module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.PowerBasis | {
"line": 126,
"column": 39
} | {
"line": 126,
"column": 70
} | {
"line": 126,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\npb : PowerBasis R S\ny : S\n⊢ y ∈ Submodule.span R (Set.range fun i ↦ pb.gen ^ ↑i)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Submodule",
"Module.Basis.m... | [] | simpa using pb.basis.mem_span y | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.PowerBasis | {
"line": 126,
"column": 39
} | {
"line": 126,
"column": 70
} | {
"line": 126,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\npb : PowerBasis R S\ny : S\n⊢ y ∈ Submodule.span R (Set.range fun i ↦ pb.gen ^ ↑i)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Submodule",
"Module.Basis.m... | [] | simpa using pb.basis.mem_span y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerBasis | {
"line": 126,
"column": 39
} | {
"line": 126,
"column": 70
} | {
"line": 126,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : Ring S\ninst✝¹ : Algebra R S\ninst✝ : Nontrivial S\npb : PowerBasis R S\ny : S\n⊢ y ∈ Submodule.span R (Set.range fun i ↦ pb.gen ^ ↑i)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Submodule",
"Module.Basis.m... | [] | simpa using pb.basis.mem_span y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerBasis | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 49
} | {
"line": 282,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝⁴ : Ring S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra A S\nS' : Type u_7\ninst✝¹ : Ring S'\ninst✝ : Algebra A S'\npb : PowerBasis A S\ny : S'\nhy : (aeval y) (minpoly A pb.gen) = 0\nf g : A[X]\n⊢ ((pb.basis.constr A) fun i ↦ y ^ ↑i) ((aeval pb.gen) f * (aeval pb.gen) g) =\n... | [] | simp only [← aeval_mul, pb.constr_pow_aeval hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerBasis | {
"line": 415,
"column": 4
} | {
"line": 415,
"column": 38
} | {
"line": 416,
"column": 2
} | [
{
"pp": "case neg\nS : Type u_2\ninst✝² : Ring S\nK : Type u_6\ninst✝¹ : Field K\ninst✝ : Algebra K S\nx : S\nh : ¬IsIntegral K x\n⊢ LinearIndependent K fun i ↦ x ^ ↑i",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"linearIndependent_empty_type",
"Algebra.toModule",
"ins... | [] | exact linearIndependent_empty_type | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.KummerPolynomial | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 73
} | {
"line": 81,
"column": 2
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nm : ℕ\nhm : m ∣ n\nhm' : m ≠ 1\nb : K\n⊢ b ^ m ≠ a",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
... | [
"K : Type u\ninst✝ : Field K\nn : ℕ\na : K\nH : Irreducible (X ^ n - C a)\nm : ℕ\nhm : m ∣ n\nhm' : m ≠ 1\nb : K\nhn : n ≠ 0\n⊢ b ^ m ≠ a"
] | have hn : n ≠ 0 := fun e ↦ not_irreducible_C
(1 - a) (by simpa only [e, pow_zero, ← C.map_one, ← map_sub] using H) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Separable | {
"line": 125,
"column": 24
} | {
"line": 125,
"column": 45
} | {
"line": 125,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np : R[X]\nh : p.Separable\nf : R →+* S\na b : R[X]\nH : a * p + b * derivative p = 1\n⊢ Polynomial.map f a * Polynomial.map f p + Polynomial.map f b * Polynomial.map f (derivative p) = 1",
"ppTerm": "?m.55",
"assigned": tr... | [
"R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np : R[X]\nh : p.Separable\nf : R →+* S\na b : R[X]\nH : a * p + b * derivative p = 1\n⊢ Polynomial.map f (a * p) + Polynomial.map f b * Polynomial.map f (derivative p) = 1"
] | ← Polynomial.map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Separable | {
"line": 125,
"column": 46
} | {
"line": 125,
"column": 67
} | {
"line": 125,
"column": 68
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np : R[X]\nh : p.Separable\nf : R →+* S\na b : R[X]\nH : a * p + b * derivative p = 1\n⊢ Polynomial.map f (a * p) + Polynomial.map f b * Polynomial.map f (derivative p) = 1",
"ppTerm": "?m.63",
"assigned": true,
"usedCo... | [
"R : Type u\ninst✝¹ : CommSemiring R\nS : Type v\ninst✝ : CommSemiring S\np : R[X]\nh : p.Separable\nf : R →+* S\na b : R[X]\nH : a * p + b * derivative p = 1\n⊢ Polynomial.map f (a * p) + Polynomial.map f (b * derivative p) = 1"
] | ← Polynomial.map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Separable | {
"line": 348,
"column": 6
} | {
"line": 351,
"column": 19
} | {
"line": 352,
"column": 4
} | [
{
"pp": "case inl\nF : Type u\ninst✝ : Field F\nf : F[X]\nhf : Irreducible f\nH : derivative f = 0\nHF : CharP F 0\n⊢ f.Separable ∨ ¬f.Separable ∧ ∃ g, Irreducible g ∧ (expand F 0) g = f",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Polynomial.derivative",
"W... | [] | haveI := CharP.charP_to_charZero F
have := derivative_eq_zero.1 H
have := (natDegree_pos_iff_degree_pos.mpr <| degree_pos_of_irreducible hf).ne'
contradiction | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 348,
"column": 6
} | {
"line": 351,
"column": 19
} | {
"line": 352,
"column": 4
} | [
{
"pp": "case inl\nF : Type u\ninst✝ : Field F\nf : F[X]\nhf : Irreducible f\nH : derivative f = 0\nHF : CharP F 0\n⊢ f.Separable ∨ ¬f.Separable ∧ ∃ g, Irreducible g ∧ (expand F 0) g = f",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Polynomial.derivative",
"W... | [] | haveI := CharP.charP_to_charZero F
have := derivative_eq_zero.1 H
have := (natDegree_pos_iff_degree_pos.mpr <| degree_pos_of_irreducible hf).ne'
contradiction | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Separable | {
"line": 506,
"column": 2
} | {
"line": 506,
"column": 50
} | {
"line": 507,
"column": 2
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni : F →+* K\nf : F[X]\nsep : f.Separable\nsp : (map i f).Splits\n⊢ ∃ s, map i f = C (i f.leadingCoeff) * ∏ a ∈ s, (X - C a)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Polynomial.splits_iff_exists_multiset",
... | [
"F : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni : F →+* K\nf : F[X]\nsep : f.Separable\nsp : (map i f).Splits\ns : Multiset K\nh : map i f = C (map i f).leadingCoeff * (Multiset.map (fun x ↦ X - C x) s).prod\n⊢ ∃ s, map i f = C (i f.leadingCoeff) * ∏ a ∈ s, (X - C a)"
] | obtain ⟨s, h⟩ := splits_iff_exists_multiset.1 sp | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.Separable | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 80
} | {
"line": 509,
"column": 2
} | [
{
