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.AdjointAction.Basic | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 48
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nh : IsNilpotent a\n⊢ IsNilpotent ((ad R A) a)",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Module.End.instRing",
"Eq.mpr",
"Algebra.to_smulCommClass",
"Module.... | rw [LieAlgebra.ad_eq_lmul_left_sub_lmul_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.AdjointAction.Basic | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nK : LieSubalgebra R L\nx : ↥K\nn : ℕ\nhn : (LieAlgebra.ad R L) ↑x ^ n = 0\n⊢ IsNilpotent ((LieAlgebra.ad R ↥K) x)",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
"Module.End.instRing... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.AdjointAction.Basic | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 48
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝⁴ : Field K\ninst✝³ : PerfectField K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\na : Module.End K V\nha : a.IsSemisimple\n⊢ ((ad K (Module.End K V)) a).IsSemisimple",
"usedConstants": [
"LieHom",
"LieAlgebra.toModule",
... | rw [LieAlgebra.ad_eq_lmul_left_sub_lmul_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Determinant | {
"line": 543,
"column": 63
} | {
"line": 543,
"column": 86
} | [
{
"pp": "R : 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] M\nh : I... | by rwa [det_toMatrix b] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 72,
"column": 68
} | {
"line": 79,
"column": 90
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : Ring A\ninst✝³ : Module ℚ A\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Module ℚ M\na : A\nm : M\nk : ℕ\nh : a ^ k • m = 0\nhn : IsNilpotent a\n⊢ exp a • m = ∑ i ∈ range k, (↑i !)⁻¹ • a ^ i • m",
"usedConstants": [
"Eq.mpr",
"Finset.sum_sm... | by
rcases le_or_gt (nilpotencyClass a) k with h₀ | h₀
· simp_rw [exp_eq_sum (pow_eq_zero_of_le h₀ (pow_nilpotencyClass hn)), sum_smul, smul_assoc]
rw [exp, sum_smul, ← sum_range_add_sum_Ico _ (Nat.le_of_succ_le h₀)]
suffices ∑ i ∈ Ico k (nilpotencyClass a), ((i.factorial : ℚ)⁻¹ • (a ^ i)) • m = 0 by
simp_rw... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdjoinRoot | {
"line": 578,
"column": 64
} | {
"line": 578,
"column": 84
} | [
{
"pp": "case ih\nR : Type u_1\ninst✝ : CommRing R\ng : R[X]\nhg : g.Monic\np✝ : R[X]\n⊢ g ∣ p✝ - g * (p✝ /ₘ g) - p✝",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemiring.toSemiring",
... | sub_sub_cancel_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 65
} | [
{
"pp": "case calc_2\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 := range (N +... | refine sum_congr rfl fun i hi ↦ sum_congr rfl fun j hj ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 94,
"column": 4
} | {
"line": 124,
"column": 54
} | [
{
"pp": "A : 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 := range (N + 1)\nhRN : RN... | calc ∑ i ∈ R2N, (i ! : ℚ)⁻¹ • (a + b) ^ i
= ∑ i ∈ R2N, (i ! : ℚ)⁻¹ • ∑ j ∈ range (i + 1), a ^ j * b ^ (i - j) * i.choose j := ?_
_ = ∑ i ∈ R2N, (∑ j ∈ range (i + 1),
((j ! : ℚ)⁻¹ * ((i - j) ! : ℚ)⁻¹) • (a ^ j * b ^ (i - j))) := ?_
_ = ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N,
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 94,
"column": 4
} | {
"line": 124,
"column": 54
} | [
{
"pp": "A : 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 := range (N + 1)\nhRN : RN... | calc ∑ i ∈ R2N, (i ! : ℚ)⁻¹ • (a + b) ^ i
= ∑ i ∈ R2N, (i ! : ℚ)⁻¹ • ∑ j ∈ range (i + 1), a ^ j * b ^ (i - j) * i.choose j := ?_
_ = ∑ i ∈ R2N, (∑ j ∈ range (i + 1),
((j ! : ℚ)⁻¹ * ((i - j) ! : ℚ)⁻¹) • (a ^ j * b ^ (i - j))) := ?_
_ = ∑ ij ∈ R2N ×ˢ R2N with ij.1 + ij.2 ≤ 2 * N,
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Semisimple | {
"line": 144,
"column": 2
} | {
"line": 147,
"column": 51
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nhn : IsNilpotent f\nhs : f.IsFinitelySemisimple\n⊢ f = 0",
"usedConstants": [
"Sublattice",
"Submodule",
"Module.End.instMonoid",
"CommSemiring.toSemiring",
"AddC... | have (p) (hp₁ : p ∈ f.invtSubmodule) (hp₂ : Module.Finite R p) : f.restrict hp₁ = 0 := by
specialize hs p hp₁ hp₂
replace hn : IsNilpotent (f.restrict hp₁) := isNilpotent.restrict hp₁ hn
exact eq_zero_of_isNilpotent_isSemisimple hn hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 186,
"column": 2
} | {
"line": 188,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na : A\nh : IsNilpotent a\n⊢ IsUnit (exp a)",
"usedConstants": [
"Iff.mpr",
"isUnit_iff_exists",
"NegZeroClass.toNeg",
"MulOne.toOne",
"HMul.hMul",
"IsNilpotent.exp_mul_exp_neg_self",
"Monoid.toMulOneClass",... | apply isUnit_iff_exists.2
use exp (-a)
exact ⟨exp_mul_exp_neg_self h, exp_neg_mul_exp_self h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Nilpotent.Exp | {
"line": 186,
"column": 2
} | {
"line": 188,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Module ℚ A\na : A\nh : IsNilpotent a\n⊢ IsUnit (exp a)",
"usedConstants": [
"Iff.mpr",
"isUnit_iff_exists",
"NegZeroClass.toNeg",
"MulOne.toOne",
"HMul.hMul",
"IsNilpotent.exp_mul_exp_neg_self",
"Monoid.toMulOneClass",... | apply isUnit_iff_exists.2
