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.Subalgebra | {
"line": 635,
"column": 2
} | {
"line": 637,
"column": 17
} | [
{
"pp": "case mpr\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\ns : Set L\nK : LieSubalgebra R L\n⊢ s ⊆ ↑K → lieSpan R L s ≤ K",
"usedConstants": [
"LieSubalgebra.instSetLike",
"congrArg",
"Membership.mem",
"Eq.mp",
"HasSubset.Subset"... | · intro hs m hm
rw [mem_lieSpan] at hm
exact hm _ hs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Ideal | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 13
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nh₁ : Function.Surjective ⇑f\nx : L\nhx : x ∈ ↑I\n⊢ ∃ x_1, f ↑x_1 = ↑f x",
"usedConstants": [
"LieHom",
... | use ⟨x, hx⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.Ideal | {
"line": 385,
"column": 2
} | {
"line": 385,
"column": 48
} | [
{
"pp": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\ninst✝ : LieAlgebra R L\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nh₁ : comap f ⊥ = ⊥\nh₂ : map f I = ⊥\n⊢ I = ⊥",
"usedConstants": [
"LieAlgebra.toModule",
"LieRing.toAd... | rw [eq_bot_iff, map_le_iff_le_comap, h₁] at h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Jordan.Basic | {
"line": 225,
"column": 12
} | {
"line": 225,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : NonUnitalNonAssocCommRing A\ninst✝ : IsCommJordan A\na b c : A\n⊢ ⁅L a, L (b * b)⁆ + ⁅L b, L (a * a)⁆ + 2 • (⁅L a, L (a * b)⁆ + ⁅L b, L (b * a)⁆) +\n (⁅L a, L (c * c)⁆ + ⁅L c, L (a * a)⁆ + 2 • (⁅L a, L (c * a)⁆ + ⁅L c, L (c * a)⁆)) +\n (⁅L b, L (c * c)⁆ + ⁅L c, L (b *... | mul_comm c a | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Submodule | {
"line": 694,
"column": 48
} | {
"line": 694,
"column": 56
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nm : M\ns : Set (LieSubmodule R L M)\nhne : s.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\nhsup : m ∈ ↑(sSup s)\nthis : ↑(sSup s) = ⋃ N ∈ s, ↑N\n⊢... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Submodule | {
"line": 694,
"column": 48
} | {
"line": 694,
"column": 56
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nm : M\ns : Set (LieSubmodule R L M)\nhne : s.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\nhsup : m ∈ ↑(sSup s)\nthis : ↑(sSup s) = ⋃ N ∈ s, ↑N\n⊢... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Submodule | {
"line": 694,
"column": 48
} | {
"line": 694,
"column": 56
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nm : M\ns : Set (LieSubmodule R L M)\nhne : s.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) s\nhsup : m ∈ ↑(sSup s)\nthis : ↑(sSup s) = ⋃ N ∈ s, ↑N\n⊢... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Submodule | {
"line": 976,
"column": 49
} | {
"line": 978,
"column": 6
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : LieRingModule L M\nN : LieSubmodule R L M\nN' : LieSubmodule R L ↥N\n⊢ map N.incl N' < N ↔ N' < ⊤",
"usedConstants": [
"LieSubmodule.map",
"LieSubmodule.ins... | by
convert! (LieSubmodule.mapOrderEmbedding (f := N.incl) Subtype.coe_injective).lt_iff_lt
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Minpoly.Field | {
"line": 199,
"column": 6
} | {
"line": 199,
"column": 14
} | [
{
"pp": "case pos.refine_2\nA : Type u_1\ninst✝² : Field A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nx : B\nhx : IsIntegral A x\nq : A[X]\nqmo : q.Monic\nhq : (Polynomial.aeval (-x)) q = 0\nthis : (Polynomial.aeval x) ((-1) ^ q.natDegree * q.comp (-X)) = 0\nH : (minpoly A x).degree ≤ ((-1) ^ q.natDeg... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.Ideal | {
"line": 36,
"column": 2
} | {
"line": 38,
"column": 81
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : CommRing R\na : R\nb : R[X]\nP : R[X][X]\nh : eval a (eval b P) = 0\n⊢ ∃ a_1 b_1, a_1 * C (X - C a) + b_1 * (X - C b) = P",
"usedConstants": [
"Iff.mpr",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Polynomial.C",
... | · rcases dvd_iff_isRoot.mpr h with ⟨p, hp⟩
rcases @X_sub_C_dvd_sub_C_eval _ b _ P with ⟨q, hq⟩
exact ⟨C p, q, by rw [mul_comm, mul_comm q, eq_add_of_sub_eq' hq, hp, C_mul]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Quotient | {
"line": 235,
"column": 2
} | {
"line": 237,
"column": 85
} | [
{
"pp": "case mul_X\nR : Type u_1\nσ : Type u_2\ninst✝ : CommRing R\nI : Ideal R\nf : MvPolynomial σ (R ⧸ I)\n⊢ ∀ (p : MvPolynomial σ (R ⧸ I)) (n : σ),\n (Ideal.Quotient.lift (Ideal.map C I) (eval₂Hom (C.comp (Ideal.Quotient.mk I)) X) ⋯)\n (eval₂ (Ideal.Quotient.lift I ((Ideal.Quotient.mk (Ideal.map... | · intro p i hp
simp only at hp
