module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 815,
"column": 30
} | {
"line": 815,
"column": 38
} | [
{
"pp": "case pos\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhg' : g = 0\n⊢ (g * ↑(⟨p, hp⟩, ⟨q, hq⟩).1).degree < ↑(m + n)",
"usedConstants": [
"WithBot.addMonoidWithOne",
"Polynomial.de... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 815,
"column": 30
} | {
"line": 815,
"column": 38
} | [
{
"pp": "case pos\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\np : R[X]\nhp : p ∈ R[X]_m\nq : R[X]\nhq : q ∈ R[X]_n\nhg' : g = 0\n⊢ (g * ↑(⟨p, hp⟩, ⟨q, hq⟩).1).degree < ↑(m + n)",
"usedConstants": [
"WithBot.addMonoidWithOne",
"Polynomial.de... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 829,
"column": 8
} | {
"line": 829,
"column": 15
} | [
{
"pp": "case a.inl\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\ni : Fin (m + n)\nj : Fin m\n⊢ (if ↑j ≤ ↑i then g.coeff (↑i - ↑j) else 0) = if ↑j ≤ ↑i ∧ ↑i ≤ ↑j + n then g.coeff (↑i - ↑j) else 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | ite_and | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 835,
"column": 8
} | {
"line": 835,
"column": 15
} | [
{
"pp": "case a.inr\nm n : ℕ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhf : f.natDegree ≤ m\nhg : g.natDegree ≤ n\ni : Fin (m + n)\nj : Fin n\n⊢ (if ↑j ≤ ↑i then f.coeff (↑i - ↑j) else 0) = if ↑j ≤ ↑i ∧ ↑i ≤ ↑j + m then f.coeff (↑i - ↑j) else 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | ite_and | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 12
} | [
{
"pp": "R : Type u\nS : Type v\nT : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra.FiniteType R T\ninst✝¹ : Algebra.IsIntegral R S\nf : S →ₐ[R] T\ng : S\nhg : Function.Surjective ⇑(awayMapₐ f g)\np : Ideal R\ninst✝ : p.IsPri... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 258,
"column": 4
} | {
"line": 259,
"column": 84
} | [
{
"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\ns : S\nhsq : s ∉ q\nhRs : IsIntegral R s\nhs : ∀ (q' : Ideal S), q'.IsPrim... | simpa using show IsScalarTower.toAlgHom R' _ P'.1.ResidueField (aeval s' a') ≠ 0 by
rw [← aeval_algHom_apply, ← aeval_map_algebraMap P.ResidueField, ← ha']; simpa | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 13
} | [
{
"pp": "case refine_2.H₂\nR : 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\ns : S\nhsq : s ∉ q\nhRs : IsIntegral R s\nhs : ∀ (q' : I... | rw [hP'q] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 316,
"column": 83
} | {
"line": 332,
"column": 10
} | [
{
"pp": "R : Type u_2\nS : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing S\ninst✝⁹ : Algebra R S\ninst✝⁸ : FiniteType R S\ninst✝⁷ : CommRing R'\ninst✝⁶ : Algebra R R'\ninst✝⁵ : CommRing R''\ninst✝⁴ : Algebra R R''\ninst✝³ : Algebra R'' S\ninst✝² : Algebra.IsIntegral R R''\nin... | by
apply Localization.awayMap_awayMap_surjective
refine Localization.awayMap_surjective_iff.mpr fun a ↦ ?_
induction a with
| zero => use 0; simp
| tmul a b =>
obtain ⟨b', m, e : _ = _⟩ := Localization.awayMap_surjective_iff.mp hg b
refine ⟨e₀ ^ m * a ⊗ₜ b', m, ?_⟩
simp [e, mul_pow... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Group.Abelian | {
"line": 64,
"column": 69
} | {
"line": 68,
"column": 81
} | [
{
"pp": "K : 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✝⁹ : Nonempty ↥G.left\nth... | by
rw [Set.diff_eq_compl_inter, ← Set.image_singleton, ← Set.image_singleton];
refine (IsOpen.isLocallyClosed ?_).inter (IsClosed.isLocallyClosed ?_)
· exact (((lift η[G] η[G]).left.isClosedMap _ hpoint).preimage γ.left.continuous).isOpen_compl
· exact (η[G].left.isClosedMap _ hpoint).preimage (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 268,
"column": 4
} | {
"line": 270,
"column": 48
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : LocallyOfFiniteType f\ninst✝¹ : IsSeparated f\ninst✝ : QuasiCompact f\nV : { x // QuasiFiniteAt f x } → (normalization f).Opens\nhxV✝ : ∀ (x : { x // QuasiFiniteAt f x }), (toNormalization f) ↑x ∈ V x\nhV : ∀ (x : { x // QuasiFiniteAt f x }), IsIso (toNormalization f ∣... | convert! this <;>
simp [Scheme.Hom.image_preimage_eq_opensRange_inf, -Scheme.preimage_basicOpen,
