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.LinearAlgebra.Matrix.Transvection | {
"line": 485,
"column": 6
} | {
"line": 488,
"column": 13
} | [
{
"pp": "case neg\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nn : ℕ\nIH : n ≤ r → (M * (List.take n (listTransvecRow M)).prod) (inr ()) (inl i) = if n ≤ ↑i then M (inr ()) (inl i) else 0\nhk : n + 1 ≤ r\nhnr : n < r\nn' : Fin r :... | have hni : n ≠ i := by
rintro rfl
cases i
tauto | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 452,
"column": 73
} | {
"line": 454,
"column": 5
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : CommRing α\nA : Matrix n n α\ninst✝ : Invertible A.det\n⊢ A.unitOfDetInvertible = A.nonsingInvUnit ⋯",
"usedConstants": [
"Units.val",
"Matrix.nonsingInvUnit",
"isUnit_of_invertible",
"CommSemiring... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MatrixPolynomialAlgebra | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 62
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝² : CommSemiring R\nn : Type w\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\ni j : n\np : R[X]\nk✝ k : ℕ\nx : R\n⊢ (matPolyEquiv (single i j ((monomial k) x))).coeff k✝ = single i j (((monomial k) x).coeff k✝)",
"usedConstants": [
"Eq.mpr",
"Semiring.toMo... | simp only [matPolyEquiv_coeff_apply_aux_1, coeff_monomial] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.LinearCombination.Lemmas | {
"line": 194,
"column": 29
} | {
"line": 194,
"column": 66
} | [
{
"pp": "α : Type u_1\na a' b b' : α\ninst✝¹ : Ring α\ninst✝ : NoZeroDivisors α\nn : ℕ\np : a - b = 0\nH : (a' - b') ^ n - (a - b) = 0\n⊢ a' - b' = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"AddGroupWithOne.toAddGroup",
"HSub.hSub",
"SubtractionMonoid.toSubNegZeroMonoid"... | apply eq_zero_of_pow_eq_zero (n := n) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.Identities | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 24
} | [
{
"pp": "case e_a.e_a\nR : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ (f.sum fun e a ↦ a * ↑e * x ^ (e - 1) * y) = eval x (derivative f) * y",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomi... | rw [derivative_eval] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Identities | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 34
} | [
{
"pp": "case e_a.e_a\nR : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ (f.sum fun e a ↦ a * ↑e * x ^ (e - 1) * y) = eval x (derivative f) * y",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomi... | · rw [derivative_eval]
exact (Finset.sum_mul ..).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Nilpotent.Basic | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 7
} | [
{
"pp": "R : Type u_1\nx : R\ninst✝ : Ring R\nn : ℕ\nhn : x ^ n = 0\n⊢ IsNilpotent (-x)",
"usedConstants": [
"NegZeroClass.toNeg",
"SubtractionMonoid.toSubNegZeroMonoid",
"SubNegZeroMonoid.toNegZeroClass",
"SubtractionCommMonoid.toSubtractionMonoid",
"Monoid.toPow",
"HPow... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Polynomial.Nilpotent | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 28
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommRing R\nP : R[X]\nh : ∀ (i : ℕ), IsNilpotent (P.coeff i)\n⊢ IsNilpotent P",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Polynomial.sum",
"LinearMap.instFunLike",
"Polynom... | rw [← sum_monomial_eq P] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENat.Lattice | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 30
} | [
{
"pp": "ι : Sort u_2\nf : ι → ℕ∞\na : ℕ∞\ninst✝ : Nonempty ι\n⊢ (⨅ i, f i) * a = ⨅ i, f i * a",
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrderENat",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemiring",
"id... | simp_rw [mul_comm, mul_iInf] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.ENat.Lattice | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 30
} | [
{
"pp": "ι : Sort u_2\nf : ι → ℕ∞\na : ℕ∞\ninst✝ : Nonempty ι\n⊢ (⨅ i, f i) * a = ⨅ i, f i * a",
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrderENat",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemiring",
"id... | simp_rw [mul_comm, mul_iInf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENat.Lattice | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 30
} | [
{
"pp": "ι : Sort u_2\nf : ι → ℕ∞\na : ℕ∞\ninst✝ : Nonempty ι\n⊢ (⨅ i, f i) * a = ⨅ i, f i * a",
"usedConstants": [
"Eq.mpr",
"iInf",
"instCompleteLinearOrderENat",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemiring",
"id... | simp_rw [mul_comm, mul_iInf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENat.Lattice | {
"line": 189,
"column": 46
} | {
"line": 189,
"column": 61
} | [
{
"pp": "ι : Sort u_2\na : ℕ∞\ninst✝ : Nonempty ι\nf : ι → ℕ∞\nha : a ≠ ⊤\ni : ι\n⊢ a + f i ≤ a + ⨆ i, f i",
"usedConstants": [
"le_refl",
"instCompleteLinearOrderENat",
"CommSemiring.toSemiring",
"iSup",
"CompletelyDistribLattice.toCompleteLattice",
"Preorder.toLE",
... | grw [← le_iSup] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.Data.ENat.Lattice | {
"line": 189,
"column": 46
} | {
"line": 189,
"column": 61
} | [
{
"pp": "ι : Sort u_2\na : ℕ∞\ninst✝ : Nonempty ι\nf : ι → ℕ∞\nha : a ≠ ⊤\ni : ι\n⊢ a + f i ≤ a + ⨆ i, f i",
"usedConstants": [
"le_refl",
"instCompleteLinearOrderENat",
"CommSemiring.toSemiring",
"iSup",
"CompletelyDistribLattice.toCompleteLattice",
"Preorder.toLE",
... | grw [← le_iSup] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENat.Lattice | {
"line": 189,
"column": 46
} | {
"line": 189,
"column": 61
} | [
{
"pp": "ι : Sort u_2\na : ℕ∞\ninst✝ : Nonempty ι\nf : ι → ℕ∞\nha : a ≠ ⊤\ni : ι\n⊢ a + f i ≤ a + ⨆ i, f i",
"usedConstants": [
"le_refl",
"instCompleteLinearOrderENat",
"CommSemiring.toSemiring",
"iSup",
"CompletelyDistribLattice.toCompleteLattice",