simp only [hp, coe_eval₂Hom, Ideal.Quotient.lift_mk, eval₂_mul, map_mul, eval₂_X] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.Separable | {
"line": 220,
"column": 2
} | {
"line": 224,
"column": 69
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nι : Type u_1\nf : ι → R\ns : Finset ι\nhfs : (∏ i ∈ s, (X - C (f i))).Separable\nx y : ι\nhx : x ∈ s\nhy : y ∈ s\nhfxy : f x = f y\n⊢ x = y",
"usedConstants": [
"Polynomial.C",
"False",
"Nat.instMulZeroClass",
"Semigroup... | by_contra hxy
rw [← insert_erase hx, prod_insert (notMem_erase _ _), ←
insert_erase (mem_erase_of_ne_of_mem (Ne.symm hxy) hy), prod_insert (notMem_erase _ _), ←
mul_assoc, hfxy, ← sq] at hfs
cases (hfs.of_mul_left.of_pow (not_isUnit_X_sub_C _) two_ne_zero).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Separable | {
"line": 220,
"column": 2
} | {
"line": 224,
"column": 69
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\nι : Type u_1\nf : ι → R\ns : Finset ι\nhfs : (∏ i ∈ s, (X - C (f i))).Separable\nx y : ι\nhx : x ∈ s\nhy : y ∈ s\nhfxy : f x = f y\n⊢ x = y",
"usedConstants": [
"Polynomial.C",
"False",
"Nat.instMulZeroClass",
"Semigroup... | by_contra hxy
rw [← insert_erase hx, prod_insert (notMem_erase _ _), ←
insert_erase (mem_erase_of_ne_of_mem (Ne.symm hxy) hy), prod_insert (notMem_erase _ _), ←
mul_assoc, hfxy, ← sq] at hfs
cases (hfs.of_mul_left.of_pow (not_isUnit_X_sub_C _) two_ne_zero).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Perfect | {
"line": 296,
"column": 66
} | {
"line": 302,
"column": 49
} | [
{
"pp": "K : Type u_1\np : ℕ\ninst✝² : Field K\ninst✝¹ : ExpChar K p\ninst✝ : PerfectRing K p\n⊢ PerfectField K",
"usedConstants": [
"PerfectRing",
"PerfectField",
"False",
"Nat.Prime",
"ExpChar.zero",
"ExpChar.casesOn",
"PerfectField.mk",
"CommSemiring.toSemi... | by
obtain hp | ⟨hp⟩ := ‹ExpChar K p›
· exact ⟨Irreducible.separable⟩
refine PerfectField.mk fun hf ↦ ?_
rcases separable_or p hf with h | ⟨-, g, -, rfl⟩
· assumption
· exfalso; revert hf; haveI := Fact.mk hp; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Separable | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 62
} | [
{
"pp": "F : Type u\ninst✝² : Field F\nK : Type v\ninst✝¹ : Field K\ninst✝ : Algebra F K\np : F[X]\nhsep : p.Separable\nhsplit : (map (algebraMap F K) p).Splits\n⊢ Fintype.card ↑(p.rootSet K) = p.natDegree",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"congrArg",
"Finset",
... | simp_rw [rootSet_def, Finset.coe_sort_coe, Fintype.card_coe] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.Determinant | {
"line": 400,
"column": 53
} | {
"line": 400,
"column": 76
} | [
{
"pp": "case a\nR : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nι : Type u_4\ninst✝² : Fintype ι\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nf : ι → M →ₗ[R] M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\nB : Basis (Free.ChooseBasisIndex R... | Equiv.symm_trans_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 518,
"column": 30
} | {
"line": 518,
"column": 45
} | [
{
"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\nf : M →ₗ[R] M'\nv : Basis ι R M\nv' : Basis ι R M'\nh : IsUnit ((toMatr... | toLin_toMatrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Determinant | {
"line": 542,
"column": 4
} | {
"line": 542,
"column": 55
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nM' : Type u_3\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : Module R M'\nι✝ : Type u_4\ninst✝³ : DecidableEq ι✝\ninst✝² : Fintype ι✝\ne : Basis ι✝ R M\ninst✝¹ : Free R M\ninst✝ : Module.Finite R M\nf : M →ₗ[R... | have : DecidableEq ι := Classical.typeDecidableEq ι | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AdjoinRoot | {
"line": 238,
"column": 19
} | {
"line": 238,
"column": 27
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : S[X]\np q : R[X]\na✝¹ : (aeval (root f)) p = (mk f) (Polynomial.map (algebraMap R S) p)\na✝ : (aeval (root f)) q = (mk f) (Polynomial.map (algebraMap R S) q)\n⊢ (aeval (root f)) (p + q) = (mk f) (Po... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.AdjoinRoot | {
"line": 238,
"column": 19
} | {
"line": 238,
"column": 27
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : S[X]\np q : R[X]\na✝¹ : (aeval (root f)) p = (mk f) (Polynomial.map (algebraMap R S) p)\na✝ : (aeval (root f)) q = (mk f) (Polynomial.map (algebraMap R S) q)\n⊢ (aeval (root f)) (p + q) = (mk f) (Po... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdjoinRoot | {
"line": 238,
"column": 19
} | {
"line": 238,
"column": 27
} | [
{