"pp": "case h\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni : F →+* K\nf : F[X]\nsep : f.Separable\nsp : (map i f).Splits\ns : Multiset K\nh : map i f = C (map i f).leadingCoeff * (Multiset.map (fun x ↦ X - C x) s).prod\n⊢ map i f = C (i f.leadingCoeff) * ∏ a ∈ s.toFinset, (X - C a)",
"ppT... | [
"case h\nF : Type u\ninst✝¹ : Field F\nK : Type v\ninst✝ : Field K\ni : F →+* K\nf : F[X]\nsep : f.Separable\nsp : (map i f).Splits\ns : Multiset K\nh : map i f = C (map i f).leadingCoeff * (Multiset.map (fun x ↦ X - C x) s).prod\n⊢ s.Nodup"
] | rw [h, Finset.prod_eq_multiset_prod, ← Multiset.toFinset_eq, leadingCoeff_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Determinant | {
"line": 615,
"column": 4
} | {
"line": 615,
"column": 32
} | {
"line": 616,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\ni j : ι\nh : v i = v j\nhij : i ≠ j\n⊢ ((e.toMatrix v)ᵀ.upda... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\ni j : ι\nh : v i = v j\nhij : i ≠ j\n⊢ (e.toMatrix v)ᵀ.updateRow i ((e.t... | apply det_zero_of_row_eq hij | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.Determinant | {
"line": 650,
"column": 35
} | {
"line": 650,
"column": 65
} | {
"line": 650,
"column": 66
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit ((LinearMap.toMatrix e e) ((e.constr ℕ) v)).det\nv' : Basis ι R M := e.map (LinearEquiv.ofIsUnitDet h)\nv'_def :... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nι : Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ne : Basis ι R M\nv : ι → M\nh : IsUnit ((LinearMap.toMatrix e e) ((e.constr ℕ) v)).det\nv' : Basis ι R M := e.map (LinearEquiv.ofIsUnitDet h)\nv'_def : v' = e.map ... | LinearEquiv.ofIsUnitDet_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdjoinRoot | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 21
} | {
"line": 244,
"column": 22
} | [
{
"pp": "case monomial\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : S[X]\nn : ℕ\na : R\na✝ : (aeval (root f)) (C a * X ^ n) = (mk f) (Polynomial.map (algebraMap R S) (C a * X ^ n))\n⊢ (aeval (root f)) (C a * X ^ (n + 1)) = (mk f) (Polynomial.map (algebraMap R S... | [] | | monomial n a _ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 27
} | {
"line": 67,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na : A\nk : ℕ\nh : a ^ k = 0\nh₁ :\n ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i =\n ∑ i ∈ range (nilpotencyClass a), (↑i !)⁻¹ • a ^ i + ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i\nthis : ∑ i ∈ Ico (nilpotencyClass a) k, (↑i !)⁻¹ • a ^ i = 0\n⊢ ∑ i ∈ ran... | [] | rw [h₁, this, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AdjoinRoot | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 7
} | {
"line": 430,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : CommRing U\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Algebra R U\nf : S →ₐ[R] T\ng : T →ₐ[R] U\np : S[X]\nq : T[X]\nr : U[X]\nhf : q ∣ Polynomial.map (↑f) p\nhg : r... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.AdjoinRoot | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 7
} | {
"line": 430,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : CommRing U\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Algebra R U\nf : S →ₐ[R] T\ng : T →ₐ[R] U\np : S[X]\nq : T[X]\nr : U[X]\nhf : q ∣ Polynomial.map (↑f) p\nhg : r... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdjoinRoot | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 7
} | {
"line": 430,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nU : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : CommRing U\ninst✝² : Algebra R S\ninst✝¹ : Algebra R T\ninst✝ : Algebra R U\nf : S →ₐ[R] T\ng : T →ₐ[R] U\np : S[X]\nq : T[X]\nr : U[X]\nhf : q ∣ Polynomial.map (↑f) p\nhg : r... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Semisimple | {
"line": 299,
"column": 16
} | {
"line": 301,
"column": 82
} | {
"line": 302,
"column": 2
} | [
{
"pp": "case refine_2.mk\nM : Type u_2\ninst✝⁴ : AddCommGroup M\nK : Type u_3\ninst✝³ : Field K\ninst✝² : Module K M\nf g : End K M\ninst✝¹ : FiniteDimensional K M\ninst✝ : PerfectField K\ncomm : Commute f g\nhf : f.IsSemisimple\nhg : g.IsSemisimple\na : End K M\nha : a ∈ K[f, g]\nR : Type u_3 := K[X] ⧸ Ideal.... | [] | exact p.induction_on (fun k ↦ by simp [R, commute_algebraMap_left])
(fun p q hp hq ↦ by simpa [R] using! hp.add_left hq)
fun n k ↦ by simpa [R, pow_succ, ← mul_assoc _ _ X] using! (·.mul_left comm) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.JordanChevalley | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 81
} | {
"line": 101,
"column": 2
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : FiniteDimensional K V\ninst✝ : PerfectField K\nn₁ s₁ n₂ s₂ : End K V\nhn₁ : IsNilpotent n₁\nhs₁ : s₁.IsSemisimple\nhn₂ : IsNilpotent n₂\nhs₂ : s₂.IsSemisimple\nhc₁ : Commute n₁ s₁\nhc₂ : Commute n₂ s₂\n... | [
"K : Type u_1\nV : Type u_2\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : FiniteDimensional K V\ninst✝ : PerfectField K\nn₁ s₁ n₂ s₂ : End K V\nhn₁ : IsNilpotent n₁\nhs₁ : s₁.IsSemisimple\nhn₂ : IsNilpotent n₂\nhs₂ : s₂.IsSemisimple\nhc₁ : Commute n₁ s₁\nhc₂ : Commute n₂ s₂\nh : n₁ + s₁ ... | have hss : (s - s₀).IsSemisimple :=
hs.sub_of_commute (commute_of_mem_adjoin_singleton_of_commute hs₀ hsf) hs₀_ss | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Solvable | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 7
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nz : L\n⊢ z ∈ Submodule.span R {x | ∃ x_1 ∈ ⊤, ∃ n ∈ ⊤, ⁅x_1, n⁆ = x} ↔ z ∈ Submodule.span R {x | ∃ x_1 y, ⁅x_1, y⁆ = x}",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModu... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Solvable | {
"line": 201,
"column": 17
} | {
"line": 201,
"column": 97
} | {
"line": 203,
"column": 0
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L' →ₗ⁅R⁆ L\nh : Function.Surjective ⇑f\nk : ℕ\nih : map f (derivedSeries R L' k) = derivedSeries R L k\n⊢ map f (derivedSeries R L' (k + 1... | [] | simp only [derivedSeries_def, map_bracket_eq f h, ih, derivedSeriesOfIdeal_succ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Solvable | {
"line": 201,
"column": 17
} | {
"line": 201,
"column": 97
} | {
"line": 203,
"column": 0
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L' →ₗ⁅R⁆ L\nh : Function.Surjective ⇑f\nk : ℕ\nih : map f (derivedSeries R L' k) = derivedSeries R L k\n⊢ map f (derivedSeries R L' (k + 1... | [] | simp only [derivedSeries_def, map_bracket_eq f h, ih, derivedSeriesOfIdeal_succ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Solvable | {
"line": 201,
"column": 17
} | {
"line": 201,
"column": 97
} | {
