module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 159,
"column": 44
} | {
"line": 159,
"column": 76
} | {
"line": 159,
"column": 76
} | [
{
"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... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 25
} | {
"line": 412,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : QuasiFiniteAt R q\nR' : Type u\nw✝⁶ : CommRing R'\nw✝⁵ : Algebra R R'\nw✝⁴ : Etale R R'\nP : ... | [
"R : Type u\nS : Type v\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : FiniteType R S\np : Ideal R\ninst✝³ : p.IsPrime\nq : Ideal S\ninst✝² : q.IsPrime\ninst✝¹ : q.LiesOver p\ninst✝ : QuasiFiniteAt R q\nR' : Type u\nw✝⁶ : CommRing R'\nw✝⁵ : Algebra R R'\nw✝⁴ : Etale R R'\nP : Ideal R'\nw✝... | change f ∉ P'.under _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.RingTheory.Ideal.CotangentBaseChange | {
"line": 79,
"column": 4
} | {
"line": 82,
"column": 48
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case tmul\nR : 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\nt : T\nx : ↥(Submodule.restrictScalars R I)\n⊢ ∃ a,\n (tenso... | [] | use t ⊗ₜ I.toCotangent x
apply Ideal.cotangentToQuotientSquare_injective
simp [-AlgHom.toRingHom_eq_coe, tensorCotangentHom_tmul, Algebra.smul_def,
← Ideal.Quotient.mk_algebraMap, ← map_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.CotangentBaseChange | {
"line": 79,
"column": 4
} | {
"line": 82,
"column": 48
} | {
"line": 84,
"column": 0
} | [
{
"pp": "case tmul\nR : 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\nt : T\nx : ↥(Submodule.restrictScalars R I)\n⊢ ∃ a,\n (tenso... | [] | use t ⊗ₜ I.toCotangent x
apply Ideal.cotangentToQuotientSquare_injective
simp [-AlgHom.toRingHom_eq_coe, tensorCotangentHom_tmul, Algebra.smul_def,
← Ideal.Quotient.mk_algebraMap, ← map_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 435,
"column": 2
} | {
"line": 435,
"column": 77
} | {
"line": 436,
"column": 2
} | [
{
"pp": "X Y S : Scheme\nf : X ⟶ Y\ng : Y ⟶ S\ns : ↥S\nH : (⇑f '' ⇑(f ≫ g) ⁻¹' {s}).Finite\ninst✝² : IsProper (f ≫ g)\ninst✝¹ : IsSeparated g\ninst✝ : LocallyOfFiniteType g\nthis✝ : IsProper f\nthis : IsProper (Scheme.Hom.imageι f ≫ g)\n⊢ ∃ U, s ∈ U ∧ IsFinite ((Scheme.Hom.imageι f ≫ g) ∣_ U)",
"ppTerm": "?... | [
"X Y S : Scheme\nf : X ⟶ Y\ng : Y ⟶ S\ns : ↥S\nH : (⇑f '' ⇑(f ≫ g) ⁻¹' {s}).Finite\ninst✝² : IsProper (f ≫ g)\ninst✝¹ : IsSeparated g\ninst✝ : LocallyOfFiniteType g\nthis✝ : IsProper f\nthis : IsProper (Scheme.Hom.imageι f ≫ g)\n⊢ (⇑(Scheme.Hom.imageι f ≫ g) ⁻¹' {s}).Finite"
] | refine exists_isFinite_morphismRestrict_of_finite_preimage_singleton _ _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Extension.Cotangent.BaseChange | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 61
} | {
"line": 76,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Extension R S\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nt : T\ns : S\nx : Ω[P.Ring⁄R]\n⊢ (P.tensorCotangentSpace T) (t ⊗ₜ[R] (s ⊗ₜ[P.Ring] x)) =\n t ⊗ₜ[R] s ⊗ₜ[P.baseChange.Ring] (KaehlerDi... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Extension R S\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nt : T\ns : S\nx : Ω[P.Ring⁄R]\n⊢ (AlgebraTensorModule.congr (LinearEquiv.refl P.baseChange.Ring (T ⊗[R] S))\n (KaehlerDifferential.tensorKaeh... | simp only [tensorCotangentSpace, LinearEquiv.trans_apply, LinearEquiv.restrictScalars_apply,
← mk_apply s x, IsTensorProduct.assocOfMapSMul_symm_tmul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Etale.Descent | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 63
} | {
"line": 54,
"column": 0
} | [
{
"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)\nx✝ : Algebra S (T ⊗[R] S) := TensorProduct.rightAlgebra\nthis : Subsingleton ... | [] | exact Module.FaithfullyFlat.lTensor_reflects_triviality R T _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Etale.Descent | {
"line": 70,
"column": 54
} | {
"line": 70,
"column": 92
} | {
"line": 71,
"column": 4
} | [
{
"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✝ : Smooth T (T ⊗[R] S)\nthis : FinitePresentation R S\nx✝ : Algebra T (S ⊗[R] T) := TensorProduct.rightAlgebra\n⊢... | [] | by simp [RingHom.algebraMap_toAlgebra] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 581,
"column": 6
} | {
"line": 584,
"column": 62
} | {
"line": 584,
"column": 62
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝³ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (max... | [] | have : P'' ⊔ Ideal.span {φ e} = P'' := by simpa [Ideal.span_le]
rw [← Ideal.under_under (B := R'' ⊗[R] S)]
simpa [Ideal.under, Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective,
← RingHom.ker_eq_comap_bot, this] using P''.over_def Q | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 581,
"column": 6
} | {
"line": 584,
"column": 62
} | {
"line": 584,
"column": 62
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Finite R S\np : Ideal R\ninst✝ : p.IsPrime\nthis✝³ : IsArtinianRing (p.ResidueField ⊗[R] S)\nhpSfin : (p.primesOver S).Finite\nn✝ : ℕ\nIH :\n ∀ m < n✝ + 1,\n ∀ {R : Type u} {S : Type (max... | [] | have : P'' ⊔ Ideal.span {φ e} = P'' := by simpa [Ideal.span_le]
rw [← Ideal.under_under (B := R'' ⊗[R] S)]
simpa [Ideal.under, Ideal.comap_map_of_surjective _ Ideal.Quotient.mk_surjective,
← RingHom.ker_eq_comap_bot, this] using P''.over_def Q | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 153,
"column": 6
} | {
"line": 153,
"column": 10
} | {
"line": 153,
"column": 11
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ ord R (a * x) = ord R x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Ring.ord.eq_1",
"HMul.hMul",
"Ring.ord",
"Submodule.Quotient.addCommGroup",
... | [
"R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ Module.length R (R ⧸ span {a * x}) = ord R x"
] | ord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 153,
"column": 11
} | {
"line": 153,
"column": 15
} | {
"line": 153,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ Module.length R (R ⧸ span {a * x}) = ord R x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Ring.ord.eq_1",
"HMul.hMul",
"Ring.ord",
"Submodule.Quot... | [
"R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ Module.length R (R ⧸ span {a * x}) = Module.length R (R ⧸ span {x})"
] | ord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 10
} | {
"line": 157,
"column": 11
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ ord R (x * a) = ord R x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Ring.ord.eq_1",
"HMul.hMul",
"Ring.ord",
"Submodule.Quotient.addCommGroup",
... | [
"R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ Module.length R (R ⧸ span {x * a}) = ord R x"