use exp (-a)
exact ⟨exp_mul_exp_neg_self h, exp_neg_mul_exp_self h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nB : BilinForm R M\nr : S\nx y : M\n⊢ (B (r • x)) y = r • (B x) y",
"usedConstants": [
... | rw [← IsScalarTower.algebraMap_smul R r, smul_left, Algebra.smul_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.BilinearForm.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nB : BilinForm R M\nr : S\nx y : M\n⊢ (B (r • x)) y = r • (B x) y",
"usedConstants": [
... | rw [← IsScalarTower.algebraMap_smul R r, smul_left, Algebra.smul_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.BilinearForm.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\nB : BilinForm R M\nr : S\nx y : M\n⊢ (B (r • x)) y = r • (B x) y",
"usedConstants": [
... | rw [← IsScalarTower.algebraMap_smul R r, smul_left, Algebra.smul_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 155,
"column": 45
} | {
"line": 157,
"column": 5
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nr : M →ₗ[R] M\n⊢ B.comp id r = B.compRight r",
"usedConstants": [
"LinearMap.id",
"Semiring.toModule",
"LinearMap.BilinForm",
"CommSemiring.toSemiring",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 161,
"column": 44
} | {
"line": 163,
"column": 5
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\nl : M →ₗ[R] M\n⊢ B.comp l id = B.compLeft l",
"usedConstants": [
"LinearMap.id",
"Semiring.toModule",
"LinearMap.BilinForm",
"CommSemiring.toSemiring",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 166,
"column": 73
} | {
"line": 168,
"column": 5
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\n⊢ B.compLeft id = B",
"usedConstants": [
"LinearMap.id",
"Semiring.toModule",
"LinearMap.BilinForm",
"CommSemiring.toSemiring",
"LinearMap.instFunLike"... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 171,
"column": 75
} | {
"line": 173,
"column": 5
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\n⊢ B.compRight id = B",
"usedConstants": [
"LinearMap.id",
"Semiring.toModule",
"LinearMap.BilinForm",
"CommSemiring.toSemiring",
"LinearMap.instFunLike... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.BilinearForm.Hom | {
"line": 178,
"column": 81
} | {
"line": 180,
"column": 5
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nB : BilinForm R M\n⊢ B.comp id id = B",
"usedConstants": [
"LinearMap.id",
"Semiring.toModule",
"LinearMap.BilinForm",
"CommSemiring.toSemiring",
"LinearMap.instFunLike",... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.InvariantForm | {
"line": 203,
"column": 60
} | {
"line": 203,
"column": 82
} | [
{
"pp": "case refine_2.h\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nΦ : LinearMap.BilinForm K L\nhΦ_nondeg : Φ.Nondegenerate\nhΦ_inv : LinearMap.BilinForm.lieInvariant L Φ\nhΦ_refl : Φ.IsRefl\nhL : ∀ (I : LieIdeal K L), IsAtom I → ¬IsLi... | lieIdeal_oper_eq_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 34
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSimple R L\n⊢ Nontrivial (LieIdeal R L)",
"usedConstants": [
"LieAlgebra.toModule",
"Nontrivial",
"Eq.mpr",
"LieRing.toAddCommGroup",
"congrArg",
"LieSubmodule... | LieSubmodule.nontrivial_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Quotient | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 49
} | [
{
"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\nx y : L ⧸ I\nx₁ x₂ y₁ y₂ : L\nh₁ : x₁ - y₁ ∈ ↑I\nh₂ : x... | simp [-lie_skew, sub_eq_add_neg, add_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 249,
"column": 4
} | {
"line": 252,
"column": 50
} | [
{
"pp": "case inr.left\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\ns : Finset (LieIdeal R L)\nhs : ↑s ⊆ {I | IsAtom I}\nI : LieIdeal R L\nhI✝ : I ≤ s.sup id\nS : Set (LieIdeal R L) := ⋯\nhI : I < s.sup id\nJ : LieIdeal R L\nhJs : J ∈ s... | apply (sSupIndep_isAtom.disjoint_sSup (hs hJs) hs'S (Finset.notMem_erase _ _)).le_bot
apply LieSubmodule.lie_le_inf
apply LieSubmodule.lie_mem_lie j.2
simpa only [K, Finset.sup_id_eq_sSup] using hz | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 249,
"column": 4
} | {
"line": 252,
"column": 50
} | [
{
"pp": "case inr.left\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsSemisimple R L\ns : Finset (LieIdeal R L)\nhs : ↑s ⊆ {I | IsAtom I}\nI : LieIdeal R L\nhI✝ : I ≤ s.sup id\nS : Set (LieIdeal R L) := ⋯\nhI : I < s.sup id\nJ : LieIdeal R L\nhJs : J ∈ s... | apply (sSupIndep_isAtom.disjoint_sSup (hs hJs) hs'S (Finset.notMem_erase _ _)).le_bot
apply LieSubmodule.lie_le_inf
apply LieSubmodule.lie_mem_lie j.2
simpa only [K, Finset.sup_id_eq_sSup] using hz | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 205,
"column": 64
} | {
"line": 263,
"column": 19
} | [
{
"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\n⊢ ∀ (s : Finset (LieIdeal R L)), ↑s ⊆ {I | IsAtom I} → ∀ I ≤ s.sup id, ∃ t ⊆ s, I = t.sup id",
"usedConstants": [
"LieAlgebra.toModule",
"Mathlib.Tactic.Push.not_exist... | by
intro s hs I hI
let S := {I : LieIdeal R L | IsAtom I}
obtain rfl | hI := hI.eq_or_lt
· exact ⟨s, Finset.Subset.rfl, rfl⟩
-- We assume that `I` is strictly smaller than the supremum of `s`.