"line": 203,
"column": 0
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\nL' : Type w₁\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L' →ₗ⁅R⁆ L\nh : Function.Surjective ⇑f\nk : ℕ\nih : map f (derivedSeries R L' k) = derivedSeries R L k\n⊢ map f (derivedSeries R L' (k + 1... | [] | simp only [derivedSeries_def, map_bracket_eq f h, ih, derivedSeriesOfIdeal_succ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Properties | {
"line": 320,
"column": 8
} | {
"line": 320,
"column": 20
} | {
"line": 320,
"column": 21
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type u_8\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R M\ne : M ≃ₗ[R] M'\nh : ((congr e) B).Nondegenerate\n⊢ B = (congr e.symm) ((congr e) B)",
"ppTerm": "?m.145",
"assign... | [
"R : Type u_1\nM : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nM' : Type u_8\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nB : BilinForm R M\ne : M ≃ₗ[R] M'\nh : ((congr e) B).Nondegenerate\n⊢ B = (congr (e ≪≫ₗ e.symm)) B"
] | congr_congr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 446,
"column": 51
} | {
"line": 446,
"column": 60
} | {
"line": 448,
"column": 0
} | [
{
"pp": "case succ\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI : LieIdeal R L\nk : ℕ\nh : IsLieAbelian ↥(derivedSeriesOfIdeal R L k I) ∧ derivedSeriesOfIdeal R L k I ≠ ⊥\n⊢ IsLieAbelian\n ↥(match k + 1 with\n | 0 => ⊥\n | k.succ => derivedSeriesOfIdea... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Solvable | {
"line": 452,
"column": 2
} | {
"line": 454,
"column": 76
} | {
"line": 455,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nI : LieIdeal R L\ninst✝ : IsSolvable ↥I\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\n⊢ sInf s = 0 ↔ I = ⊥",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",... | [
"R : Type u\nL : Type v\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nI : LieIdeal R L\ninst✝ : IsSolvable ↥I\ns : Set ℕ := {k | derivedSeriesOfIdeal R L k I = ⊥}\nhne : s.Nonempty\n⊢ sInf s = 0 ↔ I = ⊥"
] | have hne : s.Nonempty :=
have ⟨k, hk⟩ := IsSolvable.solvable R I
⟨k, by rwa [derivedSeries_def, LieIdeal.derivedSeries_eq_bot_iff] at hk⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.BaseChange | {
"line": 204,
"column": 12
} | {
"line": 204,
"column": 19
} | {
"line": 204,
"column": 19
} | [
{
"pp": "case refine_2\nR : Type u_1\nA : 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 : CommRing A\ninst✝ : Algebra R A\nN : LieSubmodule R L M\n... | [
"case refine_2\nR : Type u_1\nA : 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 : CommRing A\ninst✝ : Algebra R A\nN : LieSubmodule R L M\nx : A ⊗[R] L... | add_lie | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 59,
"column": 2
} | {
"line": 66,
"column": 14
} | {
"line": 68,
"column": 0
} | [
{
"pp": "R : 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\n⊢ LinearMap.BilinForm.lieInvariant L Φ ↔ Φ ∈ LieModule.maxTrivSubmo... | [] | refine ⟨fun h x ↦ ?_, fun h x y z ↦ ?_⟩
· ext y z
rw [LieHom.lie_apply, LinearMap.sub_apply, Module.Dual.lie_apply, LinearMap.zero_apply,
LinearMap.zero_apply, h, sub_self]
· replace h := LinearMap.congr_fun₂ (h x) y z
simp only [LieHom.lie_apply, LinearMap.sub_apply, Module.Dual.lie_apply,
Line... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 59,
"column": 2
} | {
"line": 66,
"column": 14
} | {
"line": 68,
"column": 0
} | [
{
"pp": "R : 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\n⊢ LinearMap.BilinForm.lieInvariant L Φ ↔ Φ ∈ LieModule.maxTrivSubmo... | [] | refine ⟨fun h x ↦ ?_, fun h x y z ↦ ?_⟩
· ext y z
rw [LieHom.lie_apply, LinearMap.sub_apply, Module.Dual.lie_apply, LinearMap.zero_apply,
LinearMap.zero_apply, h, sub_self]
· replace h := LinearMap.congr_fun₂ (h x) y z
simp only [LieHom.lie_apply, LinearMap.sub_apply, Module.Dual.lie_apply,
Line... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 76,
"column": 4
} | {
"line": 82,
"column": 29
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nΦ : LinearMap.BilinForm R M\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nN : LieSubmodule R L M\nx : L\ny : M\n⊢ y ... | [] | suffices (∀ n ∈ N, Φ n y = 0) → ∀ n ∈ N, Φ n ⁅x, y⁆ = 0 by
simpa only [
AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup, Submodule.mem_toAddSubmonoid,
LinearMap.BilinForm.mem_orthogonal_iff, LieSubmodule.mem_toSubmodule]
intro H a ha
rw [← neg_eq_zero, ← hΦ_inv]
exact H _ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 76,
"column": 4
} | {
"line": 82,
"column": 29
} | {
"line": 84,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nΦ : LinearMap.BilinForm R M\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nN : LieSubmodule R L M\nx : L\ny : M\n⊢ y ... | [] | suffices (∀ n ∈ N, Φ n y = 0) → ∀ n ∈ N, Φ n ⁅x, y⁆ = 0 by
simpa only [
AddSubsemigroup.mem_carrier, AddSubmonoid.mem_toSubsemigroup, Submodule.mem_toAddSubmonoid,
LinearMap.BilinForm.mem_orthogonal_iff, LieSubmodule.mem_toSubmodule]
intro H a ha
rw [← neg_eq_zero, ← hΦ_inv]
exact H _ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 61
} | {
"line": 374,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nb₁ : B.Nondegenerate\nb₂ : B.IsRefl\nx : V\nhx : (B x) x ≠ 0\nthis : ∀ (n : V), n ∈ K ∙ x ⊔ B.orthogonal (K ∙ x)\nm : ↥(B.orthogonal (K ∙ x))\nhm : ∀ (y : ↥(B.orthogonal (K ∙ x))... | [
"case refine_1\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nb₁ : B.Nondegenerate\nb₂ : B.IsRefl\nx : V\nhx : (B x) x ≠ 0\nthis : ∀ (n : V), n ∈ K ∙ x ⊔ B.orthogonal (K ∙ x)\nm : ↥(B.orthogonal (K ∙ x))\nhm : ∀ (y : ↥(B.orthogonal (K ∙ x))), ((B.restr... | obtain ⟨y, hy, z, hz, rfl⟩ := Submodule.mem_sup.1 <| this n | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 61
} | {
"line": 374,
"column": 2
} | [
{
"pp": "case refine_2\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nb₁ : B.Nondegenerate\nb₂ : B.IsRefl\nx : V\nhx : (B x) x ≠ 0\nthis : ∀ (n : V), n ∈ K ∙ x ⊔ B.orthogonal (K ∙ x)\nm : ↥(B.orthogonal (K ∙ x))\nhm : ∀ (x_1 : ↥(B.orthogonal (K ∙ x... | [
"case refine_2\nV : Type u_5\nK : Type u_6\ninst✝² : Field K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nB : BilinForm K V\nb₁ : B.Nondegenerate\nb₂ : B.IsRefl\nx : V\nhx : (B x) x ≠ 0\nthis : ∀ (n : V), n ∈ K ∙ x ⊔ B.orthogonal (K ∙ x)\nm : ↥(B.orthogonal (K ∙ x))\nhm : ∀ (x_1 : ↥(B.orthogonal (K ∙ x))), ((B.res... | obtain ⟨y, hy, z, hz, rfl⟩ := Submodule.mem_sup.1 <| this n | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Lie.Normalizer | {
"line": 102,
"column": 43
} | {