] | ord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 157,
"column": 11
} | {
"line": 157,
"column": 15
} | {
"line": 157,
"column": 16
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ Module.length R (R ⧸ span {x * a}) = ord R x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Ring.ord.eq_1",
"HMul.hMul",
"Ring.ord",
"Submodule.Quot... | [
"R : Type u_1\ninst✝ : CommRing R\na : R\nh : IsUnit a\nx : R\n⊢ Module.length R (R ⧸ span {x * a}) = Module.length R (R ⧸ span {x})"
] | ord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OrderOfVanishing.Basic | {
"line": 218,
"column": 2
} | {
"line": 219,
"column": 50
} | {
"line": 220,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nϖ : R\nhϖ : Irreducible ϖ\n⊢ IsSimpleModule R (R ⧸ span {ϖ})",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"Set.instSingletonSet",
"CommRing.toCommSemiring",
"Pr... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsPrincipalIdealRing R\nϖ : R\nhϖ : Irreducible ϖ\nthis : (span {ϖ}).IsMaximal\n⊢ IsSimpleModule R (R ⧸ span {ϖ})"
] | have : (Ideal.span {ϖ}).IsMaximal :=
PrincipalIdealRing.isMaximal_of_irreducible hϖ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.OrderOfVanishing | {
"line": 110,
"column": 2
} | {
"line": 117,
"column": 75
} | {
"line": 119,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝² : IsIntegral X\ninst✝¹ : IsLocallyNoetherian X\nx : ↥X\ninst✝ : IsDiscreteValuationRing ↑(X.presheaf.stalk x)\nf g : ↑X.functionField\nhfg : f + g ≠ 0\n⊢ min (ord f x) (ord g x) ≤ ord (f + g) x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"AlgebraicGeom... | [] | by_cases hf : f = 0
· simp [hf]
by_cases hg : g = 0
· simp [hg]
by_cases! hx : coheight x ≠ 1
· simp [hx]
rw [inf_le_iff, ord_le_ord_iff hx hx hf hfg, ord_le_ord_iff hx hx hg hfg]
exact inf_le_iff.mp <| Ring.ordFrac_add (R := X.presheaf.stalk x) _ _ hfg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.OrderOfVanishing | {
"line": 110,
"column": 2
} | {
"line": 117,
"column": 75
} | {
"line": 119,
"column": 0
} | [
{
"pp": "X : Scheme\ninst✝² : IsIntegral X\ninst✝¹ : IsLocallyNoetherian X\nx : ↥X\ninst✝ : IsDiscreteValuationRing ↑(X.presheaf.stalk x)\nf g : ↑X.functionField\nhfg : f + g ≠ 0\n⊢ min (ord f x) (ord g x) ≤ ord (f + g) x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"AlgebraicGeom... | [] | by_cases hf : f = 0
· simp [hf]
by_cases hg : g = 0
· simp [hg]
by_cases! hx : coheight x ≠ 1
· simp [hx]
rw [inf_le_iff, ord_le_ord_iff hx hx hf hfg, ord_le_ord_iff hx hx hg hfg]
exact inf_le_iff.mp <| Ring.ordFrac_add (R := X.presheaf.stalk x) _ _ hfg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 24
} | {
"line": 292,
"column": 6
} | [
{
"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\nZs : Set (Set (ProjectiveSpectrum 𝒜))\nh : Zs ⊆ Set.range (zeroLocus 𝒜)\nf : ↑Zs → Set A := fun i ↦ Classical.choose ⋯\nH : ⋂ i, zeroLocus 𝒜 (f i) ∈ Set.range (ze... | [
"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\nZs : Set (Set (ProjectiveSpectrum 𝒜))\nh : Zs ⊆ Set.range (zeroLocus 𝒜)\nf : ↑Zs → Set A := fun i ↦ Classical.choose ⋯\nH : ⋂ i, zeroLocus 𝒜 (f i) ∈ Set.range (zeroLocus 𝒜)\... | convert! H using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 385,
"column": 14
} | {
"line": 385,
"column": 51
} | {
"line": 385,
"column": 51
} | [
{
"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\nH : ∀ (i : ℕ), (GradedRing.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ f ∈ z.asHomogeneousIdeal",
"ppTerm": "?m.111",
"assigned"... | [
"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\nH : ∀ (i : ℕ), (GradedRing.proj 𝒜 i) f ∈ z.asHomogeneousIdeal\n⊢ ∑ i ∈ DFinsupp.support ((decompose 𝒜) f), ↑(((decompose 𝒜) f) i) ∈ z.asHomo... | ← DirectSum.sum_support_decompose 𝒜 f | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Topology | {
"line": 387,
"column": 70
} | {
"line": 387,
"column": 83
} | {
"line": 387,
"column": 83
} | [
{
"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... | [
"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⊢ (GradedRing.proj 𝒜 i) f ∉ z.asHomogeneousIdeal"
... | mem_basicOpen | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 89
} | {
"line": 43,
"column": 0
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\nF : Finset S\nhF : Algebra.adjoin ↥(𝒜 0) ↑F = ⊤\nι₀ : Type (... | [] | exact sum_mem fun n hn ↦ Algebra.subset_adjoin (by simpa using ⟨⟨⟨s, hs⟩, n, hn⟩, rfl⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.GradedAlgebra.FiniteType | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 13
} | {
"line": 50,
"column": 2
} | [
{
"pp": "S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\n... | [
"S : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : CommRing S\ninst✝³ : SetLike σ S\ninst✝² : AddSubgroupClass σ S\n𝒜 : ι → σ\ninst✝¹ : GradedRing 𝒜\ninst✝ : Algebra.FiniteType (↥(𝒜 0)) S\ns : Finset S\nh₁ : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤\nn : S → ι\nhn : ∀ i ∈ s... | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.AlgebraicGeometry.Modules.Sheaf | {
"line": 294,
"column": 2
} | {
"line": 300,
"column": 15
} | {
"line": 302,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ (pullbackComp f (𝟙 Y)).inv ≫ Functor.whiskerRight (pullbackId Y).hom (pullback f) ≫ (pullback f).leftUnitor.hom =\n eqToHom ⋯",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Algebraic... | [] | let e₁ := pullbackComp f (𝟙 _)
let e₂ := Functor.isoWhiskerRight (pullbackId Y) (pullback f)
let e₃ := (pullback f).leftUnitor
change e₁.inv ≫ e₂.hom ≫ e₃.hom = _
have : e₁.hom = e₂.hom ≫ e₃.hom :=
congr_arg Iso.hom (SheafOfModules.pullback_id_comp.{u} f.toRingCatSheafHom)
simp [← this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Modules.Sheaf | {
"line": 294,
"column": 2
} | {
"line": 300,
"column": 15
} | {
"line": 302,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ (pullbackComp f (𝟙 Y)).inv ≫ Functor.whiskerRight (pullbackId Y).hom (pullback f) ≫ (pullback f).leftUnitor.hom =\n eqToHom ⋯",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"Algebraic... | [] | let e₁ := pullbackComp f (𝟙 _)
let e₂ := Functor.isoWhiskerRight (pullbackId Y) (pullback f)
let e₃ := (pullback f).leftUnitor
change e₁.inv ≫ e₂.hom ≫ e₃.hom = _
have : e₁.hom = e₂.hom ≫ e₃.hom :=
congr_arg Iso.hom (SheafOfModules.pullback_id_comp.{u} f.toRingCatSheafHom)
simp [← this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 53
} | {
"line": 149,
"column": 0
} | [
{
"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\n⊢ Scheme.Hom.app (basicOpenToSpec 𝒜 f) ⊤ =\n (Scheme.ΓSpecIso (CommRingCat.of (Away 𝒜 f))).hom ≫ awayToSection 𝒜 f ≫ (basicOpen 𝒜 f).topIso.inv",
