module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 51
} | {
"line": 190,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\nx : ModelOfHasCoeff... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\nx : MvPolynomial ι R₀\n⊢ ((AlgH... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 65
} | {
"line": 242,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : Finite σ\nP : SubmersivePresentation R S ι σ\ninst✝¹ : DecidableEq σ\ninst✝ : Fintype σ\n⊢ P.jacobiMatrix.det * P.σ ↑⋯.unit⁻¹ - 1 ∈ P.ker",
"ppTerm": "?m.126",
"assig... | [] | simp [PreSubmersivePresentation.jacobian_eq_jacobiMatrix_det] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 50
} | {
"line": 368,
"column": 2
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ W'.negDblY P = -P y ^ 4",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"WeierstrassCurve.Projective.... | [
"R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Equation P\nhPz : P z = 0\n⊢ ⋯ - ⋯ - ⋯ * 0 ^ 4 + ⋯ * P y ^ 2 * 0 + ⋯ * 0 * P y * 0 ^ 2 + 9 * W'.a₂ ^ 2 * 0 ^ 4 - 8 * W'.a₂ ^ 2 * 0 * P y ^ 2 * 0 -\n ... | rw [negDblY, hPz, X_eq_zero_of_Z_eq_zero hP hPz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 572,
"column": 80
} | {
"line": 575,
"column": 74
} | {
"line": 577,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nhP : W'.Equation P\nhQ : W'.Equation Q\n⊢ W'.addX P Q * (P z * Q z) ^ 2 =\n ((P y * Q z - Q y * P z) ^ 2 * P z * Q z + W'.a₁ * (P y * Q z - Q y * P z) * P z * Q z * (P x * Q z - Q x * P z) -\n W'.a₂ * P z * Q z * (P x * Q... | [] | by
linear_combination (norm := (rw [addX]; ring1))
(2 * Q x * P z * Q z ^ 3 - P x * Q z ^ 4) * (equation_iff _).mp hP
+ (Q x * P z ^ 4 - 2 * P x * P z ^ 3 * Q z) * (equation_iff _).mp hQ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 176,
"column": 72
} | {
"line": 178,
"column": 75
} | {
"line": 180,
"column": 0
} | [
{
"pp": "⊢ HoldsForLocalizationAway fun {R S} [CommRing R] [CommRing S] ↦ IsStandardSmoothOfRelativeDimension 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"CommRing",
"CommSemiring.toSemiring",
"IsLocalization.Away",
"Algebra",
"CommRing.toCommSemiring",
... | [] | by
introv R h
exact IsStandardSmoothOfRelativeDimension.algebraMap_isLocalizationAway r | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 91,
"column": 4
} | {
"line": 94,
"column": 57
} | {
"line": 95,
"column": 2
} | [
{
"pp": "case refine_1\nR : 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⊢ Function.Surjective ⇑(algebraMap D.T S)",
"ppTerm": "?refine_1",
... | [] | refine .of_comp (g := algebraMap P.Ring D.T) ?_
convert! P.algebraMap_surjective
ext x
exact (IsScalarTower.algebraMap_apply _ D.T S x).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 91,
"column": 4
} | {
"line": 94,
"column": 57
} | {
"line": 95,
"column": 2
} | [
{
"pp": "case refine_1\nR : 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⊢ Function.Surjective ⇑(algebraMap D.T S)",
"ppTerm": "?refine_1",
... | [] | refine .of_comp (g := algebraMap P.Ring D.T) ?_
convert! P.algebraMap_surjective
ext x
exact (IsScalarTower.algebraMap_apply _ D.T S x).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 227,
"column": 6
} | {
"line": 227,
"column": 21
} | {
"line": 228,
"column": 4
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝¹ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.Submers... | [] | exact congr($H) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 60
} | {
"line": 162,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : 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\n⊢ Submodule.span D.T (Set.range fun i ↦ Extension.Cotangent.mk (D.kerGen i)) = ⊤",
"ppTerm": "?m.68... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : 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\n⊢ Ideal.span (Set.range fun i ↦ ↑(D.f (b i))) = D.presLeft.toExtension.ker"
] | refine Extension.Cotangent.span_eq_top_of_span_eq_ker _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 260,
"column": 8
} | {
"line": 260,
"column": 18
} | {
"line": 260,
"column": 19
} | [
{
"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⊢ D.basis (Sum.inr r) = Extension.Cotangent.mk ⟨D.pres.relation... | [
"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⊢ D.cotangentEquivProd.symm (0, D.basisLeft r) = Extension.Cotangent.mk ⟨D.... | basis_inr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 45
} | {
"line": 95,
"column": 0
} | [
{
"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 ⊆ ↑(subalgebra R D)",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"_private.Mathlib.RingTheory... | [] | grind [subalgebra, Algebra.subset_adjoin] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 43
} | {
"line": 112,
"column": 0
} | [
{
"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 ⊆ ↑(subalgebra R D)",
"ppTerm": "?m.73",
"assigned":... | [] | grind [subalgebra, Algebra.subset_adjoin] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 43
} | {
"line": 128,
"column": 0
} | [
{
"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.rels\np : MvPolynomial D.vars A\nhp : p ∈ ↑(D.p i).coeffs\nthis : ↑p.coeffs ⊆ ⋃ i, ⋃ x ∈ (D.p i).coeffs, ↑x.coeffs\n⊢ ↑p.coeffs ⊆ ↑... | [] | grind [subalgebra, Algebra.subset_adjoin] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 43
} | {
"line": 144,
"column": 0
} | [
{
"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\nq : MvPolynomial D.vars A\nhq : q ∈ ↑(D.q i).coeffs\nthis : ↑q.coeffs ⊆ ⋃ i, ⋃ x ∈ (D.q i).coeffs, ↑(coeffs x)\n⊢ ↑q.coeffs ⊆... | [] | grind [subalgebra, Algebra.subset_adjoin] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 700,
"column": 2
} | {
"line": 701,
"column": 57
} | {
"line": 703,
"column": 0
} | [
{
"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\n⊢ W.negAddY P Q / W.addZ P Q =\n W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z)\n (W.toAffine.slope (P ... | [] | rw [negAddY_eq hP hQ hPz hQz, addZ_eq hP hQ hPz hQz, toAffine_slope_of_ne hPz hQz hx,
toAffine_negAddY_of_ne hPz hQz <| sub_ne_zero.mpr hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 700,
"column": 2
} | {
"line": 701,
"column": 57
} | {
"line": 703,
"column": 0
} | [
{
