module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Lie.InvariantForm | {
"line": 61,
"column": 8
} | {
"line": 61,
"column": 25
} | [
{
"pp": "case refine_1.H\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nΦ : LinearMap.BilinForm R M\ninst✝¹ : LieAlgebra R L\ninst✝ : LieModule R L M\nh : LinearMap.BilinForm.lieInvariant L Φ\nx : L\ny... | LieHom.lie_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 15
} | [
{
"pp": "case h.refine_2.h\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np q : Submodule R M\nhpq : Codisjoint p q\nB : BilinForm R M\nhB : ∀ x ∈ p, ∀ y ∈ q, (B x) y = 0\nz : M\nhz : z ∈ p\nh : B z = 0\nx : M\nhx : x ∈ p\n⊢ ((B.restrict p) ⟨z, hz⟩) ⟨x, hx⟩ =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 341,
"column": 47
} | {
"line": 341,
"column": 58
} | [
{
"pp": "V : Type u_5\nK : Type u_6\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nB : BilinForm K V\nW : Submodule K V\nb₁ : B.IsRefl\nb₂ : (B.restrict W).Nondegenerate\nb₃ : B.Nondegenerate\nh : W = ⊤\nx : V\nhx : x ∈ B.orthogonal ⊤\ny : V\n⊢ (B y) x = 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 177,
"column": 22
} | {
"line": 177,
"column": 75
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieAbelian ↥I\nI : ... | by rw [← sup_inf_assoc_of_le _ hJI, this, top_inf_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 15
} | [
{
"pp": "case refine_1\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLieA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 15
} | [
{
"pp": "case h\nR : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : IsIrreducible R L M\naux : ∀ (x y : M), x = y\nm : M\n⊢ m ∈ ⊥ ↔ m ∈ ⊤",
"usedConstants": [
"LieSubmodule.instSetLike",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Normalizer | {
"line": 86,
"column": 56
} | {
"line": 86,
"column": 81
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN N' : LieSubmodule R L M\n⊢ ⁅⊤, N⁆ ≤ N' ↔ N ≤ N'.normalizer",
"usedConstants": [
"Lie... | by rw [lie_le_iff]; tauto | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 47
} | [
{
"pp": "case inr.left.a.a\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\ns : Finset (LieIdeal R L)\nhs : ↑s ⊆ {I | IsAtom I}\nI : LieIdeal R L\nhI✝ : I ≤ s.sup id\nS : Set (LieIdeal R L) := ⋯\nhI : I < s.sup id\nJ : LieIdeal R L\nhJs : J... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 301,
"column": 4
} | {
"line": 301,
"column": 15
} | [
{
"pp": "case sSupIndep_isAtom\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSimple R L\n⊢ sSupIndep {I | IsAtom I}",
"usedConstants": [
"LieAlgebra.toModule",
"sSupIndep",
"Eq.mpr",
"isAtom_iff_eq_top._simp_1",
"LieRin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Normalizer | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 19
} | [
{
"pp": "case refine_2\nR : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nH : LieSubalgebra R L\nh : ∀ m ∈ LieModule.maxTrivSubmodule R (↥H) (L ⧸ H.toLieSubmodule), m = 0\nx : L\nhx : x ∈ H.normalizer\ny : L ⧸ H.toLieSubmodule := (LieSubmodule.Quotient.mk' H.toLieSubmo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 37
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : Nontrivial L\nH : LieSubalgebra R L\ninst✝ : H.IsCartanSubalgebra\ne : H = ⊥\n⊢ False",
"usedConstants": [
"LieRing.toAddCommGroup",
"exists_ne",
"SubtractionMonoid.toSubNegZeroMonoi... | obtain ⟨x, hx⟩ := exists_ne (0 : L) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.CartanSubalgebra | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 86
} | [
{
"pp": "case h\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nx : L\n⊢ x ∈ (toLieSubalgebra R L I).normalizer ↔ x ∈ ⊤",
"usedConstants": [
"Eq.mpr",
"LieRing.toAddCommGroup",
"LieSubalgebra.instSetLike",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 49,
"column": 32
} | {
"line": 49,
"column": 43
} | [
{