"ppT... | [] | simp [basicOpenToSpec, Scheme.Opens.toSpecΓ_appTop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 53
} | {
"line": 149,
"column": 0
} | [
{
"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\n⊢ Scheme.Hom.app (basicOpenToSpec 𝒜 f) ⊤ =\n (Scheme.ΓSpecIso (CommRingCat.of (Away 𝒜 f))).hom ≫ awayToSection 𝒜 f ≫ (basicOpen 𝒜 f).topIso.inv",
"ppT... | [] | simp [basicOpenToSpec, Scheme.Opens.toSpecΓ_appTop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 53
} | {
"line": 149,
"column": 0
} | [
{
"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\n⊢ Scheme.Hom.app (basicOpenToSpec 𝒜 f) ⊤ =\n (Scheme.ΓSpecIso (CommRingCat.of (Away 𝒜 f))).hom ≫ awayToSection 𝒜 f ≫ (basicOpen 𝒜 f).topIso.inv",
"ppT... | [] | simp [basicOpenToSpec, Scheme.Opens.toSpecΓ_appTop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 11
} | {
"line": 346,
"column": 2
} | [
{
"pp": "case e'_5\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\nf g✝ : 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\nz : A\nhz : z ∈ (HomogeneousIdeal.irrel... | [
"case e'_5\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\nf g✝ : 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\nz : A\nhz : z ∈ (HomogeneousIdeal.irrelevant 𝒜).to... | subst hc0 | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 408,
"column": 10
} | {
"line": 410,
"column": 43
} | {
"line": 411,
"column": 10
} | [
{
"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\nx : A\nhx : x ∈ carrier f_deg q\nn : ℕ\na : A\nha : a ∈ 𝒜 n\ni : ℕ\nproduct : A⁰_ f... | [
"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\nx : A\nhx : x ∈ carrier f_deg q\nn : ℕ\na : A\nha : a ∈ 𝒜 n\ni : ℕ\nproduct : A⁰_ f :=\n Homog... | rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk,
HomogeneousLocalization.val_mul, HomogeneousLocalization.val_mk,
HomogeneousLocalization.val_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 122,
"column": 4
} | {
"line": 122,
"column": 31
} | {
"line": 123,
"column": 4
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nA : LocalSubring K\nthis : ∃ B, A ≤ B ∧ IsMax B\n⊢ ∃ B, A ≤ B.toLocalSubring",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"LocalSubring.instPartialOrder",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Exists",
"LE.le",... | [
"K : Type u_3\ninst✝ : Field K\nA B : LocalSubring K\nhB : A ≤ B\nhB' : IsMax B\n⊢ ∃ B, A ≤ B.toLocalSubring"
] | obtain ⟨B, hB, hB'⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 494,
"column": 2
} | {
"line": 494,
"column": 25
} | {
"line": 495,
"column": 2
} | [
{
"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\nx : ↑↑(Spec A⁰_ f).toPresheafedSpace\n⊢ (ConcreteCategory.hom (toSpec 𝒜 f)) (FromSpec.toFun f_deg hm x) = x",
"ppTerm... | [
"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\nx : ↑↑(Spec A⁰_ f).toPresheafedSpace\n⊢ ((ConcreteCategory.hom (toSpec 𝒜 f)) (FromSpec.toFun f_deg hm x)).asIdeal = x.asIdeal"
] | apply PrimeSpectrum.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 513,
"column": 39
} | {
"line": 516,
"column": 48
} | {
"line": 518,
"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\n⊢ Function.Injective ⇑(ConcreteCategory.hom (toSpec 𝒜 f))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstan... | [] | by
intro x₁ x₂ h
have := congr_arg (FromSpec.toFun f_deg hm) h
rwa [fromSpec_toSpec, fromSpec_toSpec] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 727,
"column": 10
} | {
"line": 732,
"column": 36
} | {
"line": 733,
"column": 10
} | [
{
"pp": "case refine_1\nR : CommRingCat\nM : (Spec R).Modules\nV : (Spec R).Opens\nι : Type u_1\ninst✝ : Finite ι\ng : ι → ↑R\nhg : V = ⨆ i, basicOpen (g i)\nh₁ : ∀ (i : ι), Aux M (basicOpen (g i))\nh₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))\nhgle : ∀ (i : ι), basicOpen (g i) ≤ V\nhug : ∀ (i : ι) (m : ℕ), ... | [
"case refine_1\nR : CommRingCat\nM : (Spec R).Modules\nV : (Spec R).Opens\nι : Type u_1\ninst✝ : Finite ι\ng : ι → ↑R\nhg : V = ⨆ i, basicOpen (g i)\nh₁ : ∀ (i : ι), Aux M (basicOpen (g i))\nh₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))\nhgle : ∀ (i : ι), basicOpen (g i) ≤ V\nhug : ∀ (i : ι) (m : ℕ), IsUnit ((alg... | rw [map_sub, ← M.presheaf.map_comp_apply, ← op_comp, ← M.presheaf.map_comp_apply,
← op_comp, homOfLE_comp, homOfLE_comp, ← homOfLE_comp hfgigi (hfgi i),
← homOfLE_comp hfgigj (hfgi j), op_comp, M.presheaf.map_comp_apply, ← ht i,
M.map_smul_Spec, ← M.presheaf.map_comp_apply, ← op_comp... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 775,
"column": 4
} | {
"line": 775,
"column": 26
} | {
"line": 776,
"column": 4
} | [
{
"pp": "case refine_2\nR : CommRingCat\nM : (Spec R).Modules\nV : (Spec R).Opens\nι : Type u_1\ninst✝ : Finite ι\ng : ι → ↑R\nhg : V = ⨆ i, basicOpen (g i)\nh₁ : ∀ (i : ι), Aux M (basicOpen (g i))\nh₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))\nhgle : ∀ (i : ι), basicOpen (g i) ≤ V\nhug : ∀ (i : ι) (m : ℕ), ... | [
"case refine_2\nR : CommRingCat\nM : (Spec R).Modules\nV : (Spec R).Opens\nι : Type u_1\ninst✝ : Finite ι\ng : ι → ↑R\nhg : V = ⨆ i, basicOpen (g i)\nh₁ : ∀ (i : ι), Aux M (basicOpen (g i))\nh₂ : ∀ (i j : ι), Aux M (basicOpen (g i * g j))\nhgle : ∀ (i : ι), basicOpen (g i) ≤ V\nhug : ∀ (i : ι) (m : ℕ), IsUnit ((alg... | choose n hn using this | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.Sites.Affine | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 27
} | {
"line": 64,
"column": 4
} | [
{
"pp": "S : Scheme\nP : MorphismProperty Scheme\ninst✝³ : P.IsMultiplicative\ninst✝² : IsZariskiLocalAtSource P\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.HasOfPostcompProperty P\nU : P.Over ⊤ S\n𝒰 : Cover (precoverage P) U.left := Cover.changeProp U.left.affineCover ⋯\nx✝² : (i : 𝒰.I₀) → (𝒰.X i).Over S... | [
"S : Scheme\nP : MorphismProperty Scheme\ninst✝³ : P.IsMultiplicative\ninst✝² : IsZariskiLocalAtSource P\ninst✝¹ : P.IsStableUnderBaseChange\ninst✝ : P.HasOfPostcompProperty P\nU : P.Over ⊤ S\n𝒰 : Cover (precoverage P) U.left := Cover.changeProp U.left.affineCover ⋯\nx✝² : (i : 𝒰.I₀) → (𝒰.X i).Over S := fun i ↦ ... | rw [Sieve.coverByImage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 36
} | {
"line": 236,
"column": 6
} | [
{
"pp": "case hR\nC₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\ni₁ i₂ : data.I₀\nW : C\np₁ : W ⟶ F.obj (data.X i₁)\np₂ : W ⟶ ... | [
"case hR.rj\nC₀ : Type u₀\nC : Type u\ninst✝² : Category.{v₀, u₀} C₀\ninst✝¹ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝ : IsDenseSubsite J₀ J F\nX : C\ndata : F.OneHypercoverDenseData J₀ J X\ni₁ i₂ : data.I₀\nW : C\np₁ : W ⟶ F.obj (data.X i₁)\np₂ : W ⟶ F.obj (da... | apply J₀.intersection_covering | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 863,
"column": 14
} | {
"line": 863,
