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.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 173,
"column": 2
} | {
"line": 175,
"column": 64
} | [
{
"pp": "case mpr\nX : Scheme\nU : X.AffineZariskiSite\ns : Set ↑Γ(X, U.toOpens)\nV : X.Opens\nhV : IsAffineOpen V\nf : ⟨V, hV⟩ ⟶ U\n⊢ (∃ f ∈ s, ∃ g, X.basicOpen (f * g) = toOpens ⟨V, hV⟩) → (Sieve.generate (U.presieveOfSections s)).arrows f",
"usedConstants": [
"le_refl",
"AlgebraicGeometry.She... | · rintro ⟨f₁, hf₁s, f₂, rfl⟩
refine ⟨U.basicOpen f₁, ⟨f₂ |_ _, ?_⟩, ⟨f₁, rfl⟩, ⟨f₁, hf₁s, rfl⟩, rfl⟩
exact (X.basicOpen_res _ _).trans (X.basicOpen_mul _ _).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 676,
"column": 4
} | {
"line": 677,
"column": 94
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\ny : ↥K[X]⁰\n⊢ IsUnit ((algebraMap K[X] K⟮X⟯) ↑y)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"CommRing.toNonUnitalCommRing",
"Algebra.algebraMap",
"OreLo... | rw [← ofFractionRing_algebraMap]
exact (toFractionRingRingEquiv K).symm.toRingHom.isUnit_map (IsLocalization.map_units _ y) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 676,
"column": 4
} | {
"line": 677,
"column": 94
} | [
{
"pp": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\ny : ↥K[X]⁰\n⊢ IsUnit ((algebraMap K[X] K⟮X⟯) ↑y)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"CommRing.toNonUnitalCommRing",
"Algebra.algebraMap",
"OreLo... | rw [← ofFractionRing_algebraMap]
exact (toFractionRingRingEquiv K).symm.toRingHom.isUnit_map (IsLocalization.map_units _ y) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 959,
"column": 44
} | {
"line": 960,
"column": 52
} | [
{
"pp": "K : Type u\ninst✝ : Field K\n⊢ denom 1 = 1",
"usedConstants": [
"Polynomial.monic_one._simp_1",
"Eq.mpr",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"False",
"Polynomial.ins... | by
convert! denom_div (1 : K[X]) one_ne_zero <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 156,
"column": 2
} | {
"line": 157,
"column": 98
} | [
{
"pp": "R : 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 ((Ideal.Quotient.mk I) x)) P.g * (Ideal.Quotient.mk I) a = 1\n⊢ ∃! ... | simp_rw [← Ideal.Quotient.mkₐ_eq_mk R, aeval_algHom_apply, ← map_mul, ← map_one
(Ideal.Quotient.mkₐ R I), Ideal.Quotient.mkₐ_eq_mk, Ideal.Quotient.mk_eq_mk_iff_sub_mem] at ha | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 398,
"column": 16
} | {
"line": 398,
"column": 35
} | [
{
"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✝¹ : StandardEtalePair R\nP✝ : StandardEtalePresentation R S\nP : StandardEtalePair R := { f := X, monic_f := ⋯, g := 1, cond := ⋯ }\nthis : P.X = 0\n⊢ (o... | by ext; simp [this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 23
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal S\nr : S\nhrp : r ∉ p\nH : ∀ (x : S), ∃ m, IsIntegral R (r ^ m * x)\nn : ℕ\nhn : IsIntegral R (r ^ n * r)\n⊢ IsIntegral R r",
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
... | rw [← pow_succ] at hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 153,
"column": 64
} | {
"line": 153,
"column": 96
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm n : ℕ\n⊢ ↑↑(Equiv.Perm.sign\n ((finSumFinEquiv.symm.trans ((Equiv.sumComm (Fin n) (Fin m)).trans finSumFinEquiv)).trans\n (finCongr ⋯).symm)) *\n (g.sylvester f n m).det =\n (-1) ^ (m * n) * (g.sylvester f n m).det",
"u... | Equiv.Perm.sign_eq_prod_prod_Ioi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 147,
"column": 2
} | {
"line": 175,
"column": 65
} | [
{
"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\n⊢ IsIntegral R t",
"usedConstants": [
"Subalgebra.instSetLike",
"N... | let St := Localization.Away t
let t' : St := IsLocalization.Away.invSelf t
have ht't : t' * algebraMap S St t = 1 := by rw [mul_comm, IsLocalization.Away.mul_invSelf]
let R₁ := Algebra.adjoin R {t'}
let R₂ := Algebra.adjoin R₁ {algebraMap S St (φ X)}
letI : Algebra R₁ R₂ := R₂.algebra
letI : Algebra R₂ St :... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 147,
"column": 2
} | {
"line": 175,
"column": 65
} | [
{
"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\n⊢ IsIntegral R t",
"usedConstants": [
"Subalgebra.instSetLike",
"N... | let St := Localization.Away t
let t' : St := IsLocalization.Away.invSelf t
have ht't : t' * algebraMap S St t = 1 := by rw [mul_comm, IsLocalization.Away.mul_invSelf]
let R₁ := Algebra.adjoin R {t'}
let R₂ := Algebra.adjoin R₁ {algebraMap S St (φ X)}
letI : Algebra R₁ R₂ := R₂.algebra
letI : Algebra R₂ St :... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 9
} | [
{
"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 | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 563,
"column": 6
} | {
"line": 563,
"column": 100
} | [
{
"pp": "R S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal S\ninst✝¹ : p.IsPrime\ninst✝ : WeaklyQuasiFiniteAt R p\nf : MvPolynomial (Fin 0) R →ₐ[R] S\nhf : f.Finite\n⊢ (algebraMap R S).Finite",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Comm... | convert! RingHom.Finite.comp hf (RingHom.Finite.of_surjective _ (MvPolynomial.C_surjective _)) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 569,
"column": 4
} | {
"line": 569,
"column": 90
} | [
{