"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\n⊢ W.negAddY P Q / W.addZ P Q =\n W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z)\n (W.toAffine.slope (P ... | [] | rw [negAddY_eq hP hQ hPz hQz, addZ_eq hP hQ hPz hQz, toAffine_slope_of_ne hPz hQz hx,
toAffine_negAddY_of_ne hPz hQz <| sub_ne_zero.mpr hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 700,
"column": 2
} | {
"line": 701,
"column": 57
} | {
"line": 703,
"column": 0
} | [
{
"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\n⊢ W.negAddY P Q / W.addZ P Q =\n W.toAffine.negAddY (P x / P z) (Q x / Q z) (P y / P z)\n (W.toAffine.slope (P ... | [] | rw [negAddY_eq hP hQ hPz hQz, addZ_eq hP hQ hPz hQz, toAffine_slope_of_ne hPz hQz hx,
toAffine_negAddY_of_ne hPz hQz <| sub_ne_zero.mpr hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 893,
"column": 25
} | {
"line": 893,
"column": 36
} | {
"line": 893,
"column": 37
} | [
{
"pp": "R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Projective R\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... | [
"R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Projective R\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✝ : IsScalarTower R... | ← map_addX, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.LocalClosure | {
"line": 98,
"column": 57
} | {
"line": 102,
"column": 53
} | {
"line": 104,
"column": 0
} | [
{
"pp": "W P Q : MorphismProperty Scheme\nX Y : Scheme\ninst✝⁴ : W.IsStableUnderBaseChange\ninst✝³ : Scheme.IsJointlySurjectivePreserving W\ninst✝² : W.IsStableUnderComposition\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.IsStableUnderComposition\n⊢ (sourceLocalClosure W P).IsStableUnderComposition",
"ppT... | [] | by
refine ⟨fun {X Y Z} f g ⟨𝒰, hf⟩ ⟨𝒱, hg⟩ ↦ ?_⟩
refine ⟨𝒰.bind fun i ↦ (𝒱.pullback₁ (𝒰.f i ≫ f)), fun ⟨l, r⟩ ↦ ?_⟩
simpa [← pullbackRightPullbackFstIso_inv_snd_fst_assoc, pullback.condition_assoc] using
P.comp_mem _ _ (P.pullback_snd _ _ (hf _)) (hg r) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 251,
"column": 2
} | {
"line": 252,
"column": 70
} | {
"line": 253,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [
"k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn : n ∉ s\ne₁ :... | obtain ⟨i, hi, hi'⟩ := exists_isTranscendenceBasis_and_isSeparable_of_linearIndepOn_pow p hp H
(a := fun i : ↥(insert n s) ↦ a i) ⟨n, by simp⟩ (ha.comp_equiv e₁) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 66
} | {
"line": 57,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ns : S\nH : IsIntegral R s\nt : R\nht : t ∈ R⁰\nht' : t • s ∈ (algebraMap R S).range\ni : ℕ\nhi : i < (minpoly R s).natDegree\n⊢ t ^ (minpoly R s).natDegree • s ^ i ∈ (algebraMap R S).range",
"ppTerm": "?m.109... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ns : S\nH : IsIntegral R s\nt : R\nht : t ∈ R⁰\nht' : t • s ∈ (algebraMap R S).range\ni : ℕ\nhi : i < (minpoly R s).natDegree\n⊢ t ^ ((minpoly R s).natDegree - i) • (t • s) ^ i ∈ (algebraMap R S).range"
] | rw [← Nat.sub_add_cancel hi.le, pow_add, mul_smul, ← smul_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 81
} | {
"line": 74,
"column": 0
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\n⊢ Function.Injective (⇑(algebraMap K[X] K⟮X⟯) ∘ ⇑Polynomial.C)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Polynomial.C",
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"RingHom",
"Algebra.id... | [] | exact Function.Injective.comp (algebraMap_injective K) (Polynomial.C_injective) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 122,
"column": 78
} | {
"line": 156,
"column": 60
} | {
"line": 158,
"column": 0
} | [
{
"pp": "R : 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\n⊢ IsIntegral R (p.coeff i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Polynomial.taylor_eval",
"IsRightCancelAdd.addRightStrictMono... | [] | by
nontriviality R
nontriviality S
obtain rfl | hp0 := eq_or_ne p 0; · simp [isIntegral_zero]
let q := minpoly R[X] p
let m := (q.support.sup fun i ↦ (q.coeff i).natDegree) + p.natDegree + 1
have hm₁ (i) : (q.coeff i).natDegree < m := by
by_cases hi : i ∈ q.support
· exact (Finset.le_sup (f := fun i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 150,
"column": 5
} | {
"line": 150,
"column": 45
} | {
"line": 150,
"column": 45
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP Q : Ideal S\ninst✝² : P.IsPrime\ninst✝¹ : Q.IsPrime\nh₁ : P ≤ Q\nh₂ : Ideal.under R P = Ideal.under R Q\ninst✝ : WeaklyQuasiFiniteAt R Q\n⊢ RingHom.ker (Ideal.Quotient.mk (Ideal.map (algebraMap R S) (Ideal.und... | [] | by simpa [← h₂] using Ideal.map_comap_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 45,
"column": 15
} | {
"line": 45,
"column": 34
} | {
"line": 45,
"column": 34
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesO... | [
"case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nI : Ideal R\ninst✝⁴ : I.IsPrime\nJ : Ideal R[X]\ninst✝³ : J.IsPrime\ninst✝² : J.LiesOver I\ninst✝¹ : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝ : Localization.AtPrime.IsLiesOverAlgebra I... | Ideal.mem_map_C_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 260,
"column": 4
} | {
"line": 260,
"column": 53
} | {
"line": 261,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFiniteAt p.Residu... | obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 27
} | {
"line": 264,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\nins... | [
"case refine_2\nR : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFi... | · simpa [← hQ] using hs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 465,
"column": 2
} | {
"line": 465,
"column": 79
} | {
"line": 466,
"column": 2
} | [
{
"pp": "L : Type u_2\nR : Type u_3\ninst✝¹ : Field L\ninst✝ : CommRing R\nφ : R[X] →+* L\nhφ : R[X]⁰ ≤ Submonoid.comap φ L⁰\nx : R[X]\n⊢ (liftRingHom φ hφ) { toFractionRing := (algebraMap R[X] (FractionRing R[X])) x } = φ x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"L : Type u_2\nR : Type u_3\ninst✝¹ : Field L\ninst✝ : CommRing R\nφ : R[X] →+* L\nhφ : R[X]⁰ ≤ Submonoid.comap φ L⁰\nx : R[X]\n⊢ φ x / φ ↑1 = φ x"
] | rw [← Localization.mk_one_eq_algebraMap, liftRingHom_apply_ofFractionRing_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 172,
"column": 8
} | {
"line": 172,
"column": 24
} | {
"line": 172,
"column": 25
} | [
{