-- Hence there must exist an atom `J` that is not contained in `I`.
obtain ⟨J, hJs, hJI⟩ : ∃ J ∈ s, ¬ J ≤ I := by... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 320,
"column": 4
} | {
"line": 322,
"column": 59
} | [
{
"pp": "case mp\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\n⊢ IsLieAbelian ↥(radical R L) → ∀ (I : LieIdeal R L), IsSolvable ↥I → IsLieAbelian ↥I",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
... | rintro h₁ I h₂
rw [LieIdeal.solvable_iff_le_radical] at h₂
exact (LieIdeal.inclusion_injective h₂).isLieAbelian h₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Semisimple.Basic | {
"line": 320,
"column": 4
} | {
"line": 322,
"column": 59
} | [
{
"pp": "case mp\nR : Type u_1\nL : Type u_2\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : IsNoetherian R L\n⊢ IsLieAbelian ↥(radical R L) → ∀ (I : LieIdeal R L), IsSolvable ↥I → IsLieAbelian ↥I",
"usedConstants": [
"LieAlgebra.toModule",
"LieSubmodule.instSetLike",
... | rintro h₁ I h₂
rw [LieIdeal.solvable_iff_le_radical] at h₂
exact (LieIdeal.inclusion_injective h₂).isLieAbelian h₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 159,
"column": 43
} | {
"line": 159,
"column": 72
} | [
{
"pp": "case refine_1\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\nk : ℕ\nN : LieSubmodule R L M\ninst✝ : LieModule R L M\nh : lowerCentralSeries R L (↥N) k = ⊥\n⊢ map N.incl (low... | ← LieModuleHom.le_ker_iff_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Nilpotent | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 28
} | [
{
"pp": "R : Type u\nL : Type v\nM : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nN : LieSubmodule R L M\n⊢ IsNilpotent L ↥N ↔ ∃ k, LieSubmodule.lcs k N = ⊥",
"usedConstants": [
... | rw [isNilpotent_iff R L N] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Fixed | {
"line": 125,
"column": 38
} | {
"line": 125,
"column": 43
} | [
{
"pp": "M : Type u\ninst✝² : Monoid M\nF : Type v\ninst✝¹ : Field F\ninst✝ : MulSemiringAction M F\nm : M\np : Polynomial ↥(subfield M F)\nn : ℕ\nx : ↥(subfield M F)\nx✝ : m • (Polynomial.C x * Polynomial.X ^ n) = Polynomial.C x * Polynomial.X ^ n\n⊢ Polynomial.C (m • x) * m • Polynomial.X ^ (n + 1) = Polynomi... | smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Fixed | {
"line": 223,
"column": 25
} | {
"line": 223,
"column": 63
} | [
{
"pp": "G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\nx : F\nf : Polynomial ↥(subfield G F)\nhf : Polynomial.eval₂ (subfield G F).subtype x f = 0\ny : G ⧸ stabilizer G x\ng : G\n⊢ g • Polynomial.eval₂ (subfield G F).subtype x f = 0",
"usedCon... | Subfield.toSubring_subtype_eq_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Fixed | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 55
} | [
{
"pp": "G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\nx : F\nf g : Polynomial ↥(subfield G F)\nhf : f.Monic\nhg : g.Monic\nhfg : f * g = minpoly G F x\nhf2 : f ∣ minpoly G F x\nhg2 : g ∣ minpoly G F x\nthis : Polynomial.eval₂ (subfield G F).subty... | rw [← hfg, Polynomial.eval₂_mul, mul_eq_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Fixed | {
"line": 299,
"column": 31
} | {
"line": 299,
"column": 46
} | [
{
"pp": "G : Type u\ninst✝³ : Group G\nF : Type v\ninst✝² : Field F\ninst✝¹ : MulSemiringAction G F\ninst✝ : Fintype G\n⊢ ↑(finrank (↥(subfield G F)) F) ≤ ↑(Fintype.card G)",
"usedConstants": [
"FixedPoints.instFiniteDimensionalSubtypeMemSubfieldSubfield",
"Eq.mpr",
"NonAssocSemiring.toAdd... | finrank_eq_rank | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Fixed | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 50
} | [
{
"pp": "case intro.left\nG : Type u_2\nF : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Field F\ninst✝² : MulSemiringAction G F\ninst✝¹ : Finite G\ninst✝ : FaithfulSMul G F\nval✝ : Fintype G\n⊢ Function.Injective (MulSemiringAction.toAlgHom (↥(subfield G F)) F)",
"usedConstants": [
"Subfield.instSubfieldClas... | exact MulSemiringAction.toAlgHom_injective _ F | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.Fixed | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 50
} | [
{
"pp": "case intro.left\nG : Type u_2\nF : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Field F\ninst✝² : MulSemiringAction G F\ninst✝¹ : Finite G\ninst✝ : FaithfulSMul G F\nval✝ : Fintype G\n⊢ Function.Injective (MulSemiringAction.toAlgHom (↥(subfield G F)) F)",
"usedConstants": [
"Subfield.instSubfieldClas... | exact MulSemiringAction.toAlgHom_injective _ F | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Fixed | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 50
} | [
{
"pp": "case intro.left\nG : Type u_2\nF : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Field F\ninst✝² : MulSemiringAction G F\ninst✝¹ : Finite G\ninst✝ : FaithfulSMul G F\nval✝ : Fintype G\n⊢ Function.Injective (MulSemiringAction.toAlgHom (↥(subfield G F)) F)",
"usedConstants": [
"Subfield.instSubfieldClas... | exact MulSemiringAction.toAlgHom_injective _ F | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.IntermediateField.Algebraic | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 54
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nF E : IntermediateField K L\ninst✝ : FiniteDimensional (↥F) L\nH : F < E\n⊢ finrank (↥E) L < finrank (↥F) L",
"usedConstants": [
"IntermediateField.instPartialOrder",