"column": 16
} | {
"line": 863,
"column": 16
} | [
{
"pp": "R : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec R).Modules\ninst✝ : SheafOfModules.IsQuasicoherent M\nι : Type u\nU : ι → (Spec R).Opens\npres : (i : ι) → SheafOfModules.Presentation (M.restrict (U i).ι)\nhU : IsOpenCover U\nhU' : ∀ (i : ι), IsAffineOpen (U i)\ns : Finset ι\nhs : IsOpenCover fun i ↦ U ↑i... | [
"R : CommRingCat\nM✝ : ModuleCat ↑R\nM : (Spec R).Modules\ninst✝ : SheafOfModules.IsQuasicoherent M\nι : Type u\nU : ι → (Spec R).Opens\npres : (i : ι) → SheafOfModules.Presentation (M.restrict (U i).ι)\nhU : IsOpenCover U\nhU' : ∀ (i : ι), IsAffineOpen (U i)\ns : Finset ι\nhs : IsOpenCover fun i ↦ U ↑i\nκ : ↥s → T... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 76
} | {
"line": 69,
"column": 4
} | [
{
"pp": "case refine_2\nS : Scheme\nX Y : Scheme\nf : X ⟶ Y\nE : PreZeroHypercover Y\nh : ∀ (i : E.I₀), HasPullback f (E.f i)\nhE : qcCoverFamily.property E\n⊢ qcCoverFamily.property (PreZeroHypercover.pullback₁ f E)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_2\nS : Scheme\nX Y : Scheme\nf : X ⟶ Y\nE : PreZeroHypercover Y\nh : ∀ (i : E.I₀), HasPullback f (E.f i)\nhE : QuasiCompactCover E\n⊢ QuasiCompactCover (PreZeroHypercover.pullback₁ f E)"
] | simp only [qcCoverFamily_property, Scheme.quasiCompactCover_iff] at hE ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 556,
"column": 6
} | {
"line": 556,
"column": 77
} | {
"line": 557,
"column": 6
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize... | [
"C₀ : Type u₀\nC : Type u\ninst✝⁴ : Category.{v₀, u₀} C₀\ninst✝³ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝² : Category.{v', u'} A\ninst✝¹ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\ninst✝ : HasLimitsOfSize.{w, w, v', ... | simp only [assoc, restriction.res, IsDenseSubsite.mapPreimage_comp_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 119,
"column": 65
} | {
"line": 123,
"column": 26
} | {
"line": 125,
"column": 0
} | [
{
"pp": "⊢ Monotone propQCPrecoverage",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"le_refl",
"AlgebraicGeometry.Scheme",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CategoryTheory.Precoverage",
... | [] | by
intro P Q h
rw [propQCPrecoverage, propQCPrecoverage]
gcongr
exact precoverage_mono h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 184,
"column": 2
} | {
"line": 188,
"column": 16
} | {
"line": 190,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\nR : Presieve S\n⊢ R ∈ (propQCPrecoverage P).coverings S ↔ ∃ 𝒰, QuasiCompactCover 𝒰.toPreZeroHypercover ∧ R = 𝒰.presieve₀",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.QuasiCompactCover",
"... | [] | rw [Precoverage.mem_iff_exists_zeroHypercover]
refine ⟨fun ⟨𝒰, h⟩ ↦ ⟨𝒰.weaken propQCPrecoverage_le_precoverage, ?_, h⟩,
fun ⟨𝒰, _, h⟩ ↦ ⟨⟨𝒰.1, ⟨by simpa, 𝒰.mem₀⟩⟩, h⟩⟩
rw [← Scheme.presieve₀_mem_qcPrecoverage_iff]
exact 𝒰.mem₀.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 184,
"column": 2
} | {
"line": 188,
"column": 16
} | {
"line": 190,
"column": 0
} | [
{
"pp": "P : MorphismProperty Scheme\nS : Scheme\nR : Presieve S\n⊢ R ∈ (propQCPrecoverage P).coverings S ↔ ∃ 𝒰, QuasiCompactCover 𝒰.toPreZeroHypercover ∧ R = 𝒰.presieve₀",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.QuasiCompactCover",
"... | [] | rw [Precoverage.mem_iff_exists_zeroHypercover]
refine ⟨fun ⟨𝒰, h⟩ ↦ ⟨𝒰.weaken propQCPrecoverage_le_precoverage, ?_, h⟩,
fun ⟨𝒰, _, h⟩ ↦ ⟨⟨𝒰.1, ⟨by simpa, 𝒰.mem₀⟩⟩, h⟩⟩
rw [← Scheme.presieve₀_mem_qcPrecoverage_iff]
exact 𝒰.mem₀.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 276,
"column": 6
} | {
"line": 313,
"column": 43
} | {
"line": 315,
"column": 0
} | [
{
"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\... | [] | _ = (∏ i, ψ i ^ (d i * ai i)) * ψ i₀ ^ (d i₀ * a * (d j - 1)) := by
simp only [ψ, ← map_pow, ← map_prod, ← map_mul]
congr 2
apply (show Function.Injective (algebraMap (Away 𝒜 (x j)) (Localization.Away (x j)))
from val_injective _)
simp only [map_pow, map_prod, map_mu... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact | {
"line": 66,
"column": 10
} | {
"line": 70,
"column": 100
} | {
"line": 71,
"column": 8
} | [
{
"pp": "case refine_1\nP : MorphismProperty Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nF : Schemeᵒᵖ ⥤ Type u_1\ninst✝ : IsZariskiLocalAtSource P\nx✝ :\n Presieve.IsSheaf zariskiTopology F ∧\n ∀ {R S : CommRingCat} (f : R ⟶ S),\n P (Spec.map f) → Surjective (Spec.map f) → ... | [] | rw [← Sieve.pullbackArrows_comm, ← Presieve.ofArrows_pullback,
← Presieve.isSheafFor_iff_generate]
let 𝒰' := (𝒱.cover.pullback₂ f).pullback₂ (V.affineCover.f i)
exact this _ (.ofQuasiCompactCover 𝒰' (qc := by dsimp [𝒰']; infer_instance))
⟨fun j ↦ .of_isIso (pullbackLeftPu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.SheafQuasiCompact | {
"line": 66,
"column": 10
} | {
"line": 70,
"column": 100
} | {
"line": 71,
"column": 8
} | [
{
"pp": "case refine_1\nP : MorphismProperty Scheme\ninst✝² : P.IsStableUnderBaseChange\ninst✝¹ : P.IsMultiplicative\nF : Schemeᵒᵖ ⥤ Type u_1\ninst✝ : IsZariskiLocalAtSource P\nx✝ :\n Presieve.IsSheaf zariskiTopology F ∧\n ∀ {R S : CommRingCat} (f : R ⟶ S),\n P (Spec.map f) → Surjective (Spec.map f) → ... | [] | rw [← Sieve.pullbackArrows_comm, ← Presieve.ofArrows_pullback,
← Presieve.isSheafFor_iff_generate]
let 𝒰' := (𝒱.cover.pullback₂ f).pullback₂ (V.affineCover.f i)
exact this _ (.ofQuasiCompactCover 𝒰' (qc := by dsimp [𝒰']; infer_instance))
⟨fun j ↦ .of_isIso (pullbackLeftPu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.EffectiveEpimorphic | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 44
} | {
"line": 222,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nH : EffectiveEpiFamilyStruct X π\nD : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom\nF : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram\n⊢ ∀ (s : Cocone (... | [
"C : Type u\ninst✝ : Category.{v, u} C\nB : C\nα : Type u_1\nX : α → C\nπ : (a : α) → X a ⟶ B\nH : EffectiveEpiFamilyStruct X π\nD : Type (max u v) := ObjectProperty.FullSubcategory fun T ↦ (Sieve.generateFamily X π).arrows T.hom\nF : D ⥤ C := (Sieve.generateFamily X π).arrows.diagram\nS : Cocone (Sieve.generateFam... | intro S ⟨T, a, (g : T.left ⟶ X a), hT⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Abelian.CommSq | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 12
} | {
"line": 155,
"column": 2
} | [
{
"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 : IsPushout t l r b\n⊢ ∀ ⦃A : C⦄ (y : A ⟶ kernel b), ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ y = x ≫ kernel.map t b l r ⋯",
"ppTerm": "?m.58",
"assigned": true,
"usedCo... | [