"pp": "case mp\nX : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nf₂ : ↑Γ(X, U.toOpens)\nhf₂s : f₂ ∈ s\nhW : IsAffineOpen (X.basicOpen f₂)\nf₁ : ↑Γ(X, toOpens ⟨X.basicOpen f₂, hW⟩)\nhV : IsAffineOpen (X.basicOpen f₁)\nf₃ : ↑Γ(X, ↑U)\nhf₃ : X.basicOpen f₃ = X.basicOpen f₁\n⊢ X.basicOpen (f₂ * f₃) ... | [
"case mp\nX : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nf₂ : ↑Γ(X, U.toOpens)\nhf₂s : f₂ ∈ s\nhW : IsAffineOpen (X.basicOpen f₂)\nf₁ : ↑Γ(X, toOpens ⟨X.basicOpen f₂, hW⟩)\nhV : IsAffineOpen (X.basicOpen f₁)\nf₃ : ↑Γ(X, ↑U)\nhf₃ : X.basicOpen f₃ = X.basicOpen f₁\n⊢ X.basicOpen f₂ ⊓ X.basicOpen f₃ = ... | X.basicOpen_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 57
} | {
"line": 254,
"column": 2
} | [
{
"pp": "case refine_2\nR S : CommRingCat\nφ : R ⟶ S\nhf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)\nalgInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra\nx : ↥(Spec R)\nhP : x.asIdeal.IsPrime\n⊢ (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).right ≅\n... | [] | exact asIso (Spec.map (Spec.residueFieldIso _ x).inv) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 57
} | {
"line": 254,
"column": 2
} | [
{
"pp": "case refine_2\nR S : CommRingCat\nφ : R ⟶ S\nhf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)\nalgInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra\nx : ↥(Spec R)\nhP : x.asIdeal.IsPrime\n⊢ (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).right ≅\n... | [] | exact asIso (Spec.map (Spec.residueFieldIso _ x).inv) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 57
} | {
"line": 254,
"column": 2
} | [
{
"pp": "case refine_2\nR S : CommRingCat\nφ : R ⟶ S\nhf : ∀ (x : ↥(Spec R)), LocallyQuasiFinite (Hom.fiberToSpecResidueField (Spec.map φ) x)\nalgInst✝ : Algebra ↑R ↑S := (CommRingCat.Hom.hom φ).toAlgebra\nx : ↥(Spec R)\nhP : x.asIdeal.IsPrime\n⊢ (Arrow.mk (Hom.fiberToSpecResidueField (Spec.map φ) x)).right ≅\n... | [] | exact asIso (Spec.map (Spec.residueFieldIso _ x).inv) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 30
} | {
"line": 90,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\nf : S →ₐ[R] T\n⊢ (aeval (f x)) P.f = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": ... | [] | simp [aeval_algHom, h.1] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 30
} | {
"line": 90,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\nf : S →ₐ[R] T\n⊢ (aeval (f x)) P.f = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": ... | [] | simp [aeval_algHom, h.1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 30
} | {
"line": 90,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP : StandardEtalePair R\nx : S\nh : P.HasMap x\nf : S →ₐ[R] T\n⊢ (aeval (f x)) P.f = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": ... | [] | simp [aeval_algHom, h.1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiFinite | {
"line": 363,
"column": 2
} | {
"line": 365,
"column": 70
} | {
"line": 366,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf✝ : X ⟶ Y\ninst✝² : LocallyQuasiFinite f✝\nx : ↥X\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.QuasiFinite\nJ : Ideal S\nx✝ : J.IsPrime\nalgInst✝ : Algebra R S := f.toAlgebra\nalgebraizeInst✝ : Algebra.QuasiFinite R S\n⊢ ((Localization.localRingHom (Ideal.c... | [
"X Y : Scheme\nf✝ : X ⟶ Y\ninst✝² : LocallyQuasiFinite f✝\nx : ↥X\nR S : Type u\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nhf : f.QuasiFinite\nJ : Ideal S\nx✝ : J.IsPrime\nalgInst✝ : Algebra R S := f.toAlgebra\nalgebraizeInst✝ : Algebra.QuasiFinite R S\n⊢ (Localization.localRingHom (Ideal.comap f J) J f... | convert!
RingHom.quasiFinite_algebraMap.mpr
(inferInstance : Algebra.QuasiFinite R (Localization.AtPrime J)) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 174,
"column": 4
} | {
"line": 177,
"column": 9
} | {
"line": 180,
"column": 0
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\nhf : (aeval x) P.f ∈ I\na : S\nha : (aeval x) P.g * a - 1 ∈ I\np₁ p₂ : R[X]\nn : ℕ\ne : (aeval ... | [] | rintro ε' ⟨hε'I, hε', hε''⟩
rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (hI.le (Ideal.pow_mem_pow hε'I 2))] at hε'
have : ε * ε' = 0 := ((pow_two _).symm.trans hI).le (Ideal.mul_mem_mul hεI hε'I)
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 174,
"column": 4
} | {
"line": 177,
"column": 9
} | {
"line": 180,
"column": 0
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\nhf : (aeval x) P.f ∈ I\na : S\nha : (aeval x) P.g * a - 1 ∈ I\np₁ p₂ : R[X]\nn : ℕ\ne : (aeval ... | [] | rintro ε' ⟨hε'I, hε', hε''⟩
rw [Polynomial.aeval_add_of_sq_eq_zero _ _ _ (hI.le (Ideal.pow_mem_pow hε'I 2))] at hε'
have : ε * ε' = 0 := ((pow_two _).symm.trans hI).le (Ideal.mul_mem_mul hεI hε'I)
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 24
} | {
"line": 297,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nP✝ : StandardEtalePair R\nP : StandardEtalePresentation R S\n⊢ Ideal.span (Set.range ![(Bivariate.equivMvPolynomial R) (C P.f), (Bivariate.equivMvPolynomia... | [] | simp [Set.pair_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 94
} | {
"line": 251,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.spa... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nφ : S[X] →ₐ[R] B\nhφ : Function.Surjective ⇑φ\nf : S[X]\nhf : f.Monic\nhf' : ∀ (i : ℕ), IsIntegral R (f.coeff i)\nhfx : RingHom.ker φ.toRingHom = Ideal.span {f}\nh✝ : ... | refine fun a ha ↦ (natDegree_mul_C_le _ _).trans ((natDegree_multiset_prod_le _).trans ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 21
} | {
"line": 256,
"column": 4
} | [
{
"pp": "case smul\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\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis... | [] | | smul a x hx IH => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 325,
"column": 8
} | {
"line": 325,
"column": 25
} | {
"line": 325,
"column": 25
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : Algebra.IsStandardEtale R S\n𝓟 : StandardEtalePresentation R S\nn : ℕ\n𝓟' : StandardEtalePresentation B (B ⊗[R] S) := ⋯\ne : B ⊗[R] S ≃ₐ[B] Loca... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : Algebra.IsStandardEtale R S\n𝓟 : StandardEtalePresentation R S\nn : ℕ\n𝓟' : StandardEtalePresentation B (B ⊗[R] S) := 𝓟.baseChange\ne : B ⊗[R] S ≃ₐ[B] Loca... | ← e.eq_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.DegreeLT | {
"line": 79,
"column": 38
} | {