"line": 102,
"column": 50
} | {
"line": 102,
"column": 50
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\nin... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\nM' : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra R L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : LieRingModule L M\ninst✝⁴ : LieModule R L M\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\ninst✝¹ : LieRingModule L M'\ninst✝ : LieMod... | add_lie | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 79
} | {
"line": 313,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : HasTrivialRadical R L\nh : ⊥ = ⊤\n⊢ Subsingleton L",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LieAlgebra.toModule",
"LieRing.toAddCommGroup",
"LieSubmodule... | [] | exact (LieSubmodule.subsingleton_iff R L L).mp (subsingleton_of_bot_eq_top h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Ring.Divisibility.Lemmas | {
"line": 136,
"column": 48
} | {
"line": 136,
"column": 63
} | {
"line": 136,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\ninst✝ : LinearOrder R\nx y : R\n⊢ Associated y x ↔ Associated x y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Associated.comm",
"Eq.mpr",
"congrArg",
"id",
"Iff",
"Associated",
"Semiring.toMonoid",
... | [
"R : Type u_1\ninst✝¹ : Ring R\ninst✝ : LinearOrder R\nx y : R\n⊢ Associated x y ↔ Associated x y"
] | Associated.comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Normal.Defs | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 64
} | {
"line": 217,
"column": 4
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝¹⁷ : Field F\ninst✝¹⁶ : Field K\ninst✝¹⁵ : Algebra F K\nK₁ : Type u_3\nK₂ : Type u_4\nK₃ : Type u_5\ninst✝¹⁴ : Field K₁\ninst✝¹³ : Field K₂\ninst✝¹² : Field K₃\ninst✝¹¹ : Algebra F K₁\ninst✝¹⁰ : Algebra F K₂\ninst✝⁹ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : ... | [
"F : Type u_1\nK : Type u_2\ninst✝¹⁷ : Field F\ninst✝¹⁶ : Field K\ninst✝¹⁵ : Algebra F K\nK₁ : Type u_3\nK₂ : Type u_4\nK₃ : Type u_5\ninst✝¹⁴ : Field K₁\ninst✝¹³ : Field K₂\ninst✝¹² : Field K₃\ninst✝¹¹ : Algebra F K₁\ninst✝¹⁰ : Algebra F K₂\ninst✝⁹ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : K₂ →ₐ[F] K₃\... | simp only [AlgHom.restrictNormal', AlgEquiv.coe_ofBijective] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Engel | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 28
} | {
"line": 174,
"column": 2
} | [
{
"pp": "R : Type u₁\nL : Type u₂\nL₂ : Type u₃\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nhf : Surjective ⇑f\nh : LieAlgebra.IsEngelian R L\n⊢ LieAlgebra.IsEngelian R L₂",
"ppTerm": "?m.29",
"assigned": true,
"use... | [
"R : Type u₁\nL : Type u₂\nL₂ : Type u₃\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nhf : Surjective ⇑f\nh : LieAlgebra.IsEngelian R L\nM : Type u₄\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L₂ M\n_i4 : LieModule R... | intro M _i1 _i2 _i3 _i4 h' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Engel | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 7
} | {
"line": 182,
"column": 0
} | [
{
"pp": "R : Type u₁\nL : Type u₂\nL₂ : Type u₃\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L₂\ninst✝ : LieAlgebra R L₂\nf : L →ₗ⁅R⁆ L₂\nhf : Surjective ⇑f\nh : LieAlgebra.IsEngelian R L\nM : Type u₄\n_i1 : AddCommGroup M\n_i2 : Module R M\n_i3 : LieRingModule L₂ M\n_i4 :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 48
} | {
"line": 368,
"column": 49
} | [
{
"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\nN : LieSubmodule R L M\nh₁ : N ≤ maxTrivSubmodule R L M\nk : ℕ\nhk : lowerCentralSeries R L (M... | [
"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\nN : LieSubmodule R L M\nh₁ : N ≤ maxTrivSubmodule R L M\nk : ℕ\nhk : lowerCentralSeries R L (M ⧸ N) k = ⊥\... | ← LieSubmodule.Quotient.map_mk'_eq_bot_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 375,
"column": 6
} | {
"line": 375,
"column": 48
} | {
"line": 375,
"column": 49
} | [
{
"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\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nk : ℕ\n⊢ lowerCentralSeries R L (M ⧸ N) k = ⊥ ↔ lowerCentralSeries R L M k ≤ N... | [
"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\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nk : ℕ\n⊢ lowerCentralSeries R L (M ⧸ N) k = ⊥ ↔ LieSubmodule.map (LieSubmodule.Quotient.mk... | ← LieSubmodule.Quotient.map_mk'_eq_bot_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 13
} | {
"line": 457,
"column": 0
} | [
{
"pp": "case succ\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\nh : ⁅⊤, lowerCentralSeries R L M k⁆ = ⊥ ∧ lowerCentralSeries R L M k ≠ ⊥\n⊢ ⁅⊤,\n ... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 687,
"column": 4
} | {
"line": 688,
"column": 50
} | {
"line": 690,
"column": 0
} | [
{
"pp": "case mpr\nR : 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\ninst✝⁶ : LieModule R L M\nL₂ : Type u_1\nM₂ : Type u_2\ninst✝⁵ : LieRing L₂\ninst✝⁴ : LieAlgebra R L₂\ninst✝³ : Ad... | [] | rw [LinearEquiv.coe_coe, LieEquiv.coe_toLieHom, ← g.symm_apply_apply ⁅f.symm x, g.symm m⁆, ←
hfg, f.apply_symm_apply, g.apply_symm_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 710,
"column": 48
} | {
"line": 710,
"column": 53
} | {
"line": 711,
"column": 2
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\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\nM₁ M₂ : LieSubmodule R L M\nh₁ : M₁ ≤ M₂\ninst✝ : IsNilpotent L ↥M₂\nf : L →ₗ⁅R⁆ L := LieHom.id\ng : ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 103,
"column": 2
} | {
"line": 112,
"column": 60
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case mpr\nF : Type u_1\ninst✝² : Field F\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\n⊢ Algebra.EssFiniteType F E → ⊤.FG",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"IntermediateField.instPartialOrder",
"Group... | [] | · intro _
use Algebra.EssFiniteType.finset F E
refine top_le_iff.mp fun x _ ↦ ?_
obtain ⟨x, s, rfl⟩ := IsLocalization.exists_mk'_eq (Algebra.EssFiniteType.submonoid F E) x
have hs : s.1.1 ≠ 0 := (IsLocalization.map_units E s).ne_zero
have H : IsLocalization.mk' E x s = x / s := by
simp [IsLoca... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 922,
"column": 48
} | {
"line": 922,
"column": 53
} | {
"line": 923,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\nI : LieIdeal R L\nM : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\nk : ℕ\ninst✝ : LieModule.IsNilpotent L ↥I\nf : ↥I →ₗ⁅R⁆ L := I.incl\ng : ↥I →ₗ⁅R⁆ ↥I := LieHom.id\n⊢ ∀ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 947,