"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 : IsPushout t l r b\nA₀ : C\nz : A₀ ⟶ kernel b\n⊢ ∃ A' π, ∃ (_ : Epi π), ∃ x, π ≫ z = x ≫ kernel.map t b l r ⋯"
] | intro A₀ z | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 125,
"column": 2
} | {
"line": 132,
"column": 56
} | {
"line": 134,
"column": 0
} | [
{
"pp": "C : 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\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhXκ : HasCardinalLT (Subobject X) κ\nj... | [] | have := isFiltered_of_isCardinalFiltered J κ
obtain ⟨j, h⟩ := exists_isIso_of_functor_from_monoOver (F y) hXκ _
(colimit.isColimit (kernel (g y))) (f y) (fun j ↦ by simpa using! hf y j)
(epi_f hc hy)
dsimp at h
refine ⟨j.right, j.hom, ?_⟩
simpa only [← cancel_epi ((kernel.ι (g y)).app j), comp_zero]... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ColimCoyoneda | {
"line": 125,
"column": 2
} | {
"line": 132,
"column": 56
} | {
"line": 134,
"column": 0
} | [
{
"pp": "C : 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\nκ : Cardinal.{w}\nhκ : Fact κ.IsRegular\ninst✝ : IsCardinalFiltered J κ\nhXκ : HasCardinalLT (Subobject X) κ\nj... | [] | have := isFiltered_of_isCardinalFiltered J κ
obtain ⟨j, h⟩ := exists_isIso_of_functor_from_monoOver (F y) hXκ _
(colimit.isColimit (kernel (g y))) (f y) (fun j ↦ by simpa using! hf y j)
(epi_f hc hy)
dsimp at h
refine ⟨j.right, j.hom, ?_⟩
simpa only [← cancel_epi ((kernel.ι (g y)).app j), comp_zero]... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Preorder.HasIterationOfShape | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 19
} | {
"line": 81,
"column": 6
} | [
{
"pp": "case neg.isMin\nJ : Type w\ninst✝⁶ : LinearOrder J\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : HasIterationOfShape J C\ninst✝³ : SuccOrder J\ninst✝² : WellFoundedLT J\nα : Type u_1\ninst✝¹ : PartialOrder α\nf : α ≤i J\ninst✝ : Nonempty α\nhf : ¬Function.Surjective ⇑f\ns : α <i J := f.toPrincipalS... | [] | | isMin i hi => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 223,
"column": 63
} | {
"line": 225,
"column": 22
} | {
"line": 227,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝² : 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✝¹ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝ : HasColimitsOfShape (Discrete (FunctorObjIndex f πY)) C\ni : I\nt : A i ⟶... | [] | by
subst hb' ht'
simp [functorMapSrc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.Iteration.Basic | {
"line": 413,
"column": 10
} | {
"line": 413,
"column": 68
} | {
"line": 414,
"column": 10
} | [
{
"pp": "case neg\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\nh₁ : Order.IsSuccLimit j\nh₂ : ∀ b < j, ∀ (iter₁ iter₂ : Φ.Iteration b), iter₁.F = iter... | [
"case neg\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 k₁ : J\nh₁ : Order.IsSuccLimit k₁\nh₂ : ∀ b < k₁, ∀ (iter₁ iter₂ : Φ.Iteration b), iter₁.F = iter₂.F\niter₁... | obtain rfl : k₁ = j := le_antisymm h₁₂ (by simpa using h₅) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone | {
"line": 154,
"column": 59
} | {
"line": 156,
"column": 61
} | {
"line": 156,
"column": 61
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nJ : Type u\ninst✝ : LinearOrder J\nj : J\nF : ↑(Set.Iio j) ⥤ C\nc : Cocone F\nhc : IsColimit c\nx✝ : ↑(Set.Iio j)\ni : J\nhi : i ∈ Set.Iio j\n⊢ c.ι.app ⟨i, hi⟩ ≫ (ofCoconeObjIsoPt c).symm.hom =\n ((Cocone.precompose (restrictionLTOfCoconeIso c).inv).obj ... | [] | by
dsimp
rw [ofCocone_map_to_top _ _ hi, Iso.inv_hom_id_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.Iteration.Nonempty | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 17
} | {
"line": 155,
"column": 6
} | [
{
"pp": "case isMin\nC : Type u_1\ninst✝⁵ : Category.{v_1, u_1} C\nΦ : SuccStruct C\nJ : Type u\ninst✝⁴ : LinearOrder J\ninst✝³ : OrderBot J\ninst✝² : SuccOrder J\ninst✝¹ : WellFoundedLT J\ninst✝ : HasIterationOfShape J C\ni : J\nhi : IsMin i\n⊢ Nonempty (Φ.Iteration i)",
"ppTerm": "?isMin",
"assigned":... | [] | | isMin i hi => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 117,
"column": 34
} | {
"line": 117,
"column": 70
} | {
"line": 117,
"column": 70
} | [
{
"pp": "J : Type u\ninst✝² : LinearOrder J\ninst✝¹ : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝ : OrderBot J\nval₀ : F.obj (op ⊥)\nj : J\ne : d.Extension val₀ j\ni : J\nhij : i ≤ j\nk : J\nhk : k < i\n⊢ (ConcreteCategory.hom (F.map (homOfLE ⋯).op)) e.val = d.succ k ⋯ ((ConcreteCategory.... | [
"J : Type u\ninst✝² : LinearOrder J\ninst✝¹ : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝ : OrderBot J\nval₀ : F.obj (op ⊥)\nj : J\ne : d.Extension val₀ j\ni : J\nhij : i ≤ j\nk : J\nhk : k < i\n⊢ d.succ k ⋯ ((ConcreteCategory.hom (F.map (homOfLE ⋯).op)) e.val) =\n d.succ k ⋯ ((ConcreteCat... | e.map_succ k (lt_of_lt_of_le hk hij) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 17
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case isMin\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\ni : J\nhi : IsMin i\n⊢ Subsingleton (d.Extension val₀ i)",
"ppTerm": "?isMin",
"assigned": true,
"usedCons... | [] | | isMin i hi => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.AlgebraicTopology.RelativeCellComplex.Basic | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 38
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case succ\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : Type w'\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nα : J → Type t\nA B : (j : J) → α j → C\nbasicCell : (j : J) → (i : α j) → A j i ⟶ B j i\nX Y : C\nf : X ⟶ Y\nc : RelativeCellComplex basicCell f\n... | [
"case succ.h₀\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : Type w'\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nα : J → Type t\nA B : (j : J) → α j → C\nbasicCell : (j : J) → (i : α j) → A j i ⟶ B j i\nX Y : C\nf : X ⟶ Y\nc : RelativeCellComplex basicCell f\nZ : C\nφ₁... | apply (c.attachCells j hj).hom_ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 17
} | {
"line": 257,
"column": 4
} | [
{
"pp": "case isMin\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\ni : J\nhi : IsMin i\n⊢ Nonempty (d.Extension val₀ i)",
"ppTerm": "?isMin",
"assigned": true,
"usedConstant... | [] | | isMin i hi => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 304,
"column": 6
} | {
"line": 310,
"column": 41
} | {
"line": 312,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq r : ℚ\nhqr : q + r ≠ 0\nh : emultiplicity (↑p) (q.num * ↑r.den) ≤ emultiplicity (↑p) (r.num * ↑q.den)\nhq : ¬q = 0\nhr : ¬r = 0\nhqn : q.num ≠ 0\nhqd : ↑q.den ≠ 0\nhrn : r.num ≠ 0\nhrd : ↑r.den ≠ 0\nhqreq : q + r = (q.num * ↑r.den + ↑q.den * r.num) /. (↑q.den * ↑r.den)... | [] | calc
_ ≤ min (emultiplicity ↑p (q.num * r.den * q.den))