"NonUnitalCommRing.toNonUnitalNonA... | letI := (IntermediateField.inclusion H.le).toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.FieldTheory.IsAlgClosed.Spectrum | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 85
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝² : Field 𝕜\ninst✝¹ : Ring A\ninst✝ : Algebra 𝕜 A\na : A\np : 𝕜[X]\nk : 𝕜\nhk : k ∈ σ a\nq : 𝕜[X] := C (eval k p) - p\n⊢ (fun x ↦ eval x p) k ∈ σ ((aeval a) p)",
"usedConstants": [
"Polynomial.C",
"Polynomial.eval",
"_private.Mathlib.FieldTheory.... | have hroot : IsRoot q k := by simp only [q, eval_C, eval_sub, sub_self, IsRoot.def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.IsAlgClosed.Basic | {
"line": 439,
"column": 45
} | {
"line": 444,
"column": 32
} | [
{
"pp": "k : Type u\ninst✝²⁶ : Field k\nK : Type u_1\nJ : Type u_2\nR : Type u\nS : Type u_3\nL : Type v\nM : Type w\ninst✝²⁵ : Field K\ninst✝²⁴ : Field J\ninst✝²³ : CommRing R\ninst✝²² : CommRing S\ninst✝²¹ : Field L\ninst✝²⁰ : Field M\ninst✝¹⁹ : Algebra R M\ninst✝¹⁸ : IsTorsionFree R M\ninst✝¹⁷ : IsAlgClosure... | by
have : IsTorsionFree R L := .trans_faithfulSMul R S L
have : IsAlgClosure R L :=
{ isAlgClosed := IsAlgClosure.isAlgClosed S
isAlgebraic := ‹_› }
exact IsAlgClosure.equiv _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Extension | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 27
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nL₁ L₂ : Lifts F E K\n⊢ L₁ = L₂ ↔ ∃ (h : L₁.carrier = L₂.carrier), L₂.emb.comp (inclusion ⋯) = L₁.emb",
"usedConstants": [
"Eq.mpr",
"IntermediateFie... | eq_iff_le_carrier_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Extension | {
"line": 83,
"column": 64
} | {
"line": 83,
"column": 85
} | [
{
"pp": "F : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Field K\ninst✝¹ : Algebra F E\ninst✝ : Algebra F K\nL₁ L₂ : Lifts F E K\nh : L₁ ≤ L₂\n⊢ ¬L₁ = L₂ ↔ ¬L₁.carrier = L₂.carrier",
"usedConstants": [
"Eq.mpr",
"congrArg",
"IntermediateField",
... | eq_iff_le_carrier_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 70,
"column": 2
} | {
"line": 73,
"column": 40
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ι → End R M\nμ : ι → R\ni : ι\nh✝ : ∀ (j : ι), MapsTo ⇑(f j) ↑((f i).maxGenEigenspace (μ i)) ↑((f i).maxGenEigenspace (μ i))\nthis✝ : Nonempty ι\np : Submodule R M := (f i).maxGenEigenspace (... | conv_rhs =>
enter [1]
ext
rw [p.inf_genEigenspace (f _) (h _)] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.LinearAlgebra.Eigenspace.Pi | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 67
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nf : ι → End R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nh : ∀ (i j : ι) (φ : R), MapsTo ⇑(f i) ↑((f j).maxGenEigenspace φ) ↑((f j).maxGenEigenspace φ)\nl : ι\nχ : ι → R\nx : M\nhx : x ∈... | simp only [iInf_eq_iInter, mem_iInter, SetLike.mem_coe] at hx ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 31
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nE : Type u_3\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Field E\ninst✝³ : Algebra K L\ninst✝² : Algebra K E\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower K L E\nE' : IntermediateField L E\ns : Finset E\nhs : adjoin K ↑s = restrictScalars K E'\n⊢ E'.FG",
"usedConstants"... | refine ⟨s, le_antisymm ?_ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 539,
"column": 2
} | {
"line": 541,
"column": 36
} | [
{
"pp": "K : Type u\ninst✝² : Field K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : IsIntegral K x\n⊢ (minpoly K x).natDegree ∣ finrank K L",
"usedConstants": [
"Eq.mpr",
"IntermediateField.adjoin.finrank",
"IntermediateField.isScalarTower_mid'",
"SubsemiringClass... | rw [dvd_iff_exists_eq_mul_left, ← IntermediateField.adjoin.finrank hx]
use finrank K⟮x⟯ L
rw [mul_comm, finrank_mul_finrank] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.IntermediateField.Adjoin.Basic | {
"line": 539,
"column": 2
} | {
"line": 541,
"column": 36
} | [
{
"pp": "K : Type u\ninst✝² : Field K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\nhx : IsIntegral K x\n⊢ (minpoly K x).natDegree ∣ finrank K L",
"usedConstants": [
"Eq.mpr",
"IntermediateField.adjoin.finrank",
"IntermediateField.isScalarTower_mid'",
"SubsemiringClass... | rw [dvd_iff_exists_eq_mul_left, ← IntermediateField.adjoin.finrank hx]
use finrank K⟮x⟯ L
rw [mul_comm, finrank_mul_finrank] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.Fintype | {
"line": 171,
"column": 87
} | {
"line": 171,
"column": 98
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\n⊢ m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype fun x ↦ x ∈ m.toEnumFinset) = m.coeEmbedding",
"usedConstants": [
"Multiset.coeEmbedding",
"Finset",
"Membership.mem",
"Subtype",
"Multiset.toEnumFinset",
... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Multiset.Fintype | {
"line": 171,
"column": 87
} | {
"line": 171,
"column": 98
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\n⊢ m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype fun x ↦ x ∈ m.toEnumFinset) = m.coeEmbedding",
"usedConstants": [
"Multiset.coeEmbedding",
"Finset",
"Membership.mem",
"Subtype",
"Multiset.toEnumFinset",
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Multiset.Fintype | {
"line": 171,
"column": 87
} | {
"line": 171,
"column": 98