"pp": "R : Type u_1\nx y : R\nn m p : ℕ\ninst✝ : Semiring R\nhp : n + m ≤ p + 1\nh_comm : Commute x y\nhy : y ^ n = 0\nx✝ : ℕ × ℕ\ni j : ℕ\nhij : (i, j) ∈ Finset.antidiagonal p\n⊢ i + j = p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Engel | {
"line": 115,
"column": 7
} | {
"line": 115,
"column": 80
} | [
{
"pp": "R : Type u₁\nL : Type u₂\nM : Type u₄\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nn i j : ℕ\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Engel | {
"line": 110,
"column": 2
} | {
"line": 115,
"column": 87
} | [
{
"pp": "R : Type u₁\nL : Type u₂\nM : Type u₄\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nn i j : ℕ\... | suffices
∀ l,
((⊤ : LieIdeal R L).lcs M (i + l) : Submodule R M) ≤
(I.lcs M j : Submodule R M).map (toEnd R L M x ^ l) ⊔
(I.lcs M (j + 1) : Submodule R M)
by simpa only [bot_sup_eq, LieIdeal.incl_coe, Submodule.map_zero, hxn] using this n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Lie.Engel | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 21
} | [
{
"pp": "case h\nR : Type u₁\nL : Type u₂\nM : Type u₄\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nI : LieIdeal R L\nx : L\nhxI : R ∙ x ⊔ (LieIdeal.toLieSubalgebra R L I).toSubmodule = ⊤\nhI... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 148,
"column": 10
} | {
"line": 148,
"column": 22
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nk✝ : ℕ\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nk : ℕ\nih : lowerCentralSeries R L (↥N) k = comap N.incl (lcs k N)\n⊢ ... | N.range_incl | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 253,
"column": 2
} | {
"line": 258,
"column": 36
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ derivedSeries R L k ≤ lowerCentralSeries R L L k",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"LieModule.lowerCentralSeries_zero",
"le_refl",
"Nat.recAux",
... | induction k with
| zero => rw [derivedSeries_def, derivedSeriesOfIdeal_zero, lowerCentralSeries_zero]
| succ k h =>
have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]
exact LieSubmodule.mono_lie h' h | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 253,
"column": 2
} | {
"line": 258,
"column": 36
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ derivedSeries R L k ≤ lowerCentralSeries R L L k",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"LieModule.lowerCentralSeries_zero",
"le_refl",
"Nat.recAux",
... | induction k with
| zero => rw [derivedSeries_def, derivedSeriesOfIdeal_zero, lowerCentralSeries_zero]
| succ k h =>
have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]
exact LieSubmodule.mono_lie h' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 253,
"column": 2
} | {
"line": 258,
"column": 36
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nk : ℕ\n⊢ derivedSeries R L k ≤ lowerCentralSeries R L L k",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"LieModule.lowerCentralSeries_zero",
"le_refl",
"Nat.recAux",
... | induction k with
| zero => rw [derivedSeries_def, derivedSeriesOfIdeal_zero, lowerCentralSeries_zero]
| succ k h =>
have h' : derivedSeries R L k ≤ ⊤ := by simp only [le_top]
rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]
exact LieSubmodule.mono_lie h' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 20
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝¹² : CommRing R\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : LieAlgebra R L\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : LieRingModule L M\nk✝ : ℕ\nN✝ : LieSubmodule R L M\nM₂✝ : Type w₁\ninst✝⁶ : AddCommGroup M₂✝\ninst✝⁵ : Module R M₂✝\ninst✝⁴ : LieRingModule L M... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 15
} | [
{
"pp": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R (↥(toEnd R L M).range) M k) = ↑⊥\n⊢ ↑(lowerCentralSeries R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 519,
"column": 4
} | {
"line": 519,
"column": 15
} | [
{
