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.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 161,
"column": 8
} | {
"line": 164,
"column": 56
} | [
{
"pp": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : weightedOrder w f < ⊤\nhg : weightedOrder w g < ⊤\np : ℕ := (weightedOrder w f).toNat\nhp : ↑p = weightedOrder w f\nq : ℕ := (weightedOrder w g).toNat\nhq : ↑q = weightedOrder w... | · refine weightedHomogeneousComponent_of_weightedOrder ?_ H
simp only [ENat.coe_toNat_eq_self, ne_eq, weightedOrder_eq_top_iff, p, q]
rw [← ne_eq, ne_zero_iff_weightedOrder_finite w]
exact ENat.coe_toNat (ne_top_of_lt (by simpa)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 46
} | [
{
"pp": "case neg\nσ : Type u_1\nk : Type u_3\ninst✝ : Field k\nφ ψ : MvPowerSeries σ k\nh : ¬constantCoeff (φ * ψ) = 0\n⊢ (φ * ψ)⁻¹ = ψ⁻¹ * φ⁻¹",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"MvPowerSeries.inv_eq_iff_mul_eq_one",
"MvPowerSeries",
"MvPowerSeries.... | rw [MvPowerSeries.inv_eq_iff_mul_eq_one h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 25
} | [
{
"pp": "R : Type u_1\nk : Type u_2\ninst✝ : Field k\nu : k⟦X⟧ˣ\n⊢ (↑u).Unit_of_divided_by_X_pow_order⁻¹ = u⁻¹",
"usedConstants": [
"Units.val",
"MvPowerSeries.instCommSemiring",
"CommSemiring.toCommMonoidWithZero",
"CommMonoidWithZero.toMonoidWithZero",
"Field.toSemifield",
... | set u₀ := u.1 with hu | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ | {
"line": 191,
"column": 4
} | {
"line": 193,
"column": 60
} | [
{
"pp": "case pos\nF : Type u_2\ninst✝¹ : Field F\nj : F\ninst✝ : DecidableEq F\nh0 : ¬j = 0\nh1728 : j = 1728\n⊢ (ofJ j).j = j",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.... | · have h2 : (2 : F) ≠ 0 := fun h ↦ h0 (by linear_combination h1728 + 864 * h)
have := Fact.mk h2.isUnit
simp_rw [h1728, ofJ_1728_of_two_ne_zero h2, ofJ1728_j] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 334,
"column": 2
} | {
"line": 336,
"column": 43
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ (eval P) W.polynomialY / P z ^ 2 = Polynomial.evalEval (P x / P z) (P y / P z) W.toAffine.polynomialY",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Fins... | linear_combination (norm := (rw [eval_polynomialY, Affine.evalEval_polynomialY]; ring1))
2 * P y / P z * div_self hPz + W.a₁ * P x / P z * div_self hPz
+ W.a₃ * div_self (pow_ne_zero 2 hPz) | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 334,
"column": 2
} | {
"line": 336,
"column": 43
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ (eval P) W.polynomialY / P z ^ 2 = Polynomial.evalEval (P x / P z) (P y / P z) W.toAffine.polynomialY",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Fins... | linear_combination (norm := (rw [eval_polynomialY, Affine.evalEval_polynomialY]; ring1))
2 * P y / P z * div_self hPz + W.a₁ * P x / P z * div_self hPz
+ W.a₃ * div_self (pow_ne_zero 2 hPz) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 334,
"column": 2
} | {
"line": 336,
"column": 43
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\n⊢ (eval P) W.polynomialY / P z ^ 2 = Polynomial.evalEval (P x / P z) (P y / P z) W.toAffine.polynomialY",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Fins... | linear_combination (norm := (rw [eval_polynomialY, Affine.evalEval_polynomialY]; ring1))
2 * P y / P z * div_self hPz + W.a₁ * P x / P z * div_self hPz
+ W.a₃ * div_self (pow_ne_zero 2 hPz) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 494,
"column": 4
} | {
"line": 495,
"column": 87
} | [
{
"pp": "case neg\nF : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : P z = 0\nhQz : ¬Q z = 0\n⊢ toAffine W (W.add P Q) = toAffine W Q",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Weierstr... | · rw [add_of_Z_eq_zero_left hP.left hPz hQz,
toAffine_smul _ <| ((isUnit_Y_of_Z_eq_zero hP hPz).pow 2).mul <| Ne.isUnit hQz] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 542,
"column": 78
} | {
"line": 544,
"column": 23
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : W.Point\n⊢ (-P).toAffineLift = -P.toAffineLift",
"usedConstants": [
"WeierstrassCurve.Projective.Point",
"WeierstrassCurve.Projective.toAffine",
"Units.instMulAction",
"WeierstrassCurve.Projective.Point.casesOn",
"Weie... | by
rcases P with @⟨⟨_⟩, hP⟩
exact toAffine_neg hP | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 89
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Projective R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsS... | rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 89