(emultiplicity ↑p (q.den * r.num * q.den)) :=
le_min
(by rw [emultiplicity_mul (a := _ * _) (Nat.prime_iff_prime_int.1 hp.1), add_comm])
(by grw [mul_assoc, emultiplicity_mul (b := _ * _) (Nat.prime_iff_pri... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 238,
"column": 23
} | {
"line": 238,
"column": 39
} | {
"line": 238,
"column": 40
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh✝ : ↑m = 0\n⊢ ↑p < 0 → 0 = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Preorder.toLT",
"congrArg",
"AddMonoid.toAddZeroClass",
... | [
"case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh✝ : ↑m = 0\n⊢ ↑p < ↑0 → ↑0 = 1"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 248,
"column": 21
} | {
"line": 248,
"column": 38
} | {
"line": 248,
"column": 39
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\n⊢ ¬padicNorm p ↑m < 1 ↔ ¬↑p ∣ m",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Dvd.dvd",
"padicNorm.int_eq_one_iff",
"congrArg",
"Rat",
"Rat.instIntCast"... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\n⊢ ¬padicNorm p ↑m < 1 ↔ padicNorm p ↑m = 1"
] | ← int_eq_one_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 256,
"column": 33
} | {
"line": 256,
"column": 50
} | {
"line": 256,
"column": 51
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\n⊢ padicNorm p ↑m = 1 ↔ ¬↑p ∣ ↑m",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Dvd.dvd",
"padicNorm.int_eq_one_iff",
"congrArg",
"Rat",
"Rat.instIntCast",... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\n⊢ padicNorm p ↑m = 1 ↔ padicNorm p ↑↑m = 1"
] | ← int_eq_one_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 62
} | {
"line": 144,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\ns : Finset α\nf : α → ℤ_[p]\n⊢ ↑(∑ z ∈ s, f z) = ∑ z ∈ s, ↑(f z)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",
"RingHom.instRingHomClass",
"Real.instLE",
"Real",
"RingHomClass.toAddMonoidHom... | [] | simp [← Coe.ringHom_apply, map_sum PadicInt.Coe.ringHom f s] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 62
} | {
"line": 144,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\ns : Finset α\nf : α → ℤ_[p]\n⊢ ↑(∑ z ∈ s, f z) = ∑ z ∈ s, ↑(f z)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",
"RingHom.instRingHomClass",
"Real.instLE",
"Real",
"RingHomClass.toAddMonoidHom... | [] | simp [← Coe.ringHom_apply, map_sum PadicInt.Coe.ringHom f s] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 62
} | {
"line": 144,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nα : Type u_1\ns : Finset α\nf : α → ℤ_[p]\n⊢ ↑(∑ z ∈ s, f z) = ∑ z ∈ s, ↑(f z)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",
"RingHom.instRingHomClass",
"Real.instLE",
"Real",
"RingHomClass.toAddMonoidHom... | [] | simp [← Coe.ringHom_apply, map_sum PadicInt.Coe.ringHom f s] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 421,
"column": 32
} | {
"line": 421,
"column": 47
} | {
"line": 421,
"column": 48
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nhx : x ≠ 0\n⊢ ↑x = ↑x * (↑p ^ (-↑x.valuation) * ↑p ^ x.valuation)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Semigroup.toMul",
"Real",
"... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nhx : x ≠ 0\n⊢ ↑x = ↑x * (↑p ^ (-↑x.valuation) * ↑p ^ ↑x.valuation)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 440,
"column": 2
} | {
"line": 442,
"column": 30
} | {
"line": 443,
"column": 2
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nh : ‖↑r‖ ≤ 1\nnorm_denom_lt : ‖↑r.den‖ < 1\nhr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖\nkey : ‖↑r.num‖ < 1\n⊢ False",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Re... | [
"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nh : ‖↑r‖ ≤ 1\nnorm_denom_lt : ‖↑r.den‖ < 1\nhr : ‖↑r‖ * ‖↑r.den‖ = ‖↑r.num‖\nkey : ‖↑r.num‖ < 1\nthis : ↑p ∣ r.num ∧ ↑p ∣ ↑r.den\n⊢ False"
] | have : ↑p ∣ r.num ∧ (p : ℤ) ∣ r.den := by
simp only [← norm_int_lt_one_iff_dvd, ← padic_norm_e_of_padicInt]
exact ⟨key, norm_denom_lt⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 484,
"column": 2
} | {
"line": 484,
"column": 58
} | {
"line": 486,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nn : ℤ\n⊢ ‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"PadicInt",
"congrArg",
"Real.instDivInvMonoid",
"Iff... | [] | rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 484,
"column": 2
} | {
"line": 484,
"column": 58
} | {
"line": 486,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nn : ℤ\n⊢ ‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"PadicInt",
"congrArg",
"Real.instDivInvMonoid",
"Iff... | [] | rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 484,
"column": 2
} | {
"line": 484,
"column": 58
} | {
"line": 486,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nn : ℤ\n⊢ ‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"PadicInt",
"congrArg",
"Real.instDivInvMonoid",
"Iff... | [] | rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 25
} | {
"line": 575,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx✝ y : ℤ_[p]\nx : ℚ_[p]\n⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"PadicInt",
"... | [
"case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx✝ y : ℤ_[p]\nx : ℚ_[p]\nhx : ‖x‖ ≤ 1\n⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p]) x_1.1",
"case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx✝ y : ℤ_[p]\nx : ℚ_[p]\nhx : ¬‖x‖ ≤ 1\n⊢ ∃ x_1, x * (algebraMap ℤ_[p] ℚ_[p]) ↑x_1.2 = (algebraMap ℤ_[p] ℚ_[p... | by_cases hx : ‖x‖ ≤ 1 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 61
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case e_a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ (id X n).b i.rev = (Q (n + 1)).f (n + 1)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Opposite",
"instDecidableT... | [] | exact Eq.trans (by simp) (decomposition_Q n (n + 1)).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 61
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case e_a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ (id X n).b i.rev = (Q (n + 1)).f (n + 1)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Opposite",
"instDecidableT... | [] | exact Eq.trans (by simp) (decomposition_Q n (n + 1)).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1232,
"column": 2
} | {
"line": 1232,
"column": 58
} | {
"line": 1234,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nn : ℤ\n⊢ ‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Real.instDivInvMonoid",
"Iff.rfl",
"AddM... | [] | rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 61
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case e_a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\n⊢ ∑ i, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ (id X n).b i.rev = (Q (n + 1)).f (n + 1)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Opposite",