"pp": "case h\nR : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nk : ℕ\nhk : ↑(lowerCentralSeries R L M k) = ↑⊥\n⊢ ↑(lowerCentralSeries R (↥(toEnd R L M).range... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 589,
"column": 75
} | {
"line": 590,
"column": 88
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\n⊢ LieModule.IsNilpotent L M ↔ ∃ k, ucs k ⊥ = ⊤",
"usedConstants": [
"Eq.mpr",
"LieSubm... | by
rw [LieModule.isNilpotent_iff R]; exact exists_congr fun k => by simp [ucs_eq_top_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 646,
"column": 2
} | {
"line": 646,
"column": 18
} | [
{
"pp": "case h.a\nR : Type u\nL : Type v\nM : Type w\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁶ : LieRing L₂\ninst✝⁵ : LieAlgebra R L₂\ninst✝⁴ : A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Fixed | {
"line": 242,
"column": 4
} | {
"line": 242,
"column": 84
} | [
{
"pp": "case inr.h\nG : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\nx : F\nf g : Polynomial ↥(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (subfiel... | rwa [← one_mul (minpoly G F x), hg3, mul_left_inj' (monic G F x).ne_zero] at hfg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.Fixed | {
"line": 264,
"column": 4
} | {
"line": 265,
"column": 24
} | [
{
"pp": "G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\ns : Finset F\nhs : LinearIndependent ↥(subfield G F) fun i ↦ ↑i\n⊢ #s ≤ Fintype.card G",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 31
} | [
{
"pp": "F : Type u\nK : Type v\nL : Type w\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Field F\ninst✝⁴ : Algebra K L\ninst✝³ : Algebra F K\ninst✝² : Algebra F L\ninst✝¹ : IsScalarTower F K L\nf : F[X]\ninst✝ : IsSplittingField K L ((mapAlg F K) f)\n⊢ (Polynomial.map (algebraMap K L) ((mapAlg F K) f)).Splits"... | apply IsSplittingField.splits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.FieldTheory.SplittingField.IsSplittingField | {
"line": 164,
"column": 19
} | {
"line": 164,
"column": 44
} | [
{
"pp": "K : Type v\nL : Type w\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\np : K[X]\nF : IntermediateField K L\nh : (Polynomial.map (algebraMap K L) p).Splits\nhF : ∀ x ∈ p.rootSet L, x ∈ F\nthis :\n (Polynomial.map (algebraMap K L) p).Splits →\n (∀ a ∈ (Polynomial.map (algebraMap K L) p).roo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Dimension | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 30
} | [
{
"pp": "case pos\nR : Type u\nS : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : StrongRankCondition R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nA B : Subalgebra R S\ninst✝¹ : Free R ↥A\ninst✝ : Free R ↥B\nleft✝ : Module.Finite R ↥A\nright✝ : Module.Finite R ↥B\n⊢ finrank R ↥(A ⊔ B) ≤ finrank R ↥A * finrank R ↥B",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 55
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Algebra F R\nx : R\nP : F[X]\nhP₁ : (Minpoly.toAdjoin F x) ((AdjoinRoot.mk (minpoly F x)) P) = 0\n⊢ (aeval x) P = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 17
} | [
{
"pp": "case a\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nK : IntermediateField F E\nh : K.toSubalgebra = Algebra.adjoin F S\nx : E\n⊢ x ∈ K.toSubalgebra → x⁻¹ ∈ K.toSubalgebra",
"usedConstants": [
"IntermediateField.inv_mem"
]
}
] | exact K.inv_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Adjoin.Field | {
"line": 118,
"column": 25
} | {
"line": 118,
"column": 47
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) ((minpoly K x).comp (-X))).Splits",
"usedConstants": [
"Eq.mpr",
"Polynomial.instNeg",
"Algebra.algebraMap",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 46
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) ((-1) ^ (minpoly K x).natDegree)).Splits",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 48