"pp": "case add\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nf : S[X]\np q : R[X]\na✝¹ : (aeval (root f)) p = (mk f) (Polynomial.map (algebraMap R S) p)\na✝ : (aeval (root f)) q = (mk f) (Polynomial.map (algebraMap R S) q)\n⊢ (aeval (root f)) (p + q) = (mk f) (Po... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 119,
"column": 8
} | {
"line": 119,
"column": 16
} | [
{
"pp": "case calc_3.i_inj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁✝ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := max n₁ n₂\nh₄✝ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := range (2 * N + 1)\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := ra... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 121,
"column": 8
} | {
"line": 123,
"column": 48
} | [
{
"pp": "case calc_3.i_surj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := ⋯\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := ⋯\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := ⋯\nhRN : RN = range (N + 1)... | simp only [mem_filter, mem_product, mem_range, mem_sigma, exists_prop, Sigma.exists,
and_imp, Prod.forall, Prod.mk.injEq]
exact fun x y _ _ _ ↦ ⟨x + y, x, by lia⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 121,
"column": 8
} | {
"line": 123,
"column": 48
} | [
{
"pp": "case calc_3.i_surj\nA : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na b : A\nh₁ : Commute a b\nn₁ : ℕ\nhn₁ : a ^ n₁ = 0\nn₂ : ℕ\nhn₂ : b ^ n₂ = 0\nN : ℕ := ⋯\nh₄ : a ^ (N + 1) = 0\nh₅ : b ^ (N + 1) = 0\nR2N : Finset ℕ := ⋯\nhR2N : R2N = range (2 * N + 1)\nRN : Finset ℕ := ⋯\nhRN : RN = range (N + 1)... | simp only [mem_filter, mem_product, mem_range, mem_sigma, exists_prop, Sigma.exists,
and_imp, Prod.forall, Prod.mk.injEq]
exact fun x y _ _ _ ↦ ⟨x + y, x, by lia⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Semisimple | {
"line": 230,
"column": 25
} | {
"line": 230,
"column": 67
} | [
{
"pp": "M : Type u_2\ninst✝² : AddCommGroup M\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Module K M\nf : End K M\np : K[X]\nhp : Squarefree p\nhpf : p ∈ RingHom.ker (aeval f)\n⊢ f.IsSemisimple",
"usedConstants": [
"Semiring.toModule",
"instSMulOfMul",
"Module.annihilator",
"AlgHom.alg... | ← AEval.annihilator_eq_ker_aeval (M := M), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Semisimple | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 78
} | [
{
"pp": "M : Type u_2\ninst✝² : AddCommGroup M\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Module K M\nf : End K M\np : K[X]\nhp : Squarefree p\nhpf : IsTorsionBySet K[X] (AEval K M f) ↑(Ideal.span {p})\nR : Type u_3 := K[X] ⧸ Ideal.span {p}\nthis✝³ : IsReduced R\nthis✝² : FiniteDimensional K R\nthis✝¹ : IsArtinia... | exact (e.isSemisimpleModule_iff_of_bijective bijective_id).mpr inferInstance | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 59
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : Module ℚ M\ninst✝ : Module ℚ N\nfM : End R M\nfN : End R N\ng : M →ₗ[R] N\nh : fN ∘ₗ g = g ∘ₗ fM\nm : M\nk l : ℕ\nkl : ℕ := max k l... | simp [exp_eq_sum hfM, exp_eq_sum hfN, this, map_rat_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Solvable | {
"line": 97,
"column": 40
} | {
"line": 97,
"column": 43
} | [
{
"pp": "case zero\nR : Type u\nL : Type v\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nI J : LieIdeal R L\nh₁ : I ≤ J\nl : ℕ\nh₂ : l = 0\n⊢ D 0 I ≤ D l J",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"Nat.instMulZeroClass",
"LieRing.toAddCommGroup",
... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Solvable | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 38
} | [
{
"pp": "case refine_2\nR : Type u\nL : Type v\nM : Type w\nL' : Type w₁\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : LieRing L'\ninst✝¹ : LieAlgebra R L'\nI✝ J✝ : LieIdeal R L\nf : L' →ₗ⁅R⁆ L\ninst✝ : IsNoetherian R L\nhwf : CompleteLattice.IsSupClosedCompact (LieSubmodule R L L)... | apply LieAlgebra.isSolvableAdd R L | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.BilinearForm.Orthogonal | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 52
} | [
{
"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\nhB : B.Nondegenerate\nW : Submodule K V\n⊢ finrank K ↥(B.orthogonal W) = finrank K V - finrank K ↥W",
"usedConstants": [
"Submodule",
"Semiring.t... | have := finrank_add_finrank_orthogonal' (B := B) W | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.BaseChange | {
"line": 72,
"column": 74
} | {
"line": 72,
"column": 77