"Preorder.toLE",
... | grw [← le_iSup] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.SpanRank | {
"line": 223,
"column": 2
} | {
"line": 224,
"column": 72
} | [
{
"pp": "case mpr\nR : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : Submodule R M\n⊢ I = ⊥ → I.spanRank = 0",
"usedConstants": [
"Cardinal.mk_eq_zero",
"Iff.mpr",
"Eq.mpr",
"Submodule",
"iInf",
"Cardinal",
"congrArg",
... | · rintro rfl; rw [spanRank]
exact Cardinal.iInf_eq_zero_iff.mpr (Or.inr ⟨⟨∅, by simp⟩, by simp⟩) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 64,
"column": 26
} | {
"line": 66,
"column": 80
} | [
{
"pp": "R : Type u_1\nS : Type u_4\nT : Type u_5\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Ring T\nf : R →+* S\ng : S →+* T\nx : T\nhx : (g.comp f).IsIntegralElem x\n⊢ g.IsIntegralElem x",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"congrArg",
"CommSemiring.toSemiring"... | by
obtain ⟨p, hp, hx⟩ := hx
exact ⟨p.map f, hp.map _, by simpa only [eval₂_eq_eval_map, map_map] using hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 30
} | [
{
"pp": "R : Type u_1\nB : Type u_3\ninst✝² : CommRing R\ninst✝¹ : Ring B\ninst✝ : Algebra R B\nx : B\nhx : IsIntegral R x\n⊢ (Subalgebra.toSubmodule R[x]).FG",
"usedConstants": []
}
] | rcases hx with ⟨f, hfm, hfx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Polynomial.Lifts | {
"line": 144,
"column": 23
} | {
"line": 144,
"column": 42
} | [
{
"pp": "case inr\nR : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf : R →+* S\nn : ℕ\na : R\nhl : (monomial n) (f a) ∈ lifts f\nh : f a ≠ 0\n⊢ ↑n = ((monomial n) (f a)).degree",
"usedConstants": [
"Eq.mpr",
"WithBot",
"Semiring.toModule",
"congrArg",
"LinearMa... | degree_monomial n h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Choose.Vandermonde | {
"line": 35,
"column": 47
} | {
"line": 35,
"column": 62
} | [
{
"pp": "m n k : ℕ\n⊢ ((X + 1) ^ (m + n)).coeff k = ((X + 1) ^ m * (X + 1) ^ n).coeff k",
"usedConstants": [
"Eq.mpr",
"Polynomial.instOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"pow_add",
"id",
"MulOne.toMul",
"Polynomial.instAdd",
"Po... | by rw [pow_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Taylor | {
"line": 55,
"column": 54
} | {
"line": 55,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ (taylor 0) f = f",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"Polynomial.C_0",
"AddMonoid.toAddZeroClass",
"Polynomial.taylor"... | rw [taylor_apply, C_0, add_zero, comp_X] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Taylor | {
"line": 55,
"column": 54
} | {
"line": 55,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ (taylor 0) f = f",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"Polynomial.C_0",
"AddMonoid.toAddZeroClass",
"Polynomial.taylor"... | rw [taylor_apply, C_0, add_zero, comp_X] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Taylor | {
"line": 55,
"column": 54
} | {
"line": 55,
"column": 94
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\n⊢ (taylor 0) f = f",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toModule",
"congrArg",
"Polynomial.C_0",
"AddMonoid.toAddZeroClass",
"Polynomial.taylor"... | rw [taylor_apply, C_0, add_zero, comp_X] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.ScaleRoots | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\np : R[X]\ns : R\nhs : s ∈ nonZeroDivisors R\ni : ℕ\n⊢ p.coeff i ≠ 0 → p.coeff i * s ^ (p.natDegree - i) ≠ 0",
"usedConstants": [
"Ne",
"Polynomial.coeff",
"Zero.toOfNat0",
"OfNat.ofNat",
"MulZeroClass.toZero",
"instMulZeroClassOf... | intro p_ne_zero | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Algebra.Polynomial.Splits | {
"line": 104,
"column": 2
} | {
"line": 109,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f.natDegree ≤ 1\nh : Invertible f.leadingCoeff\n⊢ f.Splits",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"MulOne.t... | obtain ⟨a, b, rfl⟩ := exists_eq_X_add_C_of_natDegree_le_one hf
rcases eq_or_ne a 0 with rfl | ha
· simp
· replace h : Invertible a := by simpa [leadingCoeff, ha] using h
rw [← mul_invOf_cancel_left a b, C_mul, ← mul_add]
exact (Splits.C a).mul (Splits.X_add_C _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 104,
"column": 2
} | {
"line": 109,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : f.natDegree ≤ 1\nh : Invertible f.leadingCoeff\n⊢ f.Splits",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"MulOne.t... | obtain ⟨a, b, rfl⟩ := exists_eq_X_add_C_of_natDegree_le_one hf
rcases eq_or_ne a 0 with rfl | ha
· simp
· replace h : Invertible a := by simpa [leadingCoeff, ha] using h
rw [← mul_invOf_cancel_left a b, C_mul, ← mul_add]
exact (Splits.C a).mul (Splits.X_add_C _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nhf : f.Splits\nhm : f.Monic\n⊢ f.coeff 0 = (-1) ^ f.natDegree * f.roots.prod",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.roots",
"HMul.hMul",
"congrAr... | simp [hf.coeff_zero_eq_leadingCoeff_mul_prod_roots, hm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Splits | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nhf : f.Splits\nhm : f.Monic\n⊢ f.coeff 0 = (-1) ^ f.natDegree * f.roots.prod",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.roots",
"HMul.hMul",
"congrAr... | simp [hf.coeff_zero_eq_leadingCoeff_mul_prod_roots, hm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Splits | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\nhf : f.Splits\nhm : f.Monic\n⊢ f.coeff 0 = (-1) ^ f.natDegree * f.roots.prod",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Polynomial.roots",
"HMul.hMul",
"congrAr... | simp [hf.coeff_zero_eq_leadingCoeff_mul_prod_roots, hm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Splits | {
"line": 417,
"column": 53
} | {
"line": 417,
"column": 67