"column": 2
} | {
"line": 947,
"column": 25
} | {
"line": 949,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : 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 : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : IsNilpotent L M\nk : ℕ\nhk : l... | [] | exact ⟨k, by simp [hk]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 56
} | {
"line": 456,
"column": 0
} | [
{
"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\nS : Set E\n⊢ restrictScalars F (adjoin (↥K) S) = K ⊔ adjoin F S",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"IntermediateField.restrictScalars",
"Eq.mpr"... | [] | rw [restrictScalars_adjoin, adjoin_union, adjoin_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 56
} | {
"line": 456,
"column": 0
} | [
{
"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\nS : Set E\n⊢ restrictScalars F (adjoin (↥K) S) = K ⊔ adjoin F S",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"IntermediateField.restrictScalars",
"Eq.mpr"... | [] | rw [restrictScalars_adjoin, adjoin_union, adjoin_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Adjoin.Defs | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 56
} | {
"line": 456,
"column": 0
} | [
{
"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\nS : Set E\n⊢ restrictScalars F (adjoin (↥K) S) = K ⊔ adjoin F S",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"IntermediateField.restrictScalars",
"Eq.mpr"... | [] | rw [restrictScalars_adjoin, adjoin_union, adjoin_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 19
} | {
"line": 85,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝⁷ : Field F\nE : Type u_2\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nS : Set E\nK : Type u_3\nL : Type u_4\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nE1 E2 : IntermediateField K L\ninst✝¹ : FiniteDimensional K ↥E1\ninst✝ : FiniteDimensional K ↥E2\ng : TensorProduct K ↥E... | [] | rwa [this] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 409,
"column": 8
} | {
"line": 409,
"column": 77
} | {
"line": 410,
"column": 8
} | [
{
"pp": "case right\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα : E\nK : Type u\ninst✝¹ : Field K\ninst✝ : Algebra F K\nh : IsIntegral F α\nf : AdjoinRoot (minpoly F α) →+* ↥F⟮α⟯ := AdjoinRoot.lift (↑(Algebra.ofId F ↥F⟮α⟯)) (AdjoinSimple.gen F α) ⋯\nthis : Fact (Irre... | [
"case right.refine_1\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\nα : E\nK : Type u\ninst✝¹ : Field K\ninst✝ : Algebra F K\nh : IsIntegral F α\nf : AdjoinRoot (minpoly F α) →+* ↥F⟮α⟯ := AdjoinRoot.lift (↑(Algebra.ofId F ↥F⟮α⟯)) (AdjoinSimple.gen F α) ⋯\nthis : Fact (Irreduc... | refine Subfield.closure_le.mpr (Set.union_subset (fun x hx => ?_) ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 473,
"column": 14
} | {
"line": 473,
"column": 43
} | {
"line": 474,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field J\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Field L\ninst✝¹⁰ : Field M\ninst✝⁹ : Algebra R M\ninst✝⁸ : IsTorsionFree R M\ninst✝⁷ : IsAlgClosure R ... | [
"k : Type u\ninst✝¹⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field J\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Field L\ninst✝¹⁰ : Field M\ninst✝⁹ : Algebra R M\ninst✝⁸ : IsTorsionFree R M\ninst✝⁷ : IsAlgClosure R M\ninst✝⁶ : ... | rw [← hSR.symm_apply_apply x] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 473,
"column": 14
} | {
"line": 473,
"column": 43
} | {
"line": 474,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field J\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Field L\ninst✝¹⁰ : Field M\ninst✝⁹ : Algebra R M\ninst✝⁸ : IsTorsionFree R M\ninst✝⁷ : IsAlgClosure R ... | [
"k : Type u\ninst✝¹⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field J\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Field L\ninst✝¹⁰ : Field M\ninst✝⁹ : Algebra R M\ninst✝⁸ : IsTorsionFree R M\ninst✝⁷ : IsAlgClosure R M\ninst✝⁶ : ... | rw [← hSR.symm_apply_apply x] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 473,
"column": 14
} | {
"line": 473,
"column": 43
} | {
"line": 474,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝¹⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field J\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Field L\ninst✝¹⁰ : Field M\ninst✝⁹ : Algebra R M\ninst✝⁸ : IsTorsionFree R M\ninst✝⁷ : IsAlgClosure R ... | [
"k : Type u\ninst✝¹⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝¹⁵ : Field K\ninst✝¹⁴ : Field J\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Field L\ninst✝¹⁰ : Field M\ninst✝⁹ : Algebra R M\ninst✝⁸ : IsTorsionFree R M\ninst✝⁷ : IsAlgClosure R M\ninst✝⁶ : ... | rw [← hSR.symm_apply_apply x] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 656,
"column": 8
} | {
"line": 657,
"column": 55
} | {
"line": 658,
"column": 8
} | [
{
"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... | [
"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.natDegree ≠ 0\nH₁ ... | rw [restrictScalars_top, adjoin_adjoin_left, Set.union_comm, ← adjoin_adjoin_left,
adjoin_root_eq_top p, restrictScalars_adjoin] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 93,
"column": 2
} | {
"line": 95,
"column": 71
} | {
"line": 97,
"column": 2
} | [
{
"pp": "case refine_3\nR : Type u_2\nL : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\nM₁ : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : LieRingModule L M₁\ninst✝⁸ : LieModule R L M₁\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Modul... | [
"case refine_2\nR : Type u_2\nL : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : LieRing L\ninst✝¹² : LieAlgebra R L\nM₁ : Type u_5\nM₂ : Type u_6\nM₃ : Type u_7\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : Module R M₁\ninst✝⁹ : LieRingModule L M₁\ninst✝⁸ : LieModule R L M₁\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst... | · rintro t₁ t₂ ⟨k₁, hk₁⟩ ⟨k₂, hk₂⟩; use max k₁ k₂
simp only [map_add, Module.End.pow_map_zero_of_le (le_max_left k₁ k₂) hk₁,
Module.End.pow_map_zero_of_le (le_max_right k₁ k₂) hk₂, add_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 87,
"column": 29
} | {
"line": 87,
"column": 36
} | {
"line": 87,
"column": 36
} | [
{
"pp": "case add\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\ns : Set (↥H → R)\nhs : ∀ χ₁ ∈ ... | [
"case add\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\ns : Set (↥H → R)\nhs : ∀ χ₁ ∈ s, ∀ χ₂ ∈ s,... | add_lie | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Multiset.Fintype | {
