/
 
 #print LieModule.isNilpotent_of_top_iff /-
 @[simp]
@@ -603,10 +641,12 @@ abbrev LieAlgebra.IsNilpotent (R : Type u) (L : Type v) [CommRing R] [LieRing L]
 
 open LieAlgebra
 
+#print LieAlgebra.nilpotent_ad_of_nilpotent_algebra /-
 theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
     ∃ k : ℕ, ∀ x : L, ad R L x ^ k = 0 :=
   LieModule.nilpotent_endo_of_nilpotent_module R L L
 #align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra
+-/
 
 #print LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent /-
 /-- See also `lie_algebra.zero_root_space_eq_top_of_nilpotent`. -/
@@ -662,6 +702,7 @@ theorem LieModule.coe_lowerCentralSeries_ideal_le {I : LieIdeal R L} (k : ℕ) :
 #align lie_module.coe_lower_central_series_ideal_le LieModule.coe_lowerCentralSeries_ideal_le
 -/
 
+#print LieAlgebra.nilpotent_of_nilpotent_quotient /-
 /-- A central extension of nilpotent Lie algebras is nilpotent. -/
 theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h₁ : I ≤ center R L)
     (h₂ : IsNilpotent R (L ⧸ I)) : IsNilpotent R L :=
@@ -672,6 +713,7 @@ theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h₁ : I
   use k
   simp [← LieSubmodule.coe_toSubmodule_eq_iff, coe_lowerCentralSeries_ideal_quot_eq, hk]
 #align lie_algebra.nilpotent_of_nilpotent_quotient LieAlgebra.nilpotent_of_nilpotent_quotient
+-/
 
 #print LieAlgebra.non_trivial_center_of_isNilpotent /-
 theorem LieAlgebra.non_trivial_center_of_isNilpotent [Nontrivial L] [IsNilpotent R L] :
@@ -680,6 +722,7 @@ theorem LieAlgebra.non_trivial_center_of_isNilpotent [Nontrivial L] [IsNilpotent
 #align lie_algebra.non_trivial_center_of_is_nilpotent LieAlgebra.non_trivial_center_of_isNilpotent
 -/
 
+#print LieIdeal.map_lowerCentralSeries_le /-
 theorem LieIdeal.map_lowerCentralSeries_le (k : ℕ) {f : L →ₗ⁅R⁆ L'} :
     LieIdeal.map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k :=
   by
@@ -688,7 +731,9 @@ theorem LieIdeal.map_lowerCentralSeries_le (k : ℕ) {f : L →ₗ⁅R⁆ L'} :
   · simp only [LieModule.lowerCentralSeries_succ]
     exact le_trans (LieIdeal.map_bracket_le f) (LieSubmodule.mono_lie _ _ _ _ le_top ih)
 #align lie_ideal.map_lower_central_series_le LieIdeal.map_lowerCentralSeries_le
+-/
 
+#print LieIdeal.lowerCentralSeries_map_eq /-
 theorem LieIdeal.lowerCentralSeries_map_eq (k : ℕ) {f : L →ₗ⁅R⁆ L'} (h : Function.Surjective f) :
     LieIdeal.map f (lowerCentralSeries R L L k) = lowerCentralSeries R L' L' k :=
   by
@@ -700,7 +745,9 @@ theorem LieIdeal.lowerCentralSeries_map_eq (k : ℕ) {f : L →ₗ⁅R⁆ L'} (h
   · simp only [LieModule.lowerCentralSeries_zero]; exact h'
   · simp only [LieModule.lowerCentralSeries_succ, LieIdeal.map_bracket_eq f h, ih, h']
 #align lie_ideal.lower_central_series_map_eq LieIdeal.lowerCentralSeries_map_eq
+-/
 
+#print Function.Injective.lieAlgebra_isNilpotent /-
 theorem Function.Injective.lieAlgebra_isNilpotent [h₁ : IsNilpotent R L'] {f : L →ₗ⁅R⁆ L'}
     (h₂ : Function.Injective f) : IsNilpotent R L :=
   {
@@ -710,7 +757,9 @@ theorem Function.Injective.lieAlgebra_isNilpotent [h₁ : IsNilpotent R L'] {f :
       apply LieIdeal.bot_of_map_eq_bot h₂; rw [eq_bot_iff, ← hk]
       apply LieIdeal.map_lowerCentralSeries_le }
 #align function.injective.lie_algebra_is_nilpotent Function.Injective.lieAlgebra_isNilpotent
+-/
 
+#print Function.Surjective.lieAlgebra_isNilpotent /-
 theorem Function.Surjective.lieAlgebra_isNilpotent [h₁ : IsNilpotent R L] {f : L →ₗ⁅R⁆ L'}
     (h₂ : Function.Surjective f) : IsNilpotent R L' :=
   {
@@ -720,6 +769,7 @@ theorem Function.Surjective.lieAlgebra_isNilpotent [h₁ : IsNilpotent R L] {f :
       rw [← LieIdeal.lowerCentralSeries_map_eq k h₂, hk]
       simp only [LieIdeal.map_eq_bot_iff, bot_le] }
 #align function.surjective.lie_algebra_is_nilpotent Function.Surjective.lieAlgebra_isNilpotent
+-/
 
 #print LieEquiv.nilpotent_iff_equiv_nilpotent /-
 theorem LieEquiv.nilpotent_iff_equiv_nilpotent (e : L ≃ₗ⁅R⁆ L') :
@@ -737,6 +787,7 @@ theorem LieHom.isNilpotent_range [IsNilpotent R L] (f : L →ₗ⁅R⁆ L') : Is
 #align lie_hom.is_nilpotent_range LieHom.isNilpotent_range
 -/
 
+#print LieAlgebra.isNilpotent_range_ad_iff /-
 /-- Note that this result is not quite a special case of
 `lie_module.is_nilpotent_range_to_endomorphism_iff` which concerns nilpotency of the
 `(ad R L).range`-module `L`, whereas this result concerns nilpotency of the `(ad R L).range`-module
@@ -752,6 +803,7 @@ theorem LieAlgebra.isNilpotent_range_ad_iff : IsNilpotent R (ad R L).range ↔ I
   · intro h
     exact (ad R L).isNilpotent_range
 #align lie_algebra.is_nilpotent_range_ad_iff LieAlgebra.isNilpotent_range_ad_iff
+-/
 
 instance [h : LieAlgebra.IsNilpotent R L] : LieAlgebra.IsNilpotent R (⊤ : LieSubalgebra R L) :=
   LieSubalgebra.topEquiv.nilpotent_iff_equiv_nilpotent.mpr h
@@ -768,6 +820,7 @@ variable (M : Type _) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModu
 
 variable (k : ℕ)
 
+#print LieIdeal.lcs /-
 /-- Given a Lie module `M` over a Lie algebra `L` together with an ideal `I` of `L`, this is the
 lower central series of `M` as an `I`-module. The advantage of using this definition instead of
 `lie_module.lower_central_series R I M` is that its terms are Lie submodules of `M` as an
@@ -777,21 +830,29 @@ See also `lie_ideal.coe_lcs_eq`. -/
 def lcs : LieSubmodule R L M :=
   ((fun N => ⁅I, N⁆)^[k]) ⊤
 #align lie_ideal.lcs LieIdeal.lcs
+-/
 
+#print LieIdeal.lcs_zero /-
 @[simp]
 theorem lcs_zero : I.lcs M 0 = ⊤ :=
   rfl
 #align lie_ideal.lcs_zero LieIdeal.lcs_zero
+-/
 
+#print LieIdeal.lcs_succ /-
 @[simp]
 theorem lcs_succ : I.lcs M (k + 1) = ⁅I, I.lcs M k⁆ :=
   Function.iterate_succ_apply' (fun N => ⁅I, N⁆) k ⊤
 #align lie_ideal.lcs_succ LieIdeal.lcs_succ
+-/
 
+#print LieIdeal.lcs_top /-
 theorem lcs_top : (⊤ : LieIdeal R L).lcs M k = lowerCentralSeries R L M k :=
   rfl
 #align lie_ideal.lcs_top LieIdeal.lcs_top
+-/
 
+#print LieIdeal.coe_lcs_eq /-
 theorem coe_lcs_eq : (I.lcs M k : Submodule R M) = lowerCentralSeries R I M k :=
   by
   induction' k with k ih
@@ -807,6 +868,7 @@ theorem coe_lcs_eq : (I.lcs M k : Submodule R M) = lowerCentralSeries R I M k :=
     · rintro ⟨⟨x, hx⟩, m, hm, rfl⟩
       exact ⟨x, hx, m, hm, rfl⟩
 #align lie_ideal.coe_lcs_eq LieIdeal.coe_lcs_eq
+-/
 
 end LieIdeal
 
@@ -814,6 +876,7 @@ section OfAssociative
 
 variable (R : Type u) {A : Type v} [CommRing R] [Ring A] [Algebra R A]
 
+#print LieAlgebra.ad_nilpotent_of_nilpotent /-
 theorem LieAlgebra.ad_nilpotent_of_nilpotent {a : A} (h : IsNilpotent a) :
     IsNilpotent (LieAlgebra.ad R A a) :=
   by
@@ -823,9 +886,11 @@ theorem LieAlgebra.ad_nilpotent_of_nilpotent {a : A} (h : IsNilpotent a) :
   have := @LinearMap.commute_mulLeft_right R A _ _ _ _ _ a a
   exact this.is_nilpotent_sub hl hr
 #align lie_algebra.ad_nilpotent_of_nilpotent LieAlgebra.ad_nilpotent_of_nilpotent
+-/
 
 variable {R}
 
+#print LieSubalgebra.isNilpotent_ad_of_isNilpotent_ad /-
 theorem LieSubalgebra.isNilpotent_ad_of_isNilpotent_ad {L : Type v} [LieRing L] [LieAlgebra R L]
     (K : LieSubalgebra R L) {x : K} (h : IsNilpotent (LieAlgebra.ad R L ↑x)) :
     IsNilpotent (LieAlgebra.ad R K x) :=
@@ -834,11 +899,14 @@ theorem LieSubalgebra.isNilpotent_ad_of_isNilpotent_ad {L : Type v} [LieRing L]
   use n
   exact LinearMap.submodule_pow_eq_zero_of_pow_eq_zero (K.ad_comp_incl_eq x) hn
 #align lie_subalgebra.is_nilpotent_ad_of_is_nilpotent_ad LieSubalgebra.isNilpotent_ad_of_isNilpotent_ad
+-/
 
+#print LieAlgebra.isNilpotent_ad_of_isNilpotent /-
 theorem LieAlgebra.isNilpotent_ad_of_isNilpotent {L : LieSubalgebra R A} {x : L}
     (h : IsNilpotent (x : A)) : IsNilpotent (LieAlgebra.ad R L x) :=
   L.isNilpotent_ad_of_isNilpotent_ad <| LieAlgebra.ad_nilpotent_of_nilpotent R h
 #align lie_algebra.is_nilpotent_ad_of_is_nilpotent LieAlgebra.isNilpotent_ad_of_isNilpotent
+-/
 
 end OfAssociative
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/c471da714c044131b90c133701e51b877c246677

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module algebra.lie.nilpotent
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
+! leanprover-community/mathlib commit fd4551cfe4b7484b81c2c9ba3405edae27659676
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.RingTheory.Nilpotent
 /-!
 # Nilpotent Lie algebras
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 Like groups, Lie algebras admit a natural concept of nilpotency. More generally, any Lie module
 carries a natural concept of nilpotency. We define these here via the lower central series.
 