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nx z₁ z₂ : A ⊗[R] L\nh₁ : ((bracket' R A L L) z₁) z₁ = 0\nh₂ : ((bracket' R A L L) z₂) z₂ = 0\nthis : ((bracket' R A L L) z₁) z₂ + ((bracket' R A L L) z₂)... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.BaseChange | {
"line": 138,
"column": 70
} | {
"line": 138,
"column": 78
} | [
{
"pp": "R✝ : Type u_1\nA✝ : Type u_2\nL✝ : Type u_3\nM : Type u_4\ninst✝¹⁷ : CommRing R✝\ninst✝¹⁶ : CommRing A✝\ninst✝¹⁵ : Algebra R✝ A✝\ninst✝¹⁴ : LieRing L✝\ninst✝¹³ : LieAlgebra R✝ L✝\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R✝ M\ninst✝¹⁰ : LieRingModule L✝ M\ninst✝⁹ : LieModule R✝ L✝ M\nR : Type u_5\nA ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.BaseChange | {
"line": 138,
"column": 70
} | {
"line": 138,
"column": 78
} | [
{
"pp": "R✝ : Type u_1\nA✝ : Type u_2\nL✝ : Type u_3\nM : Type u_4\ninst✝¹⁷ : CommRing R✝\ninst✝¹⁶ : CommRing A✝\ninst✝¹⁵ : Algebra R✝ A✝\ninst✝¹⁴ : LieRing L✝\ninst✝¹³ : LieAlgebra R✝ L✝\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R✝ M\ninst✝¹⁰ : LieRingModule L✝ M\ninst✝⁹ : LieModule R✝ L✝ M\nR : Type u_5\nA ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.BaseChange | {
"line": 138,
"column": 70
} | {
"line": 138,
"column": 78
} | [
{
"pp": "R✝ : Type u_1\nA✝ : Type u_2\nL✝ : Type u_3\nM : Type u_4\ninst✝¹⁷ : CommRing R✝\ninst✝¹⁶ : CommRing A✝\ninst✝¹⁵ : Algebra R✝ A✝\ninst✝¹⁴ : LieRing L✝\ninst✝¹³ : LieAlgebra R✝ L✝\ninst✝¹² : AddCommGroup M\ninst✝¹¹ : Module R✝ M\ninst✝¹⁰ : LieRingModule L✝ M\ninst✝⁹ : LieModule R✝ L✝ M\nR : Type u_5\nA ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 12
} | [
{
"pp": "R : 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) := {I | IsAtom I}\nhI : I < s.sup id\nJ : LieIdeal R L\nhJs : J ∈ s\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Normalizer | {
"line": 177,
"column": 2
} | {
"line": 183,
"column": 72
} | [
{
"pp": "case refine_1\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 : H.normalizer = H\n⊢ ∀ m ∈ LieModule.maxTrivSubmodule R (↥H) (L ⧸ H.toLieSubmodule), m = 0",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubalgebra.lie... | · rintro ⟨x⟩ hx
suffices x ∈ H by rwa [Submodule.Quotient.quot_mk_eq_mk, Submodule.Quotient.mk_eq_zero,
coe_toLieSubmodule, mem_toSubmodule]
rw [← h, H.mem_normalizer_iff']
intro y hy
replace hx : ⁅_, LieSubmodule.Quotient.mk' _ x⁆ = 0 := hx ⟨y, hy⟩
rwa [← LieModuleHom.map_lie, LieSubmodule.Qu... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Quotient | {
"line": 161,
"column": 22
} | {
"line": 168,
"column": 35
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : LieRingModule L M\nN N' : LieSubmodule R L M\ninst✝¹ : LieAlgebra R L\ninst✝ : LieModule R L M\nI J : LieIdeal R L\nt : R\nx' y' : L ⧸ I\n⊢ ⁅x', t • y'⁆ = t • ⁅x', y'⁆",
... | by
induction x', y' using Quotient.inductionOn₂' with | _ x y
repeat'
first
| rw [is_quotient_mk]
| rw [← mk_bracket]
| rw [← Submodule.Quotient.mk_smul (R := R) (M := L)]
apply congr_arg; apply lie_smul | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Engel | {
"line": 128,
"column": 6
} | {
"line": 130,
"column": 72
} | [
{
"pp": "case succ.refine_2\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).toSubmo... | norm_cast
gcongr
exact le_trans (antitone_lowerCentralSeries R L M le_self_add) hIM | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Engel | {
"line": 128,
"column": 6
} | {
"line": 130,
"column": 72
} | [
{
"pp": "case succ.refine_2\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).toSubmo... | norm_cast
gcongr
exact le_trans (antitone_lowerCentralSeries R L M le_self_add) hIM | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Engel | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 68
} | [
{
"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 = ⊤\nhIM : LieM... | have hk' : I.lcs M k = ⊥ := by
simp only [← toSubmodule_inj, I.coe_lcs_eq, hk, bot_toSubmodule] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 722,
"column": 2
} | {
"line": 722,
"column": 10
} | [
{
"pp": "case refine_1\nR : 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\nk : ℕ\nN : LieSubmodule R L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNoetherian R M\nhwf : CompleteLattice.IsSupCl... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 722,
"column": 2
} | {
"line": 722,
"column": 10
} | [
{
"pp": "case refine_2\nR : 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\nk : ℕ\nN : LieSubmodule R L M\ninst✝¹ : LieModule R L M\ninst✝ : IsNoetherian R M\nhwf : CompleteLattice.IsSupCl... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 228,
"column": 6
} | {
"line": 228,
"column": 37
} | [
{