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Projective R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsS... | rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 89
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Projective R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsS... | rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Geometrically.Connected | {
"line": 109,
"column": 92
} | {
"line": 110,
"column": 82
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\n⊢ GeometricallyConnected f ↔ ∀ (s : ↥S), GeometricallyConnected (Scheme.Hom.fiberToSpecResidueField f s)",
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"... | by
simp only [eq_geometrically, ← geometrically_iff_forall_fiberToSpecResidueField] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 94
} | [
{
"pp": "case inr\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsIntegral Y\ninst✝ : Subsingleton ↥Y\nthis : ∀ {X : Scheme} (f : X ⟶ Y), (∃ S, X = Spec S) → UniversallyOpen f\nhX : ¬∃ S, X = Spec S\n⊢ UniversallyOpen f",
"usedConstants": [
"Iff.mpr",
"AlgebraicGeometry.Spec",... | refine (IsZariskiLocalAtSource.iff_of_openCover X.affineCover).mpr fun i ↦ this _ ⟨_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 94
} | [
{
"pp": "case inr\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsIntegral Y\ninst✝ : Subsingleton ↥Y\nthis : ∀ {X : Scheme} (f : X ⟶ Y), (∃ S, X = Spec S) → UniversallyOpen f\nhX : ¬∃ S, X = Spec S\n⊢ UniversallyOpen f",
"usedConstants": [
"Iff.mpr",
"AlgebraicGeometry.Spec",... | refine (IsZariskiLocalAtSource.iff_of_openCover X.affineCover).mpr fun i ↦ this _ ⟨_, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 94
} | [
{
"pp": "case inr\nX✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsIntegral Y\ninst✝ : Subsingleton ↥Y\nthis : ∀ {X : Scheme} (f : X ⟶ Y), (∃ S, X = Spec S) → UniversallyOpen f\nhX : ¬∃ S, X = Spec S\n⊢ UniversallyOpen f",
"usedConstants": [
"Iff.mpr",
"AlgebraicGeometry.Spec",... | refine (IsZariskiLocalAtSource.iff_of_openCover X.affineCover).mpr fun i ↦ this _ ⟨_, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 54
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nE : PreOneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nh : E.IsStronglySeparatedFor F\nx : (i : E.I₀) → F.obj (op (E.X i))\nhc :\n ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j),\n (ConcreteCategory.hom (F.map (E.p₁ k).op)) (x i) = (ConcreteCategory.hom (F.map (E.p₂ k).op)) ... | rw [← comp_apply, ← Functor.map_comp, ← op_comp, h₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 135,
"column": 2
} | {
"line": 140,
"column": 16
} | [
{
"pp": "case neg\nC : Type u\ninst✝⁴ : Category.{v, u} C\nI : C\nF : Cᵒᵖ ⥤ Type w\nhF : IsSheafFor F (ofArrows Empty.elim fun a ↦ Empty.instIsEmpty.elim a)\nhI : IsInitial I\nα : Type u_1\ninst✝³ : Small.{w, u_1} α\nX : α → C\nc : Cofan X\ninst✝² : (ofArrows X c.inj).HasPairwisePullbacks\ninst✝¹ : HasInitial C... | · have := preservesTerminal_of_isSheaf_for_empty F hF hI
apply_fun (F.mapIso ((hd hi).isoPullback).op ≪≫ F.mapIso (terminalIsoIsTerminal
(terminalOpOfInitial initialIsInitial)).symm ≪≫ (PreservesTerminal.iso F)).hom using
injective_of_mono _
ext ⟨i⟩
exact i.elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 233,
"column": 56
} | {
"line": 236,
"column": 32
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ W.toAffine.slope (P x / P z) (Q x / Q z) (P y / P z) (Q y / Q z) = -(eval P) W.polynomialX / P z / (P y - W.negY P)",
... | by
simp only [X_eq_iff hPz hQz, ne_eq, Y_eq_iff' hPz hQz] at hx hy
rw [Affine.slope_of_Y_ne hx <| negY_of_Z_ne_zero hQz ▸ hy, ← negY_of_Z_ne_zero hPz]
simp [field, eval_polynomialX] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 308,
"column": 64
} | {
"line": 308,
"column": 91
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P) W.polynomialX * P z * (P y - W.negY P)... | ← (X_eq_iff hPz hQz).mp hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 108,
"column": 56
} | {
"line": 108,
"column": 93
} | [
{
"pp": "case h2\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : (Q.comp P).Ring\nhx' : ... | Submodule.map_subtype_span_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 393,
"column": 67
} | {
"line": 393,
"column": 94
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ (-(eval P) W.polynomialX *\n ((eval P) W.polynomialX ^ 2 - W.a₁ * (eval P... | ← (X_eq_iff hPz hQz).mp hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.Unramified | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 60