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : CommRing R\nf : R[X]\ninst✝ : IsDomain R\na : R\nhf : ((X - C a) * f).Splits\nhf₀ : ¬f = 0\nthis : (X - C a) * f = C f.leadingCoeff * (Multiset.map (fun x ↦ X - C x) ((X - C a).roots + f.roots)).prod\n⊢ f.Splits",
"usedConstants": [
"Polynomial.C",
"Poly... | roots_X_sub_C, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.IsIntegralClosure.Basic | {
"line": 548,
"column": 16
} | {
"line": 552,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_4\nT : Type u_5\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : CommRing T\ninst✝⁶ : IsDomain S\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsTorsionFree S T\ninst✝¹ : Nontrivial T\ninst✝ : IsScalarTower R S T\nh : Algebra.IsIntegral R T\n⊢ ∀... | by
apply RingHom.IsIntegral.tower_bot (algebraMap R S) (algebraMap S T)
(FaithfulSMul.algebraMap_injective S T)
rw [← IsScalarTower.algebraMap_eq R S T]
exact h.isIntegral | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Spectrum.Prime.Basic | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 13
} | [
{
"pp": "case pos\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsNoetherianRing R\nM : Ideal R\nhgt : ∀ J > M, ∃ Z, (Multiset.map asIdeal Z).prod ≤ J\nh_prM : ¬M.IsPrime\nhtop : M = ⊤\n⊢ ∃ Z, (Multiset.map asIdeal Z).prod ≤ M",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Multiset.map",... | rw [htop] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 47
} | [
{
"pp": "K : Type u\nA : Type v\ninst✝² : Field K\ninst✝¹ : Ring A\ninst✝ : Algebra K A\nx : A\np : K[X]\nhp : p ≠ 0\nhpx : (aeval x) p = 0\n⊢ IsIntegral K x",
"usedConstants": [
"Polynomial.C",
"GroupWithZero.toDivisionMonoid",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Al... | refine ⟨_, monic_mul_leadingCoeff_inv hp, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Algebraic.Integral | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 53
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Ring A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra S A\ninst✝² : IsScalarTower R S A\ninst✝¹ : NoZeroDivisors S\ninst✝ : Algebra.IsAlgebraic R S\na : A\nh : IsAlgebraic S a\np : S[X]... | rw [← faithfulSMul_iff_algebraMap_injective] at hRS | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 179,
"column": 4
} | {
"line": 183,
"column": 34
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf : (algebraMap R T).SurjectiveOnStalks\ng : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom\nJ : Ideal (S ⊗[R] T)\nhJ : J.I... | intro H
simp only [Algebra.algebraMap_eq_smul_one (A := S), Algebra.TensorProduct.algebraMap_apply,
Algebra.algebraMap_self, id_apply, smul_tmul, ← Algebra.algebraMap_eq_smul_one (A := T)] at H
rw [Ideal.mem_comap, Algebra.smul_def, g.map_mul] at ht
exact ht (J.mul_mem_right _ H) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SurjectiveOnStalks | {
"line": 179,
"column": 4
} | {
"line": 183,
"column": 34
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁴ : CommRing R\nS : Type u_2\ninst✝³ : CommRing S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : Algebra R S\nhf : (algebraMap R T).SurjectiveOnStalks\ng : T →+* S ⊗[R] T := Algebra.TensorProduct.includeRight.toRingHom\nJ : Ideal (S ⊗[R] T)\nhJ : J.I... | intro H
simp only [Algebra.algebraMap_eq_smul_one (A := S), Algebra.TensorProduct.algebraMap_apply,
Algebra.algebraMap_self, id_apply, smul_tmul, ← Algebra.algebraMap_eq_smul_one (A := T)] at H
rw [Ideal.mem_comap, Algebra.smul_def, g.map_mul] at ht
exact ht (J.mul_mem_right _ H) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 80,
"column": 32
} | {
"line": 80,
"column": 87
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\nx : R\nhx✝ : x ∈ p\ny : R\nhy : y ∉ I.radical\nn : ℕ\nhx : (x * y) ^ n ∈ I\nH : ∃ m, x ^ m * y ^ n ∈ I\nthis : Nat.find H ≠ 0\n⊢ x ^ (Nat.find H - 1 + 1) * y ^ n ∈ I",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | tsub_add_cancel_of_le (Nat.one_le_iff_ne_zero.mpr this) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 75,
"column": 2
} | {
"line": 81,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\nx : R\nhx : x ∈ p\n⊢ ∃ y ∉ I, x * y ∈ I",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Nat.instOrderedSub",
"Semigroup.toMul",
"Nat.i... | obtain ⟨y, hy, n, hx⟩ := Ideal.iUnion_minimalPrimes.subset (Set.mem_biUnion hp hx)
have H : ∃ m, x ^ m * y ^ n ∈ I := ⟨n, mul_pow x y n ▸ hx⟩
have : Nat.find H ≠ 0 :=
fun h ↦ hy ⟨n, by simpa only [h, pow_zero, one_mul] using Nat.find_spec H⟩
refine ⟨x ^ (Nat.find H - 1) * y ^ n, Nat.find_min H (Nat.sub_one_lt... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.MinimalPrime.Localization | {
"line": 75,
"column": 2
} | {
"line": 81,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI p : Ideal R\nhp : p ∈ I.minimalPrimes\nx : R\nhx : x ∈ p\n⊢ ∃ y ∉ I, x * y ∈ I",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"Nat.instOrderedSub",
"Semigroup.toMul",
"Nat.i... | obtain ⟨y, hy, n, hx⟩ := Ideal.iUnion_minimalPrimes.subset (Set.mem_biUnion hp hx)
have H : ∃ m, x ^ m * y ^ n ∈ I := ⟨n, mul_pow x y n ▸ hx⟩
have : Nat.find H ≠ 0 :=
fun h ↦ hy ⟨n, by simpa only [h, pow_zero, one_mul] using Nat.find_spec H⟩
refine ⟨x ^ (Nat.find H - 1) * y ^ n, Nat.find_min H (Nat.sub_one_lt... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.KrullDimension | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 9
} | [
{
"pp": "case top\nα : Type u_1\ninst✝ : Preorder α\na : α\nn : ℕ\nhne : Nonempty { p // RelSeries.last p = a }\nha : ∀ (p : LTSeries α), RelSeries.last p = a → p.length ≠ n\n⊢ ∃ x, ∀ y ∈ Set.range fun x ↦ (↑x).length, y ≤ x",
"usedConstants": [
"Preorder.toLT",
"RelSeries.last",
"setOf",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Order.KrullDimension | {