"pp": "K : Type u_3\nL : Type u_4\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nE1 E2 : IntermediateField K L\ninst✝ : Algebra.IsAlgebraic K ↥E2\nthis :\n Subalgebra.restrictScalars K (adjoin ↥E1 ↑E2).toSubalgebra = Subalgebra.restrictScalars K (Algebra.adjoin ↥E1 ↑E2)\n⊢ (E1 ⊔ E2).toSubalgebra ... | ← restrictScalars_toSubalgebra, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 37
} | [
{
"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... | ← restrictScalars_toSubalgebra, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 267,
"column": 74
} | {
"line": 278,
"column": 60
} | [
{
"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\n⊢ (adjoin E ↑L).toSubalgeb... | by
let i := IsScalarTower.toAlgHom F E K
let E' := i.fieldRange
let i' : E ≃ₐ[F] E' := AlgEquiv.ofInjectiveField i
have hi : algebraMap E K = (algebraMap E' K) ∘ i' := by ext x; rfl
apply_fun _ using Subalgebra.restrictScalars_injective F
rw [← restrictScalars_toSubalgebra, restrictScalars_adjoin_of_algEqui... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.IntermediateField.Adjoin.Algebra | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 58
} | [
{
"pp": "case a\nF : Type u_1\nA : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁸ : Field F\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra F A\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\ninst✝³ : Algebra A K\ninst✝² : IsFractionRing A K\ninst✝¹ : Field L\ninst✝ : Algebra F L\ng : A →ₐ[F] L\nf : K →ₐ[F] L\nh : (↑f).comp (alg... | refine ringHom_fieldRange_eq_of_comp_eq_of_range_eq h ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 222,
"column": 48
} | {
"line": 222,
"column": 56
} | [
{
"pp": "case h.a.C\nk : Type u\ninst✝² : Field k\nk' : Type u\ninst✝¹ : Field k'\ne : k ≃+* k'\ninst✝ : IsAlgClosed k\nx : k\na✝ : k'\n⊢ e.symm (eval (e x) (C a✝)) = eval x (map e.symm.toRingHom (C a✝))",
"usedConstants": [
"Polynomial.C",
"Polynomial.eval",
"Polynomial.eval_C",
"co... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 222,
"column": 48
} | {
"line": 222,
"column": 56
} | [
{
"pp": "case h.a.add\nk : Type u\ninst✝² : Field k\nk' : Type u\ninst✝¹ : Field k'\ne : k ≃+* k'\ninst✝ : IsAlgClosed k\nx : k\np✝ q✝ : k'[X]\na✝¹ : e.symm (eval (e x) p✝) = eval x (map e.symm.toRingHom p✝)\na✝ : e.symm (eval (e x) q✝) = eval x (map e.symm.toRingHom q✝)\n⊢ e.symm (eval (e x) (p✝ + q✝)) = eval ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 222,
"column": 48
} | {
"line": 222,
"column": 56
} | [
{
"pp": "case h.a.monomial\nk : Type u\ninst✝² : Field k\nk' : Type u\ninst✝¹ : Field k'\ne : k ≃+* k'\ninst✝ : IsAlgClosed k\nx : k\nn✝ : ℕ\na✝¹ : k'\na✝ : e.symm (eval (e x) (C a✝¹ * X ^ n✝)) = eval x (map e.symm.toRingHom (C a✝¹ * X ^ n✝))\n⊢ e.symm (eval (e x) (C a✝¹ * X ^ (n✝ + 1))) = eval x (map e.symm.to... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 268,
"column": 35
} | {
"line": 268,
"column": 43
} | [
{
"pp": "case refine_1\nk : Type u\ninst✝³ : Field k\nK : Type v\ninst✝² : Field K\ninst✝¹ : IsAlgClosed K\ninst✝ : Algebra k K\np q : k[X]\nh : p = 0 ∧ q = 0\n⊢ 0 = 0 ∧ 0 = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"id",
"Field.toSemifield",
"And",
"... | and_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Extension | {
"line": 163,
"column": 6
} | {
"line": 164,
"column": 48
} | [
{
"pp": "case refine_2.inl\nF : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁵ : Field F\ninst✝⁴ : Field E\ninst✝³ : Field K\ninst✝² : Algebra F E\ninst✝¹ : Algebra F K\nc : Set (Lifts F E K)\nhc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nalg : Algebra.IsAlgebraic F E\ninst✝ : Nonempty ↑c\nhext : ∀ σ ∈ c, σ.IsExtendible\... | change (θ π₂).emb (inclusion (ge π₂).1 <| inclusion h.1 ⟨x, hx⟩) =
(θ π₁).emb (inclusion (ge π₁).1 ⟨x, hx⟩) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 504,
"column": 4
} | {
"line": 504,
"column": 12
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\nα : E\ninst✝ : FiniteDimensional F E\nhprim : F⟮α⟯ = ⊤\nK✝ : IntermediateField F E\ng : E[X] := Polynomial.map (algebraMap (↥K✝) E) (minpoly (↥K✝) α)\nK' : IntermediateField F E := adjoin F ↑g.coeffs\nhsub : K' ≤ K✝\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 580,
"column": 2
} | {
"line": 582,
"column": 40
} | [
{
"pp": "K : Type u\ninst✝³ : Field K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\nS : Set L\ninst✝ : Finite ↑S\nhS : ∀ x ∈ S, IsIntegral K x\n⊢ FiniteDimensional K ↥(adjoin K S)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
... | rw [← biSup_adjoin_simple, ← iSup_subtype'']
haveI (x : S) := adjoin.finiteDimensional (hS x.1 x.2)