} | [
{
"pp": "case localizationAwayPreserves\n⊢ LocalizationAwayPreserves fun {R S} [CommRing R] [CommRing S] ↦ FormallyUnramified",
"usedConstants": [
"CommRing",
"RingHom.FormallyUnramified.isStableUnderBaseChange",
"RingHom.IsStableUnderBaseChange.localizationPreserves",
"RingHom.Forma... | · exact isStableUnderBaseChange.localizationPreserves.away | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 182,
"column": 54
} | {
"line": 182,
"column": 57
} | [
{
"pp": "case tmul.tmul\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Unramified.Field | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 78
} | [
{
"pp": "K : Type u_1\nA : Type u_2\ninst✝⁶ : Field K\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra K A\ninst✝³ : FormallyUnramified K A\ninst✝² : EssFiniteType K A\ninst✝¹ : IsAlgClosed K\ninst✝ : IsLocalRing A\nthis✝¹ : Module.Finite K A\nthis✝ : IsArtinianRing A\ne : K ≃ₐ[K] A ⧸ IsLocalRing.maximalIdeal A :=\n let... | rw [eq_comm, ← sub_eq_zero, ← hf₃.pow_succ_eq n, pow_succ, this, zero_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Unramified.Field | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 43
} | [
{
"pp": "A : Type u_2\ninst✝⁵ : CommRing A\nB : Type u_4\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : EssFiniteType A B\ninst✝¹ : FormallyUnramified A B\np : Ideal A\ninst✝ : p.IsMaximal\nthis : Field (A ⧸ p) := Ideal.Quotient.field p\n⊢ (Ideal.map (algebraMap A B) p).IsRadical",
"usedConstants": [
... | rw [Ideal.isRadical_iff_quotient_reduced] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Unramified.Field | {
"line": 175,
"column": 4
} | {
"line": 175,
"column": 24
} | [
{
"pp": "K : Type u_1\nL : Type u_3\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FormallyUnramified K L\ninst✝¹ : EssFiniteType K L\ninst✝ : IsPurelyInseparable K L\nthis✝ : Nontrivial (L ⊗[K] L)\nx : L\na✝ : x ∈ ⊤\nn : ℕ\nhn : x ^ ringExpChar K ^ n ∈ (algebraMap K L).range\nthis : ExpCha... | obtain ⟨r, hr⟩ := hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Etale.Field | {
"line": 148,
"column": 4
} | {
"line": 151,
"column": 7
} | [
{
"pp": "case refine_4\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.... | intro x y
obtain ⟨α, hx, hy⟩ := H x y
simp only [← hg₃ _ _ hx, ← hg₃ _ _ hy, ← map_add, ← hg₃ _ _ (add_mem hx hy)]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.Field | {
"line": 148,
"column": 4
} | {
"line": 151,
"column": 7
} | [
{
"pp": "case refine_4\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.... | intro x y
obtain ⟨α, hx, hy⟩ := H x y
simp only [← hg₃ _ _ hx, ← hg₃ _ _ hy, ← map_add, ← hg₃ _ _ (add_mem hx hy)]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.AsTensorProduct | {
"line": 61,
"column": 8
} | {
"line": 62,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nx y : AdicCompletion I R\n⊢ ↑R ((LinearMap.lsmul (AdicCompletion I R) (AdicCompletion I M)) (x + y)) ∘ₗ of I M =\n ↑R ((LinearMap.lsm... | apply LinearMap.ext
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AdicCompletion.AsTensorProduct | {
"line": 61,
"column": 8
} | {
"line": 62,
"column": 12
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nx y : AdicCompletion I R\n⊢ ↑R ((LinearMap.lsmul (AdicCompletion I R) (AdicCompletion I M)) (x + y)) ∘ₗ of I M =\n ↑R ((LinearMap.lsm... | apply LinearMap.ext
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 90
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst✝⁴ : IsLocalHom (al... | let ePp : Pp ≃ₐ[P] P ⊗[R] 𝓀[R] := { __ := TensorProduct.comm _ _ _, commutes' _ := rfl } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 632,
"column": 2
} | {
"line": 633,
"column": 19
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\ni j : σ\n⊢ P.basisDeriv i j = (aeval P.val) ((pderiv (P.map j)) (P.relation i))",
"usedConstants": [
"Derivation",
"Pi.Fu... | classical
simp [basisDeriv] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 632,
"column": 2
} | {
"line": 633,
"column": 19
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\ni j : σ\n⊢ P.basisDeriv i j = (aeval P.val) ((pderiv (P.map j)) (P.relation i))",
"usedConstants": [
"Derivation",
"Pi.Fu... | classical
simp [basisDeriv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 632,
"column": 2
} | {
"line": 633,
"column": 19
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\ni j : σ\n⊢ P.basisDeriv i j = (aeval P.val) ((pderiv (P.map j)) (P.relation i))",
"usedConstants": [
"Derivation",
"Pi.Fu... | classical