"pp": "case injective.a\nR✝ : Type u_1\ninst✝² : CommRing R✝\nR SatisfiesM : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing SatisfiesM\nφ : R →+* SatisfiesM\nhφ : Function.Injective ⇑φ\nf : R[X]\nIH :\n ∀ (g₁ g₂ : SatisfiesM[X]),\n (map φ f).resultant (g₁ * g₂) (map φ f).natDegree (g₁.natDegree + g₂.natD... | simpa only [resultant_map_map, ← map_mul, natDegree_map_eq_of_injective hφ] using this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 575,
"column": 37
} | {
"line": 575,
"column": 41
} | [
{
"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 :\n ∀ (q g₁ g₂ : R[X]), q.resultant (g₁ * g₂) q.natDegree (g₁.natDegree + g₂.natDegree) = q.resultant g₁ * q.resultant g₂\ng₁ g₂ : S[X]\... | hg₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 567,
"column": 4
} | {
"line": 631,
"column": 50
} | [
{
"pp": "case succ\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 : CommRing S... | let f' := f.comp (MvPolynomial.finSuccEquiv _ _).symm.toAlgHom
let := (f'.toRingHom.comp C).toAlgebra
have : IsScalarTower R (MvPolynomial (Fin n) R) S := .of_algebraMap_eq fun r ↦
(f.commutes r).symm.trans congr(f ($(MvPolynomial.finSuccEquiv_comp_C_eq_C n) r)).symm
let f'' : (MvPolynomial (Fin n) R)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 567,
"column": 4
} | {
"line": 631,
"column": 50
} | [
{
"pp": "case succ\nn : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S... | let f' := f.comp (MvPolynomial.finSuccEquiv _ _).symm.toAlgHom
let := (f'.toRingHom.comp C).toAlgebra
have : IsScalarTower R (MvPolynomial (Fin n) R) S := .of_algebraMap_eq fun r ↦
(f.commutes r).symm.trans congr(f ($(MvPolynomial.finSuccEquiv_comp_C_eq_C n) r)).symm
let f'' : (MvPolynomial (Fin n) R)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 229,
"column": 31
} | {
"line": 229,
"column": 67
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\nthis : Algebra ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U) := (CommRingCat.Hom.hom (app f ↑U)).toAlgebra\n⊢ toNormalization f ⁻¹ᵁ fromNormalization f ⁻¹ᵁ ↑U = f ⁻¹ᵁ ↑U",
"usedConstants": [
"AlgebraicGeometry.Preshea... | by simp [← Scheme.Hom.comp_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 236,
"column": 31
} | {
"line": 236,
"column": 67
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\nthis : Algebra ↑Γ(Y, ↑U) ↑Γ(X, f ⁻¹ᵁ ↑U) := (CommRingCat.Hom.hom (app f ↑U)).toAlgebra\n⊢ toNormalization f ⁻¹ᵁ fromNormalization f ⁻¹ᵁ ↑U = f ⁻¹ᵁ ↑U",
"usedConstants": [
"AlgebraicGeometry.Preshea... | by simp [← Scheme.Hom.comp_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 274,
"column": 33
} | {
"line": 274,
"column": 69
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : Y.Opens\nhU : IsAffineOpen U\nthis : Algebra ↑Γ(Y, U) ↑Γ(X, f ⁻¹ᵁ U) := (CommRingCat.Hom.hom (app f U)).toAlgebra\n⊢ f ⁻¹ᵁ U ≤ toNormalization f ⁻¹ᵁ fromNormalization f ⁻¹ᵁ U",
"usedConstants": [
"AlgebraicGeometr... | by simp [← Scheme.Hom.comp_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 394,
"column": 60
} | {
"line": 403,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ : X ⟶ T\nf₂ : T ⟶ Y\ninst✝ : IsIntegralHom f₂\nH : f = f₁ ≫ f₂\n⊢ toNormalization f ≫ normalizationDesc f f₁ f₂ H = f₁",
"usedConstants": [
"Subalgebra.instSetLike",
"AlgebraicGeometry.Presheafed... | by
refine Scheme.Cover.hom_ext (X.openCoverOfIsOpenCover _
(.comap (iSup_affineOpens_eq_top Y) f.base.hom)) _ _ fun U ↦ ?_
letI := (f.app U.1).hom.toAlgebra
refine (Scheme.Hom.ι_toNormalization_assoc ..).trans ?_
dsimp [normalizationOpenCover, normalizationDesc]
simp only [colimit.ι_desc, ← Spec.map_comp_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 46
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ f₂ : normalization f ⟶ T\ng : T ⟶ Y\ninst✝ : IsAffineHom g\nH₁ : toNormalization f ≫ f₁ = toNormalization f ≫ f₂\nhf₁ : f₁ ≫ g = fromNormalization f\nhf₂ : f₂ ≫ g = fromNormalization f\nU : (normalizationOpenCov... | have hf₀ : f₀ = toNormalization f ≫ f₂ := H₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 78
} | [
{
"pp": "case a\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)\nH : ∀ (i :... | convert_to x - (tensorEquivSum _ _ _ _ (f x)).map C ∈ Ideal.span _ using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convertTo_1 | Mathlib.Tactic.convertTo |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 337,
"column": 2
} | {
"line": 338,
"column": 96
} | [
{
"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⊢ (universalFactorizationMap R n m k hn).IsIntegral",
"usedConstants": [
"Iff.mpr",... | have : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ) :=
(MvPolynomial.tensorEquivSum ℤ (Fin m) (Fin k) ℤ).toRingEquiv.isDomain_iff.mpr inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 476,
"column": 6
} | {
"line": 477,
"column": 65
} | [
{
"pp": "case refine_2\nR : 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\nf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw represe... | ext; simp [MvPolynomial.universalFactorizationMapLiftEquiv, MvPolynomial.mapEquivMonic,
UniversalFactorizationRing.factor₂, coeff_freeMonic]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 476,
"column": 6
} | {
"line": 477,
"column": 65
} | [
{