exact finiteDimensional_iSup_of_finite | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 580,
"column": 2
} | {
"line": 582,
"column": 40
} | [
{
"pp": "K : Type u\ninst✝³ : Field K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\nS : Set L\ninst✝ : Finite ↑S\nhS : ∀ x ∈ S, IsIntegral K x\n⊢ FiniteDimensional K ↥(adjoin K S)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
... | rw [← biSup_adjoin_simple, ← iSup_subtype'']
haveI (x : S) := adjoin.finiteDimensional (hS x.1 x.2)
exact finiteDimensional_iSup_of_finite | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 762,
"column": 2
} | {
"line": 765,
"column": 39
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsIntegral K x\nhy : IsIntegral K y\n⊢ FiniteDimensional K ↥K⟮x, y⟯",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | have := adjoin.finiteDimensional hx
have := adjoin.finiteDimensional hy
rw [← Set.singleton_union, adjoin_union]
exact finiteDimensional_sup K⟮x⟯ K⟮y⟯ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 762,
"column": 2
} | {
"line": 765,
"column": 39
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx y : L\nhx : IsIntegral K x\nhy : IsIntegral K y\n⊢ FiniteDimensional K ↥K⟮x, y⟯",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | have := adjoin.finiteDimensional hx
have := adjoin.finiteDimensional hy
rw [← Set.singleton_union, adjoin_union]
exact finiteDimensional_sup K⟮x⟯ K⟮y⟯ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 14
} | [
{
"pp": "case neg\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 ∈ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 14
} | [
{
"pp": "case neg\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 ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 14
} | [
{
"pp": "case neg\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 ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 16
} | [
{
"pp": "case neg\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 ∈ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 16
} | [
{
"pp": "case neg\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 ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 16
} | [
{
"pp": "case neg\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 ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Linear | {
"line": 235,
"column": 7
} | {
"line": 235,
"column": 43
} | [
{
"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... | coe_lie_shiftedGenWeightSpace_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Multiset.Fintype | {
"line": 311,
"column": 18
} | {
"line": 311,
"column": 26
} | [
{
"pp": "case isTrue\nα : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nm✝ m : Multiset α\nf : α → β\nl : List α\na : α\ns : List α\nx✝ : { v // ∀ (a : ToType ⟦s⟧), (v a).fst = f a.fst }\nv : ToType ⟦s⟧ ≃ (map f ⟦s⟧).ToType\nhv : ∀ (a : ToType ⟦s⟧), (v a).fst = f a.fst\nx : ToType ⟦a ::... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Multiset.Fintype | {
"line": 311,
"column": 18
} | {
"line": 311,
"column": 26
} | [
{
"pp": "case isFalse\nα : Type u_1\nβ : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nm✝ m : Multiset α\nf : α → β\nl : List α\na : α\ns : List α\nx✝ : { v // ∀ (a : ToType ⟦s⟧), (v a).fst = f a.fst }\nv : ToType ⟦s⟧ ≃ (map f ⟦s⟧).ToType\nhv : ∀ (a : ToType ⟦s⟧), (v a).fst = f a.fst\nx : ToType ⟦a :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | {
"line": 90,
"column": 48
} | {
"line": 90,
"column": 58
} | [
{
"pp": "k : Type u\ninst✝ : Field k\ns : Finset (Monics k)\nf : Monics k\nh : f ∈ s\nn : ℕ\n⊢ (map (↑(MvPolynomial.aeval fun fi ↦ if hf : fi.fst ∈ s then (↑((finEquivRoots ⋯) fi.snd)).1 else 37))\n (map (algebraMap k (MvPolynomial (Vars k) k)) ↑f) -\n ∏ x, (X - C (if h : f ∈ s then (↑((finE... | dif_pos h, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 83,
"column": 35
} | {
"line": 83,
"column": 43
} | [
{
"pp": "case refine_1\nK : Type v\ninst✝ : Field K\nf g : K[X]\nh : f = 0 ∧ g = 0\n⊢ 0 = 0 ∧ 0 = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"and_self",
"id",
"Field.toSemifield",
"Field.toCommRing",
"And",
"CommRing.toCommSe... | and_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 192,
"column": 62
} | {
"line": 192,
"column": 71
} | [
{
"pp": "n✝ : ℕ\nK✝ : Type u\ninst✝ : Field K✝\nn : ℕ\nih :\n (fun n ↦\n ∀ {K : Type u} [inst : Field K] (f : K[X]),\n f.natDegree = n → Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤)\n n\nK : Type u\nx✝ : Field K\nf : K[X]\nhfn : 0 = n.succ\nh : f.natDegree = 0\n⊢ False",
"usedCo... | cases hfn | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure | {