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

@@ -107,6 +107,7 @@ theorem lcs_le_self : N.lcs k ≤ N := by
     exact (LieSubmodule.mono_lie_right _ _ ⊤ ih).trans (N.lie_le_right ⊤)
 #align lie_submodule.lcs_le_self LieSubmodule.lcs_le_self
 
+#print LieSubmodule.lowerCentralSeries_eq_lcs_comap /-
 theorem lowerCentralSeries_eq_lcs_comap : lowerCentralSeries R L N k = (N.lcs k).comap N.incl :=
   by
   induction' k with k ih
@@ -117,12 +118,15 @@ theorem lowerCentralSeries_eq_lcs_comap : lowerCentralSeries R L N k = (N.lcs k)
       apply lcs_le_self
     rw [ih, LieSubmodule.comap_bracket_eq _ _ N.incl N.ker_incl this]
 #align lie_submodule.lower_central_series_eq_lcs_comap LieSubmodule.lowerCentralSeries_eq_lcs_comap
+-/
 
+#print LieSubmodule.lowerCentralSeries_map_eq_lcs /-
 theorem lowerCentralSeries_map_eq_lcs : (lowerCentralSeries R L N k).map N.incl = N.lcs k :=
   by
   rw [lower_central_series_eq_lcs_comap, LieSubmodule.map_comap_incl, inf_eq_right]
   apply lcs_le_self
 #align lie_submodule.lower_central_series_map_eq_lcs LieSubmodule.lowerCentralSeries_map_eq_lcs
+-/
 
 end LieSubmodule
 
@@ -149,6 +153,7 @@ theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries
     apply LieSubmodule.subset_lieSpan; use x, m; rfl
 #align lie_module.trivial_iff_lower_central_eq_bot LieModule.trivial_iff_lower_central_eq_bot
 
+#print LieModule.iterate_toEndomorphism_mem_lowerCentralSeries /-
 theorem iterate_toEndomorphism_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ) :
     (toEndomorphism R L M x^[k]) m ∈ lowerCentralSeries R L M k :=
   by
@@ -158,9 +163,11 @@ theorem iterate_toEndomorphism_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ)
       to_endomorphism_apply_apply]
     exact LieSubmodule.lie_mem_lie _ _ (LieSubmodule.mem_top x) ih
 #align lie_module.iterate_to_endomorphism_mem_lower_central_series LieModule.iterate_toEndomorphism_mem_lowerCentralSeries
+-/
 
 variable {R L M}
 
+#print LieModule.map_lowerCentralSeries_le /-
 theorem map_lowerCentralSeries_le {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂]
     [LieRingModule L M₂] [LieModule R L M₂] (k : ℕ) (f : M →ₗ⁅R,L⁆ M₂) :
     LieSubmodule.map f (lowerCentralSeries R L M k) ≤ lowerCentralSeries R L M₂ k :=
@@ -170,6 +177,7 @@ theorem map_lowerCentralSeries_le {M₂ : Type w₁} [AddCommGroup M₂] [Module
   · simp only [LieModule.lowerCentralSeries_succ, LieSubmodule.map_bracket_eq]
     exact LieSubmodule.mono_lie_right _ _ ⊤ ih
 #align lie_module.map_lower_central_series_le LieModule.map_lowerCentralSeries_le
+-/
 
 variable (R L M)
 
@@ -199,6 +207,7 @@ theorem isNilpotent_iff : IsNilpotent R L M ↔ ∃ k, lowerCentralSeries R L M
 
 variable {R L M}
 
+#print LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot /-
 theorem LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot (N : LieSubmodule R L M) :
     LieModule.IsNilpotent R L N ↔ ∃ k, N.lcs k = ⊥ :=
   by
@@ -207,6 +216,7 @@ theorem LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot (N : LieSubmodule R L M)
   rw [N.lower_central_series_eq_lcs_comap k, LieSubmodule.comap_incl_eq_bot,
     inf_eq_right.mpr (N.lcs_le_self k)]
 #align lie_submodule.is_nilpotent_iff_exists_lcs_eq_bot LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot
+-/
 
 variable (R L M)
 
@@ -214,6 +224,7 @@ instance (priority := 100) trivialIsNilpotent [IsTrivial L M] : IsNilpotent R L
   ⟨by use 1; change ⁅⊤, ⊤⁆ = ⊥; simp⟩
 #align lie_module.trivial_is_nilpotent LieModule.trivialIsNilpotent
 
+#print LieModule.nilpotent_endo_of_nilpotent_module /-
 theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
     ∃ k : ℕ, ∀ x : L, toEndomorphism R L M x ^ k = 0 :=
   by
@@ -223,7 +234,9 @@ theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
   rw [LinearMap.pow_apply, LinearMap.zero_apply, ← @LieSubmodule.mem_bot R L M, ← hM]
   exact iterate_to_endomorphism_mem_lower_central_series R L M x m k
 #align lie_module.nilpotent_endo_of_nilpotent_module LieModule.nilpotent_endo_of_nilpotent_module
+-/
 
+#print LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent /-
 /-- For a nilpotent Lie module, the weight space of the 0 weight is the whole module.
 
 This result will be used downstream to show that weight spaces are Lie submodules, at which time
@@ -239,7 +252,9 @@ theorem iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
   use k; rw [hk]
   exact LinearMap.zero_apply m
 #align lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
+-/
 
+#print LieModule.nilpotentOfNilpotentQuotient /-
 /-- If the quotient of a Lie module `M` by a Lie submodule on which the Lie algebra acts trivially
 is nilpotent then `M` is nilpotent.
 
@@ -259,6 +274,7 @@ theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxT
   rw [← LieSubmodule.Quotient.map_mk'_eq_bot_le, ← le_bot_iff, ← hk]
   exact map_lower_central_series_le k (LieSubmodule.Quotient.mk' N)
 #align lie_module.nilpotent_of_nilpotent_quotient LieModule.nilpotentOfNilpotentQuotient
+-/
 
 /-- Given a nilpotent Lie module `M` with lower central series `M = C₀ ≥ C₁ ≥ ⋯ ≥ Cₖ = ⊥`, this is
 the natural number `k` (the number of inclusions).
@@ -297,6 +313,7 @@ theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
   exact Nat.sInf_upward_closed_eq_succ_iff hs k
 #align lie_module.nilpotency_length_eq_succ_iff LieModule.nilpotencyLength_eq_succ_iff
 
+#print LieModule.lowerCentralSeriesLast /-
 /-- Given a non-trivial nilpotent Lie module `M` with lower central series
 `M = C₀ ≥ C₁ ≥ ⋯ ≥ Cₖ = ⊥`, this is the `k-1`th term in the lower central series (the last
 non-trivial term).
@@ -307,6 +324,7 @@ noncomputable def lowerCentralSeriesLast : LieSubmodule R L M :=
   | 0 => ⊥
   | k + 1 => lowerCentralSeries R L M k
 #align lie_module.lower_central_series_last LieModule.lowerCentralSeriesLast
+-/
 
 theorem lowerCentralSeriesLast_le_max_triv :
     lowerCentralSeriesLast R L M ≤ maxTrivSubmodule R L M :=
@@ -330,12 +348,15 @@ theorem nontrivial_lowerCentralSeriesLast [Nontrivial M] [IsNilpotent R L M] :
     exact h.2
 #align lie_module.nontrivial_lower_central_series_last LieModule.nontrivial_lowerCentralSeriesLast
 
+#print LieModule.nontrivial_max_triv_of_isNilpotent /-
 theorem nontrivial_max_triv_of_isNilpotent [Nontrivial M] [IsNilpotent R L M] :
     Nontrivial (maxTrivSubmodule R L M) :=
   Set.nontrivial_mono (lowerCentralSeriesLast_le_max_triv R L M)
     (nontrivial_lowerCentralSeriesLast R L M)
 #align lie_module.nontrivial_max_triv_of_is_nilpotent LieModule.nontrivial_max_triv_of_isNilpotent
+-/
 
+#print LieModule.coe_lcs_range_toEndomorphism_eq /-
 @[simp]
 theorem coe_lcs_range_toEndomorphism_eq (k : ℕ) :
     (lowerCentralSeries R (toEndomorphism R L M).range M k : Submodule R M) =
@@ -354,7 +375,9 @@ theorem coe_lcs_range_toEndomorphism_eq (k : ℕ) :
       exact
         ⟨⟨to_endomorphism R L M x, LieHom.mem_range_self _ x⟩, LieSubmodule.mem_top _, n, hn, rfl⟩
 #align lie_module.coe_lcs_range_to_endomorphism_eq LieModule.coe_lcs_range_toEndomorphism_eq
+-/
 
+#print LieModule.isNilpotent_range_toEndomorphism_iff /-
 @[simp]
 theorem isNilpotent_range_toEndomorphism_iff :
     IsNilpotent R (toEndomorphism R L M).range M ↔ IsNilpotent R L M := by
@@ -362,6 +385,7 @@ theorem isNilpotent_range_toEndomorphism_iff :
       rw [← LieSubmodule.coe_toSubmodule_eq_iff] at hk ⊢ <;>
     simpa using hk
 #align lie_module.is_nilpotent_range_to_endomorphism_iff LieModule.isNilpotent_range_toEndomorphism_iff
+-/
 
 end LieModule
 
@@ -369,27 +393,36 @@ namespace LieSubmodule
 
 variable {N₁ N₂ : LieSubmodule R L M}
 
+#print LieSubmodule.ucs /-
 /-- The upper (aka ascending) central series.
 
 See also `lie_submodule.lcs`. -/
 def ucs (k : ℕ) : LieSubmodule R L M → LieSubmodule R L M :=
   normalizer^[k]
 #align lie_submodule.ucs LieSubmodule.ucs
+-/
 
+#print LieSubmodule.ucs_zero /-
 @[simp]
 theorem ucs_zero : N.ucs 0 = N :=
   rfl
 #align lie_submodule.ucs_zero LieSubmodule.ucs_zero
+-/
 
+#print LieSubmodule.ucs_succ /-
 @[simp]
 theorem ucs_succ (k : ℕ) : N.ucs (k + 1) = (N.ucs k).normalizer :=
   Function.iterate_succ_apply' normalizer k N
 #align lie_submodule.ucs_succ LieSubmodule.ucs_succ
+-/
 
+#print LieSubmodule.ucs_add /-
 theorem ucs_add (k l : ℕ) : N.ucs (k + l) = (N.ucs l).ucs k :=
   Function.iterate_add_apply normalizer k l N
 #align lie_submodule.ucs_add LieSubmodule.ucs_add
+-/
 
+#print LieSubmodule.ucs_mono /-
 @[mono]
 theorem ucs_mono (k : ℕ) (h : N₁ ≤ N₂) : N₁.ucs k ≤ N₂.ucs k :=
   by
@@ -397,11 +430,15 @@ theorem ucs_mono (k : ℕ) (h : N₁ ≤ N₂) : N₁.ucs k ≤ N₂.ucs k :=
   simp only [ucs_succ]
   mono
 #align lie_submodule.ucs_mono LieSubmodule.ucs_mono
+-/
 
+#print LieSubmodule.ucs_eq_self_of_normalizer_eq_self /-
 theorem ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) : N₁.ucs k = N₁ := by
   induction' k with k ih; · simp; · rwa [ucs_succ, ih]
 #align lie_submodule.ucs_eq_self_of_normalizer_eq_self LieSubmodule.ucs_eq_self_of_normalizer_eq_self
+-/
 
+#print LieSubmodule.ucs_le_of_normalizer_eq_self /-
 /-- If a Lie module `M` contains a self-normalizing Lie submodule `N`, then all terms of the upper
 central series of `M` are contained in `N`.
 
@@ -411,7 +448,9 @@ theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
     (⊥ : LieSubmodule R L M).ucs k ≤ N₁ := by rw [← ucs_eq_self_of_normalizer_eq_self h k]; mono;
   simp
 #align lie_submodule.ucs_le_of_normalizer_eq_self LieSubmodule.ucs_le_of_normalizer_eq_self
+-/
 
+#print LieSubmodule.lcs_add_le_iff /-
 theorem lcs_add_le_iff (l k : ℕ) : N₁.lcs (l + k) ≤ N₂ ↔ N₁.lcs l ≤ N₂.ucs k :=
   by
   revert l
@@ -419,35 +458,48 @@ theorem lcs_add_le_iff (l k : ℕ) : N₁.lcs (l + k) ≤ N₂ ↔ N₁.lcs l 
   intro l
   rw [(by abel : l + (k + 1) = l + 1 + k), ih, ucs_succ, lcs_succ, top_lie_le_iff_le_normalizer]
 #align lie_submodule.lcs_add_le_iff LieSubmodule.lcs_add_le_iff
+-/
 
+#print LieSubmodule.lcs_le_iff /-
 theorem lcs_le_iff (k : ℕ) : N₁.lcs k ≤ N₂ ↔ N₁ ≤ N₂.ucs k := by convert lcs_add_le_iff 0 k;
   rw [zero_add]
 #align lie_submodule.lcs_le_iff LieSubmodule.lcs_le_iff
+-/
 
+#print LieSubmodule.gc_lcs_ucs /-
 theorem gc_lcs_ucs (k : ℕ) :
     GaloisConnection (fun N : LieSubmodule R L M => N.lcs k) fun N : LieSubmodule R L M =>
       N.ucs k :=
   fun N₁ N₂ => lcs_le_iff k
 #align lie_submodule.gc_lcs_ucs LieSubmodule.gc_lcs_ucs
+-/
 
+#print LieSubmodule.ucs_eq_top_iff /-
 theorem ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N := by
   rw [eq_top_iff, ← lcs_le_iff]; rfl
 #align lie_submodule.ucs_eq_top_iff LieSubmodule.ucs_eq_top_iff
+-/
 