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\n⊢ m.coeEquiv.toEmbedding.trans (Function.Embedding.subtype fun x ↦ x ∈ m.toEnumFinset) = m.coeEmbedding",
"usedConstants": [
"Multiset.coeEmbedding",
"Finset",
"Membership.mem",
"Subtype",
"Multiset.toEnumFinset",
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.Fintype | {
"line": 293,
"column": 29
} | {
"line": 293,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\nv : α\nx : (v ::ₘ m).ToType\nhx : x.fst = v\nhx2 : ↑x.snd = count v m\n⊢ consEquiv x = none",
"usedConstants": [
"dite_cond_eq_true",
"Multiset.consEquiv._proof_1",
"Equiv.instEquivLike",
"congrArg",
"and_self",
... | simp [consEquiv, hx, hx2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Multiset.Fintype | {
"line": 293,
"column": 29
} | {
"line": 293,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\nv : α\nx : (v ::ₘ m).ToType\nhx : x.fst = v\nhx2 : ↑x.snd = count v m\n⊢ consEquiv x = none",
"usedConstants": [
"dite_cond_eq_true",
"Multiset.consEquiv._proof_1",
"Equiv.instEquivLike",
"congrArg",
"and_self",
... | simp [consEquiv, hx, hx2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Multiset.Fintype | {
"line": 293,
"column": 29
} | {
"line": 293,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nm : Multiset α\nv : α\nx : (v ::ₘ m).ToType\nhx : x.fst = v\nhx2 : ↑x.snd = count v m\n⊢ consEquiv x = none",
"usedConstants": [
"dite_cond_eq_true",
"Multiset.consEquiv._proof_1",
"Equiv.instEquivLike",
"congrArg",
"and_self",
... | simp [consEquiv, hx, hx2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Cartan | {
"line": 333,
"column": 68
} | {
"line": 333,
"column": 88
} | [
{
"pp": "L : Type u_2\ninst✝⁵ : LieRing L\nK : Type u_4\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\nI : LieIdeal K L\nα : Weight K (↥H) L\nhα : α.IsZero\n⊢ genWeightSpace L ⇑α = H.toLieSubm... | by ext; simp [hα.eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SplittingField.Construction | {
"line": 198,
"column": 58
} | {
"line": 198,
"column": 72
} | [
{
"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 : f.natDegree = n.succ\nhndf : f.natDegree ≠ 0\nhfn0 : f... | roots_X_sub_C, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Derivation.Killing | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 38
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝³ : Field R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\ninst✝ : Module.Finite R L\nD : LieDerivation R L L\nhD : D ∈ (killingForm R (LieDerivation R L L)).orthogonal (ad R L).range.toSubmodule\nx : L\n⊢ (ad R L) (D x) ∈ Submodule.map (ad R L).range.subtype (LinearMap.... | rw [← killingForm_restrict_range_ad] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Sl2 | {
"line": 110,
"column": 25
} | {
"line": 110,
"column": 69
} | [
{
"pp": "case smul\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\nh e f : L\nt✝ : IsSl2Triple h e f\nx y : L\nhy : y ∈ span R {e, f, h}\nt : R\nu : L\n... | simpa only [smul_lie] using smul_mem _ t hu' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Lie.Sl2 | {
"line": 110,
"column": 25
} | {
"line": 110,
"column": 69
} | [
{
"pp": "case smul\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\nh e f : L\nt✝ : IsSl2Triple h e f\nx y : L\nhy : y ∈ span R {e, f, h}\nt : R\nu : L\n... | simpa only [smul_lie] using smul_mem _ t hu' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Sl2 | {
"line": 110,
"column": 25
} | {
"line": 110,
"column": 69
} | [
{
"pp": "case smul\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\nh e f : L\nt✝ : IsSl2Triple h e f\nx y : L\nhy : y ∈ span R {e, f, h}\nt : R\nu : L\n... | simpa only [smul_lie] using smul_mem _ t hu' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Basic | {
"line": 539,
"column": 2
} | {
"line": 539,
"column": 25
} | [
{
"pp": "R : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\nM₂ : Type u_5\ninst✝³ : AddCommGroup M₂\ninst✝² : Module R M₂... | rintro - ⟨hm, ⟨m, rfl⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.BilinearForm.TensorProduct | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 25
} | [
{
"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 :... | rw [isSymm_iff_eq_flip] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Sl2 | {
"line": 166,
"column": 77
} | {
"line": 166,
"column": 87
} | [
{
"pp": "case succ\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\nh e f : L\nm : M\nμ : R\nt : IsSl2Triple h e f\nP : t.HasPrimitiveVectorWith m μ\nn :... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Sl2 | {
"line": 188,
"column": 2
} | {
"line": 201,
"column": 45
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\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\nh e f : L\nm : M\nμ : R\nt : IsSl2Triple h e f\nP : t.HasPrimitiveVectorWith m μ\ninst✝³ : IsN... | suffices ∃ n : ℕ, (ψ n) = 0 by
obtain ⟨n, hn₁, hn₂⟩ := Nat.exists_not_and_succ_of_not_zero_of_exists P.ne_zero this
refine ⟨n, ?_⟩
have := lie_e_pow_succ_toEnd_f P n
rw [hn₂, lie_zero, eq_comm, smul_eq_zero_iff_left hn₁, mul_eq_zero, sub_eq_zero] at this
exact this.resolve_left <| Nat.cast_add_one_n... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Sl2 | {
"line": 188,
"column": 2
} | {
"line": 201,
"column": 45
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\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\nh e f : L\nm : M\nμ : R\nt : IsSl2Triple h e f\nP : t.HasPrimitiveVectorWith m μ\ninst✝³ : IsN... | suffices ∃ n : ℕ, (ψ n) = 0 by