"line": 169,
"column": 54
} | {
"line": 169,
"column": 76
} | [
{
"pp": "case a\nk : Type u\ninst✝ : Field k\nf : Monics k\nn✝ : ℕ\n⊢ (map (Ideal.Quotient.mk (maxIdeal k)) (map (algebraMap k (MvPolynomial (Vars k) k)) ↑f)).coeff n✝ =\n (∏ i, map (Ideal.Quotient.mk (maxIdeal k)) (X - C (MvPolynomial.X ⟨f, i⟩))).coeff n✝",
"usedConstants": [
"Ideal.Quotient.commS... | ← Polynomial.map_prod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 218,
"column": 51
} | {
"line": 218,
"column": 59
} | [
{
"pp": "case mk.mk\nK : Type u_1\nR : 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\ntoFun✝¹ : L → R\ngenWeightSpace_ne_bot... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 10
} | [
{
"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✝ : IsNilpotent L M\n⊢ genWeightSpace M 0 = ⊤",
"usedCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 10
} | [
{
"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✝ : IsNilpotent L M\n⊢ genWeightSpace M 0 = ⊤",
"usedCo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 10
} | [
{
"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✝ : IsNilpotent L M\n⊢ genWeightSpace M 0 = ⊤",
"usedCo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 313,
"column": 62
} | {
"line": 314,
"column": 10
} | [
{
"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✝ : IsNilpotent L M\n⊢ genWeightSpace M 0 = ⊤",
"usedCo... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Sl2 | {
"line": 67,
"column": 51
} | {
"line": 70,
"column": 35
} | [
{
"pp": "L : Type u_2\ninst✝ : LieRing L\nh e f : L\nt : IsSl2Triple h e f\n⊢ e ≠ 0",
"usedConstants": [
"LieRing.toAddCommGroup",
"congrArg",
"Bracket.bracket",
"Mathlib.Tactic.Contrapose.contrapose₄",
"Eq.mp",
"Ne",
"SubtractionMonoid.toSubNegZeroMonoid",
"L... | by
have := t.h_ne_zero
contrapose this
simpa [this] using t.lie_e_f.symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Sl2 | {
"line": 72,
"column": 51
} | {
"line": 75,
"column": 35
} | [
{
"pp": "L : Type u_2\ninst✝ : LieRing L\nh e f : L\nt : IsSl2Triple h e f\n⊢ f ≠ 0",
"usedConstants": [
"LieRing.toAddCommGroup",
"congrArg",
"Bracket.bracket",
"Mathlib.Tactic.Contrapose.contrapose₄",
"Eq.mp",
"Ne",
"SubtractionMonoid.toSubNegZeroMonoid",
"L... | by
have := t.h_ne_zero
contrapose this
simpa [this] using t.lie_e_f.symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.TensorProduct | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 12
} | [
{
"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 :... | revert x y | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.Algebra.Lie.Sl2 | {
"line": 141,
"column": 72
} | {
"line": 141,
"column": 98
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nh e f x : L\nt : IsSl2Triple h e f\n⊢ x ∈ Submodule.span R {e, f, h} ↔ ∃ c₁ c₂ c₃, x = c₁ • e + c₂ • f + c₃ • h",
"usedConstants": [
"LieAlgebra.toModule",
"Eq.mpr",
"Submodule",
"in... | Submodule.mem_span_triple, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Finiteness.Nilpotent | {
"line": 33,
"column": 11
} | {
"line": 33,
"column": 19
} | [
{
"pp": "case h.h.add\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nf : End R M\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ng : M → ℕ\nhg : ∀ (m : M), (f ^ g m) m = 0\nm x✝ y✝ : M\nhx✝ : x✝ ∈ Submodule.span R ↑S\nhy✝ : y✝ ∈ Submo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Finiteness.Nilpotent | {
"line": 33,
"column": 11
} | {
"line": 33,
"column": 19
} | [
{
"pp": "case h.h.add\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nf : End R M\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ng : M → ℕ\nhg : ∀ (m : M), (f ^ g m) m = 0\nm x✝ y✝ : M\nhx✝ : x✝ ∈ Submodule.span R ↑S\nhy✝ : y✝ ∈ Submo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Finiteness.Nilpotent | {
"line": 33,
"column": 11
} | {
"line": 33,
"column": 19
} | [
{
"pp": "case h.h.add\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nf : End R M\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ng : M → ℕ\nhg : ∀ (m : M), (f ^ g m) m = 0\nm x✝ y✝ : M\nhx✝ : x✝ ∈ Submodule.span R ↑S\nhy✝ : y✝ ∈ Submo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Finiteness.Nilpotent | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 10
} | [
{
"pp": "case h.h.add\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nf : End R M\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ng : M → ℕ\nhg : ∀ (m : M), (f ^ g m) m = 0\nm x✝ y✝ : M\nhx✝ : x✝ ∈ Submodule.span R ↑S\nhy✝ : y✝ ∈ Submo... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Finiteness.Nilpotent | {