"line": 108,
"column": 75
} | {
"line": 108,
"column": 80
} | {
"line": 110,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\na : α\n⊢ {x ∈ m.toEnumFinset | x.1 = a} = {a} ×ˢ Finset.range (count a m)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.mem_filter._simp_1",
"and_true",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Multiset.Fintype | {
"line": 108,
"column": 75
} | {
"line": 108,
"column": 80
} | {
"line": 110,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\na : α\n⊢ {x ∈ m.toEnumFinset | x.1 = a} = {a} ×ˢ Finset.range (count a m)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.mem_filter._simp_1",
"and_true",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Multiset.Fintype | {
"line": 108,
"column": 75
} | {
"line": 108,
"column": 80
} | {
"line": 110,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\na : α\n⊢ {x ∈ m.toEnumFinset | x.1 = a} = {a} ×ˢ Finset.range (count a m)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_range._simp_1",
"Finset.mem_filter._simp_1",
"and_true",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 29
} | {
"line": 219,
"column": 2
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝ : LieRing.IsNilpotent ↥H\nx : L\nhx : ∀ y ∈ zeroRootSubalgebra R L H, ⁅x, y⁆ ∈ zeroRootSubalgebra R L H\n⊢ x ∈ zeroRootSubalgebra R L H",
"ppTerm": "?m.57",
"assigned": tru... | [
"R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝ : LieRing.IsNilpotent ↥H\nx : L\nhx : ∀ y ∈ zeroRootSubalgebra R L H, ⁅x, y⁆ ∈ zeroRootSubalgebra R L H\n⊢ ∀ (y : ↥H), ∃ k, ((toEnd R (↥H) L) y ^ k) x = 0"
] | rw [mem_zeroRootSubalgebra] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 7
} | {
"line": 256,
"column": 0
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\nH : LieSubalgebra R L\ninst✝¹ : LieRing.IsNilpotent ↥H\ninst✝ : IsNoetherian R L\n⊢ ↑H.toLieSubmodule = ↑(rootSpace H 0) ↔ (zeroRootSubalgebra R L H).toSubmodule = H.toSubmodule",
"ppTerm": "?m.67",
"a... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Multiset.Fintype | {
"line": 264,
"column": 6
} | {
"line": 264,
"column": 27
} | {
"line": 265,
"column": 6
} | [
{
"pp": "case isTrue\nα : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nm : Multiset α\nv x : α\nhx : Fin (count x (v ::ₘ m))\nh : x = v ∧ ↑hx = count v m\n⊢ (none.elim ⟨v, ⟨count v m, ⋯⟩⟩ fun x ↦ ⟨x.fst, Fin.castLE ⋯ x.snd⟩) = ⟨x, hx⟩",
"ppTerm": "?isTrue",
"assigned": true,
... | [
"case isTrue\nα : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nm : Multiset α\nx : α\nhx : Fin (count x (x ::ₘ m))\nh2 : ↑hx = count x m\n⊢ (none.elim ⟨x, ⟨count x m, ⋯⟩⟩ fun x_1 ↦ ⟨x_1.fst, Fin.castLE ⋯ x_1.snd⟩) = ⟨x, hx⟩"
] | obtain ⟨rfl, h2⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 37
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case pos\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible (Classical.choose H)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"semigroupDvd",
"SemigroupWithZero.toSemigroup",
... | [] | exact (Classical.choose_spec H).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 37
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case pos\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible (Classical.choose H)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"semigroupDvd",
"SemigroupWithZero.toSemigroup",
... | [] | exact (Classical.choose_spec H).1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 37
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case pos\nK : Type v\ninst✝ : Field K\nf : K[X]\nH : ∃ g, Irreducible g ∧ g ∣ f\n⊢ Irreducible (Classical.choose H)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"semigroupDvd",
"SemigroupWithZero.toSemigroup",
... | [] | exact (Classical.choose_spec H).1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 176,
"column": 4
} | {
"line": 177,
"column": 70
} | {
"line": 178,
"column": 4
} | [
{
"pp": "n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih : ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → (map (algebraMap K (SplittingFieldAux n f)) f).Splits\nK : Type u\nx✝ : Field K\nf : K[X]\nhf : f.natDegree = n.succ\n⊢ (map (algebraMap K (SplittingFieldAux n.succ f)) f).Splits",
"ppTe... | [
"n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih : ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → (map (algebraMap K (SplittingFieldAux n f)) f).Splits\nK : Type u\nx✝ : Field K\nf : K[X]\nhf : f.natDegree = n.succ\n⊢ (map (algebraMap (AdjoinRoot f.factor) (SplittingFieldAux n f.removeFactor))\n ... | rw [algebraMap_succ, ← map_map,
← X_sub_C_mul_removeFactor f fun h => by rw [h] at hf; cases hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 204,
"column": 36
} | {
"line": 204,
"column": 50
} | {
"line": 204,
"column": 51
} | [
{
"pp": "n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih :\n ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤\nK : Type u\nx✝ : Field K\nf : K[X]\nhfn : f.natDegree = n.succ\nhndf : f.natDegree ≠ 0\nhfn0 : f ≠ 0\nhmf0 :\n map (algebraMap (... | [
"n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih :\n ∀ {K : Type u} [inst : Field K] (f : K[X]), f.natDegree = n → Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤\nK : Type u\nx✝ : Field K\nf : K[X]\nhfn : f.natDegree = n.succ\nhndf : f.natDegree ≠ 0\nhfn0 : f ≠ 0\nhmf0 :\n map (algebraMap (AdjoinRoot f... | ← rootSet_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 452,
"column": 53
} | {
"line": 453,
"column": 61
} | {
"line": 455,
"column": 0
} | [
{
"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\nx : L\n⊢ posFittingCompOf R M x ≤ posFittingComp R L M",
"pp... | [] | by
rw [posFittingComp]; exact le_iSup (posFittingCompOf R M) x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 465,
"column": 6
} | {
"line": 465,
"column": 48
} | {
"line": 465,
"column": 48
} | [
{
"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 : IsNoetherian R M\ninst✝ : IsArtinian R M\n⊢ IsNilpoten... | [
"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 : IsNoetherian R M\ninst✝ : IsArtinian R M\n⊢ ∀ (x : L), _root_.IsNi... | LieModule.isNilpotent_iff_forall' (R := R) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | {
"line": 160,
"column": 21
} | {
"line": 160,
"column": 100
} | {
"line": 161,
"column": 2
} | [
{
"pp": "k : Type u\ninst✝ : Field k\nq : ℚ≥0\n⊢ ↑q = ↑q.num / ↑q.den",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Semiring.toNatCast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"AlgebraicClosure.instGroupWithZero",