obtain ⟨n, hn₁, hn₂⟩ := Nat.exists_not_and_succ_of_not_zero_of_exists P.ne_zero this
refine ⟨n, ?_⟩
have := lie_e_pow_succ_toEnd_f P n
rw [hn₂, lie_zero, eq_comm, smul_eq_zero_iff_left hn₁, mul_eq_zero, sub_eq_zero] at this
exact this.resolve_left <| Nat.cast_add_one_n... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Sl2 | {
"line": 244,
"column": 75
} | {
"line": 244,
"column": 85
} | [
{
"pp": "case succ\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\nh e f : L\nm : M\nμ : R\nt : IsSl2Triple h e f\nhm : ⁅h, m⁆ = μ • m\nn : ℕ\nih : ⁅h, ... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 80,
"column": 8
} | {
"line": 80,
"column": 32
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra R L\nM : Type u_3\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : LieRingModule L M\ninst✝⁵ : LieModule R L M\ninst✝⁴ : LieRing.IsNilpotent L\nχ₁ χ₂ : L → R\ninst✝³ : IsAddTorsionFree R\ninst✝² : IsDomain... | ← Nat.cofinite_eq_atTop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 229,
"column": 52
} | {
"line": 229,
"column": 80
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝¹³ : CommRing R\ninst✝¹² : LieRing L\ninst✝¹¹ : LieAlgebra R L\nM : Type u_3\ninst✝¹⁰ : AddCommGroup M\ninst✝⁹ : Module R M\ninst✝⁸ : LieRingModule L M\ninst✝⁷ : LieModule R L M\nH : LieSubalgebra R L\nα χ : ↥H → R\ninst✝⁶ : H.IsCartanSubalgebra\ninst✝⁵ : IsNoetherian R... | by rwa [add_left_inj] at hij | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.Weights.Chain | {
"line": 406,
"column": 4
} | {
"line": 406,
"column": 9
} | [
{
"pp": "case mem\nL : Type u_2\ninst✝¹⁰ : LieRing L\nM : Type u_3\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LieRingModule L M\nK : Type u_4\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieAlgebra K L\nH : LieSubalgebra K L\ninst✝⁴ : LieRing.IsNilpotent ↥H\ninst✝³ : Module K M\ninst✝² : LieModule K L M\ninst✝¹ : I... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Algebra.Lie.TraceForm | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 13
} | [
{
"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\nhB : ∀ (x : L) (m n : M), (B ⁅x, m⁆... | intro x k | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 207,
"column": 4
} | {
"line": 208,
"column": 60
} | [
{
"pp": "K : 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✝ : IsKilling K L\nx : L\nhx : x ∈ H\nhx' : _root_.IsNilpotent ((ad K L) x)\nthis : ⟨x, hx⟩ ∈ LinearMap.ker (traceForm K (... | simp only [ker_traceForm_eq_bot_of_isCartanSubalgebra, Submodule.mem_bot] at this
exact (AddSubmonoid.mk_eq_zero H.toAddSubmonoid).mp this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 207,
"column": 4
} | {
"line": 208,
"column": 60
} | [
{
"pp": "K : 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✝ : IsKilling K L\nx : L\nhx : x ∈ H\nhx' : _root_.IsNilpotent ((ad K L) x)\nthis : ⟨x, hx⟩ ∈ LinearMap.ker (traceForm K (... | simp only [ker_traceForm_eq_bot_of_isCartanSubalgebra, Submodule.mem_bot] at this
exact (AddSubmonoid.mk_eq_zero H.toAddSubmonoid).mp this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.Killing | {
"line": 619,
"column": 2
} | {
"line": 621,
"column": 92
} | [
{
"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✝ : CharZero K\nα : Weight K (↥H) L\nhα : α.IsNonZero\ncontr... | replace hy : ⁅y, f⁆ = 0 := by
have : killingForm K L y f = 0 := by simpa [F, traceForm_comm] using hy
simpa [this] using lie_eq_killingForm_smul_of_mem_rootSpace_of_mem_rootSpace_neg hyα hfα | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.LinearAlgebra.Reflection | {
"line": 225,
"column": 4
} | {
"line": 229,
"column": 56
} | [
{
"pp": "R : 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\nm : ℕ\nS_eval_t_sub_two :\n ∀ (k : ℤ), Polynomial.eval t (S R (k - 2)) = t * Polynomial.eval t (S R (k - 1)) - Polyno... | · linear_combination (norm := skip)
g z * (S R (e - 1 + k)).eval t * S_eval_t_sub_two (e + k) +
g z * S_eval_t_sq_add_S_eval_t_sq (k - 1)
subst ht
obtain rfl | rfl : e = 0 ∨ e = 1 := he <;> ring_nf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.PerfectPairing.Restrict | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 13
} | [
{
"pp": "K : Type u_1\nL : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module L M\ninst✝⁴ : Module L N\ninst✝³ : Module K M\ninst✝² : Module K N\ninst✝¹ : IsScalarTower K L M\np : M →ₗ[L] N →ₗ[L] L\ni... | h_span, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 50,
"column": 4
} | {
"line": 53,
"column": 34
} | [
{
"pp": "case h.refine_1\nR : Type u_1\ninst✝² : Ring R\nI J : Ideal R\nH : I ≤ J\ninst✝¹ : I.IsTwoSided\ninst✝ : J.IsTwoSided\nx : R ⧸ I\nh : x ∈ RingHom.ker (factor H)\n⊢ x ∈ map (mk I) J",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"_private.Math... | rcases Ideal.Quotient.mk_surjective x with ⟨r, hr⟩
rw [← hr] at h ⊢
simp only [factor, RingHom.mem_ker, lift_mk, eq_zero_iff_mem] at h
exact Ideal.mem_map_of_mem _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 50,
"column": 4
} | {
"line": 53,
"column": 34
} | [
{
"pp": "case h.refine_1\nR : Type u_1\ninst✝² : Ring R\nI J : Ideal R\nH : I ≤ J\ninst✝¹ : I.IsTwoSided\ninst✝ : J.IsTwoSided\nx : R ⧸ I\nh : x ∈ RingHom.ker (factor H)\n⊢ x ∈ map (mk I) J",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModule",