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K (x + (algebraMap K L) r))).Splits",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Algebra.algebraM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 44
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K (x - (algebraMap K L) r))).Splits",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"AddGroupWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 29
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K ((algebraMap K L) r + x))).Splits",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.Field | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 28
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nr : K\ng : (map (algebraMap K L) (minpoly K x)).Splits\n⊢ (map (algebraMap K L) (minpoly K ((algebraMap K L) r - x))).Splits",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 122,
"column": 4
} | {
"line": 124,
"column": 88
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nK : IntermediateField F E\nthis : (∃ s, ↑s ⊆ ↑K ∧ adjoin F ↑s = K) ↔ ∃ t, adjoin F ↑t = K\n⊢ Algebra.EssFiniteType F ↥K ↔ K.FG",
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"Lattice.toSemilatti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 36
} | [
{
"pp": "case h\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : EssFiniteType F E\ninst✝ : Algebra.IsAlgebraic F E\ns : Finset E\nhs : adjoin F ↑s = ⊤\n⊢ (adjoin F ↑s).toSubalgebra = ⊤",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 54
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS T : IntermediateField F E\n⊢ (S ⊔ T).toSubfield = Subfield.closure (↑S.toSubfield ∪ ↑T.toSubfield)",
"usedConstants": [
"Lattice.toSemilatticeSup",
"Algebra.algebraMap",
"CommSemiring.toSemiring... | simp_rw [sup_def, adjoin_toSubfield, coe_toSubfield] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 58
} | [
{
"pp": "F : Type u_1\ninst✝⁶ : Field F\nE : Type u_2\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Algebra F K\ninst✝¹ : Algebra E K\ninst✝ : IsScalarTower F E K\nL : IntermediateField F K\nhalg : Algebra.IsAlgebraic F E ∨ Algebra.IsAlgebraic F ↥L\ni : E →ₐ[F] K := IsScalarT... | apply_fun _ using Subalgebra.restrictScalars_injective F | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 35
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set (IntermediateField F E)\nx : E\n⊢ x ∈ sInf S ↔ ∀ p ∈ S, x ∈ p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 13
} | [
{
"pp": "case a\nF : Type u_1\ninst✝⁵ : Field F\nE : Type u_2\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nK : Type u_3\ninst✝² : Field K\ninst✝¹ : Algebra F K\nι : Sort u_4\ninst✝ : Nonempty ι\nf : E →ₐ[F] K\ns : ι → IntermediateField F E\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"usedConstants": [
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 13
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nS : Set E\nK : Subfield E\nHF : Set.range ⇑(algebraMap F E) ⊆ ↑K\nHS : S ⊆ ↑K\n⊢ (adjoin F S).toSubfield ≤ K",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 37
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nL' : Type u_3\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Field L'\ninst✝¹ : Algebra K L\ninst✝ : Algebra K L'\nf : L →ₐ[K] L'\nS : IntermediateField K L'\nh : S ≤ f.fieldRange\n⊢ map f (comap f S) = S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 46
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\np : 𝕜[X]\nk : 𝕜\nhk : k ∈ σ a\nq : 𝕜[X] := ⋯\nhroot : (C k - X) * -(q / (X - C k)) = q\naeval_q_eq : ↑ₐ (eval k p) - (aeval a) p = (aeval a) q\nhcomm : Commute ((aeval a) (C k - X)) ((aeval a) (-(q / (X - C k))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 81,
"column": 90
} | {
"line": 91,
"column": 49
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\np : 𝕜[X]\n⊢ (fun x ↦ eval x p) '' σ a ⊆ σ ((aeval a) p)",
"usedConstants": [
"Polynomial.mul_div_eq_iff_isRoot",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | by
rintro _ ⟨k, hk, rfl⟩