simp [basisDeriv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 48,
"column": 14
} | {
"line": 48,
"column": 22
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Fun... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 48,
"column": 14
} | {
"line": 48,
"column": 22
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 48,
"column": 14
} | {
"line": 48,
"column": 22
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 21
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Func... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 21
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Func... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 21
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Func... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 12
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Func... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 50,
"column": 14
} | {
"line": 50,
"column": 22
} | [
{
"pp": "case smul\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Fun... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 50,
"column": 14
} | {
"line": 50,
"column": 22
} | [
{
"pp": "case smul\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Basis.Exact | {
"line": 50,
"column": 14
} | {
"line": 50,
"column": 22
} | [
{
"pp": "case smul\nR : Type u_1\nM : Type u_2\nK : Type u_3\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup K\ninst✝³ : AddCommGroup P\ninst✝² : Module R M\ninst✝¹ : Module R K\ninst✝ : Module R P\nf : K →ₗ[R] M\ng : M →ₗ[R] P\ns : M →ₗ[R] K\nhs : s ∘ₗ f = LinearMap.id\nhfg : Fun... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 70,
"column": 38
} | {
"line": 70,
"column": 51
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : PreSubmersivePresentation R S ι σ\nx : ↥P.ker\n⊢ (∀ (x_1 : σ), (aeval P.val) ((pderiv (P.map x_1)) ↑x) = 0 x_1) ↔ ∀ (i : σ), (aeval P.val) ((pderiv (P.map i)) ↑x... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Extension.Cotangent.Free | {
"line": 72,
"column": 4
} | {
"line": 79,
"column": 54
} | [
{
"pp": "case hker\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_3\nσ : Type u_4\nκ : Type u_5\nP : Generators R S ι\nu : σ → ι\nhu : Function.Injective u\nv : κ → ι\nhuv : IsCompl (Set.range v) (Set.range u)\nhm : Submodule.span S (Set.range fun i ↦ (D ... | simp only [disjoint_iff, g]
apply Submodule.map_injective_of_injective (f := P.cotangentSpaceBasis.repr.symm.toLinearMap)
P.cotangentSpaceBasis.repr.symm.injective
rw [Submodule.map_inf P.cotangentSpaceBasis.repr.symm.toLinearMap
P.cotangentSpaceBasis.repr.symm.injective, Submodule.map_span, ← Set... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Cotangent.Free | {
"line": 72,
"column": 4
} | {
"line": 79,
"column": 54
} | [
{
"pp": "case hker\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_3\nσ : Type u_4\nκ : Type u_5\nP : Generators R S ι\nu : σ → ι\nhu : Function.Injective u\nv : κ → ι\nhuv : IsCompl (Set.range v) (Set.range u)\nhm : Submodule.span S (Set.range fun i ↦ (D ... | simp only [disjoint_iff, g]
apply Submodule.map_injective_of_injective (f := P.cotangentSpaceBasis.repr.symm.toLinearMap)
P.cotangentSpaceBasis.repr.symm.injective
rw [Submodule.map_inf P.cotangentSpaceBasis.repr.symm.toLinearMap
P.cotangentSpaceBasis.repr.symm.injective, Submodule.map_span, ← Set... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Cotangent.Free | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 19
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nι : Type u_3\nκ : Type u_5\nP : Generators R S ι\nv : κ → ι\nh : LinearIndependent S fun k ↦ (D R S) (P.val (v k))\n⊢ ∀ x ∈ P.toExtension.toKaehler.ker ⊓ Submodule.span S (Set.range fun x ↦ P.cotangentSpaceBasis ... | intro x ⟨hx, hxs⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 72
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ IsStandardSmoothOfRelativeDimension n (algebraMap R S) ↔ Algebra.IsStandardSmoothOfRelativeDimension n R S",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"congrArg",
"CommSe... | rw [RingHom.IsStandardSmoothOfRelativeDimension, toAlgebra_algebraMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 72
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ IsStandardSmoothOfRelativeDimension n (algebraMap R S) ↔ Algebra.IsStandardSmoothOfRelativeDimension n R S",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"congrArg",
"CommSe... | rw [RingHom.IsStandardSmoothOfRelativeDimension, toAlgebra_algebraMap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 72