"instDecidableT... | [] | exact Eq.trans (by simp) (decomposition_Q n (n + 1)).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1232,
"column": 2
} | {
"line": 1232,
"column": 58
} | {
"line": 1234,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nn : ℤ\n⊢ ‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Real.instDivInvMonoid",
"Iff.rfl",
"AddM... | [] | rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1232,
"column": 2
} | {
"line": 1232,
"column": 58
} | {
"line": 1234,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nn : ℤ\n⊢ ‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Real.instDivInvMonoid",
"Iff.rfl",
"AddM... | [] | rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.Normalized | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 66
} | {
"line": 59,
"column": 4
} | [
{
"pp": "case succ\nA : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX : SimplicialObject A\nn : ℕ\n⊢ (NormalizedMooreComplex.objX X (n + 1)).Factors (PInfty.f (n + 1))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"CategoryTheory.Subobject.Factors",
"Category... | [
"case succ\nA : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX : SimplicialObject A\nn : ℕ\n⊢ ∀ i ∈ Finset.univ, (kernelSubobject (X.δ i.succ)).Factors ((P (n + 1)).f (n + 1))"
] | rw [PInfty_f, NormalizedMooreComplex.objX, finset_inf_factors] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.DoldKan.Degeneracies | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 40
} | {
"line": 130,
"column": 40
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\nΔ' : SimplexCategory\nθ : ⦋n⦌ ⟶ Δ'\nhθ : ¬Mono θ\n⊢ X.map θ.op ≫ PInfty.f n = 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"CategoryTheory.Mono",
"congrArg",
... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn : ℕ\nΔ' : SimplexCategory\nθ : ⦋n⦌ ⟶ Δ'\nhθ : ¬Function.Injective ⇑(SimplexCategory.Hom.toOrderHom θ)\n⊢ X.map θ.op ≫ PInfty.f n = 0"
] | SimplexCategory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.Normalized | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 36
} | {
"line": 71,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX✝ X : SimplicialObject A\nn : ℕ\n⊢ (NormalizedMooreComplex.objX X (n + 1)).factorThru (PInfty.f (n + 1)) ⋯ ≫\n NormalizedMooreComplex.objD X n ≫ (inclusionOfMooreComplexMap X).f n =\n K[X].d (n + 1) n ≫ PInfty.f n",
"ppTerm":... | [
"A : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX✝ X : SimplicialObject A\nn : ℕ\n⊢ (NormalizedMooreComplex.objX X (n + 1)).factorThru (PInfty.f (n + 1)) ⋯ ≫\n ((normalizedMooreComplex A).obj X).d (n + 1) n ≫ (inclusionOfMooreComplexMap X).f n =\n K[X].d (n + 1) n ≫ PInfty.f n"
] | ← normalizedMooreComplex_objD, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.EquivalencePseudoabelian | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 15
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : IsIdempotentComplete C\ninst✝ : HasFiniteCoproducts C\nX : ChainComplex C ℕ\n⊢ (N₂.map (isoΓ₀.hom.app X)).f = PInfty",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HomologicalComplex.hom_ext",
... | [] | ext
apply comp_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.EquivalencePseudoabelian | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 15
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\ninst✝¹ : IsIdempotentComplete C\ninst✝ : HasFiniteCoproducts C\nX : ChainComplex C ℕ\n⊢ (N₂.map (isoΓ₀.hom.app X)).f = PInfty",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HomologicalComplex.hom_ext",
... | [] | ext
apply comp_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.FundamentalGroupoid.Product | {
"line": 191,
"column": 45
} | {
"line": 191,
"column": 59
} | {
"line": 191,
"column": 60
} | [
{
"pp": "case h_obj\nA : TopCat\nB : TopCat\nX✝ : ↑(π.obj (TopCat.of (↑A × ↑B)))\n⊢ (((projLeft A B).prod' (projRight A B) ⋙ prodToProdTop A B).obj X✝).as =\n ((𝟭 ↑(π.obj (TopCat.of (↑A × ↑B)))).obj X✝).as",
"ppTerm": "?h_obj",
"assigned": true,
"usedConstants": [
"TopCat.instCategory",
... | [
"case h_obj.fst\nA : TopCat\nB : TopCat\nX✝ : ↑(π.obj (TopCat.of (↑A × ↑B)))\n⊢ (((projLeft A B).prod' (projRight A B) ⋙ prodToProdTop A B).obj X✝).as.1 =\n ((𝟭 ↑(π.obj (TopCat.of (↑A × ↑B)))).obj X✝).as.1",
"case h_obj.snd\nA : TopCat\nB : TopCat\nX✝ : ↑(π.obj (TopCat.of (↑A × ↑B)))\n⊢ (((projLeft A B).prod'... | apply Prod.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 468,
"column": 2
} | {
"line": 470,
"column": 16
} | {
"line": 472,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nX : HoCat C\n⊢ WeakEquivalence (HoCat.adj.unit.app X)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.BifibrantObject",
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteColimits"... | [] | obtain ⟨X, rfl⟩ := toHoCat_obj_surjective X
dsimp [HoCat.adj]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 468,
"column": 2
} | {
"line": 470,
"column": 16
} | {
"line": 472,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\nX : HoCat C\n⊢ WeakEquivalence (HoCat.adj.unit.app X)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.BifibrantObject",
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteColimits"... | [] | obtain ⟨X, rfl⟩ := toHoCat_obj_surjective X
dsimp [HoCat.adj]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Boundary | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 29
} | {
"line": 120,
"column": 4
} | [
{
"pp": "n : ℕ\nA : Δ[n].Subcomplex\nh : A ≠ ⊤\ni : ℕ\nhi : n ≤ i\na : Δ[n] _⦋i⦌\nha : a ∈ A.obj (op ⦋i⦌)\n⊢ ⟨a, ha⟩ ∈ A.toSSet.degenerate i ↔ ⟨a, ha⟩ ∈ ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
"Opposite",
"congrArg",
... | [
"n : ℕ\nA : Δ[n].Subcomplex\nh : A ≠ ⊤\ni : ℕ\nhi : n ≤ i\na : Δ[n] _⦋i⦌\nha : a ∈ A.obj (op ⦋i⦌)\n⊢ ↑⟨a, ha⟩ ∈ Δ[n].degenerate i ↔ ⟨a, ha⟩ ∈ ⊤"
] | rw [A.mem_degenerate_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Limits.Types.Multicoequalizer | {
"line": 97,
"column": 4
} | {
"line": 100,
"column": 27
} | {
"line": 102,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u\nι : Type w\nA : Set X\nU : ι → Set X\nV : ι → ι → Set X\nc : MulticoequalizerDiagram A U V\ne : WalkingMultispan (MultispanShape.prod ι) ⥤ Type u := (c.multispanIndex.map Set.functorToTypes).multispan\nx✝ :\n ((c.multispanIndex.map Set.functorToTypes).multispan.coconeTypesEq... | [] | simp only [MulticoequalizerDiagram.multicofork_pt, ← c.iSup_eq,
Set.iSup_eq_iUnion, Set.mem_iUnion] at hx
obtain ⟨i, hi⟩ := hx
exact ⟨i, ⟨x, hi⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Multicoequalizer | {
"line": 97,
"column": 4
} | {
"line": 100,