"line": 79,
"column": 71
} | {
"line": 79,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nm n : ℕ\ni : Fin n\n⊢ ((basis R m).prod (basis R n)) (finSumFinEquiv.symm (Fin.natAdd m i)) = (0, (basis R n) i)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.degreeLT",
"Submodule",
"Semiring.toModu... | [
"R : Type u_1\ninst✝ : Semiring R\nm n : ℕ\ni : Fin n\n⊢ ((basis R m).prod (basis R n)) (Sum.inr i) = (0, (basis R n) i)"
] | finSumFinEquiv_symm_apply_natAdd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.DegreeLT | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 75
} | {
"line": 188,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nr : R\nn : ℕ\na✝ : Nontrivial R\ni j : Fin n\nhji : id j < id i\n⊢ (LinearMap.toMatrix (degreeLT.basis R n) (degreeLT.basis R n)) (↑(taylorLinearEquiv r n)) i j = 0",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
... | [
"R : Type u_1\ninst✝ : CommRing R\nr : R\nn : ℕ\na✝ : Nontrivial R\ni j : Fin n\nhji : id j < id i\n⊢ (↑((taylorLinearEquiv r n) ((degreeLT.basis R n) j))).coeff ↑i = 0"
] | rw [LinearMap.toMatrix_apply, LinearEquiv.coe_coe, degreeLT.basis_repr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 62,
"column": 14
} | {
"line": 62,
"column": 26
} | {
"line": 63,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : Y.Opensᵒᵖ\n⊢ CommRingCat.ofHom\n ((CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)).restrict\n (integralClosure ↑Γ(Y, Opposite.unop U) ↑Γ(X, f ⁻¹ᵁ Opposite.unop U))\n (integralClosure ↑Γ(Y, Opposite.unop U) ↑Γ(X, f ⁻¹ᵁ Opposite.unop U)) ⋯) =\n 𝟙 (... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 52
} | {
"line": 205,
"column": 4
} | [
{
"pp": "case a.convert_2.C\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nf : MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R :=\n eval₂Hom (↑(universalFactorizationMap R n m k hn)) (Sum.elim (fun x ↦ X x ⊗ₜ[R] 1) fun x ↦ 1 ⊗ₜ[R] X x)... | [] | induction x using MvPolynomial.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 52
} | {
"line": 201,
"column": 4
} | [
{
"pp": "case a.convert_2\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nf : MvPolynomial (Fin m ⊕ Fin k) (MvPolynomial (Fin n) R) →+* MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R :=\n eval₂Hom (↑(universalFactorizationMap R n m k hn)) (Sum.elim (fun x ↦ X x ⊗ₜ[R] 1) fun x ↦ 1 ⊗ₜ[R] X x)\n... | [] | induction x using MvPolynomial.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 10
} | {
"line": 297,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\n⊢ (universalFactorizationMapPresentation R n m k hn).jacobiMatrix =\n -((Matrix.reindex (f... | [
"R : Type u_1\ninst✝ : CommRing R\nm k : ℕ\nthis : Algebra (MvPolynomial (Fin (m + k)) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R (m + k) m k ⋯).toAlgebra\n⊢ (universalFactorizationMapPresentation R (m + k) m k ⋯).jacobiMatrix =\n -((Matrix.reindex (finCongr ⋯) (fi... | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 300,
"column": 2
} | {
"line": 303,
"column": 97
} | {
"line": 305,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nm k : ℕ\nthis : Algebra (MvPolynomial (Fin (m + k)) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R (m + k) m k ⋯).toAlgebra\ni j : Fin (m + k)\n⊢ (pderiv ((universalFactorizationMapPresentation R (m + k) m k ⋯).map i))\n ... | [] | obtain ⟨i | i, rfl⟩ := finSumFinEquiv.surjective i <;>
induction j using Fin.addCases <;>
simp [pderiv_map, coeff_freeMonic, apply_dite (DFunLike.coe _), apply_ite (DFunLike.coe _),
pderiv_inl_universalFactorizationMap_X, pderiv_inr_universalFactorizationMap_X] <;> grind | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 541,
"column": 4
} | {
"line": 543,
"column": 36
} | {
"line": 544,
"column": 2
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : CommRing R\np : R[X]\nP : {R : Type u} → [inst : CommRing R] → R[X] → Prop\nSplits : ∀ (R : Type u) [inst : Field R] (p : R[X]), p.Splits → P p\ninjective :\n ∀ (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] (φ : R →+* S),\n Function.Injective ⇑φ → ∀ (p : R[X... | [] | exact surjective _ _ (MvPolynomial.eval₂Hom (algebraMap ℤ R) id)
(fun x ↦ ⟨.X x, by simp [MvPolynomial.eval₂Hom]⟩) p
(fun _ ↦ this _ inferInstance) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 541,
"column": 4
} | {
"line": 543,
"column": 36
} | {
"line": 544,
"column": 2
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : CommRing R\np : R[X]\nP : {R : Type u} → [inst : CommRing R] → R[X] → Prop\nSplits : ∀ (R : Type u) [inst : Field R] (p : R[X]), p.Splits → P p\ninjective :\n ∀ (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] (φ : R →+* S),\n Function.Injective ⇑φ → ∀ (p : R[X... | [] | exact surjective _ _ (MvPolynomial.eval₂Hom (algebraMap ℤ R) id)
(fun x ↦ ⟨.X x, by simp [MvPolynomial.eval₂Hom]⟩) p
(fun _ ↦ this _ inferInstance) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 541,
"column": 4
} | {
"line": 543,
"column": 36
} | {
"line": 544,
"column": 2
} | [
{
"pp": "case inr\nR : Type u\ninst✝ : CommRing R\np : R[X]\nP : {R : Type u} → [inst : CommRing R] → R[X] → Prop\nSplits : ∀ (R : Type u) [inst : Field R] (p : R[X]), p.Splits → P p\ninjective :\n ∀ (R S : Type u) [inst : CommRing R] [inst_1 : CommRing S] (φ : R →+* S),\n Function.Injective ⇑φ → ∀ (p : R[X... | [] | exact surjective _ _ (MvPolynomial.eval₂Hom (algebraMap ℤ R) id)
(fun x ↦ ⟨.X x, by simp [MvPolynomial.eval₂Hom]⟩) p
(fun _ ↦ this _ inferInstance) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 597,
"column": 4
} | {
"line": 598,
"column": 96
} | {
"line": 599,
"column": 2
} | [
{
"pp": "case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : (map φ f).resultant (map φ f) = 0 ^ (map φ f).natDegree\n⊢ f.resultant f = 0 ^ f.natDegree",
"ppTerm": "?injective",
"assigned": ... | [] | apply hφ
simpa only [resultant_map_map, natDegree_map_eq_of_injective hφ, map_zero, map_pow] using IH | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 597,
"column": 4
} | {
"line": 598,
"column": 96
} | {
"line": 599,
"column": 2
} | [
{
"pp": "case injective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH : (map φ f).resultant (map φ f) = 0 ^ (map φ f).natDegree\n⊢ f.resultant f = 0 ^ f.natDegree",
"ppTerm": "?injective",
"assigned": ... | [] | apply hφ