+#print LieModule.isNilpotent_iff_exists_ucs_eq_top /-
 theorem LieModule.isNilpotent_iff_exists_ucs_eq_top :
     LieModule.IsNilpotent R L M ↔ ∃ k, (⊥ : LieSubmodule R L M).ucs k = ⊤ := by
   rw [LieModule.isNilpotent_iff]; exact exists_congr fun k => by simp [ucs_eq_top_iff]
 #align lie_module.is_nilpotent_iff_exists_ucs_eq_top LieModule.isNilpotent_iff_exists_ucs_eq_top
+-/
 
+#print LieSubmodule.ucs_comap_incl /-
 theorem ucs_comap_incl (k : ℕ) :
     ((⊥ : LieSubmodule R L M).ucs k).comap N.incl = (⊥ : LieSubmodule R L N).ucs k := by
   induction' k with k ih; · exact N.ker_incl; · simp [← ih]
 #align lie_submodule.ucs_comap_incl LieSubmodule.ucs_comap_incl
+-/
 
+#print LieSubmodule.isNilpotent_iff_exists_self_le_ucs /-
 theorem isNilpotent_iff_exists_self_le_ucs :
     LieModule.IsNilpotent R L N ↔ ∃ k, N ≤ (⊥ : LieSubmodule R L M).ucs k := by
   simp_rw [LieModule.isNilpotent_iff_exists_ucs_eq_top, ← ucs_comap_incl, comap_incl_eq_top]
 #align lie_submodule.is_nilpotent_iff_exists_self_le_ucs LieSubmodule.isNilpotent_iff_exists_self_le_ucs
+-/
 
 end LieSubmodule
 
@@ -508,16 +560,19 @@ theorem Equiv.lieModule_isNilpotent_iff (f : L ≃ₗ⁅R⁆ L₂) (g : M ≃ₗ
       hfg, f.apply_symm_apply, g.apply_symm_apply]
 #align equiv.lie_module_is_nilpotent_iff Equiv.lieModule_isNilpotent_iff
 
+#print LieModule.isNilpotent_of_top_iff /-
 @[simp]
 theorem LieModule.isNilpotent_of_top_iff :
     IsNilpotent R (⊤ : LieSubalgebra R L) M ↔ IsNilpotent R L M :=
   Equiv.lieModule_isNilpotent_iff LieSubalgebra.topEquiv (1 : M ≃ₗ[R] M) fun x m => rfl
 #align lie_module.is_nilpotent_of_top_iff LieModule.isNilpotent_of_top_iff
+-/
 
 end Morphisms
 
 end NilpotentModules
 
+#print LieAlgebra.isSolvable_of_isNilpotent /-
 instance (priority := 100) LieAlgebra.isSolvable_of_isNilpotent (R : Type u) (L : Type v)
     [CommRing R] [LieRing L] [LieAlgebra R L] [hL : LieModule.IsNilpotent R L L] :
     LieAlgebra.IsSolvable R L :=
@@ -526,6 +581,7 @@ instance (priority := 100) LieAlgebra.isSolvable_of_isNilpotent (R : Type u) (L
   use k; rw [← le_bot_iff] at h ⊢
   exact le_trans (LieModule.derivedSeries_le_lowerCentralSeries R L k) h
 #align lie_algebra.is_solvable_of_is_nilpotent LieAlgebra.isSolvable_of_isNilpotent
+-/
 
 section NilpotentAlgebras
 
@@ -533,12 +589,14 @@ variable (R : Type u) (L : Type v) (L' : Type w)
 
 variable [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L']
 
+#print LieAlgebra.IsNilpotent /-
 /-- We say a Lie algebra is nilpotent when it is nilpotent as a Lie module over itself via the
 adjoint representation. -/
 abbrev LieAlgebra.IsNilpotent (R : Type u) (L : Type v) [CommRing R] [LieRing L] [LieAlgebra R L] :
     Prop :=
   LieModule.IsNilpotent R L L
 #align lie_algebra.is_nilpotent LieAlgebra.IsNilpotent
+-/
 
 open LieAlgebra
 
@@ -547,11 +605,13 @@ theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
   LieModule.nilpotent_endo_of_nilpotent_module R L L
 #align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra
 
+#print LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent /-
 /-- See also `lie_algebra.zero_root_space_eq_top_of_nilpotent`. -/
 theorem LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L] :
     (⨅ x : L, (ad R L x).maximalGeneralizedEigenspace 0) = ⊤ :=
   LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L
 #align lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
+-/
 
 -- TODO Generalise the below to Lie modules if / when we define morphisms, equivs of Lie modules
 -- covering a Lie algebra morphism of (possibly different) Lie algebras.
@@ -559,6 +619,7 @@ variable {R L L'}
 
 open LieModule (lowerCentralSeries)
 
+#print coe_lowerCentralSeries_ideal_quot_eq /-
 /-- Given an ideal `I` of a Lie algebra `L`, the lower central series of `L ⧸ I` is the same
 whether we regard `L ⧸ I` as an `L` module or an `L ⧸ I` module.
 
@@ -581,7 +642,9 @@ theorem coe_lowerCentralSeries_ideal_quot_eq {I : LieIdeal R L} (k : ℕ) :
       erw [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.mem_coeSubmodule] at hz 
       exact ⟨⟨y, LieSubmodule.mem_top _⟩, ⟨z, hz⟩, rfl⟩
 #align coe_lower_central_series_ideal_quot_eq coe_lowerCentralSeries_ideal_quot_eq
+-/
 
+#print LieModule.coe_lowerCentralSeries_ideal_le /-
 /-- Note that the below inequality can be strict. For example the ideal of strictly-upper-triangular
 2x2 matrices inside the Lie algebra of upper-triangular 2x2 matrices with `k = 1`. -/
 theorem LieModule.coe_lowerCentralSeries_ideal_le {I : LieIdeal R L} (k : ℕ) :
@@ -594,6 +657,7 @@ theorem LieModule.coe_lowerCentralSeries_ideal_le {I : LieIdeal R L} (k : ℕ) :
     rintro x ⟨⟨y, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩
     exact ⟨⟨y.val, LieSubmodule.mem_top _⟩, ⟨z, ih hz⟩, rfl⟩
 #align lie_module.coe_lower_central_series_ideal_le LieModule.coe_lowerCentralSeries_ideal_le
+-/
 
 /-- A central extension of nilpotent Lie algebras is nilpotent. -/
 theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h₁ : I ≤ center R L)
@@ -606,10 +670,12 @@ theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h₁ : I
   simp [← LieSubmodule.coe_toSubmodule_eq_iff, coe_lowerCentralSeries_ideal_quot_eq, hk]
 #align lie_algebra.nilpotent_of_nilpotent_quotient LieAlgebra.nilpotent_of_nilpotent_quotient
 
+#print LieAlgebra.non_trivial_center_of_isNilpotent /-
 theorem LieAlgebra.non_trivial_center_of_isNilpotent [Nontrivial L] [IsNilpotent R L] :
     Nontrivial <| center R L :=
   LieModule.nontrivial_max_triv_of_isNilpotent R L L
 #align lie_algebra.non_trivial_center_of_is_nilpotent LieAlgebra.non_trivial_center_of_isNilpotent
+-/
 
 theorem LieIdeal.map_lowerCentralSeries_le (k : ℕ) {f : L →ₗ⁅R⁆ L'} :
     LieIdeal.map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k :=
@@ -652,6 +718,7 @@ theorem Function.Surjective.lieAlgebra_isNilpotent [h₁ : IsNilpotent R L] {f :
       simp only [LieIdeal.map_eq_bot_iff, bot_le] }
 #align function.surjective.lie_algebra_is_nilpotent Function.Surjective.lieAlgebra_isNilpotent
 
+#print LieEquiv.nilpotent_iff_equiv_nilpotent /-
 theorem LieEquiv.nilpotent_iff_equiv_nilpotent (e : L ≃ₗ⁅R⁆ L') :
     IsNilpotent R L ↔ IsNilpotent R L' :=
   by
@@ -659,10 +726,13 @@ theorem LieEquiv.nilpotent_iff_equiv_nilpotent (e : L ≃ₗ⁅R⁆ L') :
   · exact e.symm.injective.lie_algebra_is_nilpotent
   · exact e.injective.lie_algebra_is_nilpotent
 #align lie_equiv.nilpotent_iff_equiv_nilpotent LieEquiv.nilpotent_iff_equiv_nilpotent
+-/
 
+#print LieHom.isNilpotent_range /-
 theorem LieHom.isNilpotent_range [IsNilpotent R L] (f : L →ₗ⁅R⁆ L') : IsNilpotent R f.range :=
   f.surjective_rangeRestrict.lieAlgebra_isNilpotent
 #align lie_hom.is_nilpotent_range LieHom.isNilpotent_range
+-/
 
 /-- Note that this result is not quite a special case of
 `lie_module.is_nilpotent_range_to_endomorphism_iff` which concerns nilpotency of the

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -4,14 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 
 ! This file was ported from Lean 3 source module algebra.lie.nilpotent
-! leanprover-community/mathlib commit 938fead7abdc0cbbca8eba7a1052865a169dc102
+! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.Lie.Solvable
 import Mathbin.Algebra.Lie.Quotient
 import Mathbin.Algebra.Lie.Normalizer
-import Mathbin.LinearAlgebra.Eigenspace
+import Mathbin.LinearAlgebra.Eigenspace.Basic
 import Mathbin.RingTheory.Nilpotent
 
 /-!

mathlib commit https://github.com/leanprover-community/mathlib/commit/13361559d66b84f80b6d5a1c4a26aa5054766725

@@ -518,13 +518,14 @@ end Morphisms
 
 end NilpotentModules
 
-instance (priority := 100) LieAlgebra.isSolvableOfIsNilpotent (R : Type u) (L : Type v) [CommRing R]
-    [LieRing L] [LieAlgebra R L] [hL : LieModule.IsNilpotent R L L] : LieAlgebra.IsSolvable R L :=
+instance (priority := 100) LieAlgebra.isSolvable_of_isNilpotent (R : Type u) (L : Type v)
+    [CommRing R] [LieRing L] [LieAlgebra R L] [hL : LieModule.IsNilpotent R L L] :
+    LieAlgebra.IsSolvable R L :=
   by
   obtain ⟨k, h⟩ : ∃ k, LieModule.lowerCentralSeries R L L k = ⊥ := hL.nilpotent
   use k; rw [← le_bot_iff] at h ⊢
   exact le_trans (LieModule.derivedSeries_le_lowerCentralSeries R L k) h
-#align lie_algebra.is_solvable_of_is_nilpotent LieAlgebra.isSolvableOfIsNilpotent
+#align lie_algebra.is_solvable_of_is_nilpotent LieAlgebra.isSolvable_of_isNilpotent
 
 section NilpotentAlgebras
 

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

@@ -265,13 +265,13 @@ the natural number `k` (the number of inclusions).
 