"pp": "case refine_2\nR : 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\nf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw represe... | ext; simp [MvPolynomial.universalFactorizationMapLiftEquiv, MvPolynomial.mapEquivMonic,
UniversalFactorizationRing.factor₂, coeff_freeMonic]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 526,
"column": 44
} | {
"line": 526,
"column": 80
} | [
{
"pp": "case inr.inr.inr\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nh✝² : Nontrivial 𝓡\nh✝¹ : Nontrivial (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R)\nh✝ : Nontrivial R\nthis✝ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k)... | (monic_freeMonic R k).natDegree_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 670,
"column": 4
} | {
"line": 673,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nP : Ideal R\ninst✝ : P.IsPrime\nf : MonicDegreeEq P.ResidueField m\ng : MonicDegreeEq P.ResidueField k\nH✝ : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g\nHpq : IsCoprime ↑f ↑g\nφ : 𝓡' →ₐ[R] P.ResidueField := ⋯\nQ :... | · rw [H]
simp [homEquiv, UniversalFactorizationRing.homEquiv, factor₁,
MonicDegreeEq.map, Polynomial.map_map]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Finiteness.Descent | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 96
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u_3\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Module.FaithfullyFlat R T\ninst✝ : FiniteType T (T ⊗[R] S)\ns : Finset (T ⊗[R] S)\nhs : adjoin T ↑s = ⊤\nk : ↥s → ℕ\nt : (x : ↥s) → Fin (k x) → T... | let f : MvPolynomial (Σ x : s, Fin (k x)) R →ₐ[R] S := MvPolynomial.aeval (fun ⟨x, i⟩ ↦ m x i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Etale.Descent | {
"line": 50,
"column": 2
} | {
"line": 51,
"column": 78
} | [
{
"pp": "case subsingleton_kaehlerDifferential\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u_3\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Module.FaithfullyFlat R T\ninst✝ : FormallyUnramified T (T ⊗[R] S)\nx✝ : Algebra S (T ⊗[R] S) := TensorPro... | have : Subsingleton (T ⊗[R] Ω[S⁄R]) :=
(KaehlerDifferential.tensorKaehlerEquivBase R T S (T ⊗[R] S)).subsingleton | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Etale.Descent | {
"line": 47,
"column": 30
} | {
"line": 52,
"column": 63
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u_3\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Module.FaithfullyFlat R T\ninst✝ : FormallyUnramified T (T ⊗[R] S)\n⊢ FormallyUnramified R S",
"usedConstants": [
"Algebra.FormallyUnra... | by
constructor
let _ : Algebra S (T ⊗[R] S) := TensorProduct.rightAlgebra
have : Subsingleton (T ⊗[R] Ω[S⁄R]) :=
(KaehlerDifferential.tensorKaehlerEquivBase R T S (T ⊗[R] S)).subsingleton
exact Module.FaithfullyFlat.lTensor_reflects_triviality R T _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 415,
"column": 4
} | {
"line": 416,
"column": 48
} | [
{
"pp": "case refine_2.h.h\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : IsProper f\ny : ↥Y\nhx : (⇑f ⁻¹' {y}).Finite\nV : Y.Opens := { carrier := (⇑f '' (↑(Scheme.Hom.quasiFiniteLocus f))ᶜ)ᶜ, is_open' := ⋯ }\nx : ↥↑(f ⁻¹ᵁ V)\nthis : Scheme.Hom.QuasiFiniteAt f ((f ⁻¹ᵁ V).ι x)\n⊢ x ∈ ↑(Scheme.Hom.quasiFiniteLocus (f ∣_ V)) ... | rw [← Scheme.Hom.quasiFiniteAt_comp_iff_of_isOpenImmersion, ← morphismRestrict_ι,
Scheme.Hom.quasiFiniteAt_comp_iff] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 122,
"column": 10
} | {
"line": 123,
"column": 31
} | [
{
"pp": "case refine_3.refine_2\nK : Type u\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nG : Over (Spec (CommRingCat.of K))\ninst✝² : IsProper G.hom\ninst✝¹ : IsIntegral (G ⊗ G).left\ninst✝ : GrpObj G\nS : Scheme := Spec (CommRingCat.of K)\npoint : ↥S := IsLocalRing.closedPoint K\nhpoint : IsClosed {point}\nthis✝... | simp only [xe, γ, ← Scheme.Hom.comp_apply, ← Over.comp_left]
congr 6; ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 122,
"column": 10
} | {
"line": 123,
"column": 31
} | [
{
"pp": "case refine_3.refine_2\nK : Type u\ninst✝⁴ : Field K\ninst✝³ : IsAlgClosed K\nG : Over (Spec (CommRingCat.of K))\ninst✝² : IsProper G.hom\ninst✝¹ : IsIntegral (G ⊗ G).left\ninst✝ : GrpObj G\nS : Scheme := Spec (CommRingCat.of K)\npoint : ↥S := IsLocalRing.closedPoint K\nhpoint : IsClosed {point}\nthis✝... | simp only [xe, γ, ← Scheme.Hom.comp_apply, ← Over.comp_left]
congr 6; ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.IrreducibleComponent | {
"line": 58,
"column": 79
} | {
"line": 60,
"column": 5
} | [
{
"pp": "X : Scheme\nZ : Set ↥X\nhZ : Z ∈ irreducibleComponents ↥X\ninst✝ : IsNoetherian X\n⊢ X.irreducibleComponentIdeal Z hZ = Hom.ker (X.irreducibleComponentOpen Z).ι",
"usedConstants": [
"Semiring.toModule",
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.ca... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 161,
"column": 40
} | {
"line": 161,
"column": 83
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nR : CommRingCat\ni : Spec R ⟶ Y\nw✝ : IsOpenImmersion i\ny' : ↥(Spec R)\n⊢ IsAffine.finrank (pullback.snd (pullback.snd f g) i) y' = IsAffine.finrank (pullback.snd f (i ≫ g)) y'",
"usedConstants": [
"CategoryTheory.Lim... | ← pullbackLeftPullbackSndIso_hom_snd f g i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 57
} | [
{