"line": 34,
"column": 12
} | {
"line": 34,
"column": 20
} | [
{
"pp": "case h.h.smul\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nf : End R M\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ng : M → ℕ\nhg : ∀ (m : M), (f ^ g m) m = 0\nm : M\na✝¹ : R\nx✝ : M\nhx✝ : x✝ ∈ Submodule.span R ↑S\na✝ :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Finiteness.Nilpotent | {
"line": 34,
"column": 12
} | {
"line": 34,
"column": 20
} | [
{
"pp": "case h.h.smul\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nf : End R M\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ng : M → ℕ\nhg : ∀ (m : M), (f ^ g m) m = 0\nm : M\na✝¹ : R\nx✝ : M\nhx✝ : x✝ ∈ Submodule.span R ↑S\na✝ :... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Finiteness.Nilpotent | {
"line": 34,
"column": 12
} | {
"line": 34,
"column": 20
} | [
{
"pp": "case h.h.smul\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nf : End R M\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ng : M → ℕ\nhg : ∀ (m : M), (f ^ g m) m = 0\nm : M\na✝¹ : R\nx✝ : M\nhx✝ : x✝ ∈ Submodule.span R ↑S\na✝ :... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Eigenspace.Semisimple | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 21
} | [
{
"pp": "case mpr.inl\nK : Type u_3\nV : Type u_4\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : End K V\nhf : f.IsSemisimple\nh : ∀ (μ : K), f.HasEigenvalue μ → μ = 0\n⊢ f.eigenspace 0 ≤ f.eigenspace 0",
"usedConstants": [
"... | · exact le_refl _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 202,
"column": 11
} | {
"line": 202,
"column": 19
} | [
{
"pp": "case add\nR : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 202,
"column": 11
} | {
"line": 202,
"column": 19
} | [
{
"pp": "case add\nR : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 202,
"column": 11
} | {
"line": 202,
"column": 19
} | [
{
"pp": "case add\nR : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 10
} | [
{
"pp": "case add\nR : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ : ... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 203,
"column": 12
} | {
"line": 203,
"column": 20
} | [
{
"pp": "case smul\nR : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ :... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 203,
"column": 12
} | {
"line": 203,
"column": 20
} | [
{
"pp": "case smul\nR : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ :... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 203,
"column": 12
} | {
"line": 203,
"column": 20
} | [
{
"pp": "case smul\nR : Type u_1\nL : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : LieRing L\ninst✝⁶ : LieAlgebra R L\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : LieRingModule L M\ninst✝² : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\np q : ℤ\ninst✝¹ : H.IsCartanSubalgebra\ninst✝ :... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.TraceForm | {
"line": 153,
"column": 53
} | {
"line": 153,
"column": 56
} | [
{
"pp": "R : Type u_1\nL : Type u_3\nM : Type u_4\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\ninst✝⁵ : Free R M\ninst✝⁴ : IsDomain R\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : LieRing.Is... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.TraceForm | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 12
} | [
{
"pp": "R : Type u_1\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\nB : LinearMap.BilinForm R M\nm₀ m₁ : M\nhm₀ : m₀ ∈ genWeightSpac... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 142,
"column": 11
} | {
"line": 142,
"column": 19
} | [
{
"pp": "case h.add\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), ((ki... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 142,
"column": 11
} | {
"line": 142,
"column": 19
} | [
{
"pp": "case h.add\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), ((ki... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 142,
"column": 11
} | {
"line": 142,
"column": 19
} | [
{
"pp": "case h.add\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), ((ki... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 10
} | [
{
"pp": "case h.add\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), ((ki... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 74
} | [
{
"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\nthis : range (Weight.toLinear K (↥H) L) ⊆ insert 0 (Weight.toLine... | simpa only [Submodule.span_insert_zero] using Submodule.span_mono this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
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