 For a non-nilpotent module, we use the junk value 0. -/
 noncomputable def nilpotencyLength : ℕ :=
-  sInf { k | lowerCentralSeries R L M k = ⊥ }
+  sInf {k | lowerCentralSeries R L M k = ⊥}
 #align lie_module.nilpotency_length LieModule.nilpotencyLength
 
 theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
     nilpotencyLength R L M = 0 ↔ Subsingleton M :=
   by
-  let s := { k | lowerCentralSeries R L M k = ⊥ }
+  let s := {k | lowerCentralSeries R L M k = ⊥}
   have hs : s.nonempty :=
     by
     obtain ⟨k, hk⟩ := (by infer_instance : IsNilpotent R L M)
@@ -288,7 +288,7 @@ theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
     nilpotencyLength R L M = k + 1 ↔
       lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥ :=
   by
-  let s := { k | lowerCentralSeries R L M k = ⊥ }
+  let s := {k | lowerCentralSeries R L M k = ⊥}
   change Inf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s
   have hs : ∀ k₁ k₂, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s :=
     by
@@ -359,7 +359,7 @@ theorem coe_lcs_range_toEndomorphism_eq (k : ℕ) :
 theorem isNilpotent_range_toEndomorphism_iff :
     IsNilpotent R (toEndomorphism R L M).range M ↔ IsNilpotent R L M := by
   constructor <;> rintro ⟨k, hk⟩ <;> use k <;>
-      rw [← LieSubmodule.coe_to_submodule_eq_iff] at hk ⊢ <;>
+      rw [← LieSubmodule.coe_toSubmodule_eq_iff] at hk ⊢ <;>
     simpa using hk
 #align lie_module.is_nilpotent_range_to_endomorphism_iff LieModule.isNilpotent_range_toEndomorphism_iff
 
@@ -471,8 +471,8 @@ theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
   induction' k with k ih
   · simp [LinearMap.range_eq_top, hg]
   · suffices
-      g '' { m | ∃ (x : L) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m } =
-        { m | ∃ (x : L₂) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m }
+      g '' {m | ∃ (x : L) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m} =
+        {m | ∃ (x : L₂) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m}
       by
       simp only [← LieSubmodule.mem_coeSubmodule] at this 
       simp [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.lieIdeal_oper_eq_linear_span',
@@ -490,7 +490,7 @@ theorem Function.Surjective.lieModuleIsNilpotent [IsNilpotent R L M] : IsNilpote
   by
   obtain ⟨k, hk⟩ := id (by infer_instance : IsNilpotent R L M)
   use k
-  rw [← LieSubmodule.coe_to_submodule_eq_iff] at hk ⊢
+  rw [← LieSubmodule.coe_toSubmodule_eq_iff] at hk ⊢
   simp [← hf.lie_module_lcs_map_eq hg hfg k, hk]
 #align function.surjective.lie_module_is_nilpotent Function.Surjective.lieModuleIsNilpotent
 
@@ -602,7 +602,7 @@ theorem LieAlgebra.nilpotent_of_nilpotent_quotient {I : LieIdeal R L} (h₁ : I
     exact LieModule.nilpotentOfNilpotentQuotient R L L h₁ this
   obtain ⟨k, hk⟩ := h₂
   use k
-  simp [← LieSubmodule.coe_to_submodule_eq_iff, coe_lowerCentralSeries_ideal_quot_eq, hk]
+  simp [← LieSubmodule.coe_toSubmodule_eq_iff, coe_lowerCentralSeries_ideal_quot_eq, hk]
 #align lie_algebra.nilpotent_of_nilpotent_quotient LieAlgebra.nilpotent_of_nilpotent_quotient
 
 theorem LieAlgebra.non_trivial_center_of_isNilpotent [Nontrivial L] [IsNilpotent R L] :

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

@@ -111,7 +111,7 @@ theorem lowerCentralSeries_eq_lcs_comap : lowerCentralSeries R L N k = (N.lcs k)
   by
   induction' k with k ih
   · simp
-  · simp only [lcs_succ, lowerCentralSeries_succ] at ih⊢
+  · simp only [lcs_succ, lowerCentralSeries_succ] at ih ⊢
     have : N.lcs k ≤ N.incl.range := by
       rw [N.range_incl]
       apply lcs_le_self
@@ -145,7 +145,7 @@ theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries
   by
   constructor <;> intro h
   · erw [eq_bot_iff, LieSubmodule.lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn, h.trivial]; simp
-  · rw [LieSubmodule.eq_bot_iff] at h; apply is_trivial.mk; intro x m; apply h
+  · rw [LieSubmodule.eq_bot_iff] at h ; apply is_trivial.mk; intro x m; apply h
     apply LieSubmodule.subset_lieSpan; use x, m; rfl
 #align lie_module.trivial_iff_lower_central_eq_bot LieModule.trivial_iff_lower_central_eq_bot
 
@@ -255,7 +255,7 @@ theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxT
   suffices lowerCentralSeries R L M k ≤ N
     by
     replace this := LieSubmodule.mono_lie_right _ _ ⊤ (le_trans this h₁)
-    rwa [ideal_oper_max_triv_submodule_eq_bot, le_bot_iff] at this
+    rwa [ideal_oper_max_triv_submodule_eq_bot, le_bot_iff] at this 
   rw [← LieSubmodule.Quotient.map_mk'_eq_bot_le, ← le_bot_iff, ← hk]
   exact map_lower_central_series_le k (LieSubmodule.Quotient.mk' N)
 #align lie_module.nilpotent_of_nilpotent_quotient LieModule.nilpotentOfNilpotentQuotient
@@ -315,7 +315,7 @@ theorem lowerCentralSeriesLast_le_max_triv :
   cases' h : nilpotency_length R L M with k
   · exact bot_le
   · rw [le_max_triv_iff_bracket_eq_bot]
-    rw [nilpotency_length_eq_succ_iff, lowerCentralSeries_succ] at h
+    rw [nilpotency_length_eq_succ_iff, lowerCentralSeries_succ] at h 
     exact h.1
 #align lie_module.lower_central_series_last_le_max_triv LieModule.lowerCentralSeriesLast_le_max_triv
 
@@ -324,9 +324,9 @@ theorem nontrivial_lowerCentralSeriesLast [Nontrivial M] [IsNilpotent R L M] :
   by
   rw [LieSubmodule.nontrivial_iff_ne_bot, lower_central_series_last]
   cases h : nilpotency_length R L M
-  · rw [nilpotency_length_eq_zero_iff, ← not_nontrivial_iff_subsingleton] at h
+  · rw [nilpotency_length_eq_zero_iff, ← not_nontrivial_iff_subsingleton] at h 
     contradiction
-  · rw [nilpotency_length_eq_succ_iff] at h
+  · rw [nilpotency_length_eq_succ_iff] at h 
     exact h.2
 #align lie_module.nontrivial_lower_central_series_last LieModule.nontrivial_lowerCentralSeriesLast
 
@@ -359,7 +359,7 @@ theorem coe_lcs_range_toEndomorphism_eq (k : ℕ) :
 theorem isNilpotent_range_toEndomorphism_iff :
     IsNilpotent R (toEndomorphism R L M).range M ↔ IsNilpotent R L M := by
   constructor <;> rintro ⟨k, hk⟩ <;> use k <;>
-      rw [← LieSubmodule.coe_to_submodule_eq_iff] at hk⊢ <;>
+      rw [← LieSubmodule.coe_to_submodule_eq_iff] at hk ⊢ <;>
     simpa using hk
 #align lie_module.is_nilpotent_range_to_endomorphism_iff LieModule.isNilpotent_range_toEndomorphism_iff
 
@@ -471,10 +471,10 @@ theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
   induction' k with k ih
   · simp [LinearMap.range_eq_top, hg]
   · suffices
-      g '' { m | ∃ (x : L)(n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m } =
-        { m | ∃ (x : L₂)(n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m }
+      g '' { m | ∃ (x : L) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m } =
+        { m | ∃ (x : L₂) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m }
       by
-      simp only [← LieSubmodule.mem_coeSubmodule] at this
+      simp only [← LieSubmodule.mem_coeSubmodule] at this 
       simp [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.lieIdeal_oper_eq_linear_span',
         Submodule.map_span, -Submodule.span_image, this]
     ext m₂
@@ -490,7 +490,7 @@ theorem Function.Surjective.lieModuleIsNilpotent [IsNilpotent R L M] : IsNilpote
   by
   obtain ⟨k, hk⟩ := id (by infer_instance : IsNilpotent R L M)
   use k
-  rw [← LieSubmodule.coe_to_submodule_eq_iff] at hk⊢
+  rw [← LieSubmodule.coe_to_submodule_eq_iff] at hk ⊢
   simp [← hf.lie_module_lcs_map_eq hg hfg k, hk]
 #align function.surjective.lie_module_is_nilpotent Function.Surjective.lieModuleIsNilpotent
 
@@ -522,7 +522,7 @@ instance (priority := 100) LieAlgebra.isSolvableOfIsNilpotent (R : Type u) (L :
     [LieRing L] [LieAlgebra R L] [hL : LieModule.IsNilpotent R L L] : LieAlgebra.IsSolvable R L :=
   by
   obtain ⟨k, h⟩ : ∃ k, LieModule.lowerCentralSeries R L L k = ⊥ := hL.nilpotent
-  use k; rw [← le_bot_iff] at h⊢
+  use k; rw [← le_bot_iff] at h ⊢
   exact le_trans (LieModule.derivedSeries_le_lowerCentralSeries R L k) h
 #align lie_algebra.is_solvable_of_is_nilpotent LieAlgebra.isSolvableOfIsNilpotent
 
@@ -574,10 +574,10 @@ theorem coe_lowerCentralSeries_ideal_quot_eq {I : LieIdeal R L} (k : ℕ) :
     ext x
     constructor
     · rintro ⟨⟨y, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩
-      erw [← LieSubmodule.mem_coeSubmodule, ih, LieSubmodule.mem_coeSubmodule] at hz
+      erw [← LieSubmodule.mem_coeSubmodule, ih, LieSubmodule.mem_coeSubmodule] at hz 
       exact ⟨⟨LieSubmodule.Quotient.mk y, LieSubmodule.mem_top _⟩, ⟨z, hz⟩, rfl⟩
     · rintro ⟨⟨⟨y⟩, -⟩, ⟨z, hz⟩, rfl : ⁅y, z⁆ = x⟩
-      erw [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.mem_coeSubmodule] at hz
+      erw [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.mem_coeSubmodule] at hz 
       exact ⟨⟨y, LieSubmodule.mem_top _⟩, ⟨z, hz⟩, rfl⟩
 #align coe_lower_central_series_ideal_quot_eq coe_lowerCentralSeries_ideal_quot_eq
 

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

@@ -111,7 +111,7 @@ theorem lowerCentralSeries_eq_lcs_comap : lowerCentralSeries R L N k = (N.lcs k)
   by
   induction' k with k ih
   · simp
-  · simp only [lcs_succ, lower_central_series_succ] at ih⊢
+  · simp only [lcs_succ, lowerCentralSeries_succ] at ih⊢
     have : N.lcs k ≤ N.incl.range := by
       rw [N.range_incl]
       apply lcs_le_self
@@ -136,7 +136,7 @@ theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M :=
   induction' k with k ih generalizing l <;> intro h
   · exact (le_zero_iff.mp h).symm ▸ le_rfl
   · rcases Nat.of_le_succ h with (hk | hk)
-    · rw [lower_central_series_succ]
+    · rw [lowerCentralSeries_succ]
       exact (LieSubmodule.mono_lie_right _ _ ⊤ (ih hk)).trans (LieSubmodule.lie_le_right _ _)
     · exact hk.symm ▸ le_rfl
 #align lie_module.antitone_lower_central_series LieModule.antitone_lowerCentralSeries
@@ -154,7 +154,7 @@ theorem iterate_toEndomorphism_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ)
   by
   induction' k with k ih
   · simp only [Function.iterate_zero]
-  · simp only [lower_central_series_succ, Function.comp_apply, Function.iterate_succ',
+  · simp only [lowerCentralSeries_succ, Function.comp_apply, Function.iterate_succ',
       to_endomorphism_apply_apply]
     exact LieSubmodule.lie_mem_lie _ _ (LieSubmodule.mem_top x) ih
 #align lie_module.iterate_to_endomorphism_mem_lower_central_series LieModule.iterate_toEndomorphism_mem_lowerCentralSeries
@@ -179,10 +179,10 @@ theorem derivedSeries_le_lowerCentralSeries (k : ℕ) :
     derivedSeries R L k ≤ lowerCentralSeries R L L k :=
   by
   induction' k with k h
-  · rw [derived_series_def, derived_series_of_ideal_zero, lower_central_series_zero]
+  · rw [derived_series_def, derived_series_of_ideal_zero, lowerCentralSeries_zero]
     exact le_rfl
   · have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
-    rw [derived_series_def, derived_series_of_ideal_succ, lower_central_series_succ]
+    rw [derived_series_def, derived_series_of_ideal_succ, lowerCentralSeries_succ]
     exact LieSubmodule.mono_lie _ _ _ _ h' h
 #align lie_module.derived_series_le_lower_central_series LieModule.derivedSeries_le_lowerCentralSeries
 