"pp": "R S : CommRingCat\nx : ↥(Spec S)\nφ : R ⟶ S\ninst✝¹ : Flat (Spec.map φ)\ninst✝ : IsFinite (Spec.map φ)\n⊢ 1 ≤ finrank (Spec.map φ) ((Spec.map φ) x)",
"usedConstants": [
"AlgebraicGeometry.Flat",
"RingHom.Flat",
"AlgebraicGeometry.Spec",
"CommRingCat.Hom.hom",
"CommRing... | simp only [IsFinite.SpecMap_iff, Flat.SpecMap_iff] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 59
} | [
{
"pp": "case surj\nR S : CommRingCat\nφ : R ⟶ S\ninst✝¹ : Flat (Spec.map φ)\ninst✝ : IsFinite (Spec.map φ)\nx : ↥(Spec R)\nh : 1 x ≤ finrank (Spec.map φ) x\n⊢ ∃ a, (Spec.map φ) a = x",
"usedConstants": [
"AlgebraicGeometry.Flat",
"RingHom.Flat",
"AlgebraicGeometry.Spec",
"CommRingCa... | simp only [IsFinite.SpecMap_iff, Flat.SpecMap_iff] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 57
} | [
{
"pp": "R S : CommRingCat\nφ : R ⟶ S\ninst✝¹ : Flat (Spec.map φ)\ninst✝ : IsFinite (Spec.map φ)\nh : finrank (Spec.map φ) = 1\n⊢ IsIso (Spec.map φ)",
"usedConstants": [
"AlgebraicGeometry.Flat",
"RingHom.Flat",
"AlgebraicGeometry.Spec",
"CommRingCat.Hom.hom",
"_private.Mathlib... | simp only [IsFinite.SpecMap_iff, Flat.SpecMap_iff] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 90,
"column": 75
} | {
"line": 90,
"column": 95
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nV : Scheme\nf : (∐ fun i ↦ Spec (R i)) ⟶ V\ninst✝¹ : CompactSpace ↥V\nI : Ideal ↑(CommRingCat.of ((i : ι) → ↑(R i)))\ne : V ≅ Spec (CommRingCat.of (↑(CommRingCat.of ((i : ι) → ↑(R i))) ⧸ I))\ninst✝ : IsImmersion (e.hom ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)... | rwa [Category.assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 90,
"column": 75
} | {
"line": 90,
"column": 95
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nV : Scheme\nf : (∐ fun i ↦ Spec (R i)) ⟶ V\ninst✝¹ : CompactSpace ↥V\nI : Ideal ↑(CommRingCat.of ((i : ι) → ↑(R i)))\ne : V ≅ Spec (CommRingCat.of (↑(CommRingCat.of ((i : ι) → ↑(R i))) ⧸ I))\ninst✝ : IsImmersion (e.hom ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)... | rwa [Category.assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.PointsPi | {
"line": 90,
"column": 75
} | {
"line": 90,
"column": 95
} | [
{
"pp": "ι : Type u\nR : ι → CommRingCat\nV : Scheme\nf : (∐ fun i ↦ Spec (R i)) ⟶ V\ninst✝¹ : CompactSpace ↥V\nI : Ideal ↑(CommRingCat.of ((i : ι) → ↑(R i)))\ne : V ≅ Spec (CommRingCat.of (↑(CommRingCat.of ((i : ι) → ↑(R i))) ⧸ I))\ninst✝ : IsImmersion (e.hom ≫ Spec.map (CommRingCat.ofHom (Ideal.Quotient.mk I)... | rwa [Category.assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 386,
"column": 62
} | {
"line": 386,
"column": 84
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nz : ProjectiveSpectrum 𝒜\nhz : f ∉ z.asHomogeneousIdeal\ni : ℕ\nhi : (GradedRing.proj 𝒜 i) f ∉ z.asHomogeneousIdeal\n⊢ z ∈ basicOpen 𝒜 ((GradedRing.proj 𝒜... | by rwa [mem_basicOpen] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 40
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : IsOrderedCancelAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nI_ne_top : I ≠ ⊤\nhomogene... | rw [← sum_support_decompose 𝒜 x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 101,
"column": 8
} | {
"line": 101,
"column": 45
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : IsOrderedCancelAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nI_ne_top : I ≠ ⊤\nhomogene... | rw [eq_sub_of_add_eq eq_add_sum.symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 132,
"column": 8
} | {
"line": 132,
"column": 31
} | [
{
"pp": "case inr\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : IsOrderedCancelAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nI_ne_top : I ≠ ⊤... | · apply neither_mem.2 h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 41
} | [
{
"pp": "case a\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : IsOrderedCancelAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\n⊢ InfSet.sInf {J |... | · exact sInf_le_sInf fun J => And.right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf | {
"line": 84,
"column": 9
} | {
"line": 84,
"column": 83
} | [
{
"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\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuive... | by rintro V U i f ⟨j, r, s, h, w⟩; exact ⟨j, r, s, (h <| i ·), (w <| i ·)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 318,
"column": 2
} | {
"line": 319,
"column": 33
} | [
{
"pp": "case e_a.a\nA : 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 * m)\n⊢ HomogeneousLocalization.val\n ... | simp only [HomogeneousLocalization.val_mk, HomogeneousLocalization.val_pow,
Localization.mk_pow, pow_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 362,
"column": 8
} | {
"line": 362,
"column": 18
} | [
{
"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\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\na b : A\nha : a ∈ carrier f_deg q\nhb : b ∈ carrier f_deg q\ni j : ℕ\nh2 : ¬m + m < j\nh1 : ¬j ≤... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 614,
"column": 61
} | {
"line": 614,
"column": 66
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nf : A\nd : ι\nn : ℕ\nhf : f ∈ 𝒜 d\nx : A\nhx : x ∈ 𝒜 (n • d)\n⊢ f ^ n ∈ Submonoid.powers f",
"us... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 614,
"column": 61
} | {
"line": 614,
"column": 66