"_private.Math... | rcases Ideal.Quotient.mk_surjective x with ⟨r, hr⟩
rw [← hr] at h ⊢
simp only [factor, RingHom.mem_ker, lift_mk, eq_zero_iff_mem] at h
exact Ideal.mem_map_of_mem _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 161,
"column": 2
} | {
"line": 162,
"column": 44
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\na b : ℕ\napos : 0 < a\nle : a ≤ b\n⊢ comap (factorPow I le) (map (mk (I ^ a)) I) = map (mk (I ^ b)) I",
"usedConstants": [
"Semiring.toModule",
"Ideal.map_mk_comap_factor",
"CommSemiring.toSemiring",
"instOfNatNat",
"Ideal... | apply Ideal.map_mk_comap_factor
exact pow_le_self (Nat.ne_zero_of_lt apos) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 161,
"column": 2
} | {
"line": 162,
"column": 44
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\na b : ℕ\napos : 0 < a\nle : a ≤ b\n⊢ comap (factorPow I le) (map (mk (I ^ a)) I) = map (mk (I ^ b)) I",
"usedConstants": [
"Semiring.toModule",
"Ideal.map_mk_comap_factor",
"CommSemiring.toSemiring",
"instOfNatNat",
"Ideal... | apply Ideal.map_mk_comap_factor
exact pow_le_self (Nat.ne_zero_of_lt apos) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Quotient.PowTransition | {
"line": 179,
"column": 34
} | {
"line": 179,
"column": 54
} | [
{
"pp": "R : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\nnpos : n > 0\na : R ⧸ I ^ (n + 1)\nh : IsUnit ((factorPow I ⋯) a)\nb : R ⧸ I ^ n\nright✝ : b * (factorPow I ⋯) a = 1\nb' : R ⧸ I ^ n.succ\nhb' : (factor ⋯) b' = b\nhb : a * b' - 1 ∈ map (mk (I ^ n.succ)) (I ^ n)\nc : R\nhc : c ∈ ↑(I ^ n)\neq : (mk (... | sub_add_sub_cancel', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 536,
"column": 69
} | {
"line": 537,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ C R (n - 2) = X * C R (n - 1) - C R n",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Math... | by
linear_combination (norm := ring_nf) C_add_two R (n - 2) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 542,
"column": 64
} | {
"line": 543,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ C R n = X * C R (n - 1) - C R (n - 2)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Math... | by
linear_combination (norm := ring_nf) C_add_two R (n - 2) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Length | {
"line": 97,
"column": 52
} | {
"line": 97,
"column": 69
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh : FiniteDimensionalOrder (Submodule R M)\n⊢ IsNoetherian R M ∧ IsArtinian R M",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Preorder.toLT",
"congrArg",
"AddCommGro... | isNoetherian_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 619,
"column": 2
} | {
"line": 622,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Invertible 2\nn : ℤ\n⊢ C R n = 2 * (T R n).comp (Polynomial.C ⅟2 * X)",
"usedConstants": [
"Polynomial.C",
"Polynomial.comp_assoc",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Polynomial.instOne",
... | have := congr_arg (·.comp (Polynomial.C ⅟2 * X)) (C_comp_two_mul_X R n)
simp_rw [comp_assoc, mul_comp, ofNat_comp, X_comp, ← mul_assoc, ← C_eq_natCast, ← C_mul,
Nat.cast_ofNat, mul_invOf_self', map_one, one_mul, comp_X, map_ofNat] at this
assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 619,
"column": 2
} | {
"line": 622,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : Invertible 2\nn : ℤ\n⊢ C R n = 2 * (T R n).comp (Polynomial.C ⅟2 * X)",
"usedConstants": [
"Polynomial.C",
"Polynomial.comp_assoc",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"Polynomial.instOne",
... | have := congr_arg (·.comp (Polynomial.C ⅟2 * X)) (C_comp_two_mul_X R n)
simp_rw [comp_assoc, mul_comp, ofNat_comp, X_comp, ← mul_assoc, ← C_eq_natCast, ← C_mul,
Nat.cast_ofNat, mul_invOf_self', map_one, one_mul, comp_X, map_ofNat] at this
assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Chebyshev | {
"line": 899,
"column": 91
} | {
"line": 904,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\n⊢ (1 - X ^ 2) * derivative (T R (n + 1)) = (↑n + 1) * (T R n - X * T R (n + 1))",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Polynomial.derivative",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Mathlib.Tactic.... | by
have H₁ := one_sub_X_sq_mul_U_eq_pol_in_T R n
have H₂ := T_derivative_eq_U (R := R) (n + 1)
have h₁ := T_add_two R n
linear_combination (norm := (push_cast; ring_nf))
(-n - 1) * h₁ + (-(X : R[X]) ^ 2 + 1) * H₂ + (n + 1) * H₁ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 251,
"column": 25
} | {
"line": 251,
"column": 35
} | [
{
"pp": "case succ\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\nf : ℕ → M\nn : ℕ\nhf : f 1 ≡ f (n + 1) [SMOD ⊥]\n⊢ f (n + 1) ≡ f 1 [SMOD ⊥ ^ (n + 1) • ⊤]",
"... | SModEq.bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 355,
"column": 47
} | {
"line": 356,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_6\ns : Finset ι\nf : ι → AdicCompletion I M\n⊢ ↑(∑ i ∈ s, f i) = ∑ i ∈ s, ↑(f i)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Submodule.Quotient.addCommMonoid",
... | by
simp_rw [← funext (incl_apply _ _ _), map_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 546,
"column": 2
} | {
"line": 551,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : AdicCauchySequence I M\nh : ∃ k, ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, ↑f m ∈ I ^ l • ⊤\n⊢ (mk I M) f = 0",
"usedConstants": [
"Eq.mpr",