"column": 27
} | {
"line": 102,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u\nι : Type w\nA : Set X\nU : ι → Set X\nV : ι → ι → Set X\nc : MulticoequalizerDiagram A U V\ne : WalkingMultispan (MultispanShape.prod ι) ⥤ Type u := (c.multispanIndex.map Set.functorToTypes).multispan\nx✝ :\n ((c.multispanIndex.map Set.functorToTypes).multispan.coconeTypesEq... | [] | simp only [MulticoequalizerDiagram.multicofork_pt, ← c.iSup_eq,
Set.iSup_eq_iUnion, Set.mem_iUnion] at hx
obtain ⟨i, hi⟩ := hx
exact ⟨i, ⟨x, hi⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 19
} | {
"line": 276,
"column": 2
} | [
{
"pp": "case inl\nX₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nx₄ : X₄\nh₁ : ∃ x₂, (ConcreteCategory.hom r) x₂ = x₄\n⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨\n ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)",
"ppT... | [] | exact Or.inl h₁ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 19
} | {
"line": 276,
"column": 2
} | [
{
"pp": "case inl\nX₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nx₄ : X₄\nh₁ : ∃ x₂, (ConcreteCategory.hom r) x₂ = x₄\n⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨\n ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)",
"ppT... | [] | exact Or.inl h₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 19
} | {
"line": 276,
"column": 2
} | [
{
"pp": "case inl\nX₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nx₄ : X₄\nh₁ : ∃ x₂, (ConcreteCategory.hom r) x₂ = x₄\n⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨\n ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategory.hom l)",
"ppT... | [] | exact Or.inl h₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.Quasicategory.InnerFibration | {
"line": 50,
"column": 2
} | {
"line": 52,
"column": 46
} | {
"line": 54,
"column": 0
} | [
{
"pp": "case refine_2\nX✝ Y✝ : SSet\nf✝ : X✝ ⟶ Y✝\nh : (⨆ n, ofHoms fun p ↦ Λ[n + 2, ↑p].ι) f✝\n⊢ innerHornInclusions f✝",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"CategoryTheory.MorphismProperty",
"instNeZeroNatHAdd_1",
"Opposit... | [] | · simp only [iSup_iff, ofHoms_iff] at h
obtain ⟨n, ⟨i, h0, hn⟩, _, _⟩ := h
exact horn_ι_mem_innerHornInclusions h0 hn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 17
} | {
"line": 91,
"column": 0
} | [
{
"pp": "X : SSet\n⊢ X.skeleton 0 = ⊥",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.ofSimplex",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"iSup",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"SSet.... | [] | simp [skeleton] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 17
} | {
"line": 91,
"column": 0
} | [
{
"pp": "X : SSet\n⊢ X.skeleton 0 = ⊥",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.ofSimplex",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"iSup",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"SSet.... | [] | simp [skeleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 17
} | {
"line": 91,
"column": 0
} | [
{
"pp": "X : SSet\n⊢ X.skeleton 0 = ⊥",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.ofSimplex",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"iSup",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"SSet.... | [] | simp [skeleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Skeleton | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 32
} | {
"line": 105,
"column": 4
} | [
{
"pp": "case a\nX : SSet\nn : ℕ\n⊢ X.skeleton (n + 1) ≤ X.skeleton n ⊔ ⨆ x, Subcomplex.ofSimplex ↑x",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SSet.Subcomplex.ofSimplex",
"Lattice.toSemilatticeSup",
"Opposite",
"iSup",
"PartialOrder.toPreo... | [
"case a\nX : SSet\nn : ℕ\n⊢ ⨆ i, ⨆ x, Subcomplex.ofSimplex ↑x ≤ X.skeleton n ⊔ ⨆ x, Subcomplex.ofSimplex ↑x"
] | conv_lhs => dsimp [skeleton] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.CategoryTheory.Bicategory.CatEnriched | {
"line": 149,
"column": 35
} | {
"line": 149,
"column": 53
} | {
"line": 150,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝ : EnrichedCategory Cat C\na✝ b✝ c✝ d✝ e✝ : CatEnriched C\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\ni : d✝ ⟶ e✝\nh1 : (f ≫ g) ≫ h = f ≫ g ≫ h\nh2 : (f ≫ g ≫ h) ≫ i = f ≫ (g ≫ h) ≫ i\nh3 : (g ≫ h) ≫ i = g ≫ h ≫ i\nh4 : ((f ≫ g) ≫ h) ≫ i = (f ≫ g) ≫ h ≫ i\npf✝ : (f ≫ g) ≫ h ≫ i = f ≫ g ≫... | [
"C : Type u_1\ninst✝ : EnrichedCategory Cat C\na✝ b✝ c✝ d✝ e✝ : CatEnriched C\nf : a✝ ⟶ b✝\ng : b✝ ⟶ c✝\nh : c✝ ⟶ d✝\ni : d✝ ⟶ e✝\npf✝ : (f ≫ g) ≫ h ≫ i = f ≫ g ≫ h ≫ i\n⊢ ∀ (h1 : (f ≫ g) ≫ h = f ≫ g ≫ h) (h2 : (f ≫ g ≫ h) ≫ i = f ≫ (g ≫ h) ≫ i) (h3 : (g ≫ h) ≫ i = g ≫ h ≫ i)\n (h4 : ((f ≫ g) ≫ h) ≫ i = (f ≫ g) ... | revert h1 h2 h3 h4 | Lean.Elab.Tactic.evalRevert | Lean.Parser.Tactic.revert |
Mathlib.CategoryTheory.Monoidal.Closed.FunctorToTypes | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 20
} | {
"line": 67,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF G H : C ⥤ Type (max w v u)\nf : G ⟶ H\nx✝⁴ : C\nx✝³ : ((𝟭 (C ⥤ Type (max w v u))).obj G).obj x✝⁴\nx✝² : C\nx✝¹ : (coyoneda.obj (Opposite.op x✝⁴)).obj x✝²\nx✝ : F.obj x✝²\n⊢ (ConcreteCategory.hom\n (((ConcreteC... | [
"case fst\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nF G H : C ⥤ Type (max w v u)\nf : G ⟶ H\nx✝⁴ : C\nx✝³ : ((𝟭 (C ⥤ Type (max w v u))).obj G).obj x✝⁴\nx✝² : C\nx✝¹ : (coyoneda.obj (Opposite.op x✝⁴)).obj x✝²\nx✝ : F.obj x✝²\n⊢ ((ConcreteCategory.hom\n (((Concrete... | apply Prod.ext | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicTopology.SimplicialSet.HoFunctorMonoidal | {
"line": 151,
"column": 51
} | {
"line": 154,
"column": 59
} | {
"line": 154,
"column": 59
} | [
{
"pp": "X X' Y Y' Z : Truncated 2\nx₀ x₁ x₂ : X.obj (Opposite.op { obj := ⦋0⦌, property := OneTruncation₂._proof_1 })\ne₀₁ : Edge x₀ x₁\ne₁₂ : Edge x₁ x₂\ne₀₂ : Edge x₀ x₂\nh : e₀₁.CompStruct e₁₂ e₀₂\n⊢ mkNatTrans (fun y ↦ homMk (e₀₁.tensor (Edge.id y))) ⋯ ≫ mkNatTrans (fun y ↦ homMk (e₁₂.tensor (Edge.id y))) ... | [] | by
ext y
obtain ⟨y, rfl⟩ := mk_surjective y
simpa using homMk_comp_homMk (h.tensor (.idCompId y)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplexCategory.SemiSimplexCategory | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 40
} | {
"line": 86,
"column": 40
} | [
{
"pp": "n m : SemiSimplexCategory\nf : n ⟶ m\n⊢ Mono (toSimplexCategory.map f)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Mono",
"congrArg",
"PartialOrder.toPreorder",
"id",
"instOfNatNat",
"SimplexCategory.mono_iff_... | [
"n m : SemiSimplexCategory\nf : n ⟶ m\n⊢ Function.Injective ⇑(SimplexCategory.Hom.toOrderHom (toSimplexCategory.map f))"
] | SimplexCategory.mono_iff_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
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