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nf : A\nd : ι\nn : ℕ\nhf : f ∈ 𝒜 d\nx : A\nhx : x ∈ 𝒜 (n • d)\n⊢ f ^ n ∈ Submonoid.powers f",
"us... | use n | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 614,
"column": 61
} | {
"line": 614,
"column": 66
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nf : A\nd : ι\nn : ℕ\nhf : f ∈ 𝒜 d\nx : A\nhx : x ∈ 𝒜 (n • d)\n⊢ f ^ n ∈ Submonoid.powers f",
"us... | use n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Functor | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 24
} | [
{
"pp": "A B σ τ : Type u\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₊ ≤ HomogeneousIdeal.map f (𝒜₊)\np : Projectiv... | ← Iso.eq_inv_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 26
} | [
{
"pp": "R : CommRingCat\nM : ModuleCat ↑R\n⊢ IsIso tilde.adjunction.unit",
"usedConstants": [
"AlgebraicGeometry.Spec",
"CategoryTheory.Functor",
"CategoryTheory.IsIso",
"AlgebraicGeometry.Scheme.Modules.instCategory",
"CommRingCat.carrier",
"ModuleCat",
"Algebraic... | dsimp [tilde.adjunction] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 844,
"column": 2
} | {
"line": 847,
"column": 37
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ne : ι\nf g : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\na : ↥(𝒜 0)\n⊢ (awayMap 𝒜 hg hx) ((fromZeroRing... | ext
simp only [fromZeroRingHom, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk,
val_awayMap, val_mk]
convert! IsLocalization.lift_eq _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 844,
"column": 2
} | {
"line": 847,
"column": 37
} | [
{
"pp": "ι : Type u_1\nA : Type u_2\nσ : Type u_3\ninst✝⁵ : CommRing A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubgroupClass σ A\ninst✝² : AddCommMonoid ι\ninst✝¹ : DecidableEq ι\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ne : ι\nf g : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\na : ↥(𝒜 0)\n⊢ (awayMap 𝒜 hg hx) ((fromZeroRing... | ext
simp only [fromZeroRingHom, RingHom.coe_mk, MonoidHom.coe_mk, OneHom.coe_mk,
val_awayMap, val_mk]
convert! IsLocalization.lift_eq _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 599,
"column": 8
} | {
"line": 599,
"column": 66
} | [
{
"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\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nx : failed to pretty print expression (use 'set_option pp.r... | obtain ⟨s, rfl⟩ := HomogeneousLocalization.mk_surjective s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 79
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\nthis✝ : Invertible x := invertibleOfNonzero hx0\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : R.toSubring < S.toSubring\np : Polynomial ↥R.toSubring\nhp : p.leadingCoeff - 1 ∈ maxi... | simpa using .inr hpx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 79
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\nthis✝ : Invertible x := invertibleOfNonzero hx0\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : R.toSubring < S.toSubring\np : Polynomial ↥R.toSubring\nhp : p.leadingCoeff - 1 ∈ maxi... | simpa using .inr hpx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 79
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\nthis✝ : Invertible x := invertibleOfNonzero hx0\nS : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x]\nthis : R.toSubring < S.toSubring\np : Polynomial ↥R.toSubring\nhp : p.leadingCoeff - 1 ∈ maxi... | simpa using .inr hpx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 184,
"column": 57
} | {
"line": 184,
"column": 77
} | [
{
"pp": "K : Type u_3\ninst✝¹ : Field K\nx : K\nR : LocalSubring K\nhxR : x ∉ R.toSubring\ninst✝ : IsIntegrallyClosedIn (↥R.toSubring) K\nhx0 : x ≠ 0\nthis : Invertible x := invertibleOfNonzero hx0\nB : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x⁻¹]\nxinv : ↥B.toSubring := ⟨x⁻¹, ⋯⟩\neq : Ideal.map (algebraM... | simpa using .inr hpx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 184,
"column": 57
} | {
"line": 184,
"column": 77
} | [
{
"pp": "K : Type u_3\ninst✝¹ : Field K\nx : K\nR : LocalSubring K\nhxR : x ∉ R.toSubring\ninst✝ : IsIntegrallyClosedIn (↥R.toSubring) K\nhx0 : x ≠ 0\nthis : Invertible x := invertibleOfNonzero hx0\nB : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x⁻¹]\nxinv : ↥B.toSubring := ⟨x⁻¹, ⋯⟩\neq : Ideal.map (algebraM... | simpa using .inr hpx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 184,
"column": 57
} | {
"line": 184,
"column": 77
} | [
{
"pp": "K : Type u_3\ninst✝¹ : Field K\nx : K\nR : LocalSubring K\nhxR : x ∉ R.toSubring\ninst✝ : IsIntegrallyClosedIn (↥R.toSubring) K\nhx0 : x ≠ 0\nthis : Invertible x := invertibleOfNonzero hx0\nB : Subalgebra (↥R.toSubring) K := (↥R.toSubring)[x⁻¹]\nxinv : ↥B.toSubring := ⟨x⁻¹, ⋯⟩\neq : Ideal.map (algebraM... | simpa using .inr hpx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Elements | {
"line": 79,
"column": 80
} | {
"line": 82,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nA : C ⥤ Type w\nI : Type u₁\ninst✝³ : Category.{v₁, u₁} I\ninst✝² : Small.{w, u₁} I\nF : I ⥤ A.Elements\ninst✝¹ : HasLimitsOfShape I C\ninst✝ : PreservesLimitsOfShape I A\ni : I\n⊢ (hom (A.map (limit.π (F ⋙ π A) i))) (liftedConeElement F) = (F.obj i).snd",
"u... | by
have := congr_hom