"AdicCompletion.AdicCauchySequence.mk_eq_mk",
... | obtain ⟨k, h⟩ := h
ext n
obtain ⟨m, hnm, l, hnl, hl⟩ := h (n + k) (by lia)
rw [mk_apply_coe, Submodule.mkQ_apply, val_zero,
← AdicCauchySequence.mk_eq_mk (show n ≤ m by lia)]
simpa using (Submodule.smul_mono_left (Ideal.pow_le_pow_right (by lia))) hl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.Basic | {
"line": 546,
"column": 2
} | {
"line": 551,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\nM : Type u_4\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : AdicCauchySequence I M\nh : ∃ k, ∀ n ≥ k, ∃ m ≥ n, ∃ l ≥ n, ↑f m ∈ I ^ l • ⊤\n⊢ (mk I M) f = 0",
"usedConstants": [
"Eq.mpr",
"AdicCompletion.AdicCauchySequence.mk_eq_mk",
... | obtain ⟨k, h⟩ := h
ext n
obtain ⟨m, hnm, l, hnl, hl⟩ := h (n + k) (by lia)
rw [mk_apply_coe, Submodule.mkQ_apply, val_zero,
← AdicCauchySequence.mk_eq_mk (show n ≤ m by lia)]
simpa using (Submodule.smul_mono_left (Ideal.pow_le_pow_right (by lia))) hl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 68
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : UniqueFactorizationMonoid R\nh₂ : ∀ ⦃p q : R⦄, Irreducible p → Irreducible q → Associated p q\nPID : IsPrincipalIdealRing R\np : R\nhp : Irreducible p\n⊢ ∃ x ∈ span {p}, x ≠ 0",
"usedConstants": [
"Iff.mpr",
"S... | exact ⟨p, Ideal.mem_span_singleton.mpr (dvd_refl p), hp.ne_zero⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 294,
"column": 22
} | {
"line": 294,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : Algebra (R[X] ⧸ P) S... | ← congr_arg φ' hq'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 7
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\ns : Ideal R\nhs : s ≠ ⊥\nϖ : R\nhirr : Irreducible ϖ\ngen_ne_zero : generator s ≠ 0\nn : ℕ\nu : Rˣ\nhnu : generator s * ↑u = ϖ ^ n\n⊢ ∃ n, s = span {ϖ ^ n}",
"usedConstants": [
"CommSemiring.toSemiring"... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Jacobson.Ring | {
"line": 309,
"column": 6
} | {
"line": 309,
"column": 35
} | [
{
"pp": "case refine_1.inr\nR : Type u_1\ninst✝⁶ : CommRing R\nRₘ : Type u_3\nSₘ : Type u_4\ninst✝⁵ : CommRing Rₘ\ninst✝⁴ : CommRing Sₘ\nP : Ideal R[X]\npX : R[X]\nhpX : pX ∈ P\ninst✝³ : Algebra (R ⧸ comap C P) Rₘ\ninst✝² : IsLocalization.Away (map (Ideal.Quotient.mk (comap C P)) pX).leadingCoeff Rₘ\ninst✝¹ : A... | rw [degree_le_zero_iff] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DiscreteValuationRing.Basic | {
"line": 634,
"column": 10
} | {
"line": 634,
"column": 26
} | [
{
"pp": "case pos\nR✝¹ : Type u\ninst✝⁵ : CommRing R✝¹\ninst✝⁴ : IsDomain R✝¹\ninst✝³ : IsDiscreteValuationRing R✝¹\nR✝ : Type u_1\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsDiscreteValuationRing R\nx y : R\nh₁ : ¬y = 0\nh₂ : y ∣ x\n⊢ y * Exists.choose h₂ + 0 = x",
"usedConstants": [... | ← h₂.choose_spec | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DiscreteValuationRing.TFAE | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsNoetherianRing R\ninst✝¹ : IsLocalRing R\ninst✝ : IsDomain R\nI : Ideal R\nhI : I ≠ ⊥\nh : ¬IsField R\nx : R\nhx : maximalIdeal R = Ideal.span {x}\nhI' : ¬I = ⊤\nH : ∀ (r : R), ¬IsUnit r ↔ x ∣ r\nthis : x ≠ 0\nhx' : Irreducible x\nH' : ∀ (r : R), r ≠ 0 → r ... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Valuation.Basic | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 18
} | [
{
"pp": "case right.e_a\nR : Type u_3\nΓ₀ : Type u_4\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ : (ValueGroup₀ v)ˣ\nu : ↥(valueGroup v) := WithZero.unzero ⋯\nhu_def : u = WithZero.unzero ⋯\na : R\nha : v a ≠ 0\nx : R\nhax : v a * ↑↑u = v x\nhx : 0 < v x\nha0 : v.restrict ... | exact Eq.refl .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Valuation.Basic | {
"line": 860,
"column": 4
} | {
"line": 860,
"column": 21
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Val... | · simp [hx0, hy0] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Flat.Tensor | {
"line": 74,
"column": 76
} | {
"line": 75,
"column": 71
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\n⊢ Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(lTensor M (Submodule.subtype I))",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"TensorProduct.comm",
"congrAr... | by
simpa [← comm_comp_rTensor_comp_comm_eq] using iff_rTensor_injective' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.Basic | {
"line": 862,
"column": 4
} | {
"line": 862,
"column": 21
} | [
{
"pp": "case pos\nK : Type u_1\nF : Type u_2\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_4\nΓ'₀ : Type u_5\nΓ''₀ : Type u_6\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : LinearOrderedCommGroupWithZero Γ'₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ''₀\ninst✝ : Ring R\nv : Valuation R Γ₀\nw : Val... | · simp [hx0, hy0] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.SumsOfSquares | {
"line": 151,
"column": 78
} | {
"line": 153,
"column": 60
} | [
{
"pp": "T : Type u_2\ninst✝ : NonUnitalCommSemiring T\n⊢ (sumSq T).toAddSubmonoid = AddSubmonoid.sumSq T",
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"setOf",
"NonUnitalCommSemiring.toCommSemigroup",
"NonUnitalSemiring.toNonUnitalNonAssocSemiring",
"AddSu... | by
simp [sumSq, ← AddSubmonoid.closure_isSquare,
Subsemigroup.nonUnitalSubsemiringClosure_toAddSubmonoid] | [anonymous] | Lean.Parser.Term.byTactic |
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