"line": 573,
"column": 79
} | {
"line": 582,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nx : α\n⊢ 1 < height x ↔ ∃ y, ∃ z < y, y < x",
"usedConstants": [
"List.isChain_pair",
"Iff.mpr",
"List.getLast",
"List.IsChain.cons_cons",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"False",
"Preorder.toLT",
"_private.Mat... | by
rw [← ENat.add_one_le_iff ENat.one_ne_top, one_add_one_eq_two]
refine ⟨fun h ↦ ?_, ?_⟩
· obtain ⟨p, hp, hlen⟩ := Order.exists_series_of_le_height x (n := 2) h
refine ⟨p 1, p 0, p.rel_of_lt ?_, hp ▸ p.rel_of_lt ?_⟩ <;> simp [Fin.lt_def, hlen]
· rintro ⟨y, z, hzy, hyx⟩
let p : LTSeries α := RelSeries.f... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.KrullDimension | {
"line": 727,
"column": 8
} | {
"line": 727,
"column": 52
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝ : Preorder α\nh : krullDim α = ⊤\nh✝¹ : Nonempty α\nh✝ : FiniteDimensionalOrder α\n⊢ InfiniteDimensionalOrder α",
"usedConstants": [
"WithBot",
"Preorder.toLT",
"ENat.instNatCast",
"instTopENat",
"congrArg",
"LTSeries.longestOf",... | krullDim_eq_length_of_finiteDimensionalOrder | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.BooleanSubalgebra | {
"line": 72,
"column": 17
} | {
"line": 72,
"column": 43
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : BooleanAlgebra α\ninst✝¹ : BooleanAlgebra β\ninst✝ : BooleanAlgebra γ\nL✝ M : BooleanSubalgebra α\nf : BoundedLatticeHom α β\ns✝ t : Set α\na b : α\nL : BooleanSubalgebra α\ns : Set α\nhs : s = ↑L\n⊢ ⊥ ∈ (L.copy s ⋯).carrier",
"usedCo... | subst hs; exact L.bot_mem' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.BooleanSubalgebra | {
"line": 72,
"column": 17
} | {
"line": 72,
"column": 43
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : BooleanAlgebra α\ninst✝¹ : BooleanAlgebra β\ninst✝ : BooleanAlgebra γ\nL✝ M : BooleanSubalgebra α\nf : BoundedLatticeHom α β\ns✝ t : Set α\na b : α\nL : BooleanSubalgebra α\ns : Set α\nhs : s = ↑L\n⊢ ⊥ ∈ (L.copy s ⋯).carrier",
"usedCo... | subst hs; exact L.bot_mem' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.QuasiSeparated | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 46
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : QuasiSeparatedSpace α\ns : Set (Set α)\nhf : s.Finite\nhne : s.Nonempty\nho : ∀ t ∈ s, IsOpen[inst✝¹] t ∨ IsClosed[inst✝¹] t\nhc : ∀ t ∈ s, IsCompact t\nthis :\n ∀ {α : Type u_1} [inst : TopologicalSpace α] [QuasiSeparatedSpace α] {s : Set (... | obtain (ha | ha) := a.eq_empty_or_nonempty | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Spectral.Prespectral | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 47
} | [
{
"pp": "case refine_2\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : PrespectralSpace X\nf : X → Y\nhfc : Continuous[inst✝², inst✝¹] f\nh : IsOpenMap f\nU : Set Y\nhs : U ⊆ Set.range f\nhU : IsOpen[inst✝¹] U\nhc : IsCompact U\nUs : Set (Opens X)\nhUs : Us ⊆ {U | ... | have := heq ▸ mem_sSup.mpr ⟨i.1, i.2, hx⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Spectral.Prespectral | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 34
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : PrespectralSpace X\nZ U₁ U₂ : Set X\nhU₁ : ∀ a ∈ U₁, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₁\nhU₂ : ∀ a ∈ U₂, ∃ t ∈ {U | IsOpen[inst✝¹] U ∧ IsCompact U}, a ∈ t ∧ t ⊆ U₂\nhU₁Z : (Z ∩ U₁).Nonempty\nhU₂Z : (Z ∩ U₂).Nonempty\nhU₁₂ : Z ∩ ... | obtain ⟨W₂, hW₂⟩ := hU₂ x₂ hx₂.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Constructible | {
"line": 112,
"column": 27
} | {
"line": 112,
"column": 46
} | [
{
"pp": "X : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nι : Type u_4\ns : Finset ι\nhs : s.Nonempty\nt : ι → Set X\nht : ∀ i ∈ s, IsRetrocompact (t i)\n⊢ IsRetrocompact (s.inf t)",
"usedConstants": [
"IsRetrocompact.finsetInf"
]
}
] | exact .finsetInf ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Constructible | {
"line": 418,
"column": 27
} | {
"line": 418,
"column": 46
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Type u_4\ns : Finset ι\nhs : s.Nonempty\nt : ι → Set X\nht : ∀ i ∈ s, IsLocallyConstructible (t i)\n⊢ IsLocallyConstructible (s.inf t)",
"usedConstants": [
"Topology.IsLocallyConstructible.finsetInf"
]
}
] | exact .finsetInf ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 42
} | [
{
"pp": "case over\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.last.asIdeal]... | ← L.toList_getElem_zero_eq_head, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.GoingDown | {
"line": 92,
"column": 10
} | {
"line": 92,
"column": 40
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Algebra.HasGoingDown R S\nl : RelSeries {(a, b) | a < b}\nq : PrimeSpectrum R\nlt : (q, l.head) ∈ {(a, b) | a < b}\nih :\n ∀ (P : Ideal S) [inst : P.IsPrime] [lo : P.LiesOver l.last.asIdeal],\n ∃ L,... | List.getElem_of_eq spec.symm _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Category.ModuleCat.Descent | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 65
} | [
{
"pp": "A B : Type u\ninst✝¹ : CommRing A\ninst✝ : CommRing B\nf : A →+* B\nhf : f.FaithfullyFlat\nthis : PreservesFiniteLimits (extendScalars f)\n⊢ ComonadicLeftAdjoint (extendScalars f)",
"usedConstants": [
"ModuleCat",
"ModuleCat.reflectsIsomorphisms_extendScalars_of_faithfullyFlat",
"... | have := reflectsIsomorphisms_extendScalars_of_faithfullyFlat hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Nat.Factorization.LCM | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 59
} | [
{
"pp": "a b : ℕ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ a.factorizationLCMLeft b * a.factorizationLCMRight b = a.lcm b",
"usedConstants": [
"Nat.lcm",
"Eq.mpr",
"Nat.factorizationLCMRight",
"Nat.instMulZeroClass",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"id",