(preservesLimitIso_inv_π A (F ⋙ π A) i) (liftedConeElement' F)
simp [liftedConeElement, ← comp_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 341,
"column": 59
} | {
"line": 355,
"column": 53
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | by
rintro ⟨⟨i₁, i₂⟩, j⟩
dsimp at i₁ i₂ j ⊢
refine Presheaf.IsSheaf.hom_ext
hG₀ ⟨_, cover_lift F J₀ _
(J.pullback_stable (F.map ((data X).p₁ j) ≫ (data X).f i₁) S.2)⟩ _ _ ?_
rintro ⟨W₀, a, ha⟩
dsimp
simp only [assoc, ← Functor.map_comp, ← op_comp]
have ha₁ : S (F.map (a ≫ (data ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 135,
"column": 10
} | {
"line": 138,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : Y ⟶ X\nHf : IsColimit (Sieve.generateSingleton f).arrows.cocone\nW : C\ne : Y ⟶ W\nh : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e\n⊢ ∀ ⦃X_1 Y_1 : (Sieve.generateSingleton f).arrows.category⦄ (f_1 : X_1 ⟶ Y_1),\n ((Sieve.generateSi... | rintro ⟨A, hA⟩ ⟨B, hB⟩ ⟨q : A ⟶ B⟩
dsimp; simp only [← Category.assoc, Category.comp_id]
apply h
rw [Category.assoc, hB.choose_spec, hA.choose_spec, Over.w] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 135,
"column": 10
} | {
"line": 138,
"column": 69
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf : Y ⟶ X\nHf : IsColimit (Sieve.generateSingleton f).arrows.cocone\nW : C\ne : Y ⟶ W\nh : ∀ {Z : C} (g₁ g₂ : Z ⟶ Y), g₁ ≫ f = g₂ ≫ f → g₁ ≫ e = g₂ ≫ e\n⊢ ∀ ⦃X_1 Y_1 : (Sieve.generateSingleton f).arrows.category⦄ (f_1 : X_1 ⟶ Y_1),\n ((Sieve.generateSi... | rintro ⟨A, hA⟩ ⟨B, hB⟩ ⟨q : A ⟶ B⟩
dsimp; simp only [← Category.assoc, Category.comp_id]
apply h
rw [Category.assoc, hB.choose_spec, hA.choose_spec, Over.w] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 44
} | [
{
"pp": "case hR\nP : MorphismProperty Scheme\nX Y : Scheme\nf : X ⟶ Y\nhf : P f\ninst✝¹ : Surjective f\ninst✝ : QuasiCompact f\n⊢ Presieve.singleton f ∈ (propQCPrecoverage P).coverings Y",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.singleton_mem_propQCPrecoverage"
]
}
] | exact f.singleton_mem_propQCPrecoverage hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 41
} | [
{
"pp": "σ : 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 O K\... | refine zero_lt_iff.mpr fun hKmax ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Abelian.CommSq | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 15
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX₁ X₂ X₃ X₄ : C\nt : X₁ ⟶ X₂\nl : X₁ ⟶ X₃\nr : X₂ ⟶ X₄\nb : X₃ ⟶ X₄\nsq : IsPullback t l r b\n⊢ ∀ (P : C) (g : P ⟶ cokernel t), g ≫ cokernel.map t b l r ⋯ = 0 → g = 0",
"usedConstants": []
}
] | intro A₀ z hz | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition | {
"line": 157,
"column": 4
} | {
"line": 158,
"column": 25
} | [
{
"pp": "case inr\nC : Type u\ninst✝⁹ : Category.{v, u} C\nD : Type u'\ninst✝⁸ : Category.{v', u'} D\nW : MorphismProperty C\nJ : Type w\ninst✝⁷ : LinearOrder J\ninst✝⁶ : SuccOrder J\ninst✝⁵ : OrderBot J\ninst✝⁴ : WellFoundedLT J\nJ' : Type w'\ninst✝³ : LinearOrder J'\ninst✝² : SuccOrder J'\ninst✝¹ : OrderBot J... | · rw [isMax_iff_eq_top] at hj
exact (hj rfl).elim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.MorphismProperty.TransfiniteComposition | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 69
} | [
{
"pp": "case h\nC : Type u\ninst✝⁸ : Category.{v, u} C\nW : MorphismProperty C\nJ : Type w\ninst✝⁷ : LinearOrder J\ninst✝⁶ : SuccOrder J\ninst✝⁵ : OrderBot J\ninst✝⁴ : WellFoundedLT J\nJ' : Type w'\ninst✝³ : LinearOrder J'\ninst✝² : SuccOrder J'\ninst✝¹ : OrderBot J'\ninst✝ : WellFoundedLT J'\ne : J ≃o J'\nX✝ ... | exact ⟨fun ⟨h⟩ ↦ ⟨h.ofOrderIso e.symm⟩, fun ⟨h⟩ ↦ ⟨h.ofOrderIso e⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 369,
"column": 6
} | {
"line": 369,
"column": 28
} | [
{
"pp": "case isMin\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\nh : IsMin j\n⊢ ∀ (iter₁ iter₂ : Φ.Iteration j), iter₁.F = iter₂.F",
"usedConstant... | obtain rfl := h.eq_bot | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 236,
"column": 4
} | {
"line": 236,
"column": 37
} | [
{
"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\nhj : Order.IsSuccLimit j\ne : (i : J) → i < j → d.Extension val₀ i\ni : J\nhi : Order.IsSuccLimit i\nhij : i ≤ j\n⊢ (Concre... | obtain hij' | rfl := hij.lt_or_eq | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 388,
"column": 6
} | {
"line": 388,
"column": 33
} | [
{
"pp": "p b : ℕ\nhp : Fact (Nat.Prime p)\ndvd : p ∣ b\n⊢ padicValNat p (b / p) = padicValNat p b - 1",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"HSub.hSub",
"id",
"padicValNat",
"HDiv.hDiv",
"instSubNat",
"instOfNatNat",
"instHSub",
... | padicValNat.div_of_dvd dvd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 399,
"column": 6
} | {
"line": 399,
"column": 33
} | [
{
"pp": "p a b : ℕ\nhp : Fact (Nat.Prime p)\ndvd : p ^ a ∣ b\n⊢ padicValNat p (b / p ^ a) = padicValNat p b - a",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"id",
"padicValNat",
"HDiv.hDiv",
"instSubNat",
"M... | padicValNat.div_of_dvd dvd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 403,
"column": 6
} | {
"line": 403,
"column": 33
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\ncpm : p.Coprime m\nb : ℕ\ndvd : m ∣ b\n⊢ padicValNat p (b / m) = padicValNat p b",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"HSub.hSub",