let q := C (eval k p) - p
have hroot : IsRoot q k := by simp only [q, eval_C, eval_sub, sub_self, IsRoot.def]
rw [← mul_div_eq_iff_isRoot, ← neg_mul_neg, neg_sub] at hroot
have aeval_q_eq : ↑ₐ (eval k p) - aeval a p = aeval a q := by
simp only [q, aeval_C, map_sub]
rw [mem_iff, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 114,
"column": 28
} | {
"line": 114,
"column": 71
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\np : 𝕜[X]\nhdeg : 0 < p.degree\nk : 𝕜\nhprod : C k - p = C (C k - p).leadingCoeff * (Multiset.map (fun x ↦ X - C x) (C k - p).roots).prod\nh_ne : C k - p ≠ 0\nlead_ne : (C k - p).leadingC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 44
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\nn : ℕ\n⊢ (fun x ↦ x ^ n) '' σ a ⊆ σ (a ^ n)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 9
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\nn : ℕ\nhn : 0 < n\n⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 44
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : IsAlgClosed 𝕜\na : A\nha : (σ a).Nonempty\nn : ℕ\n⊢ σ (a ^ n) = (fun x ↦ x ^ n) '' σ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\np : A\nhp : IsIdempotentElem p\na✝ : Nontrivial A\na : 𝕜\nha : a ∈ (fun x ↦ eval x (X ^ 2 - X)) ⁻¹' spectrum 𝕜 ((aeval p) (X ^ 2 - X))\n⊢ a ^ 2 = a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 198,
"column": 17
} | {
"line": 198,
"column": 41
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nH : ∀ (p : k[X]), p.Monic → Irreducible p → ∃ x, eval x p = 0\np : k[X]\nhp : Irreducible p\nx : k\nhx : eval x (p * C p.leadingCoeff⁻¹) = 0\n⊢ eval x p = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 19
} | [
{
"pp": "k : Type u\ninst✝² : Field k\nk' : Type u\ninst✝¹ : Field k'\ne : k ≃+* k'\ninst✝ : IsAlgClosed k\np : k'[X]\nhmp : p.Monic\nhp : Irreducible p\n⊢ (map e.symm.toRingHom p).degree ≠ 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"WithBot... | rw [degree_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 35
} | [
{
"pp": "k : Type u\ninst✝³ : Field k\nK : Type v\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Algebra k K\nx : k[X]\nhu : x ∈ nonunits k[X]\nh0 : x ≠ 0\nw✝¹ w✝ : k[X]\nh : ∀ (a : K), (aeval a) (x * w✝¹) ≠ 0 ∨ (aeval a) (x * w✝) ≠ 0\n⊢ (map (algebraMap k K) x).degree ≠ 0",
"usedConstants": [
"Eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 40
} | [
{
"pp": "M : Type w\ninst✝¹ : Field M\ninst✝ : IsAlgClosed M\np : M[X]\nhp : 0 < p.degree\nx : M\n⊢ (p - C x).degree ≠ 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"CommSemiring.toSemiring",
"WithBot.zero",
"H... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 320,
"column": 17
} | {
"line": 320,
"column": 52
} | [
{
"pp": "M : Type w\ninst✝¹ : Field M\ninst✝ : IsAlgClosed M\np : M[X]\nhp : 0 < p.degree\nx y : M\nhy : eval y (p - C x) = 0\n⊢ (fun x ↦ eval x p) y = x",
"usedConstants": [
"Polynomial.eval",
"id",
"Field.toSemifield",
"Semifield.toDivisionSemiring",
"DivisionSemiring.toSemir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Extension | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 38
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nS : Set E\nL₁ L₂ L₃ : Lifts F E K\nh₁₂ : L₁.carrier ≤ L₂.carrier\nh₁₂' : ∀ (x : ↥L₁.carrier), L₂.emb ((inclusion h₁₂) x) = L₁.emb x\nh₂₃ : L₂.carrier ≤ L₃.carrier\n... | ← inclusion_inclusion h₁₂ h₂₃, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 36
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun _i xi ↦ x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 52
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ι → End R M\nμ : ι → R\ni : ι\nh✝ : ∀ (j : ι), MapsTo ⇑(f j) ↑((f i).maxGenEigenspace (μ i)) ↑((f i).maxGenEigenspace (μ i))\nthis : Nonempty ι\np : Submodule R M := (f i).maxGenEigenspace (μ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 20