"in... | ← prod_factorization_pow_eq_self (lcm_ne_zero ha hb), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.Flat | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 69
} | [
{
"pp": "⊢ flat.IsStableUnderCobaseChange",
"usedConstants": [
"RingHom.isStableUnderCobaseChange_toMorphismProperty_iff",
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"RingHom.Flat",
"CommRing",
"congrArg",
"CommSemiring.toSemiring",
"CommRingCat",
"Cat... | rw [flat, RingHom.isStableUnderCobaseChange_toMorphismProperty_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 70
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx✝ : {I | I.IsMaximal}.Finite ∧ sInf {I | I.IsMaximal} ≤ nilradical R\nfin : {I | I.IsMaximal}.Finite\nI : Ideal R\nhI : I.IsPrime\nle : sInf {I | I.IsMaximal} ≤ I\n⊢ I.IsMaximal",
"usedConstants": [
"Semiring.toModule",
"congrArg",
"CommSemirin... | rw [← fin.coe_toFinset, ← Finset.inf_id_eq_sInf, hI.inf_le'] at le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Spectrum.Prime.Topology | {
"line": 607,
"column": 4
} | {
"line": 607,
"column": 15
} | [
{
"pp": "case h.mp\nR : Type u\ninst✝³ : CommSemiring R\nS : Type v\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nx : PrimeSpectrum R\n⊢ ↑(Submonoid.powers r) ⊓ ↑x.asIdeal ≤ ⊥ → r ∉ x.asIdeal",
"usedConstants": [
"Semiring.toModule",
"CompleteBooleanAlge... | intro h₁ h₂ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Derivation.Basic | {
"line": 277,
"column": 22
} | {
"line": 277,
"column": 61
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : CommSemiring A\ninst✝¹¹ : CommSemiring B\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra R B\ninst✝⁷ : Module A M\ninst✝⁶ : Module B M\ninst✝⁵ : Module R M\nD✝ D1 D2 : Derivation R A M\nr✝ : R... | ext; dsimp; simp only [_root_.map_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Derivation.Basic | {
"line": 277,
"column": 22
} | {
"line": 277,
"column": 61
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹³ : CommSemiring R\ninst✝¹² : CommSemiring A\ninst✝¹¹ : CommSemiring B\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra R B\ninst✝⁷ : Module A M\ninst✝⁶ : Module B M\ninst✝⁵ : Module R M\nD✝ D1 D2 : Derivation R A M\nr✝ : R... | ext; dsimp; simp only [_root_.map_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Exponent | {
"line": 512,
"column": 90
} | {
"line": 512,
"column": 96
} | [
{
"pp": "G : Type u\ninst✝¹ : CancelCommMonoid G\ninst✝ : Fintype G\n⊢ exponent G = sSup (Set.range orderOf)",
"usedConstants": [
"Eq.mpr",
"CancelCommMonoid.toCommMonoid",
"congrArg",
"iSup",
"iSup.eq_1",
"id",
"ConditionallyCompleteLinearOrder.toConditionallyCompl... | ← iSup | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PGroup | {
"line": 138,
"column": 36
} | {
"line": 139,
"column": 65
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhG : IsPGroup p G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite G\nhGnt : Nontrivial G\nk : ℕ\nhk : Nat.card G = p ^ k\nhk0 : k = 0\n⊢ False",
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"Eq... | by
rw [hk0, pow_zero] at hk; exact Finite.one_lt_card.ne' hk | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Sylow | {
"line": 278,
"column": 56
} | {
"line": 279,
"column": 84
} | [
{
"pp": "p : ℕ\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nP : Sylow p G\n⊢ P ∈ fixedPoints (↥H) (Sylow p G) ↔ H ≤ normalizer ↑P",
"usedConstants": [
"Eq.mpr",
"Sylow.instSetLike",
"instHSMul",
"Sylow",
"_private.Mathlib.GroupTheory.Sylow.0.Subgroup.sylow_mem_fixedPoints_if... | by
simp_rw [SetLike.le_def, ← Sylow.smul_eq_iff_mem_normalizer]; exact Subtype.forall | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Sylow | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 50
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\ninst✝² : Finite G\nG' : Type u_2\ninst✝¹ : Group G'\nf : G →* G'\nhf : Function.Surjective ⇑f\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nthis : Finite G'\nP : Sylow p G'\nQ₀ : Sylow p ↥(comap f ↑P) := ⋯.some\n⊢ ∃ a, mapSurjective hf a = P",
"usedConstants": [
"Sylow.t... | let Q : Subgroup G := Q₀.map (P.comap f).subtype | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.GroupTheory.SpecificGroups.Cyclic | {
"line": 516,
"column": 69
} | {
"line": 517,
"column": 31
} | [
{
"pp": "G : Type u_2\ninst✝¹ : Infinite G\ninst✝ : Group G\ng : G\nhg : zpowers g = ⊤\n⊢ (intEquivOfZPowersEqTop g hg).symm g = Multiplicative.ofAdd 1",
"usedConstants": [
"MulEquiv.instEquivLike",
"Equiv.instEquivLike",
"Monoid.toMulOneClass",
"congrArg",
"zpow_one",
"E... | by
simp [MulEquiv.symm_apply_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Sylow | {
"line": 725,
"column": 2
} | {
"line": 725,
"column": 40
} | [
{
"pp": "G : Type u\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nh : (↑P).Normal\nQ : Sylow p G\n⊢ Q = default",
"usedConstants": [
"Sylow",
"DivInvMonoid.toMonoid",
"Sylow.mulAction",
"Group.toDivInvMonoid",
"Monoid.toSemigr... | obtain ⟨x, h1⟩ := exists_smul_eq G P Q | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.JordanHolder | {
"line": 393,
"column": 20
} | {
"line": 393,
"column": 46
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih :\n ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),\n head s ≤ x →\n s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)\ns : CompositionSeries ... | rw [eq_snoc_eraseLast h0s] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Order.JordanHolder | {
"line": 393,
"column": 20
} | {
"line": 393,
"column": 46
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih :\n ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),\n head s ≤ x →\n s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)\ns : CompositionSeries ... | rw [eq_snoc_eraseLast h0s] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Order.JordanHolder | {