... | [] | by change algebraMap k _ _ = _; simp_rw [NNRat.cast_def, map_div₀, map_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.TensorProduct | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 7
} | {
"line": 86,
"column": 0
} | [
{
"pp": "R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\nN₁ : Type uN₁\nN₂ : Type uN₂\ninst✝¹⁶ : CommSemiring R\ninst✝¹⁵ : CommSemiring A\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid N₁\ninst✝¹¹ : AddCommMonoid N₂\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Module R M₁\ninst✝⁸ :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Sl2 | {
"line": 96,
"column": 4
} | {
"line": 97,
"column": 51
} | {
"line": 99,
"column": 0
} | [
{
"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' :... | [] | rw [← nsmul_lie, ← t.lie_h_e_nsmul, lie_lie, hm', lie_smul, he, lie_smul, hm',
smul_smul, smul_smul, mul_comm ρ μ, sub_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Sl2 | {
"line": 135,
"column": 10
} | {
"line": 135,
"column": 43
} | {
"line": 136,
"column": 10
} | [
{
"pp": "case mem.mem.inr.inr.inr.inl\nR : 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\ne x y u v : L\nt : IsSl2Triple u e v\n__spread✝⁻⁰ : Submodule R L ... | [
"case mem.mem.inr.inr.inr.inl\nR : 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\ne x y u v : L\nt : IsSl2Triple u e v\n__spread✝⁻⁰ : Submodule R L := span R {e... | rw [t.lie_h_f_nsmul, neg_mem_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Sl2 | {
"line": 262,
"column": 4
} | {
"line": 263,
"column": 79
} | {
"line": 264,
"column": 4
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\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\ninst✝⁶ : LieModule R L M\nh e f : L\ninst✝⁵ : IsDomain R\ninst✝⁴ : CharZero R\ninst✝³ : Nontrivial M\ninst✝² : IsTorsi... | [
"R : Type u_1\nL : Type u_2\nM : Type u_3\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\ninst✝⁶ : LieModule R L M\nh e f : L\ninst✝⁵ : IsDomain R\ninst✝⁴ : CharZero R\ninst✝³ : Nontrivial M\ninst✝² : IsTorsionFree R M\n... | have h_inj : Function.Injective evals := fun a b hab ↦ by
simpa [evals, add_right_inj, mul_eq_mul_left_iff, Nat.cast_inj] using hab | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Sequence | {
"line": 95,
"column": 25
} | {
"line": 95,
"column": 49
} | {
"line": 95,
"column": 50
} | [
{
"pp": "case h₁\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\ni : ℕ\nhi : i ∈ Set.Iio m\nhP : ↑S i ∈ ↑S '' Set.Iio m\n⊢ ↑S i ∈ degreeLT R m",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
... | [
"case h₁\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\ni : ℕ\nhi : i ∈ Set.Iio m\nhP : ↑S i ∈ ↑S '' Set.Iio m\n⊢ (↑S i).degree < ↑m"
] | Polynomial.mem_degreeLT, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Sequence | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 11
} | {
"line": 110,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\na✝ : Nontrivial R\nn : ℕ\nih : ∀ m_1 < n, ∀ ⦃P : R[X]⦄, P ∈ degreeLT R m → P.natDegree = m_1 → P ∈ span R (↑S '' Set.Iio m)\nP : R[X]\nhP : P.degree < ↑m\nhp : P.natDegree = n\np_ne_zero : P ≠ 0\nthis : P... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Polynomial.Sequence | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 34
} | {
"line": 150,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\na✝ : Nontrivial R\nn : ℕ\nih : ∀ m_1 < n, ∀ ⦃P : R[X]⦄, P ∈ degreeLT R m → P.natDegree = m_1 → P ∈ span R (↑S '' Set.Iio m)\nP : R[X]\nhP : P ∈ degreeLT R m\nhp : P.natDegree = n\np_ne_zero : P ≠ 0\nhn : ... | [
"R : Type u_1\ninst✝ : Ring R\nS : Sequence R\nm : ℕ\nhCoeff : ∀ i < m, IsUnit (↑S i).leadingCoeff\na✝ : Nontrivial R\nn : ℕ\nih : ∀ m_1 < n, ∀ ⦃P : R[X]⦄, P ∈ degreeLT R m → P.natDegree = m_1 → P ∈ span R (↑S '' Set.Iio m)\nP : R[X]\nhP : P ∈ degreeLT R m\nhp : P.natDegree = n\np_ne_zero : P ≠ 0\nhn : n < m\nu : R... | nth_rw 2 [← coeff_natDegree] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Algebra.Polynomial.Sequence | {
"line": 173,
"column": 34
} | {
"line": 173,
"column": 58
} | {
"line": 173,
"column": 59
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nhCoeff : ∀ (i : ℕ), IsUnit (↑S i).leadingCoeff\nP : R[X]\np_ne_zero : P ≠ 0\n⊢ P ∈ degreeLT R (P.natDegree + 1)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.d... | [
"case neg\nR : Type u_1\ninst✝ : Ring R\nS : Sequence R\nhCoeff : ∀ (i : ℕ), IsUnit (↑S i).leadingCoeff\nP : R[X]\np_ne_zero : P ≠ 0\n⊢ P.degree < ↑(P.natDegree + 1)"
] | Polynomial.mem_degreeLT, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 85
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_2\nL : Type u_3\ninst✝⁵ : LieRing L\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα : ↥H → K\nx : L\nhx : x ∈ rootSpace H α\nhx' : ∀ y ∈ rootSpace H (-α), ((kill... | [] | exact killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero K L H hx hy hαβ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 85
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_2\nL : Type u_3\ninst✝⁵ : LieRing L\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα : ↥H → K\nx : L\nhx : x ∈ rootSpace H α\nhx' : ∀ y ∈ rootSpace H (-α), ((kill... | [] | exact killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero K L H hx hy hαβ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 85
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case neg\nK : Type u_2\nL : Type u_3\ninst✝⁵ : LieRing L\ninst✝⁴ : Field K\ninst✝³ : LieAlgebra K L\ninst✝² : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : IsTriangularizable K (↥H) L\nα : ↥H → K\nx : L\nhx : x ∈ rootSpace H α\nhx' : ∀ y ∈ rootSpace H (-α), ((kill... | [] | exact killingForm_apply_eq_zero_of_mem_rootSpace_of_add_ne_zero K L H hx hy hαβ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 277,
"column": 2
} | {
"line": 285,
"column": 84
} | {
"line": 287,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁶ : LieRing L\ninst✝⁵ : Field K\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\n⊢ (cartanEquivDual H).symm (Weight.toLinear ... | [] | obtain ⟨e : L, he₀ : e ≠ 0, he : ∀ x, ⁅x, e⁆ = α x • e⟩ := exists_forall_lie_eq_smul K H L α
have heα : e ∈ rootSpace H α := (mem_genWeightSpace L α e).mpr fun x ↦ ⟨1, by simp [← he x]⟩
obtain ⟨f, hfα, hf⟩ : ∃ f ∈ rootSpace H (-α), killingForm K L e f ≠ 0 := by
contrapose! he₀
simpa using mem_ker_killingFor... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 277,
"column": 2
} | {
"line": 285,
"column": 84