simpa only [resultant_map_map, natDegree_map_eq_of_injective hφ, map_zero, map_pow] using IH | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 372,
"column": 6
} | {
"line": 372,
"column": 54
} | {
"line": 373,
"column": 6
} | [
{
"pp": "case tmul.refine_1\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)\n... | [] | induction x using MvPolynomial.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 414,
"column": 29
} | {
"line": 415,
"column": 57
} | {
"line": 415,
"column": 57
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"p... | [] | by simp +contextual only [Polynomial.coeff_map, p.2,
map_one, map_zero, gt_iff_lt, implies_true, and_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 725,
"column": 63
} | {
"line": 725,
"column": 69
} | {
"line": 725,
"column": 70
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.... | [
"R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n r ^ (q.natDegree * ... | ← hg', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 726,
"column": 58
} | {
"line": 726,
"column": 64
} | {
"line": 727,
"column": 6
} | [
{
"pp": "case inr.inr.surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDe... | [
"case inr.inr.surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n ... | ← hg', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 730,
"column": 15
} | {
"line": 730,
"column": 21
} | {
"line": 730,
"column": 22
} | [
{
"pp": "case inr.inr.surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDe... | [
"case inr.inr.surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH :\n ∀ (q g : R[X]) (r : R),\n q ≠ 0 →\n g ≠ 0 →\n (q.scaleRoots r).resultant (g.scaleRoots r) q.natDegree g.natDegree =\n ... | ← hg', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 748,
"column": 63
} | {
"line": 748,
"column": 69
} | {
"line": 748,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.natDegree ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),\n g.natDegree ≠ 0 →\n IsDomain R →\n (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =\n g.leadingCoeff ^ (f.n... | [
"R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.natDegree ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),\n g.natDegree ≠ 0 →\n IsDomain R →\n (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =\n g.leadingCoeff ^ (f.natDegree * (... | ← hg', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 614,
"column": 2
} | {
"line": 614,
"column": 91
} | {
"line": 615,
"column": 2
} | [
{
"pp": "X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\ninst✝⁴ : QuasiCompact f✝\ninst✝³ : QuasiSeparated f✝\nX S Y : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\ninst✝ : Smooth g\nx : ↥(pullback (fromNormalization f) g)\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S.affineOpens\nhxU : (pullback... | [
"X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\ninst✝⁴ : QuasiCompact f✝\ninst✝³ : QuasiSeparated f✝\nX S Y : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\ninst✝ : Smooth g\nx : ↥(pullback (fromNormalization f) g)\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S.affineOpens\nhxU : (pullback.snd (fromNo... | refine ⟨W, hxV, (isIso_morphismRestrict_iff_isIso_app _ (U := W) (hV.preimage _)).mpr ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 750,
"column": 61
} | {
"line": 750,
"column": 67
} | {
"line": 750,
"column": 68
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.natDegree ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),\n g.natDegree ≠ 0 →\n IsDomain R →\n (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =\n g.leadingCo... | [
"case neg\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.natDegree ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),\n g.natDegree ≠ 0 →\n IsDomain R →\n (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =\n g.leadingCoeff ^ (f.nat... | ← hg', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 792,
"column": 63
} | {
"line": 792,
"column": 69
} | {
"line": 792,
"column": 70
} | [
{
"pp": "R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH : ∀ (q g : R[X]) (r : R), ((taylor r) q).resultant ((taylor r) g) = q.resultant g\ng : S[X]\nf' : R[X]\nhf' : map φ f' = f\nef : f'.degree = f.degree\ng' ... | [
"R✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH : ∀ (q g : R[X]) (r : R), ((taylor r) q).resultant ((taylor r) g) = q.resultant g\ng : S[X]\nf' : R[X]\nhf' : map φ f' = f\nef : f'.degree = f.degree\ng' : R[X]\nhg' ... | ← hg', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 793,
"column": 65
} | {
"line": 793,
"column": 71
} | {
"line": 793,
"column": 72
} | [
{
"pp": "case surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH : ∀ (q g : R[X]) (r : R), ((taylor r) q).resultant ((taylor r) g) = q.resultant g\ng : S[X]\nf' : R[X]\nhf' : map φ f' = f\nef : f'.degre... | [
"case surjective\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR S : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nφ : R →+* S\nhφ : Function.Surjective ⇑φ\nf : S[X]\nIH : ∀ (q g : R[X]) (r : R), ((taylor r) q).resultant ((taylor r) g) = q.resultant g\ng : S[X]\nf' : R[X]\nhf' : map φ f' = f\nef : f'.degree = f.degree... | ← hg', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 100
} | {
"line": 129,
"column": 4
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nx : ↥X\nhx : QuasiFiniteAt f x\nT : Scheme\nfT : T ⟶ Y\nleft✝¹ : Etale fT\nu : ↥T\nhu : fT u = f x\nV W : (pullback f fT).Opens\nv : ↥V\nhVW : IsCompl V W\nleft✝ : IsFinite (V.ι ≫ pullback.snd f fT)\nhv₂ : (pul... | have : IsClosedImmersion V.ι := .of_isPreimmersion _ (by simp [eq_compl_comm.mp hVW', W.isOpen]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.CotangentBaseChange | {
"line": 72,
"column": 22
} | {
"line": 72,
"column": 70
} | {
"line": 72,
"column": 70
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nI : Ideal S\na : S →+* T ⊗[R] S := Algebra.TensorProduct.includeRight.toRingHom\n⊢ (tensorCotangentHom R T I) 0 =\n (map Algebra.TensorProduct.includeRi... | [] | by simp only [map_zero]; exact (map_zero _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.CotangentBaseChange | {
"line": 104,
"column": 2
} | {
"line": 106,
"column": 45
} | {
"line": 108,
"column": 0
} | [
{