"line": 393,
"column": 20
} | {
"line": 393,
"column": 46
} | [
{
"pp": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nn : ℕ\nih :\n ∀ (s : CompositionSeries X) (x : X) (hm : IsMaximal x (last s)),\n head s ≤ x →\n s.length = n → ∃ t, head t = head s ∧ t.length + 1 = n ∧ ∃ (htx : last t = x), s.Equivalent (snoc t (last s) ⋯)\ns : CompositionSeries ... | rw [eq_snoc_eraseLast h0s] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Algebra.Polynomial.Module.Basic | {
"line": 162,
"column": 13
} | {
"line": 162,
"column": 39
} | [
{
"pp": "case monomial\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ng : PolynomialModule R M\nn f_n : ℕ\nf_a : R\n⊢ (if f_n ≤ n then f_a • g (n - f_n) else 0) = ∑ k ∈ Finset.range n.succ, ((monomial f_n) f_a).coeff k • g (n - k)",
"usedConstants": [
"F... | Polynomial.coeff_monomial, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Finiteness.Nakayama | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nI : Ideal R\nN : Submodule R M\ns✝ : Set M\ni : M\ns : Set M\na✝ : i ∉ s\nhs✝ : s.Finite\nih :\n (∃ r, r - 1 ∈ I ∧ N ≤ comap ((LinearMap.lsmul R M) r) (I • span R s) ∧ s ⊆ ↑N) → ∃ r, r - 1 ∈ I ∧ ∀ n ∈ N, r • ... | specialize hrn hs.1 | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 135,
"column": 8
} | {
"line": 135,
"column": 13
} | [
{
"pp": "ι : Type u_3\nR : Type u_4\nM : Type u_5\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nh_ne_zero : ∀ (i : ι), torsionOf R M (v i) = ⊥\ni : ι\nr : R\nhi : r • v i ∈ Submodule.span R (v '' (Set.univ \\ {i}))\nhv : R ∙ v i ⊓ Submodule.span R (Set.range fun i_1 ↦ v ↑i_1) = ⊥\n⊢ ... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Torsion.Basic | {
"line": 877,
"column": 4
} | {
"line": 877,
"column": 9
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝² : Monoid R\ninst✝¹ : AddCommMonoid M\ninst✝ : DistribMulAction R M\np : R\nh : IsTorsion' M ↥(Submonoid.powers p)\nx : M\na : R\nn : ℕ\nhn : p ^ n = a\nhx : ⟨a, ⋯⟩ • x = 0\n⊢ ∃ n, p ^ n • x = 0",
"usedConstants": [
"instHSMul",
"DistribMulActi... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Filtration | {
"line": 440,
"column": 42
} | {
"line": 448,
"column": 50
} | [
{
"pp": "R : Type u_3\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nI : Ideal R\n⊢ IsIdempotentElem I ↔ I = ⊥ ∨ I = ⊤",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"MulOne.toOne",
"Ideal.one_eq_top",
"iInf",
"Semiring.toModule",
... | by
constructor
· intro H
by_cases I = ⊤; · exact Or.inr ‹_›
refine Or.inl (eq_bot_iff.mpr ?_)
rw [← Ideal.iInf_pow_eq_bot_of_isLocalRing I ‹_›]
apply le_iInf
rintro (_ | n) <;> simp [H.pow_succ_eq]
· rintro (rfl | rfl) <;> simp [IsIdempotentElem] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 201,
"column": 2
} | {
"line": 203,
"column": 79
} | [
{
"pp": "R : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Al... | refine (Ideal.quotEquivOfEq (Ideal.map_eq_submodule_map _ _).symm).trans ?_
refine (DoubleQuot.quotQuotEquivQuotSup _ _).trans ?_
exact Ideal.quotEquivOfEq (sup_eq_right.mpr <| Ideal.pow_le_self two_ne_zero) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Cotangent | {
"line": 201,
"column": 2
} | {
"line": 203,
"column": 79
} | [
{
"pp": "R : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Al... | refine (Ideal.quotEquivOfEq (Ideal.map_eq_submodule_map _ _).symm).trans ?_
refine (DoubleQuot.quotQuotEquivQuotSup _ _).trans ?_
exact Ideal.quotEquivOfEq (sup_eq_right.mpr <| Ideal.pow_le_self two_ne_zero) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomologicalComplexAbelian | {
"line": 73,
"column": 4
} | {
"line": 76,
"column": 29
} | [
{
"pp": "case mp\nC : Type u_1\nι : Type u_2\nc : ComplexShape ι\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : ShortComplex (HomologicalComplex C c)\n⊢ S.ShortExact → ∀ (i : ι), (S.map (eval C c i)).ShortExact",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory... | intro hS i
have := hS.mono_f
have := hS.epi_g
exact hS.map (eval C c i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomologicalComplexAbelian | {
"line": 73,
"column": 4
} | {
"line": 76,
"column": 29
} | [
{
"pp": "case mp\nC : Type u_1\nι : Type u_2\nc : ComplexShape ι\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Abelian C\nS : ShortComplex (HomologicalComplex C c)\n⊢ S.ShortExact → ∀ (i : ι), (S.map (eval C c i)).ShortExact",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"CategoryTheory... | intro hS i
have := hS.mono_f
have := hS.epi_g
exact hS.map (eval C c i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Category.ModuleCat.Presheaf | {
"line": 247,
"column": 52
} | {
"line": 247,
"column": 66
} | [
{
"pp": "case h\nC : Type u₁\ninst✝ : Category.{v₁, u₁} C\nR : Cᵒᵖ ⥤ RingCat\nM M₁ M₂ : PresheafOfModules R\na✝ b✝ : M₁ ⟶ M₂\nX✝ : Cᵒᵖ\n⊢ a✝.app X✝ + b✝.app X✝ = b✝.app X✝ + a✝.app X✝",
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"ModuleCat"... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 18
} | [
{
"pp": "J : Type w\nC : J → Type u\nD : J → Type u'\ninst✝¹ : (j : J) → Category.{v, u} (C j)\ninst✝ : (j : J) → Category.{v', u'} (D j)\nW : (j : J) → MorphismProperty (C j)\nF : (j : J) → C j ⥤ D j\nhF : ∀ (j : J), (W j).IsInvertedBy (F j)\nX✝ Y✝ : (j : J) → C j\nf : X✝ ⟶ Y✝\nhf : MorphismProperty.pi W f\n⊢ ... | isIso_pi_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCofiber | {
"line": 84,
"column": 22
} | {
"line": 90,
"column": 32
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nF G : HomologicalComplex C c\nφ : F ⟶ G\ninst✝¹ : HasHomotopyCofiber φ\ninst✝ : DecidableRel c.Rel\ni : ι\nhG : IsZero (G.X i)\nhF : ∀ (j : ι), c.Rel i j → IsZero (F.X j)\n⊢ IsZero (X φ i)",
"use... | by
by_cases h : c.Rel i (c.next i)