} | {
"line": 287,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁶ : LieRing L\ninst✝⁵ : Field K\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nα : Weight K (↥H) L\n⊢ (cartanEquivDual H).symm (Weight.toLinear ... | [] | obtain ⟨e : L, he₀ : e ≠ 0, he : ∀ x, ⁅x, e⁆ = α x • e⟩ := exists_forall_lie_eq_smul K H L α
have heα : e ∈ rootSpace H α := (mem_genWeightSpace L α e).mpr fun x ↦ ⟨1, by simp [← he x]⟩
obtain ⟨f, hfα, hf⟩ : ∃ f ∈ rootSpace H (-α), killingForm K L e f ≠ 0 := by
contrapose! he₀
simpa using mem_ker_killingFor... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 84
} | {
"line": 349,
"column": 4
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : PerfectField K\nx : L\nhx : x ∈ H\nN S : End K L\nhN : _... | [
"K : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : PerfectField K\nx : L\nhx : x ∈ H\nN S : End K L\nhN : _root_.IsNilp... | have hz : z ∈ ⨆ α : H → K, rootSpace H α := by simp [iSup_genWeightSpace_eq_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 21
} | {
"line": 352,
"column": 22
} | [
{
"pp": "case mem.mem\nK : Type u_2\nL : Type u_3\ninst✝⁷ : LieRing L\ninst✝⁶ : Field K\ninst✝⁵ : LieAlgebra K L\ninst✝⁴ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝³ : H.IsCartanSubalgebra\ninst✝² : IsKilling K L\ninst✝¹ : IsTriangularizable K (↥H) L\ninst✝ : PerfectField K\nx : L\nhx : x ∈ H\nN S : E... | [] | | mem β z hz => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 105,
"column": 27
} | {
"line": 105,
"column": 32
} | {
"line": 106,
"column": 4
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain S... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 106,
"column": 29
} | {
"line": 106,
"column": 34
} | {
"line": 107,
"column": 2
} | [
{
"pp": "case smul\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹⁸ : CommRing R\ninst✝¹⁷ : AddCommGroup M\ninst✝¹⁶ : Module R M\ninst✝¹⁵ : AddCommGroup N\ninst✝¹⁴ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ninst✝¹³ : p.IsPerfPair\nS : Type u_4\nM' : Type u_5\nN' : Type u_6\ninst✝¹² : CommRing S\ninst✝¹¹ : IsDomain ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 7
} | {
"line": 741,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁴ : LieRing L\ninst✝³ : Field K\ninst✝² : LieAlgebra K L\ninst✝¹ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝ : H.IsCartanSubalgebra\nα : ↥LieSubalgebra.root\n⊢ (↑α).IsNonZero",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"LieAlgeb... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 7
} | {
"line": 741,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁴ : LieRing L\ninst✝³ : Field K\ninst✝² : LieAlgebra K L\ninst✝¹ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝ : H.IsCartanSubalgebra\nα : ↥LieSubalgebra.root\n⊢ (↑α).IsNonZero",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"LieAlgeb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 7
} | {
"line": 741,
"column": 0
} | [
{
"pp": "K : Type u_2\nL : Type u_3\ninst✝⁴ : LieRing L\ninst✝³ : Field K\ninst✝² : LieAlgebra K L\ninst✝¹ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝ : H.IsCartanSubalgebra\nα : ↥LieSubalgebra.root\n⊢ (↑α).IsNonZero",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"LieAlgeb... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Reflection | {
"line": 256,
"column": 8
} | {
"line": 256,
"column": 17
} | {
"line": 256,
"column": 18
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nz : M\nt : R\nht : t = f y * g x - 2\na✝ :\n ∀ (n : ℕ),\n ((reflection hf * reflection hg) ^ ↑n) z =\n z +\n (Polynomial.eval t ... | [
"case neg\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nf g : Dual R M\nhf : f x = 2\nhg : g y = 2\nz : M\nt : R\nht : t = f y * g x - 2\na✝ :\n ∀ (n : ℕ),\n ((reflection hf * reflection hg) ^ ↑n) z =\n z +\n (Polynomial.eval t (S R ((↑n - ... | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.FiniteLength | {
"line": 114,
"column": 26
} | {
"line": 117,
"column": 70
} | {
"line": 119,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nM : Type u_2\nN : Type u_3\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nH : IsFiniteLength R N\nhf : Function.Injective ⇑f\n⊢ IsFiniteLength R M",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants... | [] | by
rw [isFiniteLength_iff_isNoetherian_isArtinian] at H ⊢
cases H
exact ⟨isNoetherian_of_injective f hf, isArtinian_of_injective f hf⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Length | {
"line": 131,
"column": 45
} | {
"line": 134,
"column": 65
} | {
"line": 136,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\ne : M ≃ₗ[R] N\n⊢ Module.length R M = Module.length R N",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submo... | [] | by
apply WithBot.coe_injective
rw [Module.coe_length, Module.coe_length,
Order.krullDim_eq_of_orderIso (Submodule.orderIsoMapComap e)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 409,
"column": 6
} | {
"line": 409,
"column": 62
} | {
"line": 410,
"column": 6
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nx✝¹ : AdicCompletion I M\nh : ∀ (n : ℕ), x✝¹ ≡ 0 [SMOD I ^ n • ⊤]\nn : ℕ\nr : R\nhr : r ∈ I ^... | [
"case refine_1.h\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_5\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nx✝¹ : AdicCompletion I M\nh : ∀ (n : ℕ), x✝¹ ≡ 0 [SMOD I ^ n • ⊤]\nn : ℕ\nr : R\nhr : r ∈ I ^ n\nx : Ad... | induction x.val n using Quotient.inductionOn' with | _ a | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 868,
"column": 27
} | {
"line": 868,
"column": 49
} | {
"line": 868,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nx : R\nhx : x ∈ I\n⊢ -x ∈ ⊥.jacobson",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"CommS... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : IsAdicComplete I R\nx : R\nhx : x ∈ I\n⊢ ∀ (y : R), IsUnit (-x * y + 1)"
] | Ideal.mem_jacobson_bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 26
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case mpr\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (IsPrincipalIdealRing R ∧ ∃! P, P ≠ ⊥ ∧ P.IsPrime) → IsDiscreteValuationRing R",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"CommSemiring.toSemiring",
"IsDiscreteValuationRing... | [
"case mpr\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nRPID : IsPrincipalIdealRing R\nPunique : ∃! P, P ≠ ⊥ ∧ P.IsPrime\n⊢ IsDiscreteValuationRing R"
] | rintro ⟨RPID, Punique⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
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