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ IsStandardSmoothOfRelativeDimension n (algebraMap R S) ↔ Algebra.IsStandardSmoothOfRelativeDimension n R S",
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"congrArg",
"CommSe... | rw [RingHom.IsStandardSmoothOfRelativeDimension, toAlgebra_algebraMap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nD : DescentAux A B\n⊢ D.P.coeffs ⊆ Set.range ⇑(algebraMap (↥(subalgebra R D)) A)",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | rw [Subalgebra.setRange_algebraMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nA : Type u\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nD : DescentAux A B\ni : D.vars\nthis : ↑(D.h i).coeffs ⊆ ⋃ i, ↑(D.h i).coeffs\n⊢ ↑(D.h i).coeffs ⊆ Set.range ⇑(algebraMap (↥(subalgebra R D)) A)",
"usedC... | rw [Subalgebra.setRange_algebraMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 169,
"column": 79
} | {
"line": 177,
"column": 19
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\n⊢ S ⊗[D.T] D.presLeft.toExtension.Cotangent ≃ₗ[S] P.toExtension.Cotangent",
"usedCon... | by
refine LinearEquiv.ofLinear D.tensorCotangentHom D.tensorCotangentInv ?_ ?_
· refine b.ext fun i ↦ ?_
simpa only [LinearMap.coe_comp, Function.comp_apply, tensorCotangentInv_apply,
tensorCotangentHom_tmul] using D.hf (b i)
· ext : 2
refine LinearMap.ext_on_range D.span_range_mk_kerGen fun i ↦ ?_
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 57
} | [
{
"pp": "case h.e_6.h.e_val\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ninst✝ : Nontrivial S\nthis : Nontrivial (MvPolynomial ι R ⧸ Ideal.span (Set.range ... | exact Generators.toAlgHom_ofComp_localizationAway _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 57
} | [
{
"pp": "case h.e_6.h.e_val\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ninst✝ : Nontrivial S\nthis : Nontrivial (MvPolynomial ι R ⧸ Ideal.span (Set.range ... | exact Generators.toAlgHom_ofComp_localizationAway _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 57
} | [
{
"pp": "case h.e_6.h.e_val\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ninst✝ : Nontrivial S\nthis : Nontrivial (MvPolynomial ι R ⧸ Ideal.span (Set.range ... | exact Generators.toAlgHom_ofComp_localizationAway _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 248,
"column": 71
} | {
"line": 249,
"column": 14
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ninst✝ : Nontrivial S\ni : σ\n⊢ D.basis (Sum.inr i) = D.cotangentEquivProd.symm (0, D.basisLeft i)",
... | by
simp [basis] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 261,
"column": 17
} | {
"line": 261,
"column": 40
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\ninst✝ : Nontrivial S\nr : σ\n⊢ (LinearMap.liftBaseChange S\n (Extension.Cotangent.map... | Module.Basis.map_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 37
} | [
{
"pp": "case inr\nk : Type u_1\nK : Type u_2\ninst✝⁴ : Field k\ninst✝³ : Field K\ninst✝² : Algebra k K\ninst✝¹ : PerfectField k\ninst✝ : Algebra.EssFiniteType k K\np : ℕ\nhp : Fact (Nat.Prime p)\nhpk : CharP k p\n⊢ ∃ s, IsTranscendenceBasis k Subtype.val ∧ Algebra.IsSeparable (↥(adjoin k ↑s)) K",
"usedCons... | have : ExpChar k p := .prime hp.out | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.MorphismProperty.Descent | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 66
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\nP : MorphismProperty C\ninst✝³ : (isomorphisms C).DescendsAlong P\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : HasEqualizers C\ninst✝ : HasPullbacks C\nX Y S T : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nv : T ⟶ S\nhv : P v\nH :... | suffices IsIso (equalizer.ι f g) from Limits.eq_of_epi_equalizer | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.Flat.FaithfullyFlat.Descent | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : FaithfullyFlat R S\nH : Function.Injective ⇑(algebraMap S (S ⊗[R] T))\nthis :\n LinearMap.lTensor S (Algebra.linearMap R T) = ↑R (Algebra.linearM... | simpa [this] using H | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Flat.FaithfullyFlat.Descent | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nT : Type u_3\ninst✝² : CommRing T\ninst✝¹ : Algebra R T\ninst✝ : FaithfullyFlat R S\nH : Function.Surjective ⇑(algebraMap S (S ⊗[R] T))\nthis :\n LinearMap.lTensor S (Algebra.linearMap R T) = ↑R (Algebra.linear... | simpa [this] using H | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 30
} | [
{