@@ -251,8 +251,8 @@ theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxT
   by
   obtain ⟨k, hk⟩ := h₂
   use k + 1
-  simp only [lower_central_series_succ]
-  suffices lower_central_series R L M k ≤ N
+  simp only [lowerCentralSeries_succ]
+  suffices lowerCentralSeries R L M k ≤ N
     by
     replace this := LieSubmodule.mono_lie_right _ _ ⊤ (le_trans this h₁)
     rwa [ideal_oper_max_triv_submodule_eq_bot, le_bot_iff] at this
@@ -271,14 +271,14 @@ noncomputable def nilpotencyLength : ℕ :=
 theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
     nilpotencyLength R L M = 0 ↔ Subsingleton M :=
   by
-  let s := { k | lower_central_series R L M k = ⊥ }
+  let s := { k | lowerCentralSeries R L M k = ⊥ }
   have hs : s.nonempty :=
     by
     obtain ⟨k, hk⟩ := (by infer_instance : IsNilpotent R L M)
     exact ⟨k, hk⟩
   change Inf s = 0 ↔ _
   rw [← LieSubmodule.subsingleton_iff R L M, ← subsingleton_iff_bot_eq_top, ←
-    lower_central_series_zero, @eq_comm (LieSubmodule R L M)]
+    lowerCentralSeries_zero, @eq_comm (LieSubmodule R L M)]
   refine' ⟨fun h => h ▸ Nat.sInf_mem hs, fun h => _⟩
   rw [Nat.sInf_eq_zero]
   exact Or.inl h
@@ -288,11 +288,11 @@ theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
     nilpotencyLength R L M = k + 1 ↔
       lowerCentralSeries R L M (k + 1) = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥ :=
   by
-  let s := { k | lower_central_series R L M k = ⊥ }
+  let s := { k | lowerCentralSeries R L M k = ⊥ }
   change Inf s = k + 1 ↔ k + 1 ∈ s ∧ k ∉ s
   have hs : ∀ k₁ k₂, k₁ ≤ k₂ → k₁ ∈ s → k₂ ∈ s :=
     by
-    rintro k₁ k₂ h₁₂ (h₁ : lower_central_series R L M k₁ = ⊥)
+    rintro k₁ k₂ h₁₂ (h₁ : lowerCentralSeries R L M k₁ = ⊥)
     exact eq_bot_iff.mpr (h₁ ▸ antitone_lower_central_series R L M h₁₂)
   exact Nat.sInf_upward_closed_eq_succ_iff hs k
 #align lie_module.nilpotency_length_eq_succ_iff LieModule.nilpotencyLength_eq_succ_iff
@@ -315,7 +315,7 @@ theorem lowerCentralSeriesLast_le_max_triv :
   cases' h : nilpotency_length R L M with k
   · exact bot_le
   · rw [le_max_triv_iff_bracket_eq_bot]
-    rw [nilpotency_length_eq_succ_iff, lower_central_series_succ] at h
+    rw [nilpotency_length_eq_succ_iff, lowerCentralSeries_succ] at h
     exact h.1
 #align lie_module.lower_central_series_last_le_max_triv LieModule.lowerCentralSeriesLast_le_max_triv
 
@@ -343,8 +343,8 @@ theorem coe_lcs_range_toEndomorphism_eq (k : ℕ) :
   by
   induction' k with k ih
   · simp
-  · simp only [lower_central_series_succ, LieSubmodule.lieIdeal_oper_eq_linear_span', ←
-      (lower_central_series R (to_endomorphism R L M).range M k).mem_coe_submodule, ih]
+  · simp only [lowerCentralSeries_succ, LieSubmodule.lieIdeal_oper_eq_linear_span', ←
+      (lowerCentralSeries R (to_endomorphism R L M).range M k).mem_coe_submodule, ih]
     congr
     ext m
     constructor
@@ -471,8 +471,8 @@ theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
   induction' k with k ih
   · simp [LinearMap.range_eq_top, hg]
   · suffices
-      g '' { m | ∃ (x : L)(n : _), n ∈ lower_central_series R L M k ∧ ⁅x, n⁆ = m } =
-        { m | ∃ (x : L₂)(n : _), n ∈ lower_central_series R L M k ∧ ⁅x, g n⁆ = m }
+      g '' { m | ∃ (x : L)(n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m } =
+        { m | ∃ (x : L₂)(n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m }
       by
       simp only [← LieSubmodule.mem_coeSubmodule] at this
       simp [← LieSubmodule.mem_coeSubmodule, ← ih, LieSubmodule.lieIdeal_oper_eq_linear_span',
@@ -722,7 +722,7 @@ theorem coe_lcs_eq : (I.lcs M k : Submodule R M) = lowerCentralSeries R I M k :=
   by
   induction' k with k ih
   · simp
-  · simp_rw [lower_central_series_succ, lcs_succ, LieSubmodule.lieIdeal_oper_eq_linear_span', ←
+  · simp_rw [lowerCentralSeries_succ, lcs_succ, LieSubmodule.lieIdeal_oper_eq_linear_span', ←
       (I.lcs M k).mem_coe_submodule, ih, LieSubmodule.mem_coeSubmodule, LieSubmodule.mem_top,
       exists_true_left, (I : LieSubalgebra R L).coe_bracket_of_module]
     congr

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

@@ -144,17 +144,9 @@ theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M :=
 theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries R L M 1 = ⊥ :=
   by
   constructor <;> intro h
-  · erw [eq_bot_iff, LieSubmodule.lieSpan_le]
-    rintro m ⟨x, n, hn⟩
-    rw [← hn, h.trivial]
-    simp
-  · rw [LieSubmodule.eq_bot_iff] at h
-    apply is_trivial.mk
-    intro x m
-    apply h
-    apply LieSubmodule.subset_lieSpan
-    use x, m
-    rfl
+  · erw [eq_bot_iff, LieSubmodule.lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn, h.trivial]; simp
+  · rw [LieSubmodule.eq_bot_iff] at h; apply is_trivial.mk; intro x m; apply h
+    apply LieSubmodule.subset_lieSpan; use x, m; rfl
 #align lie_module.trivial_iff_lower_central_eq_bot LieModule.trivial_iff_lower_central_eq_bot
 
 theorem iterate_toEndomorphism_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ) :
@@ -219,10 +211,7 @@ theorem LieSubmodule.isNilpotent_iff_exists_lcs_eq_bot (N : LieSubmodule R L M)
 variable (R L M)
 
 instance (priority := 100) trivialIsNilpotent [IsTrivial L M] : IsNilpotent R L M :=
-  ⟨by
-    use 1
-    change ⁅⊤, ⊤⁆ = ⊥
-    simp⟩
+  ⟨by use 1; change ⁅⊤, ⊤⁆ = ⊥; simp⟩
 #align lie_module.trivial_is_nilpotent LieModule.trivialIsNilpotent
 
 theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
@@ -409,11 +398,8 @@ theorem ucs_mono (k : ℕ) (h : N₁ ≤ N₂) : N₁.ucs k ≤ N₂.ucs k :=
   mono
 #align lie_submodule.ucs_mono LieSubmodule.ucs_mono
 
-theorem ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) : N₁.ucs k = N₁ :=
-  by
-  induction' k with k ih
-  · simp
-  · rwa [ucs_succ, ih]
+theorem ucs_eq_self_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) : N₁.ucs k = N₁ := by
+  induction' k with k ih; · simp; · rwa [ucs_succ, ih]
 #align lie_submodule.ucs_eq_self_of_normalizer_eq_self LieSubmodule.ucs_eq_self_of_normalizer_eq_self
 
 /-- If a Lie module `M` contains a self-normalizing Lie submodule `N`, then all terms of the upper
@@ -422,10 +408,7 @@ central series of `M` are contained in `N`.
 An important instance of this situation arises from a Cartan subalgebra `H ⊆ L` with the roles of
 `L`, `M`, `N` played by `H`, `L`, `H`, respectively. -/
 theorem ucs_le_of_normalizer_eq_self (h : N₁.normalizer = N₁) (k : ℕ) :
-    (⊥ : LieSubmodule R L M).ucs k ≤ N₁ :=
-  by
-  rw [← ucs_eq_self_of_normalizer_eq_self h k]
-  mono
+    (⊥ : LieSubmodule R L M).ucs k ≤ N₁ := by rw [← ucs_eq_self_of_normalizer_eq_self h k]; mono;
   simp
 #align lie_submodule.ucs_le_of_normalizer_eq_self LieSubmodule.ucs_le_of_normalizer_eq_self
 
@@ -437,9 +420,7 @@ theorem lcs_add_le_iff (l k : ℕ) : N₁.lcs (l + k) ≤ N₂ ↔ N₁.lcs l 
   rw [(by abel : l + (k + 1) = l + 1 + k), ih, ucs_succ, lcs_succ, top_lie_le_iff_le_normalizer]
 #align lie_submodule.lcs_add_le_iff LieSubmodule.lcs_add_le_iff
 
-theorem lcs_le_iff (k : ℕ) : N₁.lcs k ≤ N₂ ↔ N₁ ≤ N₂.ucs k :=
-  by
-  convert lcs_add_le_iff 0 k
+theorem lcs_le_iff (k : ℕ) : N₁.lcs k ≤ N₂ ↔ N₁ ≤ N₂.ucs k := by convert lcs_add_le_iff 0 k;
   rw [zero_add]
 #align lie_submodule.lcs_le_iff LieSubmodule.lcs_le_iff
 
@@ -449,25 +430,18 @@ theorem gc_lcs_ucs (k : ℕ) :
   fun N₁ N₂ => lcs_le_iff k
 #align lie_submodule.gc_lcs_ucs LieSubmodule.gc_lcs_ucs
 
-theorem ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N :=
-  by
-  rw [eq_top_iff, ← lcs_le_iff]
-  rfl
+theorem ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N := by
+  rw [eq_top_iff, ← lcs_le_iff]; rfl
 #align lie_submodule.ucs_eq_top_iff LieSubmodule.ucs_eq_top_iff
 
 theorem LieModule.isNilpotent_iff_exists_ucs_eq_top :
-    LieModule.IsNilpotent R L M ↔ ∃ k, (⊥ : LieSubmodule R L M).ucs k = ⊤ :=
-  by
-  rw [LieModule.isNilpotent_iff]
-  exact exists_congr fun k => by simp [ucs_eq_top_iff]
+    LieModule.IsNilpotent R L M ↔ ∃ k, (⊥ : LieSubmodule R L M).ucs k = ⊤ := by
+  rw [LieModule.isNilpotent_iff]; exact exists_congr fun k => by simp [ucs_eq_top_iff]
 #align lie_module.is_nilpotent_iff_exists_ucs_eq_top LieModule.isNilpotent_iff_exists_ucs_eq_top
 
 theorem ucs_comap_incl (k : ℕ) :
-    ((⊥ : LieSubmodule R L M).ucs k).comap N.incl = (⊥ : LieSubmodule R L N).ucs k :=
-  by
-  induction' k with k ih
-  · exact N.ker_incl
-  · simp [← ih]
+    ((⊥ : LieSubmodule R L M).ucs k).comap N.incl = (⊥ : LieSubmodule R L N).ucs k := by
+  induction' k with k ih; · exact N.ker_incl; · simp [← ih]
 #align lie_submodule.ucs_comap_incl LieSubmodule.ucs_comap_incl
 
 theorem isNilpotent_iff_exists_self_le_ucs :
@@ -653,8 +627,7 @@ theorem LieIdeal.lowerCentralSeries_map_eq (k : ℕ) {f : L →ₗ⁅R⁆ L'} (h
     rw [← f.ideal_range_eq_map]
     exact f.ideal_range_eq_top_of_surjective h
   induction' k with k ih
-  · simp only [LieModule.lowerCentralSeries_zero]
-    exact h'
+  · simp only [LieModule.lowerCentralSeries_zero]; exact h'
   · simp only [LieModule.lowerCentralSeries_succ, LieIdeal.map_bracket_eq f h, ih, h']
 #align lie_ideal.lower_central_series_map_eq LieIdeal.lowerCentralSeries_map_eq
 

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

@@ -239,17 +239,17 @@ theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
 
 This result will be used downstream to show that weight spaces are Lie submodules, at which time
 it will be possible to state it in the language of weight spaces. -/
-theorem infᵢ_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
+theorem iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
     (⨅ x : L, (toEndomorphism R L M x).maximalGeneralizedEigenspace 0) = ⊤ :=
   by
   ext m
   simp only [Module.End.mem_maximalGeneralizedEigenspace, Submodule.mem_top, sub_zero, iff_true_iff,
-    zero_smul, Submodule.mem_infᵢ]
+    zero_smul, Submodule.mem_iInf]
   intro x
   obtain ⟨k, hk⟩ := nilpotent_endo_of_nilpotent_module R L M
   use k; rw [hk]
   exact LinearMap.zero_apply m
-#align lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieModule.infᵢ_max_gen_zero_eigenspace_eq_top_of_nilpotent
+#align lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
 
 /-- If the quotient of a Lie module `M` by a Lie submodule on which the Lie algebra acts trivially
 is nilpotent then `M` is nilpotent.
@@ -276,7 +276,7 @@ the natural number `k` (the number of inclusions).
 