"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\nI : Ideal S\ninst✝ : Module.Flat R T\na : S →+* T ⊗[R] S := ⋯\nf : (map a I).Cotangent →ₗ[T] T ⊗[R] S ⧸ map a I ^ 2 := ⋯\ng : T ⊗[R] I.Cotangent →ₗ[T] T ⊗... | [] | · apply Module.Flat.lTensor_preserves_injective_linearMap (M := T)
(I.cotangentToQuotientSquare.restrictScalars R)
apply cotangentToQuotientSquare_injective | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Bicategory.Adjunction.Basic | {
"line": 148,
"column": 2
} | {
"line": 159,
"column": 42
} | {
"line": 161,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝ : Bicategory B\na b c : B\nf₁ : a ⟶ b\ng₁ : b ⟶ a\nf₂ : b ⟶ c\ng₂ : c ⟶ b\nadj₁ : f₁ ⊣ g₁\nadj₂ : f₂ ⊣ g₂\n⊢ rightZigzag (adj₁.compUnit adj₂) (adj₁.compCounit adj₂) = (ρ_ (g₂ ≫ g₁)).hom ≫ (λ_ (g₂ ≫ g₁)).inv",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"... | [] | calc
_ = 𝟙 _ ⊗≫
(g₂ ≫ g₁) ◁ adj₁.unit ⊗≫
g₂ ◁ ((g₁ ≫ f₁) ◁ adj₂.unit ≫ adj₁.counit ▷ (f₂ ≫ g₂)) ▷ g₁ ⊗≫
adj₂.counit ▷ (g₂ ≫ g₁) ⊗≫ 𝟙 _ := by
dsimp only [compUnit, compCounit]; bicategory
_ = 𝟙 _ ⊗≫
g₂ ◁ (rightZigzag adj₁.unit adj₁.counit) ⊗≫
(ri... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.CategoryTheory.Bicategory.Adjunction.Basic | {
"line": 148,
"column": 2
} | {
"line": 159,
"column": 42
} | {
"line": 161,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝ : Bicategory B\na b c : B\nf₁ : a ⟶ b\ng₁ : b ⟶ a\nf₂ : b ⟶ c\ng₂ : c ⟶ b\nadj₁ : f₁ ⊣ g₁\nadj₂ : f₂ ⊣ g₂\n⊢ rightZigzag (adj₁.compUnit adj₂) (adj₁.compCounit adj₂) = (ρ_ (g₂ ≫ g₁)).hom ≫ (λ_ (g₂ ≫ g₁)).inv",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"... | [] | calc
_ = 𝟙 _ ⊗≫
(g₂ ≫ g₁) ◁ adj₁.unit ⊗≫
g₂ ◁ ((g₁ ≫ f₁) ◁ adj₂.unit ≫ adj₁.counit ▷ (f₂ ≫ g₂)) ▷ g₁ ⊗≫
adj₂.counit ▷ (g₂ ≫ g₁) ⊗≫ 𝟙 _ := by
dsimp only [compUnit, compCounit]; bicategory
_ = 𝟙 _ ⊗≫
g₂ ◁ (rightZigzag adj₁.unit adj₁.counit) ⊗≫
(ri... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Adjunction.Basic | {
"line": 148,
"column": 2
} | {
"line": 159,
"column": 42
} | {
"line": 161,
"column": 0
} | [
{
"pp": "B : Type u₁\ninst✝ : Bicategory B\na b c : B\nf₁ : a ⟶ b\ng₁ : b ⟶ a\nf₂ : b ⟶ c\ng₂ : c ⟶ b\nadj₁ : f₁ ⊣ g₁\nadj₂ : f₂ ⊣ g₂\n⊢ rightZigzag (adj₁.compUnit adj₂) (adj₁.compCounit adj₂) = (ρ_ (g₂ ≫ g₁)).hom ≫ (λ_ (g₂ ≫ g₁)).inv",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"... | [] | calc
_ = 𝟙 _ ⊗≫
(g₂ ≫ g₁) ◁ adj₁.unit ⊗≫
g₂ ◁ ((g₁ ≫ f₁) ◁ adj₂.unit ≫ adj₁.counit ▷ (f₂ ≫ g₂)) ▷ g₁ ⊗≫
adj₂.counit ▷ (g₂ ≫ g₁) ⊗≫ 𝟙 _ := by
dsimp only [compUnit, compCounit]; bicategory
_ = 𝟙 _ ⊗≫
g₂ ◁ (rightZigzag adj₁.unit adj₁.counit) ⊗≫
(ri... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.KrullDimension.NonZeroDivisors | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 60
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : CommRing R\nσ : Type u_3\na✝ : Nontrivial R\nh✝ : Infinite σ\n⊢ ringKrullDim R + ⊤ ≤ ringKrullDim (MvPolynomial σ R)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Nat.instMulZeroClass",
"WithBot",
"i... | [
"case inr\nR : Type u_1\ninst✝ : CommRing R\nσ : Type u_3\na✝ : Nontrivial R\nh✝ : Infinite σ\n⊢ ringKrullDim (MvPolynomial σ R) = ⊤"
] | suffices ringKrullDim (MvPolynomial σ R) = ⊤ by simp_all | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.AlgebraicGeometry.Modules.Sheaf | {
"line": 520,
"column": 2
} | {
"line": 521,
"column": 19
} | {
"line": 522,
"column": 2
} | [
{
"pp": "X Y : Scheme\nU : X.Opens\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nx : ↥X\nM : Y.Modules\nhxU : x ∈ U\n⊢ M.presheaf.germ (f ''ᵁ Opposite.unop (Opposite.op U)) (f x) ⋯ ≫ (restrictStalkNatIso f x).inv.app M =\n ((restrictFunctor f).obj M).presheaf.germ U x hxU",
"ppTerm": "?m.57",
"assigned": tr... | [
"X Y : Scheme\nU : X.Opens\nf : X ⟶ Y\ninst✝ : IsOpenImmersion f\nx : ↥X\nM : Y.Modules\nhxU : x ∈ U\n⊢ ((restrictFunctor f).obj M).presheaf.germ U x hxU ≫\n (𝟙 (restrictFunctor f ⋙ toPresheaf X ⋙ TopCat.Presheaf.stalkFunctor Ab x)).app M =\n ((restrictFunctor f).obj M).presheaf.germ U x hxU"
] | rw [← germ_restrictStalkNatIso_hom_app f x M hxU, Category.assoc, ← NatTrans.comp_app,
Iso.hom_inv_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 22
} | {
"line": 115,
"column": 2
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝¹ : CommRing A\ninst✝ : SetLike σ A\n𝒜 : ι → σ\nx : Submonoid A\ni1 : ι\nn1 : A\nhn1 : n1 ∈ 𝒜 i1\nd1 : A\nhd1 : d1 ∈ 𝒜 i1\nh1 : ↑⟨d1, hd1⟩ ∈ x\ni2 : ι\nn2 : A\nhn2 : n2 ∈ 𝒜 i2\nd2 : A\nhd2 : d2 ∈ 𝒜 i2\nh2 : ↑⟨d2, hd2⟩ ∈ x\nhdeg : i1 = i2\nhnum : n1 = ... | [
"ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝¹ : CommRing A\ninst✝ : SetLike σ A\n𝒜 : ι → σ\nx : Submonoid A\ni1 : ι\nn1 : A\nhn1 : n1 ∈ 𝒜 i1\nd1 : A\nhd1 : d1 ∈ 𝒜 i1\nh1 : ↑⟨d1, hd1⟩ ∈ x\nhn2 : n1 ∈ 𝒜 i1\nhd2 : d1 ∈ 𝒜 i1\nh2 : ↑⟨d1, hd2⟩ ∈ x\n⊢ { deg := i1, num := ⟨n1, hn1⟩, den := ⟨d1, hd1⟩, den_mem := h1... | subst hdeg hnum hden | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 276,
"column": 20
} | {
"line": 276,
"column": 41
} | {
"line": 276,
"column": 41
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫\n... | [
"σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ awayι 𝒜 x ⋯ ... | SpecMap_awayMap_awayι | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 285,
"column": 20
} | {
"line": 285,
"column": 41
} | {
"line": 285,
"column": 41
} | [
{
"pp": "σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫\n... | [
"σ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nm' : ℕ\ng : A\ng_deg : g ∈ 𝒜 m'\nhm' : 0 < m'\nx : A\nhx : x = f * g\n⊢ (pullbackAwayιIso 𝒜 f_deg hm g_deg hm' hx).hom ≫ awayι 𝒜 x ⋯ ... | SpecMap_awayMap_awayι | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 394,
"column": 12
} | {
"line": 394,
"column": 97
} | {
"line": 395,
"column": 4
} | [
{
"pp": "case e_a\nσ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nx x' : ↑Γ(X, ⊤)\nt : A\nd : ℕ\nH : f t = x\nh0d : 0 < d\nhd : t ∈ 𝒜 d\ns : A\nn : ℕ\nhs : s ∈ 𝒜 n\nhx'x : PrimeSpectrum.basi... | [
"case e_a\nσ : Type u_1\nA : Type u\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nX : Scheme\nf : A →+* ↑Γ(X, ⊤)\nx x' : ↑Γ(X, ⊤)\nt : A\nd : ℕ\nH : f t = x\nh0d : 0 < d\nhd : t ∈ 𝒜 d\ns : A\nn : ℕ\nhs : s ∈ 𝒜 n\nhx'x : PrimeSpectrum.basicOpen x' ≤ P... | basicOpenIsoSpecAway_inv_homOfLE_assoc (R := Γ(X, ⊤)) x (f s) x' (by simp [← H', H]), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 54