"pp": "case H\nX Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : HasPullback f g\nhf : (@Surjective ⊓ @Flat ⊓ @QuasiCompact) f\nhg : IsOpenImmersion (pullback.fst f g)\nthis : UniversallyOpen g\nU : Z.Opens := { carrier := Set.range ⇑g, is_open' := ⋯ }\n⊢ IsOpenImmersion g",
"usedConstants": [
"Category... | let f' := pullback.snd f U.ι | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 84,
"column": 36
} | {
"line": 84,
"column": 44
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : ¬y = 0\nhy : ¬v y = 1\nhlt : 1 < v y\nthis✝ : IsAlgebraic A y\np : A[X]\nhpnt : p ≠ 0\nhp : (aeval ((alge... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 84,
"column": 36
} | {
"line": 84,
"column": 44
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : ¬y = 0\nhy : ¬v y = 1\nhlt : 1 < v y\nthis✝ : IsAlgebraic A y\np : A[X]\nhpnt : p ≠ 0\nhp : (aeval ((alge... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 84,
"column": 36
} | {
"line": 84,
"column": 44
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : ¬y = 0\nhy : ¬v y = 1\nhlt : 1 < v y\nthis✝ : IsAlgebraic A y\np : A[X]\nhpnt : p ≠ 0\nhp : (aeval ((alge... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 85,
"column": 86
} | {
"line": 85,
"column": 94
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : ¬y = 0\nhy : ¬v y = 1\nhlt : 1 < v y\nthis : IsAlgebraic A y\np : A[X]\nhpnt : p ≠ 0\nhp : (aeval ((algeb... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 85,
"column": 86
} | {
"line": 85,
"column": 94
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : ¬y = 0\nhy : ¬v y = 1\nhlt : 1 < v y\nthis : IsAlgebraic A y\np : A[X]\nhpnt : p ≠ 0\nhp : (aeval ((algeb... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 89,
"column": 49
} | {
"line": 89,
"column": 57
} | [
{
"pp": "case add\nK : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np✝ q✝ : K[X]\na✝¹ : (Polynomial.aeval X) p✝ = (algebraMap K[X] K⟮X⟯) p✝\na✝ : (Polynomial.aeval X) q✝ = (algebraMap K[X] K⟮X⟯) q✝\n⊢ (Polynomial.aeval X) (p✝ + q✝) = (algebraMap K[X] K⟮X⟯) (p✝ + q✝)",
"usedConstants": [
"NonAssocS... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 89,
"column": 49
} | {
"line": 89,
"column": 57
} | [
{
"pp": "case monomial\nK : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nn✝ : ℕ\na✝ : K\n⊢ (Polynomial.aeval X) ((Polynomial.monomial n✝) a✝) = (algebraMap K[X] K⟮X⟯) ((Polynomial.monomial n✝) a✝)",
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"Algebra.algebraMap",
"congr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 185,
"column": 94
} | {
"line": 195,
"column": 21
} | [
{
"pp": "K : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx y : K⟮X⟯\nhx : Polynomial.eval₂ f a x.denom ≠ 0\nhy : Polynomial.eval₂ f a y.denom ≠ 0\n⊢ eval f a (x + y) = eval f a x + eval f a y",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"D... | by
unfold eval
by_cases hxy : Polynomial.eval₂ f a (denom (x + y)) = 0
· have := Polynomial.eval₂_eq_zero_of_dvd_of_eval₂_eq_zero f a (denom_add_dvd x y) hxy
rw [Polynomial.eval₂_mul] at this
cases mul_eq_zero.mp this <;> contradiction
rw [div_add_div _ _ hx hy, eq_div_iff (mul_ne_zero hx hy), div_eq_mu... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 49
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhpmon : IsIntegral R p.leadingCoeff\nhp : p.Splits\nhpr : ∀ (x : S), p.IsRoot x → IsIntegral R x\ni : ℕ\nm : Multiset S\nhm : p = C p.leadingCoeff * (Multiset.map (fun x ↦ X - C x) m).prod\n⊢ IsIntegral... | rw [hm, Multiset.prod_eq_prod_coe, coeff_C_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 72,
"column": 81
} | {
"line": 72,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R[X]\nq : S[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ map (algebraMap R S) p\ni : ℕ\na✝ : Nontrivial S\nT : Type u_2\nw✝⁴ : CommRing T\nw✝³ : Algebra S T\nw✝² : Module.Finite S T\nw✝¹ : Module.Free S T\nw✝ : No... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 72,
"column": 81
} | {
"line": 72,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R[X]\nq : S[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ map (algebraMap R S) p\ni : ℕ\na✝ : Nontrivial S\nT : Type u_2\nw✝⁴ : CommRing T\nw✝³ : Algebra S T\nw✝² : Module.Finite S T\nw✝¹ : Module.Free S T\nw✝ : No... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 72,
"column": 81
} | {
"line": 72,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R[X]\nq : S[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ map (algebraMap R S) p\ni : ℕ\na✝ : Nontrivial S\nT : Type u_2\nw✝⁴ : CommRing T\nw✝³ : Algebra S T\nw✝² : Module.Finite S T\nw✝¹ : Module.Free S T\nw✝ : No... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 12