 For a non-nilpotent module, we use the junk value 0. -/
 noncomputable def nilpotencyLength : ℕ :=
-  infₛ { k | lowerCentralSeries R L M k = ⊥ }
+  sInf { k | lowerCentralSeries R L M k = ⊥ }
 #align lie_module.nilpotency_length LieModule.nilpotencyLength
 
 theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
@@ -290,8 +290,8 @@ theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
   change Inf s = 0 ↔ _
   rw [← LieSubmodule.subsingleton_iff R L M, ← subsingleton_iff_bot_eq_top, ←
     lower_central_series_zero, @eq_comm (LieSubmodule R L M)]
-  refine' ⟨fun h => h ▸ Nat.infₛ_mem hs, fun h => _⟩
-  rw [Nat.infₛ_eq_zero]
+  refine' ⟨fun h => h ▸ Nat.sInf_mem hs, fun h => _⟩
+  rw [Nat.sInf_eq_zero]
   exact Or.inl h
 #align lie_module.nilpotency_length_eq_zero_iff LieModule.nilpotencyLength_eq_zero_iff
 
@@ -305,7 +305,7 @@ theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
     by
     rintro k₁ k₂ h₁₂ (h₁ : lower_central_series R L M k₁ = ⊥)
     exact eq_bot_iff.mpr (h₁ ▸ antitone_lower_central_series R L M h₁₂)
-  exact Nat.infₛ_upward_closed_eq_succ_iff hs k
+  exact Nat.sInf_upward_closed_eq_succ_iff hs k
 #align lie_module.nilpotency_length_eq_succ_iff LieModule.nilpotencyLength_eq_succ_iff
 
 /-- Given a non-trivial nilpotent Lie module `M` with lower central series
@@ -573,10 +573,10 @@ theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
 #align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra
 
 /-- See also `lie_algebra.zero_root_space_eq_top_of_nilpotent`. -/
-theorem LieAlgebra.infᵢ_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L] :
+theorem LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L] :
     (⨅ x : L, (ad R L x).maximalGeneralizedEigenspace 0) = ⊤ :=
-  LieModule.infᵢ_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L
-#align lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieAlgebra.infᵢ_max_gen_zero_eigenspace_eq_top_of_nilpotent
+  LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L
+#align lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
 
 -- TODO Generalise the below to Lie modules if / when we define morphisms, equivs of Lie modules
 -- covering a Lie algebra morphism of (possibly different) Lie algebras.

mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25

@@ -189,7 +189,7 @@ theorem derivedSeries_le_lowerCentralSeries (k : ℕ) :
   induction' k with k h
   · rw [derived_series_def, derived_series_of_ideal_zero, lower_central_series_zero]
     exact le_rfl
-  · have h' : derived_series R L k ≤ ⊤ := by simp only [le_top]
+  · have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
     rw [derived_series_def, derived_series_of_ideal_succ, lower_central_series_succ]
     exact LieSubmodule.mono_lie _ _ _ _ h' h
 #align lie_module.derived_series_le_lower_central_series LieModule.derivedSeries_le_lowerCentralSeries

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

Mathlib.RingTheory.Nilpotent has a few very simple definitions (Mathlib.Data.Nat.Lattice is sufficient to state them), but needs some pretty heavy imports (ideals, linear algebra) towards the end. This change moves the heavier parts into a new file.

@@ -10,7 +10,7 @@ import Mathlib.Algebra.Lie.Normalizer
 import Mathlib.LinearAlgebra.Eigenspace.Basic
 import Mathlib.Order.Filter.AtTopBot
 import Mathlib.RingTheory.Artinian
-import Mathlib.RingTheory.Nilpotent
+import Mathlib.RingTheory.Nilpotent.Lemmas
 import Mathlib.Tactic.Monotonicity
 
 #align_import algebra.lie.nilpotent from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"

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

@@ -134,7 +134,7 @@ variable (R L M)
 theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M := by
   intro l k
   induction' k with k ih generalizing l <;> intro h
-  · exact (le_zero_iff.mp h).symm ▸ le_rfl
+  · exact (Nat.le_zero.mp h).symm ▸ le_rfl
   · rcases Nat.of_le_succ h with (hk | hk)
     · rw [lowerCentralSeries_succ]
       exact (LieSubmodule.mono_lie_right _ _ ⊤ (ih hk)).trans (LieSubmodule.lie_le_right _ _)

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -36,11 +36,8 @@ universe u v w w₁ w₂
 section NilpotentModules
 
 variable {R : Type u} {L : Type v} {M : Type w}
-
 variable [CommRing R] [LieRing L] [LieAlgebra R L] [AddCommGroup M] [Module R M]
-
 variable [LieRingModule L M] [LieModule R L M]
-
 variable (k : ℕ) (N : LieSubmodule R L M)
 
 namespace LieSubmodule
@@ -559,11 +556,8 @@ section Morphisms
 open LieModule Function
 
 variable {L₂ M₂ : Type*} [LieRing L₂] [LieAlgebra R L₂]
-
 variable [AddCommGroup M₂] [Module R M₂] [LieRingModule L₂ M₂] [LieModule R L₂ M₂]
-
 variable {f : L →ₗ⁅R⁆ L₂} {g : M →ₗ[R] M₂}
-
 variable (hf : Surjective f) (hg : Surjective g) (hfg : ∀ x m, ⁅f x, g m⁆ = g ⁅x, m⁆)
 
 theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
@@ -633,7 +627,6 @@ instance (priority := 100) LieAlgebra.isSolvable_of_isNilpotent (R : Type u) (L
 section NilpotentAlgebras
 
 variable (R : Type u) (L : Type v) (L' : Type w)
-
 variable [CommRing R] [LieRing L] [LieAlgebra R L] [LieRing L'] [LieAlgebra R L']
 
 /-- We say a Lie algebra is nilpotent when it is nilpotent as a Lie module over itself via the
@@ -780,9 +773,7 @@ namespace LieIdeal
 open LieModule
 
 variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L)
-
 variable (M : Type*) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
-
 variable (k : ℕ)
 
 /-- Given a Lie module `M` over a Lie algebra `L` together with an ideal `I` of `L`, this is the

I ran tryAtEachStep on all files under Mathlib to find all locations where omega succeeds. For each that was a linarith without an only, I tried replacing it with omega, and I verified that elaboration time got smaller. (In almost all cases, there was a noticeable speedup.) I also replaced some slow aesops along the way.

@@ -273,7 +273,7 @@ theorem isNilpotent_toEndomorphism_of_isNilpotent₂ [IsNilpotent R L M] (x y :
     _root_.IsNilpotent (toEndomorphism R L M x ∘ₗ toEndomorphism R L M y) := by
   obtain ⟨k, hM⟩ := exists_lowerCentralSeries_eq_bot_of_isNilpotent R L M
   replace hM : lowerCentralSeries R L M (2 * k) = ⊥ := by
-    rw [eq_bot_iff, ← hM]; exact antitone_lowerCentralSeries R L M (by linarith)
+    rw [eq_bot_iff, ← hM]; exact antitone_lowerCentralSeries R L M (by omega)
   use k
   ext m
   rw [LinearMap.pow_apply, LinearMap.zero_apply, ← LieSubmodule.mem_bot (R := R) (L := L), ← hM]

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -151,7 +151,7 @@ theorem eventually_iInf_lowerCentralSeries_eq [IsArtinian R M] :
   obtain ⟨n, hn : ∀ m, n ≤ m → lowerCentralSeries R L M n = lowerCentralSeries R L M m⟩ :=
     WellFounded.monotone_chain_condition.mp h_wf ⟨_, antitone_lowerCentralSeries R L M⟩
   refine Filter.eventually_atTop.mpr ⟨n, fun l hl ↦ le_antisymm (iInf_le _ _) (le_iInf fun m ↦ ?_)⟩
-  cases' le_or_lt l m with h h
+  rcases le_or_lt l m with h | h
   · rw [← hn _ hl, ← hn _ (hl.trans h)]
   · exact antitone_lowerCentralSeries R L M (le_of_lt h)
 

For consistency, we also rename Submodule.extendScalars to Submodule.baseChange and likewise for LieSubmodule.

@@ -866,11 +866,11 @@ variable (R A L M : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
   [CommRing A] [Algebra R A]
 
 @[simp]
-lemma LieSubmodule.lowerCentralSeries_tensor_eq_extendScalars (k : ℕ) :
+lemma LieSubmodule.lowerCentralSeries_tensor_eq_baseChange (k : ℕ) :
     lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) k =
-    (lowerCentralSeries R L M k).extendScalars A := by
+    (lowerCentralSeries R L M k).baseChange A := by
   induction' k with k ih; simp
-  simp only [lowerCentralSeries_succ, ih, ← extendScalars_top, lie_extendScalars]
+  simp only [lowerCentralSeries_succ, ih, ← baseChange_top, lie_baseChange]
 
 instance LieModule.instIsNilpotentTensor [IsNilpotent R L M] :
     IsNilpotent A (A ⊗[R] L) (A ⊗[R] M) := by

@@ -3,6 +3,7 @@ Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
 -/
+import Mathlib.Algebra.Lie.BaseChange
 import Mathlib.Algebra.Lie.Solvable
 import Mathlib.Algebra.Lie.Quotient
 import Mathlib.Algebra.Lie.Normalizer
@@ -30,7 +31,6 @@ carries a natural concept of nilpotency. We define these here via the lower cent
 lie algebra, lower central series, nilpotent
 -/
 
-
 universe u v w w₁ w₂
 
 section NilpotentModules
@@ -856,3 +856,25 @@ theorem _root_.LieAlgebra.isNilpotent_ad_of_isNilpotent {L : LieSubalgebra R A}
 #align lie_algebra.is_nilpotent_ad_of_is_nilpotent LieAlgebra.isNilpotent_ad_of_isNilpotent
 
 end OfAssociative
+
+section ExtendScalars
+
+open LieModule TensorProduct
+
+variable (R A L M : Type*) [CommRing R] [LieRing L] [LieAlgebra R L]
+  [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
+  [CommRing A] [Algebra R A]
+
+@[simp]
+lemma LieSubmodule.lowerCentralSeries_tensor_eq_extendScalars (k : ℕ) :
+    lowerCentralSeries A (A ⊗[R] L) (A ⊗[R] M) k =
+    (lowerCentralSeries R L M k).extendScalars A := by
+  induction' k with k ih; simp
+  simp only [lowerCentralSeries_succ, ih, ← extendScalars_top, lie_extendScalars]
+
+instance LieModule.instIsNilpotentTensor [IsNilpotent R L M] :
+    IsNilpotent A (A ⊗[R] L) (A ⊗[R] M) := by
+  obtain ⟨k, hk⟩ := inferInstanceAs (IsNilpotent R L M)
+  exact ⟨k, by simp [hk]⟩
+
+end ExtendScalars

Note: the proof (due to Zassenhaus) makes no assumption about the characteristic of the coefficients.

@@ -325,6 +325,7 @@ noncomputable def nilpotencyLength : ℕ :=
   sInf {k | lowerCentralSeries R L M k = ⊥}
 #align lie_module.nilpotency_length LieModule.nilpotencyLength
 
+@[simp]
 theorem nilpotencyLength_eq_zero_iff [IsNilpotent R L M] :
     nilpotencyLength R L M = 0 ↔ Subsingleton M := by
   let s := {k | lowerCentralSeries R L M k = ⊥}
@@ -350,6 +351,19 @@ theorem nilpotencyLength_eq_succ_iff (k : ℕ) :
   exact Nat.sInf_upward_closed_eq_succ_iff hs k
 #align lie_module.nilpotency_length_eq_succ_iff LieModule.nilpotencyLength_eq_succ_iff
 
+@[simp]
+theorem nilpotencyLength_eq_one_iff [Nontrivial M] :
+    nilpotencyLength R L M = 1 ↔ IsTrivial L M := by
+  rw [nilpotencyLength_eq_succ_iff, ← trivial_iff_lower_central_eq_bot]
+  simp
+
+theorem isTrivial_of_nilpotencyLength_le_one [IsNilpotent R L M] (h : nilpotencyLength R L M ≤ 1) :
+    IsTrivial L M := by
+  nontriviality M
+  cases' Nat.le_one_iff_eq_zero_or_eq_one.mp h with h h
+  · rw [nilpotencyLength_eq_zero_iff] at h; infer_instance
+  · rwa [nilpotencyLength_eq_one_iff] at h
+
 /-- Given a non-trivial nilpotent Lie module `M` with lower central series
 `M = C₀ ≥ C₁ ≥ ⋯ ≥ Cₖ = ⊥`, this is the `k-1`th term in the lower central series (the last
 non-trivial term).
@@ -381,6 +395,35 @@ theorem nontrivial_lowerCentralSeriesLast [Nontrivial M] [IsNilpotent R L M] :
     exact h.2
 #align lie_module.nontrivial_lower_central_series_last LieModule.nontrivial_lowerCentralSeriesLast
 