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : ι → End R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nχ₁ : ι → R\na✝ : χ₁ ∈ {χ | ⨅ i, (f i).maxGenEigenspace (χ i) ≠ ⊥}\nχ₂ : ι → R\nhχ₁₂ : ⨅ i, (f i).maxGenEigenspace (χ₁ i) = ⨅ i, (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 47
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : ι → End R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nh : ∀ (l : ι) (χ : ι → R), MapsTo (⇑(f l)) (⨅ i, ↑((f i).maxGenEigenspace (χ i))) (⨅ i, ↑((f i).maxGenEigenspace (χ i)))\nthis :\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Triangularizable | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 50
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\np : Submodule K V\nf : End K V\ninst✝ : FiniteDimensional K V\nh : ∀ x ∈ p, f x ∈ p\nk : ℕ∞\nm : K →₀ V\nhm₂ : ∀ (i : K), m i ∈ (f.genEigenspace i) k\nhm₀ : (m.sum fun _i xi ↦ xi) ∈ p\nhm₁ : (m.sum fun _i xi ↦ x... | rw [LinearMap.sub_apply, algebraMap_end_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 46
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : ι → End R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nh : ∀ (l : ι) (χ : ι → R), MapsTo (⇑(f l)) (⨅ i, ↑((f i).maxGenEigenspace (χ i))) (⨅ i, ↑((f i).maxGenEigenspace (χ i)))\nχ₁ χ₂ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 97
} | [
{
"pp": "F : Type u_1\ninst✝⁵ : Field F\nE : Type u_2\ninst✝⁴ : Field E\ninst✝³ : Algebra F E\nS : Set E\nM : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MulSemiringAction M E\ninst✝ : SMulCommClass M F E\nm : M\n⊢ (∀ x ∈ adjoin F S, m • x = x) ↔ ∀ x ∈ S, m • x = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 176,
"column": 66
} | {
"line": 176,
"column": 77
} | [
{
"pp": "ι : Type u_1\nK : Type u_3\ninst✝³ : Field K\nn : ℕ\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : FiniteDimensional K M\nf : ι → End K M\nh : ∀ (i j : ι) (φ : K), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)\nh' : ∀ (i : ι), ⨆ μ, (f i).maxGenEigenspace μ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 36
} | [
{
"pp": "K : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nι : Type u_5\nt : ι → IntermediateField K L\np : ι → K[X]\ns : Finset ι\nh0 : ∏ i ∈ s, p i ≠ 0\nF : IntermediateField K L := ⨆ i ∈ s, t i\nhF : ∀ i ∈ s, t i ≤ F\nh : ∀ i ∈ s, (Polynomial.map (algebraMap K ↥(t i)) (p i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 67
} | [
{
"pp": "case ind.inr\nι : Type u_1\nK : Type u_3\ninst✝³ : Field K\nn : ℕ\nM : Type u_4\ninst✝² : AddCommGroup M\ninst✝¹ : Module K M\ninst✝ : FiniteDimensional K M\nf : ι → End K M\nh : ∀ (i j : ι) (φ : K), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)\nh' : ∀ (i : ι), ⨆ μ, (f i).maxGe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 325,
"column": 4
} | {
"line": 326,
"column": 11
} | [
{
"pp": "case refine_2\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhp : Nat.Prime (finrank F E)\nK : IntermediateField F E\n⊢ K = ⊥ ∨ K = ⊤",
"usedConstants": [
"Eq.mpr",
"IntermediateField.instPartialOrder",
"Lattice.toSemilatticeSup",
"Comp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 632,
"column": 39
} | {
"line": 632,
"column": 77
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 634,
"column": 7
} | {
"line": 634,
"column": 45
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 651,
"column": 59
} | {
"line": 651,
"column": 92
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nf g : K[X]\nhfm : f.Monic\nhgm : g.Monic\nhf : Irreducible f\nhg :\n ∀ (E : Type u) [inst : Field E] [inst_1 : Algebra K E] (x : E),\n minpoly K x = f → Irreducible (Polynomial.map (algebraMap K ↥K⟮x⟯) g - C (AdjoinSimple.gen K x))\nhf' : f.natDegree ≠ 0\nhg' : g.natDeg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 75,
"column": 12
} | {
"line": 75,
"column": 23
} | [
{
"pp": "case zero\nR : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nχ₁ χ₂ : ↥H → R\nx : L\nm : M\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 96