· haveI := HasHomotopyCofiber.hasBinaryBiproduct φ _ _ h
refine IsZero.of_iso ?_ (XIsoBiprod φ _ _ h)
simp only [biprod_isZero_iff]
exact ⟨hF _ h, hG⟩
· exact hG.of_iso (XIso φ i h) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory | {
"line": 87,
"column": 31
} | {
"line": 87,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nι : Type u_2\nV : Type u\ninst✝² : Category.{v, u} V\ninst✝¹ : Preadditive V\nc : ComplexShape ι\ninst✝ : HasZeroObject V\n⊢ 𝟙 ((quotient V c).obj 0) = 0",
"usedConstants": [
"Eq.mpr",
"HomologicalComplex.instCategory",
"CategoryTheory.CategoryS... | ← (quotient V c).map_id, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 741,
"column": 43
} | {
"line": 744,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nF G K : CochainComplex C ℤ\nn : ℤ\nz₁ : Cochain F G n\nz₂ : Cocycle G K 0\nm : ℤ\n⊢ δ n m (z₁.comp ↑z₂ ⋯) = (δ n m z₁).comp ↑z₂ ⋯",
"usedConstants": [
"CochainComplex.HomComplex.instAddCommGroupCochain",
"CochainComplex.HomC... | by
by_cases hnm : n + 1 = m
· simp [δ_comp_zero_cochain _ _ _ hnm]
· simp [δ_shape _ _ hnm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 791,
"column": 8
} | {
"line": 796,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nφ₁ φ₂ : F ⟶ G\nz : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }\np : ℤ\n⊢ φ₁.f p =\n (((dNext p) fun i j ↦ if hij : i + -1 = j then (↑z).v i j hij el... | have eq := Cochain.congr_v z.2 p p (add_zero p)
have h₁ : (ComplexShape.up ℤ).Rel (p - 1) p := by simp
have h₂ : (ComplexShape.up ℤ).Rel p (p + 1) := by simp
simp only [δ_neg_one_cochain, Cochain.ofHom_v, ComplexShape.up_Rel, Cochain.add_v,
Homotopy.nullHomotopicMap'_f h₁ h₂] at eq
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 791,
"column": 8
} | {
"line": 796,
"column": 63
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Preadditive C\nR : Type u_1\ninst✝¹ : Ring R\ninst✝ : Linear R C\nF G K L : CochainComplex C ℤ\nn m : ℤ\nφ₁ φ₂ : F ⟶ G\nz : { z // ofHom φ₁ = δ (-1) 0 z + ofHom φ₂ }\np : ℤ\n⊢ φ₁.f p =\n (((dNext p) fun i j ↦ if hij : i + -1 = j then (↑z).v i j hij el... | have eq := Cochain.congr_v z.2 p p (add_zero p)
have h₁ : (ComplexShape.up ℤ).Rel (p - 1) p := by simp
have h₂ : (ComplexShape.up ℤ).Rel p (p + 1) := by simp
simp only [δ_neg_one_cochain, Cochain.ofHom_v, ComplexShape.up_Rel, Cochain.add_v,
Homotopy.nullHomotopicMap'_f h₁ h₂] at eq
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplex | {
"line": 866,
"column": 4
} | {
"line": 866,
"column": 10
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\np q : ℤ\nf : K.X p ⟶ L.X q\nn m : ℤ\nhm : n + 1 = m\np' q' : ℤ\nhp' : p' + 1 = p\nhq' : q + 1 = q'\np'' q'' : ℤ\nhpq'' : p'' + m = q''\n⊢ (single f n).v p'' (q'' - 1) ⋯ ≫ L.d (q'' - 1) q'' +\n m.negOne... | add_v, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Shift.Quotient | {
"line": 117,
"column": 41
} | {
"line": 117,
"column": 75
} | [
{
"pp": "case w.h.w.h\nC : Type u\ninst✝⁶ : Category.{v, u} C\nD : Type u'\ninst✝⁵ : Category.{v', u'} D\nF : C ⥤ D\nr : HomRel C\nA : Type w\ninst✝⁴ : AddMonoid A\ninst✝³ : HasShift C A\ninst✝² : HasShift D A\ninst✝¹ : r.IsCompatibleWithShift A\ninst✝ : F.CommShift A\nhF : ∀ (x y : C) (f₁ f₂ : x ⟶ y), r f₁ f₂ ... | Functor.CommShift.isoZero_inv_app, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 85,
"column": 66
} | {
"line": 103,
"column": 89
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower R S M\nD : Derivation R S M\nx y : S ⊗[R] S\n⊢ D.tensorProductTo (x * y) =\n (TensorProduct.lmul' R) x • D.te... | by
refine TensorProduct.induction_on x ?_ ?_ ?_
· rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero]
swap
· intro x₁ y₁ h₁ h₂
rw [add_mul, map_add, map_add, map_add, add_smul, smul_add, h₁, h₂, add_add_add_comm]
intro x₁ x₂
refine TensorProduct.induction_on y ?_ ?_ ?_
· rw [mul_zero, ma... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Shift.CommShift | {
"line": 258,
"column": 61
} | {
"line": 265,
"column": 31
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Category.{v_2, u_2} D\nF : C ⥤ D\nA : Type u_4\ninst✝³ : AddMonoid A\ninst✝² : HasShift C A\ninst✝¹ : HasShift D A\ninst✝ : F.CommShift A\nX : C\na b : A\nh : a + b = 0\n⊢ F.map ((shiftFunctorCompIsoId C a b h).hom.app X) =\n (comm... | by
dsimp [shiftFunctorCompIsoId]
have eq := NatTrans.congr_app (congr_arg Iso.hom (F.commShiftIso_add' h)) X
simp only [commShiftIso_zero, comp_obj, CommShift.isoZero_hom_app,
CommShift.isoAdd'_hom_app] at eq
rw [← cancel_epi (F.map ((shiftFunctorAdd' C a b 0 h).hom.app X)), ← reassoc_of% eq, F.map_comp]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomotopyCategory.Shift | {
"line": 252,
"column": 4
} | {
"line": 254,
"column": 47
} | [
{
"pp": "case w.w.h.h\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Preadditive C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : Preadditive D\nF : C ⥤ D\ninst✝ : F.Additive\na b : ℤ\nx✝ : HomologicalComplex C (ComplexShape.up ℤ)\ni✝ : ℤ\n⊢ 𝟙 (F.obj (x✝.X (i✝ + (a + b)))) =\n F.map (((shiftFunctor... | simp only [CochainComplex.shiftFunctorAdd_hom_app_f,
CochainComplex.shiftFunctorAdd_inv_app_f, HomologicalComplex.XIsoOfEq, eqToIso,
eqToHom_map, eqToHom_trans, eqToHom_refl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Homology.HomotopyCategory.HomComplexShift | {
"line": 146,
"column": 2
} | {
"line": 148,
"column": 66
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nK L : CochainComplex C ℤ\nn : ℤ\nγ : Cochain K L n\na n' : ℤ\nhn' : n + a = n'\np q : ℤ\nhpq : p + n = q\n⊢ ((γ.leftShift a n' hn').leftUnshift n hn').v p q hpq = γ.v p q hpq",
"usedConstants": [
"Eq.mpr",
"MulOne.to... | rw [(γ.leftShift a n' hn').leftUnshift_v n hn' p q hpq (q - n') (by lia),
γ.leftShift_v a n' hn' (q - n') q (by lia) p hpq, Linear.comp_units_smul,