+theorem lowerCentralSeriesLast_le_of_not_isTrivial [IsNilpotent R L M] (h : ¬ IsTrivial L M) :
+    lowerCentralSeriesLast R L M ≤ lowerCentralSeries R L M 1 := by
+  rw [lowerCentralSeriesLast]
+  replace h : 1 < nilpotencyLength R L M := by
+    by_contra contra
+    have := isTrivial_of_nilpotencyLength_le_one R L M (not_lt.mp contra)
+    contradiction
+  cases' hk : nilpotencyLength R L M with k <;> rw [hk] at h
+  · contradiction
+  · exact antitone_lowerCentralSeries _ _ _ (Nat.lt_succ.mp h)
+
+/-- For a nilpotent Lie module `M` of a Lie algebra `L`, the first term in the lower central series
+of `M` contains a non-zero element on which `L` acts trivially unless the entire action is trivial.
+
+Taking `M = L`, this provides a useful characterisation of Abelian-ness for nilpotent Lie
+algebras. -/
+lemma disjoint_lowerCentralSeries_maxTrivSubmodule_iff [IsNilpotent R L M] :
+    Disjoint (lowerCentralSeries R L M 1) (maxTrivSubmodule R L M) ↔ IsTrivial L M := by
+  refine ⟨fun h ↦ ?_, fun h ↦ by simp⟩
+  nontriviality M
+  by_contra contra
+  have : lowerCentralSeriesLast R L M ≤ lowerCentralSeries R L M 1 ⊓ maxTrivSubmodule R L M :=
+    le_inf_iff.mpr ⟨lowerCentralSeriesLast_le_of_not_isTrivial R L M contra,
+      lowerCentralSeriesLast_le_max_triv R L M⟩
+  suffices ¬ Nontrivial (lowerCentralSeriesLast R L M) by
+    exact this (nontrivial_lowerCentralSeriesLast R L M)
+  rw [h.eq_bot, le_bot_iff] at this
+  exact this ▸ not_nontrivial _
+
 theorem nontrivial_max_triv_of_isNilpotent [Nontrivial M] [IsNilpotent R L M] :
     Nontrivial (maxTrivSubmodule R L M) :=
   Set.nontrivial_mono (lowerCentralSeriesLast_le_max_triv R L M)

@@ -506,6 +506,9 @@ theorem isNilpotent_iff_exists_self_le_ucs :
   simp_rw [LieModule.isNilpotent_iff_exists_ucs_eq_top, ← ucs_comap_incl, comap_incl_eq_top]
 #align lie_submodule.is_nilpotent_iff_exists_self_le_ucs LieSubmodule.isNilpotent_iff_exists_self_le_ucs
 
+theorem ucs_bot_one : (⊥ : LieSubmodule R L M).ucs 1 = LieModule.maxTrivSubmodule R L M := by
+  simp [LieSubmodule.normalizer_bot_eq_maxTrivSubmodule]
+
 end LieSubmodule
 
 section Morphisms

These are all motivated by a result I've proved but I believe they make sense in their own right so I have split them out in the hopes of simplifying review.

@@ -7,6 +7,8 @@ import Mathlib.Algebra.Lie.Solvable
 import Mathlib.Algebra.Lie.Quotient
 import Mathlib.Algebra.Lie.Normalizer
 import Mathlib.LinearAlgebra.Eigenspace.Basic
+import Mathlib.Order.Filter.AtTopBot
+import Mathlib.RingTheory.Artinian
 import Mathlib.RingTheory.Nilpotent
 import Mathlib.Tactic.Monotonicity
 
@@ -68,6 +70,13 @@ theorem lcs_succ : N.lcs (k + 1) = ⁅(⊤ : LieIdeal R L), N.lcs k⁆ :=
   Function.iterate_succ_apply' (fun N' => ⁅⊤, N'⁆) k N
 #align lie_submodule.lcs_succ LieSubmodule.lcs_succ
 
+@[simp]
+lemma lcs_sup {N₁ N₂ : LieSubmodule R L M} {k : ℕ} :
+    (N₁ ⊔ N₂).lcs k = N₁.lcs k ⊔ N₂.lcs k := by
+  induction' k with k ih
+  · simp
+  · simp only [LieSubmodule.lcs_succ, ih, LieSubmodule.lie_sup]
+
 end LieSubmodule
 
 namespace LieModule
@@ -135,6 +144,17 @@ theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M := by
     · exact hk.symm ▸ le_rfl
 #align lie_module.antitone_lower_central_series LieModule.antitone_lowerCentralSeries
 
+theorem eventually_iInf_lowerCentralSeries_eq [IsArtinian R M] :
+    ∀ᶠ l in Filter.atTop, ⨅ k, lowerCentralSeries R L M k = lowerCentralSeries R L M l := by
+  have h_wf : WellFounded ((· > ·) : (LieSubmodule R L M)ᵒᵈ → (LieSubmodule R L M)ᵒᵈ → Prop) :=
+    LieSubmodule.wellFounded_of_isArtinian R L M
+  obtain ⟨n, hn : ∀ m, n ≤ m → lowerCentralSeries R L M n = lowerCentralSeries R L M m⟩ :=
+    WellFounded.monotone_chain_condition.mp h_wf ⟨_, antitone_lowerCentralSeries R L M⟩
+  refine Filter.eventually_atTop.mpr ⟨n, fun l hl ↦ le_antisymm (iInf_le _ _) (le_iInf fun m ↦ ?_)⟩
+  cases' le_or_lt l m with h h
+  · rw [← hn _ hl, ← hn _ (hl.trans h)]
+  · exact antitone_lowerCentralSeries R L M (le_of_lt h)
+
 theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries R L M 1 = ⊥ := by
   constructor <;> intro h
   · erw [eq_bot_iff, LieSubmodule.lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn, h.trivial]; simp

@@ -122,6 +122,7 @@ end LieSubmodule
 
 namespace LieModule
 
+variable {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂] [LieRingModule L M₂] [LieModule R L M₂]
 variable (R L M)
 
 theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M := by
@@ -166,15 +167,23 @@ theorem iterate_toEndomorphism_mem_lowerCentralSeries₂ (x y : L) (m : M) (k :
 
 variable {R L M}
 
-theorem map_lowerCentralSeries_le {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂]
-    [LieRingModule L M₂] [LieModule R L M₂] (k : ℕ) (f : M →ₗ⁅R,L⁆ M₂) :
-    LieSubmodule.map f (lowerCentralSeries R L M k) ≤ lowerCentralSeries R L M₂ k := by
+theorem map_lowerCentralSeries_le (f : M →ₗ⁅R,L⁆ M₂) :
+    (lowerCentralSeries R L M k).map f ≤ lowerCentralSeries R L M₂ k := by
   induction' k with k ih
   · simp only [Nat.zero_eq, lowerCentralSeries_zero, le_top]
   · simp only [LieModule.lowerCentralSeries_succ, LieSubmodule.map_bracket_eq]
     exact LieSubmodule.mono_lie_right _ _ ⊤ ih
 #align lie_module.map_lower_central_series_le LieModule.map_lowerCentralSeries_le
 
+lemma map_lowerCentralSeries_eq {f : M →ₗ⁅R,L⁆ M₂} (hf : Function.Surjective f) :
+    (lowerCentralSeries R L M k).map f = lowerCentralSeries R L M₂ k := by
+  apply le_antisymm (map_lowerCentralSeries_le k f)
+  induction' k with k ih
+  · rwa [lowerCentralSeries_zero, lowerCentralSeries_zero, top_le_iff, f.map_top, f.range_eq_top]
+  · simp only [lowerCentralSeries_succ, LieSubmodule.map_bracket_eq]
+    apply LieSubmodule.mono_lie_right
+    assumption
+
 variable (R L M)
 
 open LieAlgebra
@@ -198,6 +207,12 @@ theorem exists_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent R L M] :
     ∃ k, lowerCentralSeries R L M k = ⊥ :=
   IsNilpotent.nilpotent
 
+@[simp] lemma iInf_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent R L M] :
+    ⨅ k, lowerCentralSeries R L M k = ⊥ := by
+  obtain ⟨k, hk⟩ := exists_lowerCentralSeries_eq_bot_of_isNilpotent R L M
+  rw [eq_bot_iff, ← hk]
+  exact iInf_le _ _
+
 /-- See also `LieModule.isNilpotent_iff_exists_ucs_eq_top`. -/
 theorem isNilpotent_iff : IsNilpotent R L M ↔ ∃ k, lowerCentralSeries R L M k = ⊥ :=
   ⟨fun h => h.nilpotent, fun h => ⟨h⟩⟩
@@ -270,6 +285,18 @@ theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxT
   exact map_lowerCentralSeries_le k (LieSubmodule.Quotient.mk' N)
 #align lie_module.nilpotent_of_nilpotent_quotient LieModule.nilpotentOfNilpotentQuotient
 
+theorem isNilpotent_quotient_iff :
+    IsNilpotent R L (M ⧸ N) ↔ ∃ k, lowerCentralSeries R L M k ≤ N := by
+  rw [LieModule.isNilpotent_iff]
+  refine exists_congr fun k ↦ ?_
+  rw [← LieSubmodule.Quotient.map_mk'_eq_bot_le, map_lowerCentralSeries_eq k
+    (LieSubmodule.Quotient.surjective_mk' N)]
+
+theorem iInf_lcs_le_of_isNilpotent_quot (h : IsNilpotent R L (M ⧸ N)) :
+    ⨅ k, lowerCentralSeries R L M k ≤ N := by
+  obtain ⟨k, hk⟩ := (isNilpotent_quotient_iff R L M N).mp h
+  exact iInf_le_of_le k hk
+
 /-- Given a nilpotent Lie module `M` with lower central series `M = C₀ ≥ C₁ ≥ ⋯ ≥ Cₖ = ⊥`, this is
 the natural number `k` (the number of inclusions).
 
@@ -521,6 +548,10 @@ theorem LieModule.isNilpotent_of_top_iff :
   Equiv.lieModule_isNilpotent_iff LieSubalgebra.topEquiv (1 : M ≃ₗ[R] M) fun _ _ => rfl
 #align lie_module.is_nilpotent_of_top_iff LieModule.isNilpotent_of_top_iff
 
+@[simp] lemma LieModule.isNilpotent_of_top_iff' :
+    IsNilpotent R L {x // x ∈ (⊤ : LieSubmodule R L M)} ↔ IsNilpotent R L M :=
+  Equiv.lieModule_isNilpotent_iff 1 (LinearEquiv.ofTop ⊤ rfl) fun _ _ ↦ rfl
+
 end Morphisms
 
 end NilpotentModules

The key change is the new definition LieModule.weightSpaceOf which shows that the eigenspaces of the action of a single element of a Lie algebra on a representation are Lie submodules. I need to use this fact to develop the theory further.