} | [
{
"pp": "case a\nk : Type u_1\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : LieRing.IsNilpotent L\nχ : L → R\ninst✝ : LinearWeights R L M\nt... | simp only [smul_lie, LinearWeights.map_smul χ (aux R L M χ), smul_assoc t, SetLike.val_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 749,
"column": 2
} | {
"line": 749,
"column": 13
} | [
{
"pp": "F : Type u\ninst✝² : Field F\nE : Type u\ninst✝¹ : Field E\ninst✝ : Algebra F E\ns : Set E\n⊢ #↥(adjoin F s) ≤ max (max #F #↑s) ℵ₀",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"IntermediateField",
"PartialOrder.toPreorder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 91,
"column": 42
} | {
"line": 91,
"column": 59
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁴ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\ns : Set (↥H → R)\nhs : ∀ χ₁ ∈ s, ∀ χ₂ ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Fintype | {
"line": 127,
"column": 6
} | {
"line": 128,
"column": 58
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\ns : Finset (α × ℕ)\nhsm : s ⊆ m.toEnumFinset\na : α\nha : (filter (fun x ↦ a = x.1) s.val).card > 0\nh : ∀ n ≥ (filter (fun x ↦ a = x.1) s.val).card - 1, (a, n) ∉ s\n⊢ {x ∈ s | x.1 = a} ⊆ {a} ×ˢ Finset.range ((filter (fun x ↦ a = x.1) s.val).card - 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Fintype | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\ns : Finset (α × ℕ)\nhsm : s ⊆ m.toEnumFinset\na : α\nha : (filter (fun x ↦ a = x.1) s.val).card > 0\nh : {x ∈ s | x.1 = a} ⊆ {a} ×ˢ Finset.range ((filter (fun x ↦ a = x.1) s.val).card - 1)\n⊢ (filter (fun x ↦ a = x.1) s.val).card ≤ (filter (fun x ↦ a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 232,
"column": 16
} | {
"line": 232,
"column": 53
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : LinearWeights R L M\ninst✝ : IsNoetherian R M\nχ : Wei... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Fintype | {
"line": 132,
"column": 46
} | {
"line": 132,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\ns : Finset (α × ℕ)\nhsm : s ⊆ m.toEnumFinset\na : α\nha : (filter (fun x ↦ a = x.1) s.val).card > 0\nn : ℕ\nhan : n ≥ (filter (fun x ↦ a = x.1) s.val).card - 1\nhn : (a, n) ∈ s\n⊢ n < count a m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Fintype | {
"line": 144,
"column": 14
} | {
"line": 144,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm₁ m₂ : Multiset α\nh : m₁.toEnumFinset ⊆ m₂.toEnumFinset\n⊢ m₁ ≤ m₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 245,
"column": 4
} | {
"line": 245,
"column": 48
} | [
{
"pp": "k : Type u_1\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : LieRing L\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : LieRingModule L M\ninst✝⁸ : LieRing.IsNilpotent L\ninst✝⁷ : Field k\ninst✝⁶ : LieAlgebra k L\ninst✝⁵ : Module k M\ninst✝⁴ : Module.Finite k M\ninst✝³ : LieModule k L M\ninst✝² : LinearWeights k L M\ninst✝¹... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Fintype | {
"line": 190,
"column": 2
} | {
"line": 191,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\nthis : map Prod.fst (Finset.map m.coeEmbedding Finset.univ).val = m\n⊢ map (fun x ↦ x.fst) Finset.univ.val = m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 205,
"column": 55
} | {
"line": 205,
"column": 66
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝⁵ : LieRing.IsNilpotent ↥H\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : Nontrivial ↥H\nx : L\nhx : x ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 41
} | [
{
"pp": "k : Type u\ninst✝ : Field k\ns : Finset (Monics k)\nf : Monics k\nhf : f ∈ s\n⊢ (∏ x ∈ s, map (algebraMap k (∏ f ∈ s, ↑f).SplittingField) ↑x).Splits",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.PID | {
"line": 47,
"column": 71
} | {
"line": 47,
"column": 82