} | {
"line": 310,
"column": 0
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\na : A\nn : ℕ\nhn : a ∈ 𝒜 n\n⊢ HomogeneousLocalization.mk { deg := m * n, num := ⟨(p... | [] | · simp only [proj_apply, decompose_of_mem_same _ hn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 611,
"column": 57
} | {
"line": 615,
"column": 55
} | {
"line": 617,
"column": 0
} | [
{
"pp": "X : Scheme\nM : X.Modules\ninst✝ : SheafOfModules.IsQuasicoherent M\n⊢ ∃ 𝒰, ∀ (i : 𝒰.I₀), Nonempty (SheafOfModules.Presentation (M.restrict (𝒰.f i)))",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.IsAffineOpen.isoSpec",
"CategoryTheory.Grothendie... | [] | by
obtain ⟨ι, U, pres, hU, hU'⟩ := M.exists_isOpenCover_presentation
refine ⟨Scheme.AffineOpenCover.ofIsOpenCover _ hU hU', fun i ↦ ⟨?_⟩⟩
exact SheafOfModules.Presentation.ofIsIso.{u, u, u} ((restrictFunctorComp _ _).app M).inv <|
(presentationRestrict (hU' i).isoSpec.inv (pres i)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor | {
"line": 63,
"column": 74
} | {
"line": 71,
"column": 35
} | {
"line": 73,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nσ : Type u_4\nτ : Type u_5\ninst✝⁷ : CommRing A\ninst✝⁶ : SetLike σ A\ninst✝⁵ : AddSubgroupClass σ A\ninst✝⁴ : CommRing B\ninst✝³ : SetLike τ B\ninst✝² : AddSubgroupClass τ B\n𝒜 : ℕ → σ\nℬ : ℕ → τ\ninst✝¹ : GradedRing 𝒜\ninst✝ : GradedRing ℬ\nf : 𝒜 →+*ᵍ ℬ\nhf : ℬ₊ ≤ Homog... | [] | by
rintro ⟨p, hpV⟩
rcases hs ⟨.comap f hf p, hUV hpV⟩ with ⟨W, m, iWU, i, a, b, hb, h_frac⟩
refine ⟨W.comap (ProjectiveSpectrum.comap f hf) ⊓ V, ⟨m, hpV⟩, Opens.infLERight _ _, i,
f.gradedAddHom i a, f.gradedAddHom i b, fun ⟨q, ⟨hqW, hqV⟩⟩ ↦ hb ⟨_, hqW⟩,
fun ⟨q, ⟨hqW, hqV⟩⟩ ↦ ?_⟩
ext
specialize h_frac... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 24
} | {
"line": 94,
"column": 0
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : ValuationSubring K\nS : LocalSubring K\nhS : R.toLocalSubring ≤ S\nx : K\nhx : x ∈ S.toSubring\nh : x⁻¹ ∈ R.carrier\nh' : x ∉ R.toLocalSubring.toSubring\nhx0 : x ≠ 0\nthis✝ : IsUnit ((Subring.inclusion ⋯) ⟨x⁻¹, h⟩)\nx' : ↥R.toLocalSubring.toSubring\nhx' : ⟨x⁻¹, h⟩ * x... | [] | exact h' (this ▸ x'.2) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 619,
"column": 62
} | {
"line": 624,
"column": 5
} | {
"line": 626,
"column": 0
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↑(CommRingCat.of (A⁰_ f))\np : ↥(unop (op (pbo f)))\n⊢ HomogeneousLocalization.val (↑((awayToSection 𝒜 f).hom' x) p) =\n (IsLocalization.map (Localizat... | [] | by
obtain ⟨x, rfl⟩ := HomogeneousLocalization.mk_surjective x
change (HomogeneousLocalization.mapId 𝒜 _ _).val = _
dsimp [HomogeneousLocalization.mapId, HomogeneousLocalization.map]
rw [Localization.mk_eq_mk', Localization.mk_eq_mk', IsLocalization.map_mk']
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 783,
"column": 2
} | {
"line": 785,
"column": 13
} | {
"line": 787,
"column": 0
} | [
{
"pp": "case refine_4\nR : CommRingCat\nM : (Spec R).Modules\nφ : (f : ↑R) →\n ↑((modulesSpecToSheaf.obj M).obj.obj (op ⊤)) →ₗ[↑R] ↑((modulesSpecToSheaf.obj M).obj.obj (op (basicOpen f))) :=\n fun f ↦ ModuleCat.Hom.hom ((modulesSpecToSheaf.obj M).obj.map (basicOpen f).leTop.op)\nh : Aux M ⊤\nf : ↑R\n⊢ ∀ (x :... | [] | · intro x hx
obtain ⟨n, hn⟩ := h.uniqueness _ _ _ hx
use n, hn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 116,
"column": 4
} | {
"line": 117,
"column": 91
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case h₀\nσ : Type u_1\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\ni j : (affineOpenCover 𝒜).openCover.I₀\ne₁ : pullback ((affineOpenCover 𝒜).f i ≫ toSpecZero 𝒜) ((affineOpenCover 𝒜).f j ≫ toSpecZero 𝒜) ≅\n Spec (CommR... | [] | exact DFunLike.congr_fun (Algebra.TensorProduct.lift_comp_includeLeft
(awayMapₐ 𝒜 j.2.2 rfl) (awayMapₐ 𝒜 i.2.2 (mul_comm _ _)) (fun _ _ ↦ .all _ _)).symm x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 270,
"column": 48
} | {
"line": 271,
"column": 68
} | {
"line": 271,
"column": 68
} | [
{
"pp": "case right.mem\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsF... | [
"case right.mem\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFractionRing ... | ← pow_le_pow_iff_left₀ zero_le zero_le
(mul_pos (hdi j) (Finset.prod_pos fun i _ => hdi i)).ne.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 22
} | {
"line": 196,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nY : C\ng : Y ⟶ B\n⊢ (generateFamily X π).arrows g → (generate (Presieve.ofArrows X π)).arrows g",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStru... | [
"case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nY : C\na : α\ng : Y ⟶ X a\n⊢ (generate (Presieve.ofArrows X π)).arrows (g ≫ π a)"
] | rintro ⟨a, g, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 111,
"column": 13
} | {
"line": 114,
"column": 65
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝ : J.Subcanonical\nP Q : Sheaf J (Type v)\nf g : P ⟶ Q\nh : ∀ (X : C) (p : J.yoneda.obj X ⟶ P), p ≫ f = p ≫ g\n⊢ f = g",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Eq... | [] | by
ext X x
simpa only [yonedaEquiv_comp, Equiv.apply_symm_apply]
using! congr_arg (J.yonedaEquiv) (h _ (J.yonedaEquiv.symm x)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 195,
"column": 8
} | {
"line": 195,
"column": 25
} | {
"line": 195,
"column": 26
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nz : X ⟶ c.pt\ninst✝ : IsFiltered J\nj : J\n⊢ colimit.ι (pullback c.ι ((Functor.const J).map z)) ... | [
"case refine_1\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : Abelian C\ninst✝² : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝¹ : SmallCategory J\nY : J ⥤ C\nc : Cocone Y\nhc : IsColimit c\nz : X ⟶ c.pt\ninst✝ : IsFiltered J\nj : J\n⊢ colimit.ι (pullback c.ι ((Functor.const J).map z)) j ≫ f z =\n ... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Limit | {
"line": 37,
"column": 4
} | {
"line": 42,
"column": 42
} | {
"line": 42,
"column": 42
} | [
{
"pp": "case neg\nJ : Type u\ninst✝ : LinearOrder J\nj : J\nm : ↑(Ici j)\nhm : Order.IsSuccLimit m\nb : J\nhb : b < ↑m\nhb' : ¬j ≤ b\n⊢ ∃ x, ∃ (_ : b < x), x < ↑m",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Set.Ici",
"PartialOrder.toPreorder",
... | [] | · simp only [not_le] at hb'
refine ⟨j, hb', ?_⟩
by_contra!