Iso.inv_hom_id_assoc, smul_smul, Int.units_mul_self, one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Kaehler.Basic | {
"line": 750,
"column": 6
} | {
"line": 750,
"column": 22
} | [
{
"pp": "case a.convert_2.inr\nR : Type u\ninst✝⁶ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra A B\ninst✝¹ : Algebra R B\ninst✝ : IsScalarTower R A B\nx : A\n⊢ (fun x ↦ single ((algebraMap A B) x) 1) x ∈\n ↑(Submodule.comap (linear... | use 1 ⊗ₜ D _ _ x | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.CategoryTheory.Triangulated.Basic | {
"line": 292,
"column": 29
} | {
"line": 292,
"column": 43
} | [
{
"pp": "case h₁\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasShift C ℤ\nT₁ T₂ T₃ : Triangle C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nf g : T₁ ⟶ T₂\n⊢ (f + g).hom₁ = (g + f).hom₁",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom"... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Triangulated.Basic | {
"line": 292,
"column": 29
} | {
"line": 292,
"column": 43
} | [
{
"pp": "case h₂\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasShift C ℤ\nT₁ T₂ T₃ : Triangle C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nf g : T₁ ⟶ T₂\n⊢ (f + g).hom₂ = (g + f).hom₂",
"usedConstants": [
"CategoryTheory.Pretriangulated.TriangleMorphism.hom₂",
... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Triangulated.Basic | {
"line": 292,
"column": 29
} | {
"line": 292,
"column": 43
} | [
{
"pp": "case h₃\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasShift C ℤ\nT₁ T₂ T₃ : Triangle C\ninst✝¹ : Preadditive C\ninst✝ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nf g : T₁ ⟶ T₂\n⊢ (f + g).hom₃ = (g + f).hom₃",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom"... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Triangulated.Pretriangulated | {
"line": 642,
"column": 74
} | {
"line": 642,
"column": 76
} | [
{
"pp": "case h\nC : Type u\ninst✝⁸ : Category.{v, u} C\ninst✝⁷ : HasZeroObject C\ninst✝⁶ : HasShift C ℤ\ninst✝⁵ : Preadditive C\ninst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive\nhC : Pretriangulated C\nJ : Type u_1\nT : J → Triangle C\nhT : ∀ (j : J), T j ∈ distinguishedTriangles\ninst✝³ : HasProduct fun j ↦ (... | hα | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Homology.HomotopyCategory.DegreewiseSplit | {
"line": 247,
"column": 70
} | {
"line": 258,
"column": 33
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : HasZeroObject C\ninst✝ : HasBinaryBiproducts C\nT : Triangle (HomotopyCategory C (ComplexShape.up ℤ))\n⊢ T ∈ distinguishedTriangles ↔ ∃ S σ, Nonempty (T ≅ CochainComplex.trianglehOfDegreewiseSplit S σ)",
"usedConstants": [... | by
constructor
· intro hT
obtain ⟨K, L, φ, ⟨e⟩⟩ := inv_rot_of_distTriang _ hT
exact ⟨_, _, ⟨(triangleRotation _).counitIso.symm.app _ ≪≫ (rotate _).mapIso e ≪≫
CochainComplex.mappingCone.trianglehRotateIsoTrianglehOfDegreewiseSplit φ⟩⟩
· rintro ⟨S, σ, ⟨e⟩⟩
rw [rotate_distinguished_triangle, rota... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Homology.HomologySequence | {
"line": 208,
"column": 4
} | {
"line": 210,
"column": 59
} | [
{
"pp": "C : Type u_1\nι : Type u_2\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : Abelian C\nc : ComplexShape ι\nS : ShortComplex (HomologicalComplex C c)\nhS : S.Exact\ninst✝³ : Mono S.f\ni : ι\ninst✝² : S.X₁.HasHomology i\ninst✝¹ : S.X₂.HasHomology i\ninst✝ : S.X₃.HasHomology i\nthis : Mono (S.map (eval C c i)).f... | have H := KernelFork.IsLimit.lift' hi (k ≫ S.X₂.iCycles i) (by
dsimp
rw [assoc, ← cyclesMap_i, reassoc_of% hk, zero_comp]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 20
} | [
{
"pp": "V : Type u₁\ninst✝¹ : Quiver V\nC : Type ?u.2697\ninst✝ : Category.{v_1, ?u.2697} C\nφ : V ⥤q C\nX✝ Y✝ Z✝ : Paths V\nf : X✝ ⟶ Y✝\ng : Y✝ ⟶ Z✝\n⊢ Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) (f ≫ g) =\n Quiver.Path.rec (𝟙 (φ.obj X✝)) (fun {b c} x f ihp ↦ ihp ≫ φ.map f) f ≫\n ... | induction g with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.PathCategory.Basic | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 18
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nX Y Z : C\nf : Path X Y\ng : Path Y Z\n⊢ composePath (f.comp g) = composePath f ≫ composePath g",
"usedConstants": [
"CategoryTheory.composePath",
"CategoryTheory.Category.assoc",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom... | induction g with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.Localization.Construction | {
"line": 214,
"column": 6
} | {
"line": 214,
"column": 25
} | [
{
"pp": "C : Type uC\ninst✝¹ : Category.{uC', uC} C\nW : MorphismProperty C\nP : MorphismProperty W.Localization\ninst✝ : P.IsStableUnderComposition\nhP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f)\nhP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (wInv w hw)\nX Y : W.Localization\nf : X ⟶ Y\na✝ : ⊤ f\nG : Paths (LocQu... | rcases X with ⟨⟨X⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 17
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\ninst✝ : L.IsLocalization W\nX : C\nhX : W (𝟙 X)\n⊢ 𝟙 (L.obj X) = (isoOfHom L W (𝟙 X) hX).hom",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.to... | isoOfHom_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Localization.Predicate | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 68
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Category.{v_2, u_2} D\nL : C ⥤ D\nW : MorphismProperty C\nhW : W ≤ MorphismProperty.isomorphisms C\ninst✝ : L.IsEquivalence\nthis : (𝟭 C).IsLocalization W\n⊢ L.IsLocalization W",
"usedConstants": [
"CategoryTheory.Functor.i... | exact of_equivalence_target (𝟭 C) W L L.asEquivalence L.leftUnitor | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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