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Free R M\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\np : Submodule R M\nf : M →ₗ[R] M\nhf : ∀ (x : M), f x ∈ p\nhf' : ∀ x ∈ p, f x ∈ p\nι : Type u_2 := Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Killing | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 26
} | [
{
"pp": "R : Type u_1\nL : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nL' : Type u_4\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : IsKilling R L\ne : L ≃ₗ⁅R⁆ L'\nx' : L'\nhx' : ∀ y ∈ ⊤, ((LieModule.traceForm R L' L') x') y = 0\nthis : e.symm x' ∈ ⊥\n⊢ x' ∈ ⊥",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Killing | {
"line": 130,
"column": 55
} | {
"line": 130,
"column": 66
} | [
{
"pp": "R : Type u_1\nL : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nL' : Type u_4\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : IsKilling R L\ne : L ≃ₗ⁅R⁆ L'\nx' : L'\nhx' : ∀ y ∈ ⊤, ((LieModule.traceForm R L' L') x') y = 0\ny : L\n⊢ ∀ (y' : L'), ((killingForm R L') ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 39
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : LieRing.IsNilpotent L\nχ : Weight R L M\n⊢ ∃ x ∈ genWeightSpace M ⇑χ, x ≠ 0",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 70
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : LieRing.IsNilpotent L\nχ : Weight R L M\nx : L\n⊢ ?m.37",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 13
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : LieRing.IsNilpotent L\ninst✝ : IsNoetherian R M\nx : L\n⊢ ∃ k, ↑(genWeightSpace M 0) ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 13
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\ninst✝¹ : LieRing.IsNilpotent L\ninst✝ : IsNoetherian R M\nx : L\n⊢ _root_.IsNilpotent ((toEnd R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 13
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\ninst✝ : LieRing.IsNilpotent L\n⊢ ⨆ k, LieSubmodule.ucs k ⊥ ≤ genWeightSpace M 0",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Derivation.Killing | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 13
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁴ : Field R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : Module.Finite R L\ninst✝ : LieAlgebra.IsKilling R L\nD : LieDerivation R L L\nhD : D ∈ (LieModule.traceForm R (LieDerivation R L L) (LieDerivation R L L)).orthogonal (ad R L).range.toSubmodule\nx : L\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 53,
"column": 18
} | {
"line": 53,
"column": 29
} | [
{
"pp": "L : Type u_2\ninst✝ : LieRing L\nh e f : L\nht : IsSl2Triple h e f\n⊢ -h ≠ 0",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"LieRing.toAddCommGroup",
"congrArg",
"neg_eq_zero._simp_1",
"id",
"Ne",
"SubtractionMonoid.toSubNegZeroMonoid",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 13
} | [
{
"pp": "case h.h.mk\nR : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : LieRing L\ninst✝⁷ : LieAlgebra R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : LieRingModule L M\ninst✝³ : LieModule R L M\ninst✝² : LieRing.IsNilpotent L\ninst✝¹ : IsNoetherian R M\ninst✝ : IsArtinian R M\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 20
} | [
{
"pp": "L : Type u_2\ninst✝ : LieRing L\nh e f : L\nt : IsSl2Triple h e f\nthis : e = 0\n⊢ h = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 20
} | [
{
"pp": "L : Type u_2\ninst✝ : LieRing L\nh e f : L\nt : IsSl2Triple h e f\nthis : f = 0\n⊢ h = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 95,
"column": 31
} | {
"line": 95,
"column": 42
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : LieRing L\ninst✝⁵ : LieAlgebra R L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\ninst✝¹ : LieModule R L M\nh e f : L\ninst✝ : IsAddTorsionFree M\nt : IsSl2Triple h e f\nm : M\nμ ρ : R\nhm : m ≠ 0\nhm' :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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