"id",
"padicValNat",
"HDiv.hDiv",
"instSubNat",
"instHSub... | padicValNat.div_of_dvd dvd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 235,
"column": 31
} | {
"line": 235,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nm : J\nhm : Order.IsSuccLimit m\nthis : Non... | dsimp [c] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nm : J\nhm : Order.IsSuccLimit m\nthis✝ : No... | dsimp [c] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 281,
"column": 2
} | {
"line": 282,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nG : C\ninst✝⁶ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁵ : IsGrothendieckAbelian.{w, v, u} C\nA : C\nf : A ⟶ X\ninst✝⁴ : Mono f\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nhj : transfiniteIte... | rw [assoc, Subobject.underlyingIso_hom_comp_eq_mk, Subobject.ofLE_arrow,
Subobject.ofLE_arrow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 256,
"column": 51
} | {
"line": 256,
"column": 67
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\n⊢ padicNorm p ↑m = 1 ↔ padicNorm p ↑↑m = 1",
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrArg",
"Rat",
"AddGroupWithOne.toAddMonoidWithOne",
"Rat.instIntCast",
"id",
... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 259,
"column": 51
} | {
"line": 259,
"column": 67
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\n⊢ padicNorm p ↑m < 1 ↔ padicNorm p ↑↑m < 1",
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"congrArg",
"Rat",
"AddGroupWithOne.toAddMonoidWithOne",
"Rat.instIntCast",
"id",
... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 41
} | [
{
"pp": "case inr\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\nh : (Int.padicValuation p) ↑x.den < 1\n⊢ (Int.padicValuation p) x.num ≤ (Int.padicValuation p) ↑x.den ↔ (Int.padicValuation p) ↑x.den = 1",
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"_private.Mathlib.NumberTheory.Padi... | simp only [h.ne, iff_false, not_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.Compatibility | {
"line": 103,
"column": 87
} | {
"line": 105,
"column": 21
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\nB' : Type u_4\ninst✝² : Category.{v_1, u_1} A\ninst✝¹ : Category.{v_2, u_2} A'\ninst✝ : Category.{v_4, u_4} B'\neA : A ≌ A'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\n⊢ (equivalence₁ hF).unitIso = equivalence₁UnitIso hF",
"usedConstants": [
"Alge... | by
ext X
simp [equivalence₁] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.DoldKan.Compatibility | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 35
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{v_1, u_1} A\ninst✝² : Category.{v_2, u_2} A'\ninst✝¹ : Category.{v_3, u_3} B\ninst✝ : Category.{v_4, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\n⊢ (equivalence₂ eB hF).unitIso... | ext X
simp [equivalence₂, equivalence₁] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Compatibility | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 35
} | [
{
"pp": "A : Type u_1\nA' : Type u_2\nB : Type u_3\nB' : Type u_4\ninst✝³ : Category.{v_1, u_1} A\ninst✝² : Category.{v_2, u_2} A'\ninst✝¹ : Category.{v_3, u_3} B\ninst✝ : Category.{v_4, u_4} B'\neA : A ≌ A'\neB : B ≌ B'\ne' : A' ≌ B'\nF : A ⥤ B'\nhF : eA.functor ⋙ e'.functor ≅ F\n⊢ (equivalence₂ eB hF).unitIso... | ext X
simp [equivalence₂, equivalence₁] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialObject.Split | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 27
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\nZ : C\nΔ : SimplexCategoryᵒᵖ\nF : (A : IndexSet Δ) → s.N (unop A.fst).len ⟶ Z\nA : IndexSet Δ\n⊢ (s.cofan Δ).inj A ≫ s.desc Δ F = F A",
"usedConstants": [
"CategoryTheory.SimplicialObject.Splitting.isColimit... | apply Cofan.IsColimit.fac | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicTopology.SimplicialObject.Split | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 27
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\nZ : C\nΔ : SimplexCategoryᵒᵖ\nF : (A : IndexSet Δ) → s.N (unop A.fst).len ⟶ Z\nA : IndexSet Δ\n⊢ (s.cofan Δ).inj A ≫ s.desc Δ F = F A",
"usedConstants": [
"CategoryTheory.SimplicialObject.Splitting.isColimit... | apply Cofan.IsColimit.fac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.Split | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 27
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\nZ : C\nΔ : SimplexCategoryᵒᵖ\nF : (A : IndexSet Δ) → s.N (unop A.fst).len ⟶ Z\nA : IndexSet Δ\n⊢ (s.cofan Δ).inj A ≫ s.desc Δ F = F A",
"usedConstants": [
"CategoryTheory.SimplicialObject.Splitting.isColimit... | apply Cofan.IsColimit.fac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 230,
"column": 19
} | {
"line": 231,
"column": 52
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\nK✝ K' : ChainComplex C ℕ\nf : K✝ ⟶ K'\nΔ✝ Δ' Δ'' : SimplexCategory\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nΔ : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\n⊢ Cofan.inj (colimit.cocone (Discrete.functor (Splitting.summan... | Γ₀.Obj.map_on_summand₀ K (SimplicialObject.Splitting.IndexSet.id A.1)
(show A.e ≫ 𝟙 _ = A.e.op.unop ≫ 𝟙 _ by rfl), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 247,