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\nΓ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\nhv : IsTrivialOn K v\nhle : v RatFunc.X ≤ 1\np : K[X]\ni : ℕ\nh0 : p.coeff i = 0\n⊢ i ∈ Finset.range (p.natDegree + 1) → v ((algebraMap K[X] K⟮X⟯) ((monomial i) (p.coeff i))) ≤ 1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 12
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\nΓ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\nhv : IsTrivialOn K v\nhle : v RatFunc.X ≤ 1\np : K[X]\ni : ℕ\nh0 : p.coeff i = 0\n⊢ i ∈ Finset.range (p.natDegree + 1) → v ((algebraMap K[X] K⟮X⟯) ((monomial i) (p.coeff i))) ≤ 1",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 12
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\nΓ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\nhv : IsTrivialOn K v\nhle : v RatFunc.X ≤ 1\np : K[X]\ni : ℕ\nh0 : p.coeff i = 0\n⊢ i ∈ Finset.range (p.natDegree + 1) → v ((algebraMap K[X] K⟮X⟯) ((monomial i) (p.coeff i))) ≤ 1",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 85
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : S[X]\nhp : IsIntegral R[X] p\ni✝ : ℕ\na✝¹ : Nontrivial R\na✝ : Nontrivial S\nhp0 : p ≠ 0\nq : R[X][X] := minpoly R[X] p\nm : ℕ := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1\ni :... | · exact (Finset.le_sup (f := fun i ↦ (q.coeff i).natDegree) hi).trans_lt (by lia) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 214,
"column": 4
} | {
"line": 215,
"column": 56
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\nhf : ∀ (x : ↥Y), LocallyQuasiFinite (Hom.fiberToSpecResidueField f x)\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y),\n (∀ (x : ↥Y), LocallyQuasiFinite (Hom.fiberToSpecResidueField f x)) → (∃ R, Y = Spec R) → id (LocallyQuasiFinite f)\nhY : ¬∃ R, Y = Spec R\n⊢ id (Locally... | refine (IsZariskiLocalAtTarget.iff_of_openCover Y.affineCover).mpr fun i ↦
this (f := pullback.snd _ _) (fun x ↦ ?_) ⟨_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 276,
"column": 4
} | {
"line": 277,
"column": 56
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\nH : ∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})\nthis :\n ∀ {X Y : Scheme} {f : X ⟶ Y} [LocallyOfFiniteType f],\n (∀ (x : ↥Y), IsDiscrete (⇑f ⁻¹' {x})) → (∃ R, Y = Spec R) → id (LocallyQuasiFinite f)\nhY : ¬∃ R, Y = Spec R\n⊢ id (LocallyQuas... | refine (IsZariskiLocalAtTarget.iff_of_openCover Y.affineCover).mpr fun i ↦
this (f := pullback.snd _ _) (fun x ↦ ?_) ⟨_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 300,
"column": 4
} | {
"line": 301,
"column": 56
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\nhf : ∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) [LocallyOfFiniteType f],\n (∀ (x : ↥Y), (⇑f ⁻¹' {x}).Finite) → (∃ R, Y = Spec R) → id (LocallyQuasiFinite f)\nhY : ¬∃ R, Y = Spec R\n⊢ id (LocallyQuasiFinite... | refine (IsZariskiLocalAtTarget.iff_of_openCover Y.affineCover).mpr fun i ↦
this (f := pullback.snd _ _) (fun x ↦ ?_) ⟨_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 38
} | [
{
"pp": "X : Scheme\n⊢ { I₀ := X.AffineZariskiSite, X := fun U ↦ ↑↑U, f := fun U ↦ (↑U).ι }.presieve₀ ∈\n (precoverage IsOpenImmersion).coverings X",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.mk",
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.PreZeroHypercover.f... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 161,
"column": 46
} | {
"line": 161,
"column": 65
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\nhp : p = 0\n⊢ { toFractionRing := 0 } = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"FractionRing",
"nonZeroDivisors",
"RatFun... | ofFractionRing_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 88
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nf g : P.Ring →ₐ[R] S\nH✝ : f P.X = g P.X\nH :\n (f.comp (Ideal.Quotient.mkₐ R (Ideal.span {C P.f, Y * C P.g - 1}))).comp CAlgHom =\n (g.comp (Ideal.Quotient.mkₐ R (Ideal.span {C P.f, ... | change f.toMonoidHom (Ideal.Quotient.mk _ .X) = g.toMonoidHom (Ideal.Quotient.mk _ .X) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 22
} | [
{
"pp": "R : Type u_1\nB : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nσ : Type u_4\nx : MvPolynomial σ R ⊗[R] B\nhx : x ∈ integralClosure (MvPolynomial σ R) (MvPolynomial σ R ⊗[R] B)\ne₀ : MvPolynomial σ R ⊗[R] B ≃ₐ[R] MvPolynomial σ B := MvPolynomial.scalarRTensorAlgEquiv\ne : MvP... | exact congr($this y) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 33