@@ -244,20 +244,13 @@ theorem isNilpotent_toEndomorphism_of_isNilpotent₂ [IsNilpotent R L M] (x y :
   rw [LinearMap.pow_apply, LinearMap.zero_apply, ← LieSubmodule.mem_bot (R := R) (L := L), ← hM]
   exact iterate_toEndomorphism_mem_lowerCentralSeries₂ R L M x y m k
 
-/-- For a nilpotent Lie module, the weight space of the 0 weight is the whole module.
-
-This result will be used downstream to show that weight spaces are Lie submodules, at which time
-it will be possible to state it in the language of weight spaces. -/
-theorem iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
-    ⨅ x : L, (toEndomorphism R L M x).maximalGeneralizedEigenspace 0 = ⊤ := by
+@[simp] lemma maxGenEigenSpace_toEndomorphism_eq_top [IsNilpotent R L M] (x : L) :
+    ((toEndomorphism R L M x).maximalGeneralizedEigenspace 0) = ⊤ := by
   ext m
-  simp only [Module.End.mem_maximalGeneralizedEigenspace, Submodule.mem_top, sub_zero, iff_true_iff,
-    zero_smul, Submodule.mem_iInf]
-  intro x
+  simp only [Module.End.mem_maximalGeneralizedEigenspace, zero_smul, sub_zero, Submodule.mem_top,
+    iff_true]
   obtain ⟨k, hk⟩ := exists_forall_pow_toEndomorphism_eq_zero R L M
-  use k; rw [hk]
-  exact LinearMap.zero_apply m
-#align lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
+  exact ⟨k, by simp [hk x]⟩
 
 /-- If the quotient of a Lie module `M` by a Lie submodule on which the Lie algebra acts trivially
 is nilpotent then `M` is nilpotent.
@@ -560,12 +553,6 @@ theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
   LieModule.exists_forall_pow_toEndomorphism_eq_zero R L L
 #align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra
 
-/-- See also `LieAlgebra.zero_rootSpace_eq_top_of_nilpotent`. -/
-theorem LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L] :
-    ⨅ x : L, (ad R L x).maximalGeneralizedEigenspace 0 = ⊤ :=
-  LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L
-#align lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
-
 -- TODO Generalise the below to Lie modules if / when we define morphisms, equivs of Lie modules
 -- covering a Lie algebra morphism of (possibly different) Lie algebras.
 variable {R L L'}

@@ -154,6 +154,16 @@ theorem iterate_toEndomorphism_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ)
     exact LieSubmodule.lie_mem_lie _ _ (LieSubmodule.mem_top x) ih
 #align lie_module.iterate_to_endomorphism_mem_lower_central_series LieModule.iterate_toEndomorphism_mem_lowerCentralSeries
 
+theorem iterate_toEndomorphism_mem_lowerCentralSeries₂ (x y : L) (m : M) (k : ℕ) :
+    (toEndomorphism R L M x ∘ₗ toEndomorphism R L M y)^[k] m ∈
+      lowerCentralSeries R L M (2 * k) := by
+  induction' k with k ih; simp
+  have hk : 2 * k.succ = (2 * k + 1) + 1 := rfl
+  simp only [lowerCentralSeries_succ, Function.comp_apply, Function.iterate_succ', hk,
+      toEndomorphism_apply_apply, LinearMap.coe_comp, toEndomorphism_apply_apply]
+  refine' LieSubmodule.lie_mem_lie _ _ (LieSubmodule.mem_top x) _
+  exact LieSubmodule.lie_mem_lie _ _ (LieSubmodule.mem_top y) ih
+
 variable {R L M}
 
 theorem map_lowerCentralSeries_le {M₂ : Type w₁} [AddCommGroup M₂] [Module R M₂]
@@ -184,6 +194,10 @@ class IsNilpotent : Prop where
   nilpotent : ∃ k, lowerCentralSeries R L M k = ⊥
 #align lie_module.is_nilpotent LieModule.IsNilpotent
 
+theorem exists_lowerCentralSeries_eq_bot_of_isNilpotent [IsNilpotent R L M] :
+    ∃ k, lowerCentralSeries R L M k = ⊥ :=
+  IsNilpotent.nilpotent
+
 /-- See also `LieModule.isNilpotent_iff_exists_ucs_eq_top`. -/
 theorem isNilpotent_iff : IsNilpotent R L M ↔ ∃ k, lowerCentralSeries R L M k = ⊥ :=
   ⟨fun h => h.nilpotent, fun h => ⟨h⟩⟩
@@ -205,14 +219,30 @@ instance (priority := 100) trivialIsNilpotent [IsTrivial L M] : IsNilpotent R L
   ⟨by use 1; change ⁅⊤, ⊤⁆ = ⊥; simp⟩
 #align lie_module.trivial_is_nilpotent LieModule.trivialIsNilpotent
 
-theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
+theorem exists_forall_pow_toEndomorphism_eq_zero [hM : IsNilpotent R L M] :
     ∃ k : ℕ, ∀ x : L, toEndomorphism R L M x ^ k = 0 := by
   obtain ⟨k, hM⟩ := hM
   use k
   intro x; ext m
   rw [LinearMap.pow_apply, LinearMap.zero_apply, ← @LieSubmodule.mem_bot R L M, ← hM]
   exact iterate_toEndomorphism_mem_lowerCentralSeries R L M x m k
-#align lie_module.nilpotent_endo_of_nilpotent_module LieModule.nilpotent_endo_of_nilpotent_module
+#align lie_module.nilpotent_endo_of_nilpotent_module LieModule.exists_forall_pow_toEndomorphism_eq_zero
+
+theorem isNilpotent_toEndomorphism_of_isNilpotent [IsNilpotent R L M] (x : L) :
+    _root_.IsNilpotent (toEndomorphism R L M x) := by
+  change ∃ k, toEndomorphism R L M x ^ k = 0
+  have := exists_forall_pow_toEndomorphism_eq_zero R L M
+  tauto
+
+theorem isNilpotent_toEndomorphism_of_isNilpotent₂ [IsNilpotent R L M] (x y : L) :
+    _root_.IsNilpotent (toEndomorphism R L M x ∘ₗ toEndomorphism R L M y) := by
+  obtain ⟨k, hM⟩ := exists_lowerCentralSeries_eq_bot_of_isNilpotent R L M
+  replace hM : lowerCentralSeries R L M (2 * k) = ⊥ := by
+    rw [eq_bot_iff, ← hM]; exact antitone_lowerCentralSeries R L M (by linarith)
+  use k
+  ext m
+  rw [LinearMap.pow_apply, LinearMap.zero_apply, ← LieSubmodule.mem_bot (R := R) (L := L), ← hM]
+  exact iterate_toEndomorphism_mem_lowerCentralSeries₂ R L M x y m k
 
 /-- For a nilpotent Lie module, the weight space of the 0 weight is the whole module.
 
@@ -224,7 +254,7 @@ theorem iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
   simp only [Module.End.mem_maximalGeneralizedEigenspace, Submodule.mem_top, sub_zero, iff_true_iff,
     zero_smul, Submodule.mem_iInf]
   intro x
-  obtain ⟨k, hk⟩ := nilpotent_endo_of_nilpotent_module R L M
+  obtain ⟨k, hk⟩ := exists_forall_pow_toEndomorphism_eq_zero R L M
   use k; rw [hk]
   exact LinearMap.zero_apply m
 #align lie_module.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
@@ -527,7 +557,7 @@ open LieAlgebra
 
 theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
     ∃ k : ℕ, ∀ x : L, ad R L x ^ k = 0 :=
-  LieModule.nilpotent_endo_of_nilpotent_module R L L
+  LieModule.exists_forall_pow_toEndomorphism_eq_zero R L L
 #align lie_algebra.nilpotent_ad_of_nilpotent_algebra LieAlgebra.nilpotent_ad_of_nilpotent_algebra
 
 /-- See also `LieAlgebra.zero_rootSpace_eq_top_of_nilpotent`. -/

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

This has nice performance benefits.

@@ -442,7 +442,7 @@ section Morphisms
 
 open LieModule Function
 
-variable {L₂ M₂ : Type _} [LieRing L₂] [LieAlgebra R L₂]
+variable {L₂ M₂ : Type*} [LieRing L₂] [LieAlgebra R L₂]
 
 variable [AddCommGroup M₂] [Module R M₂] [LieRingModule L₂ M₂] [LieModule R L₂ M₂]
 
@@ -665,9 +665,9 @@ namespace LieIdeal
 
 open LieModule
 
-variable {R L : Type _} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L)
+variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L] (I : LieIdeal R L)
 
-variable (M : Type _) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
+variable (M : Type*) [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L M]
 
 variable (k : ℕ)
 

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2021 Oliver Nash. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Oliver Nash
-
-! This file was ported from Lean 3 source module algebra.lie.nilpotent
-! leanprover-community/mathlib commit 6b0169218d01f2837d79ea2784882009a0da1aa1
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.Lie.Solvable
 import Mathlib.Algebra.Lie.Quotient
@@ -15,6 +10,8 @@ import Mathlib.LinearAlgebra.Eigenspace.Basic
 import Mathlib.RingTheory.Nilpotent
 import Mathlib.Tactic.Monotonicity
 
+#align_import algebra.lie.nilpotent from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
+
 /-!
 # Nilpotent Lie algebras
 

This is the second half of the changes originally in #5699, removing all occurrences of ; after a space and implementing a linter rule to enforce it.

In most cases this 2-character substring has a space after it, so the following command was run first:

find . -type f -name "*.lean" -exec sed -i -E 's/ ; /; /g' {} \;

The remaining cases were few enough in number that they were done manually.

@@ -140,7 +140,7 @@ theorem antitone_lowerCentralSeries : Antitone <| lowerCentralSeries R L M := by
 theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries R L M 1 = ⊥ := by
   constructor <;> intro h
   · erw [eq_bot_iff, LieSubmodule.lieSpan_le]; rintro m ⟨x, n, hn⟩; rw [← hn, h.trivial]; simp
-  · rw [LieSubmodule.eq_bot_iff] at h ; apply IsTrivial.mk; intro x m; apply h
+  · rw [LieSubmodule.eq_bot_iff] at h; apply IsTrivial.mk; intro x m; apply h
     apply LieSubmodule.subset_lieSpan
     -- Porting note: was `use x, m; rfl`
     simp only [LieSubmodule.top_coe, Subtype.exists, LieSubmodule.mem_top, exists_prop, true_and,

@@ -222,7 +222,7 @@ theorem nilpotent_endo_of_nilpotent_module [hM : IsNilpotent R L M] :
 This result will be used downstream to show that weight spaces are Lie submodules, at which time
 it will be possible to state it in the language of weight spaces. -/
 theorem iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L M] :
-    (⨅ x : L, (toEndomorphism R L M x).maximalGeneralizedEigenspace 0) = ⊤ := by
+    ⨅ x : L, (toEndomorphism R L M x).maximalGeneralizedEigenspace 0 = ⊤ := by
   ext m
   simp only [Module.End.mem_maximalGeneralizedEigenspace, Submodule.mem_top, sub_zero, iff_true_iff,
     zero_smul, Submodule.mem_iInf]
@@ -535,7 +535,7 @@ theorem LieAlgebra.nilpotent_ad_of_nilpotent_algebra [IsNilpotent R L] :
 
 /-- See also `LieAlgebra.zero_rootSpace_eq_top_of_nilpotent`. -/
 theorem LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent [IsNilpotent R L] :
-    (⨅ x : L, (ad R L x).maximalGeneralizedEigenspace 0) = ⊤ :=
+    ⨅ x : L, (ad R L x).maximalGeneralizedEigenspace 0 = ⊤ :=
   LieModule.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent R L L
 #align lie_algebra.infi_max_gen_zero_eigenspace_eq_top_of_nilpotent LieAlgebra.iInf_max_gen_zero_eigenspace_eq_top_of_nilpotent
 

@@ -149,7 +149,7 @@ theorem trivial_iff_lower_central_eq_bot : IsTrivial L M ↔ lowerCentralSeries
 #align lie_module.trivial_iff_lower_central_eq_bot LieModule.trivial_iff_lower_central_eq_bot
 
 theorem iterate_toEndomorphism_mem_lowerCentralSeries (x : L) (m : M) (k : ℕ) :
-    (toEndomorphism R L M x^[k]) m ∈ lowerCentralSeries R L M k := by
+    (toEndomorphism R L M x)^[k] m ∈ lowerCentralSeries R L M k := by
   induction' k with k ih
   · simp only [Nat.zero_eq, Function.iterate_zero, lowerCentralSeries_zero, LieSubmodule.mem_top]
   · simp only [lowerCentralSeries_succ, Function.comp_apply, Function.iterate_succ',
@@ -681,7 +681,7 @@ lower central series of `M` as an `I`-module. The advantage of using this defini
 
 See also `LieIdeal.coe_lcs_eq`. -/
 def lcs : LieSubmodule R L M :=
-  ((fun N => ⁅I, N⁆)^[k]) ⊤
+  (fun N => ⁅I, N⁆)^[k] ⊤
 #align lie_ideal.lcs LieIdeal.lcs
 
 @[simp]

I was looking on https://github.com/leanprover-community/mathlib4/pull/4933 to see what simp related porting notes I could improve after https://github.com/leanprover/lean4/pull/2266 lands in Lean 4. Mostly things I found could be cleaned up in any case, and so I've moved those into this PR.

There is lots more work to do diagnosing all the simp-related porting notes!

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -456,10 +456,7 @@ variable (hf : Surjective f) (hg : Surjective g) (hfg : ∀ x m, ⁅f x, g m⁆
 theorem Function.Surjective.lieModule_lcs_map_eq (k : ℕ) :
     (lowerCentralSeries R L M k : Submodule R M).map g = lowerCentralSeries R L₂ M₂ k := by
   induction' k with k ih
-  · -- Porting note: was `simp [LinearMap.range_eq_top, hg]`
-    simp only [Nat.zero_eq, lowerCentralSeries_zero, LieSubmodule.top_coeSubmodule,
-      Submodule.map_top, LinearMap.range_eq_top]
-    exact hg
+  · simpa [LinearMap.range_eq_top]
   · suffices
       g '' {m | ∃ (x : L) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, n⁆ = m} =
         {m | ∃ (x : L₂) (n : _), n ∈ lowerCentralSeries R L M k ∧ ⁅x, g n⁆ = m} by