apply hm.1
rintro ⟨k, hk⟩ _
exact this.trans (by simpa using hk) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Shapes.Preorder.WellOrderContinuous | {
"line": 100,
"column": 6
} | {
"line": 105,
"column": 25
} | {
"line": 106,
"column": 6
} | [
{
"pp": "case pos\nC : Type u\ninst✝³ : Category.{v, u} C\nJ✝ : Type w\ninst✝² : PartialOrder J✝\nJ : Type w\ninst✝¹ : LinearOrder J\nF : J ⥤ C\ninst✝ : F.IsWellOrderContinuous\nj : J\nm : { x // x ∈ Set.Ici j }\nhm : Order.IsSuccLimit m\nf : ↑(Set.Iio m) → ↑(Set.Iio ↑m) :=\n fun x ↦\n match x with\n | ⟨... | [
"case neg\nC : Type u\ninst✝³ : Category.{v, u} C\nJ✝ : Type w\ninst✝² : PartialOrder J✝\nJ : Type w\ninst✝¹ : LinearOrder J\nF : J ⥤ C\ninst✝ : F.IsWellOrderContinuous\nj : J\nm : { x // x ∈ Set.Ici j }\nhm : Order.IsSuccLimit m\nf : ↑(Set.Iio m) → ↑(Set.Iio ↑m) :=\n fun x ↦\n match x with\n | ⟨⟨a, ha⟩, ha'... | · refine ⟨⟨⟨j, le_refl j⟩, ?_⟩, h⟩
by_contra h'
simp only [Set.mem_Iio, not_lt] at h'
apply hm.1
rintro ⟨k, hk⟩ hkm
exact h'.trans hk | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.RelativeCellComplex.AttachCells | {
"line": 82,
"column": 39
} | {
"line": 82,
"column": 71
} | {
"line": 82,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nα : Type t\nA B : α → C\ng : (a : α) → A a ⟶ B a\nX₁ X₂ : C\nf : X₁ ⟶ X₂\nc : AttachCells g f\nx✝ : Discrete c.ι\ni : c.ι\n⊢ c.cofan₁.ι.app { as := i } ≫ c.m =\n c.cofan₁.ι.app { as := i } ≫\n c.isColimit₁.desc\n { pt := c.cofan₂.pt,\n ι :=... | [] | rw [IsColimit.fac]; exact c.hm i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.RelativeCellComplex.AttachCells | {
"line": 82,
"column": 39
} | {
"line": 82,
"column": 71
} | {
"line": 82,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nα : Type t\nA B : α → C\ng : (a : α) → A a ⟶ B a\nX₁ X₂ : C\nf : X₁ ⟶ X₂\nc : AttachCells g f\nx✝ : Discrete c.ι\ni : c.ι\n⊢ c.cofan₁.ι.app { as := i } ≫ c.m =\n c.cofan₁.ι.app { as := i } ≫\n c.isColimit₁.desc\n { pt := c.cofan₂.pt,\n ι :=... | [] | rw [IsColimit.fac]; exact c.hm i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc | {
"line": 157,
"column": 2
} | {
"line": 158,
"column": 81
} | {
"line": 160,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nJ : Type u\ninst✝¹ : LinearOrder J\ninst✝ : SuccOrder J\nj : J\nhj : ¬IsMax j\nF : ↑(Set.Iic j) ⥤ C\nX : C\nτ : F.obj ⟨j, ⋯⟩ ⟶ X\ni₁ i₂ : J\nhi : i₁ ≤ i₂\nhi₂ : i₂ ≤ j\n⊢ (extendToSucc hj F τ).map (homOfLE hi) ≫ (extendToSuccObjIso hj F τ i₂ hi₂).hom =\n ... | [] | dsimp [extendToSucc, extendToSuccObjIso]
rw [extendToSucc.map_eq _ _ _ _ _ _ hi₂, assoc, assoc, Iso.inv_hom_id, comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.SmallObject.Iteration.ExtendToSucc | {
"line": 157,
"column": 2
} | {
"line": 158,
"column": 81
} | {
"line": 160,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nJ : Type u\ninst✝¹ : LinearOrder J\ninst✝ : SuccOrder J\nj : J\nhj : ¬IsMax j\nF : ↑(Set.Iic j) ⥤ C\nX : C\nτ : F.obj ⟨j, ⋯⟩ ⟶ X\ni₁ i₂ : J\nhi : i₁ ≤ i₂\nhi₂ : i₂ ≤ j\n⊢ (extendToSucc hj F τ).map (homOfLE hi) ≫ (extendToSuccObjIso hj F τ i₂ hi₂).hom =\n ... | [] | dsimp [extendToSucc, extendToSuccObjIso]
rw [extendToSucc.map_eq _ _ _ _ _ _ hi₂, assoc, assoc, Iso.inv_hom_id, comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 387,
"column": 10
} | {
"line": 387,
"column": 72
} | {
"line": 388,
"column": 8
} | [
{
"pp": "case succ.inr.inl\nC : Type u\ninst✝⁵ : Category.{v, u} C\nJ : Type w\nΦ : SuccStruct C\ninst✝⁴ : LinearOrder J\ninst✝³ : SuccOrder J\ninst✝² : OrderBot J\ninst✝¹ : HasIterationOfShape J C\ninst✝ : WellFoundedLT J\nj✝ j : J\nhj₁ : ¬IsMax j\niter₁ iter₂ : Φ.Iteration (Order.succ j)\nk₁ k₂ : J\nh₁₂ : k₁ ... | [] | exact this _ _ _ _ ((Order.lt_succ_iff_of_not_isMax hj₁).1 h₂) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 387,
"column": 10
} | {
"line": 387,
"column": 72
} | {
"line": 388,
"column": 8
} | [
{
"pp": "case succ.inr.inl\nC : Type u\ninst✝⁵ : Category.{v, u} C\nJ : Type w\nΦ : SuccStruct C\ninst✝⁴ : LinearOrder J\ninst✝³ : SuccOrder J\ninst✝² : OrderBot J\ninst✝¹ : HasIterationOfShape J C\ninst✝ : WellFoundedLT J\nj✝ j : J\nhj₁ : ¬IsMax j\niter₁ iter₂ : Φ.Iteration (Order.succ j)\nk₁ k₂ : J\nh₁₂ : k₁ ... | [] | exact this _ _ _ _ ((Order.lt_succ_iff_of_not_isMax hj₁).1 h₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 387,
"column": 10
} | {
"line": 387,
"column": 72
} | {
"line": 388,
"column": 8
} | [
{
"pp": "case succ.inr.inl\nC : Type u\ninst✝⁵ : Category.{v, u} C\nJ : Type w\nΦ : SuccStruct C\ninst✝⁴ : LinearOrder J\ninst✝³ : SuccOrder J\ninst✝² : OrderBot J\ninst✝¹ : HasIterationOfShape J C\ninst✝ : WellFoundedLT J\nj✝ j : J\nhj₁ : ¬IsMax j\niter₁ iter₂ : Φ.Iteration (Order.succ j)\nk₁ k₂ : J\nh₁₂ : k₁ ... | [] | exact this _ _ _ _ ((Order.lt_succ_iff_of_not_isMax hj₁).1 h₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 293,
"column": 69
} | {
"line": 294,
"column": 32
} | {
"line": 296,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS T X Y : C\nπX : X ⟶ S\nπY : Y ⟶ T\nτ : Arrow.mk πX ⟶ Arrow.mk πY\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C\ninst✝¹ : HasPus... | [] | by
simp [ιFunctorObj, functorMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 163,
"column": 43
} | {
"line": 165,
"column": 5
} | {
"line": 167,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝¹ : OrderBot J\nval₀ : F.obj (op ⊥)\ninst✝ : WellFoundedLT J\nj : J\ne : d.Extension val₀ j\ni : J\ne' : d.Extension val₀ i\nh : i ≤ j\n⊢ (ConcreteCategory.hom (F.map (homOfLE h).op)) e.val = ... | [] | by
obtain rfl : e' = e.ofLE h := Subsingleton.elim _ _
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 230,
"column": 18
} | {
"line": 230,
"column": 27
} | {
"line": 230,
"column": 28
} | [
{
"pp": "case e'_2\nJ : Type u\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝¹ : OrderBot J\nval₀ : F.obj (op ⊥)\ninst✝ : WellFoundedLT J\nj : J\nhj : Order.IsSuccLimit j\ne : (i : J) → i < j → d.Extension val₀ i\ni : J\nhi : i < j\n⊢ (ConcreteCategory.hom (... | [
"case e'_2\nJ : Type u\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝¹ : OrderBot J\nval₀ : F.obj (op ⊥)\ninst✝ : WellFoundedLT J\nj : J\nhj : Order.IsSuccLimit j\ne : (i : J) → i < j → d.Extension val₀ i\ni : J\nhi : i < j\n⊢ (ConcreteCategory.hom (F.map (homOf... | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 272,
"column": 22
} | {
"line": 272,
"column": 54
} | {
"line": 274,
"column": 0
} | [
{
"pp": "J : Type u\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝¹ : OrderBot J\ninst✝ : WellFoundedLT J\nval₀ : F.obj (op ⊥)\nj✝ j'✝ : Jᵒᵖ\nf : j✝ ⟶ j'✝\n⊢ (ConcreteCategory.hom (F.map f)) ((fun j ↦ default.val) j✝) = (fun j ↦ default.val) j'✝",
"ppTer... | [] | by apply Extension.compatibility | [anonymous] | Lean.Parser.Term.byTactic |
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