} | [
{
"pp": "case neg.a\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : 0 < f.natDegree\nhn : ¬f.natDegree = 0\ni : ℕ\nhi : i < f.natDegree - 1 + f.natDegree\nj : ℕ\nhj : j < f.natDegree - 1 + f.natDegree\n⊢ f.sylvesterDeriv.updateRow ⟨2 * f.natDegree - 2, ⋯⟩ (f.leadingCoeff • f.sylvesterDeriv ⟨2 * f.natDegree - ... | rw [sylvesterDeriv, dif_neg hn] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 46
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np r : R[X]\nht : φ.IsIntegralElem t\nhpm : p.Monic\nhpr : r.natDegree < p.natDegree\nhp : φ p * t = φ r\nSt : Type u_2 := Localization.Away t\n⊢ IsIntegral R t",
"usedConstants": [
... | let t' : St := IsLocalization.Away.invSelf t | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 199,
"column": 25
} | {
"line": 200,
"column": 71
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nr : R\nm n k : ℕ\nhk : k + m ≤ n\nhf : f.natDegree ≤ m\nhm : m ≠ 0\nM₁ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester (g + f * (monomial k) r) m n\nM₂ : Matrix (Fin (m + n)) (Fin (m + n)) R := f.sylvester g m n\nM : ℕ → Matrix (Fin (m + n)... | ← Matrix.det_updateCol_add_smul_self (i := ⟨i, by lia⟩)
(j := ⟨i + k + m, by lia⟩) (c := -r) (M (i + 1)) (by simp; lia) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 73
} | [
{
"pp": "case inr.h.zero.inr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\ni : ℕ\nthis : ∀ (p : R[X]), φ p * t ∈ (conductor R (φ X)).radical → p.leadingCoeff • t ∈ (conductor R (φ X)).radical\np ... | · simpa [← coeff_natDegree, hpn, show i = 0 by lia] using this _ hp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 42
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np✝ : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\np : R[X]\ni : ℕ\nhi : i = p.natDegree\nn : ℕ\nhn : (φ p * t) ^ n ∈ conductor R (φ X)\nk : ℕ\nhk : (p ^ n).leadingCoeff ^... | use n; simp [_root_.smul_pow, hpn, hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 42
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np✝ : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\np : R[X]\ni : ℕ\nhi : i = p.natDegree\nn : ℕ\nhn : (φ p * t) ^ n ∈ conductor R (φ X)\nk : ℕ\nhk : (p ^ n).leadingCoeff ^... | use n; simp [_root_.smul_pow, hpn, hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 409,
"column": 2
} | {
"line": 413,
"column": 10
} | [
{
"pp": "case inr\nK : Type u_3\ninst✝ : Field K\nf g : K[X]\nhf : f.Monic\nhg : g.Monic\nhf' : f.Splits\nhg' : g.Splits\nthis :\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree → f.resultant g = (... | · trans ((f.roots ×ˢ g.roots).map fun ij ↦ (-1) * (ij.2 - ij.1)).prod
· rw [resultant_comm, this g f hg hf hg' hf' (le_of_not_ge hfg), ← Multiset.map_swap_product,
Multiset.map_map, Multiset.prod_map_mul]
simp [hf'.natDegree_eq_card_roots, hg'.natDegree_eq_card_roots]
· simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 598,
"column": 8
} | {
"line": 598,
"column": 27
} | [
{
"pp": "case refine_1\nn : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRi... | letI := φ.toAlgebra | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 812,
"column": 30
} | {
"line": 812,
"column": 38
} | [
{
"pp": "case pos\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhf' : f = 0\n⊢ (f * ↑(⟨p, hp⟩, ⟨q, hq⟩).2).degree < ↑(m + n)",
"usedConstants": [
"WithBot.addMonoidWithOne",
"Polynomial.de... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 812,
"column": 30
} | {
"line": 812,
"column": 38
} | [
{
"pp": "case pos\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhf' : f = 0\n⊢ (f * ↑(⟨p, hp⟩, ⟨q, hq⟩).2).degree < ↑(m + n)",
"usedConstants": [
"WithBot.addMonoidWithOne",
"Polynomial.de... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 812,
"column": 30
} | {
"line": 812,
"column": 38
} | [
{
"pp": "case pos\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhf' : f = 0\n⊢ (f * ↑(⟨p, hp⟩, ⟨q, hq⟩).2).degree < ↑(m + n)",
"usedConstants": [
"WithBot.addMonoidWithOne",
"Polynomial.de... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 815,
"column": 30
} | {
"line": 815,
"column": 38
} | [
{
"pp": "case pos\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhg' : g = 0\n⊢ (g * ↑(⟨p, hp⟩, ⟨q, hq⟩).1).degree < ↑(m + n)",
"usedConstants": [
"WithBot.addMonoidWithOne",
"Polynomial.de... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
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