"column": 2
} | {
"line": 248,
"column": 90
} | [
{
"pp": "C : Type u_1\ninst✝⁴ : Category.{v_1, u_1} C\ninst✝³ : Preadditive C\nK : ChainComplex C ℕ\ninst✝² : HasFiniteCoproducts C\nΔ Δ' : SimplexCategoryᵒᵖ\nA : Splitting.IndexSet Δ\nθ : Δ ⟶ Δ'\nΔ'' : SimplexCategory\ne : unop Δ' ⟶ Δ''\ni : Δ'' ⟶ unop A.fst\ninst✝¹ : Epi e\ninst✝ : Mono i\nfac : e ≫ i = θ.uno... | rw [Γ₀.Obj.map_on_summand₀ K (Splitting.IndexSet.id A.1)
(show e ≫ i = ((Splitting.IndexSet.e A).op ≫ θ).unop ≫ 𝟙 _ by rw [comp_id, fac]; rfl)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 31
} | [
{
"pp": "case h.h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nn : ℕ\ni : Fin (n + 1)\nA : IndexSet (op ⦋n⦌)\n⊢ ¬(A.epiComp (SimplexCategory.σ i).op).EqId",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.op_mono_of_epi",
"Cate... | rw [IndexSet.eqId_iff_len_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.DoldKan.NCompGamma | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 26
} | [
{
"pp": "case pos.hj₂\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nm k : ℕ\nh₁ : ⦋m⦌.len ≠ m + k + 1 + 1\nj₂ : Fin (m + k + 2)\ni : ⦋m⦌ ⟶ ⦋m + k⦌\n⊢ m + k + 1 + 2 ≤ ↑j₂.succ + (m + k + 1 + 1)",
"usedConstants": [
"Fin.succ",
"id",
"instOfNat... | simp only [Fin.succ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.NReflectsIso | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 66
} | [
{
"pp": "case refine_1.refine_2.h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP : Karoubi (SimplicialObject C)\nn : ℕ\n⊢ (((N₂ ⋙ (karoubiChainComplexEquivalence C ℕ).functor).obj P).d (n + 1) n ≫ eqToHom ⋯).f =\n (eqToHom ⋯ ≫\n ((karoubiFunctorCategoryEmbedding SimplexCategor... | have h := (AlternatingFaceMapComplex.map P.p).comm (n + 1) n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 200,
"column": 28
} | {
"line": 200,
"column": 41
} | [
{
"pp": "case h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nn : ℕ\n⊢ (s.cofan (op ⦋n⦌)).inj (IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n ≫ PInfty.f n =\n (s.cofan (op ⦋n⦌)).inj (IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n",
"usedConstants": [
"Eq.mpr... | PInfty_f_idem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Groupoid.Grpd.Basic | {
"line": 111,
"column": 5
} | {
"line": 115,
"column": 11
} | [
{
"pp": "J : Type u\nF : J → Grpd\n⊢ ∀ (s : Limits.Fan F) (m : s.pt ⟶ (piLimitFan F).pt),\n (∀ (j : J), m ≫ (piLimitFan F).proj j = s.proj j) → m = (fun s ↦ Functor.pi' fun j ↦ s.proj j) s",
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.Limits.Fan.proj",
"CategoryTheory.C... | by
intro s m w
apply Functor.pi_ext
intro j; specialize w j
simpa | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.FundamentalGroupoid.Product | {
"line": 188,
"column": 4
} | {
"line": 192,
"column": 98
} | [
{
"pp": "A : TopCat\nB : TopCat\n⊢ (projLeft A B).prod' (projRight A B) ≫ prodToProdTop A B = 𝟙 (π.obj (TopCat.of (↑A × ↑B)))",
"usedConstants": [
"Eq.mpr",
"Path.Homotopic.projLeft",
"CategoryTheory.Functor.hext",
"FundamentalGroupoid.casesOn",
"Path.Homotopic.prod_projLeft_p... | change (projLeft A B).prod' (projRight A B) ⋙ prodToProdTop A B = 𝟭 _
apply CategoryTheory.Functor.hext
· intros; apply FundamentalGroupoid.ext; apply Prod.ext <;> simp <;> rfl
rintro ⟨x₀, x₁⟩ ⟨y₀, y₁⟩ f
simpa [-Path.Homotopic.prod_projLeft_projRight] using Path.Homotopic.prod_projLeft_projRight f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.FundamentalGroupoid.Product | {
"line": 188,
"column": 4
} | {
"line": 192,
"column": 98
} | [
{
"pp": "A : TopCat\nB : TopCat\n⊢ (projLeft A B).prod' (projRight A B) ≫ prodToProdTop A B = 𝟙 (π.obj (TopCat.of (↑A × ↑B)))",
"usedConstants": [
"Eq.mpr",
"Path.Homotopic.projLeft",
"CategoryTheory.Functor.hext",
"FundamentalGroupoid.casesOn",
"Path.Homotopic.prod_projLeft_p... | change (projLeft A B).prod' (projRight A B) ⋙ prodToProdTop A B = 𝟭 _
apply CategoryTheory.Functor.hext
· intros; apply FundamentalGroupoid.ext; apply Prod.ext <;> simp <;> rfl
rintro ⟨x₀, x₁⟩ ⟨y₀, y₁⟩ f
simpa [-Path.Homotopic.prod_projLeft_projRight] using Path.Homotopic.prod_projLeft_projRight f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Localization.DerivabilityStructure.Constructor | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 19
} | [
{
"pp": "case h\nC₁ : Type u_1\nC₂ : Type u_2\ninst✝⁶ : Category.{v_1, u_1} C₁\ninst✝⁵ : Category.{v_2, u_2} C₂\nW₁ : MorphismProperty C₁\nW₂ : MorphismProperty C₂\nΦ : LocalizerMorphism W₁ W₂\ninst✝⁴ : ∀ (X₂ : C₂), IsConnected (Φ.RightResolution X₂)\ninst✝³ : Φ.arrow.HasRightResolutions\ninst✝² : W₂.ContainsId... | isoOfHom_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.FundamentalLemma | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 40
} | [
{
"pp": "C : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\ninst✝⁴ : ModelCategory C\nH : Type u_2\ninst✝³ : Category.{v_2, u_2} H\nL : C ⥤ H\ninst✝² : L.IsLocalization (weakEquivalences C)\nX Y : C\ninst✝¹ : IsCofibrant X\ninst✝ : IsFibrant Y\nf g : X ⟶ Y\nh : L.map f = L.map g\n⊢ RightHomotopyRel f g",
"usedCo... | rw [← RightHomotopyClass.mk_eq_mk_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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