f.toNormalization.preimage_mono, hrV, H] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Finiteness.Descent | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Module.FaithfullyFlat R S\nI : Ideal R\nhI : (Ideal.map (algebraMap R S) I).FG\nf : S ⊗[R] ↥I →ₗ[S] S := ↑(AlgebraTensorModule.rid R S S) ∘ₗ (AlgebraTensorModule.lTensor S S) (Submodule.subtype I)\nhf : ... | apply Module.Finite.of_finite_tensorProduct_of_faithfullyFlat S | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.CotangentBaseChange | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nI : Ideal S\na : S →+* T ⊗[R] S := Algebra.TensorProduct.includeRight.toRingHom\nx : T ⊗[R] S\nhx : x ∈ map Algebra.TensorProduct.includeRight.toRingHom I\... | obtain ⟨y, rfl⟩ := I.map_includeRight_eq.le hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Etale.Descent | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 46
} | [
{
"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✝ : Etale T (T ⊗[R] S)\n⊢ Etale R S",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Algebra.Etal... | rw [Etale.iff_formallyUnramified_and_smooth] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Etale.Descent | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 52
} | [
{
"pp": "R S T : Type u_1\nx✝⁴ : CommRing R\nx✝³ : CommRing S\nx✝² : CommRing T\nx✝¹ : Algebra R S\nx✝ : Algebra R T\nh : Module.FaithfullyFlat R S\nh' : Algebra.Smooth S (S ⊗[R] T)\n⊢ Algebra.Smooth R T",
"usedConstants": [
"Algebra.Smooth.of_smooth_tensorProduct_of_faithfullyFlat"
]
}
] | exact .of_smooth_tensorProduct_of_faithfullyFlat S | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Group.Smooth | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 93
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nG : Scheme\nf : G ⟶ Spec (CommRingCat.of K)\ninst✝³ : LocallyOfFiniteType f\ninst✝² : GrpObj (Over.mk f)\ninst✝¹ : IsReduced G\ninst✝ : IsAlgClosed K\nthis✝ : JacobsonSpace ↥G\nthis : Nonempty ↥G\nH : (↑(Scheme.Hom.smoothLocus f))ᶜ.Nonempty\nx : ↥G\nhx : x ∈ (↑(Scheme.Hom.... | simpa [x', y', pointEquivClosedPoint] using congr(($hα).left (IsLocalRing.closedPoint K)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Tactic.CategoryTheory.Bicategory.Normalize | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 10
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf g h i : a ⟶ b\nj : b ⟶ c\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ≫ j ⟶ i ≫ j\nη₁ : g ≫ j ⟶ h ≫ j\nη₂ : g ≫ j ⟶ i ≫ j\nη₃ : f ≫ j ⟶ i ≫ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ η ≫ ηs) ▷ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Tactic.CategoryTheory.Bicategory.Normalize | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 10
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf g h i : a ⟶ b\nj : b ⟶ c\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ≫ j ⟶ i ≫ j\nη₁ : g ≫ j ⟶ h ≫ j\nη₂ : g ≫ j ⟶ i ≫ j\nη₃ : f ≫ j ⟶ i ≫ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ η ≫ ηs) ▷ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Bicategory.Normalize | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 10
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf g h i : a ⟶ b\nj : b ⟶ c\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ≫ j ⟶ i ≫ j\nη₁ : g ≫ j ⟶ h ≫ j\nη₂ : g ≫ j ⟶ i ≫ j\nη₃ : f ≫ j ⟶ i ≫ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ η ≫ ηs) ▷ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Bicategory.Normalize | {
"line": 121,
"column": 33
} | {
"line": 122,
"column": 10
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\nf g h i : a ⟶ b\nj : b ⟶ c\nα : f ≅ g\nη : g ⟶ h\nηs : h ⟶ i\nηs₁ : h ≫ j ⟶ i ≫ j\nη₁ : g ≫ j ⟶ h ≫ j\nη₂ : g ≫ j ⟶ i ≫ j\nη₃ : f ≫ j ⟶ i ≫ j\ne_ηs₁ : ηs ▷ j = ηs₁\ne_η₁ : η ▷ j = η₁\ne_η₂ : η₁ ≫ ηs₁ = η₂\ne_η₃ : (α ▷ᵢ j).hom ≫ η₂ = η₃\n⊢ (α.hom ≫ η ≫ ηs) ▷ ... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 60
} | [
{
"pp": "X S T : Scheme\nf : X ⟶ S\ng : T ⟶ S\ninst✝² : IsAffine T\nt : ↥T\ninst✝¹ : Flat f\ninst✝ : IsFinite f\n⊢ finrank f (g t) = IsAffine.finrank (pullback.snd f g) t",
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.PresheafedSpace.carrier",... | let i := S.affineOpenCover.f (S.affineOpenCover.idx (g t)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 98
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nx : ↥X\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f] (x : ↥X), (∃ R, Y = Spec R) → 1 ≤ finrank f (f x)\nhY : ¬∃ R, Y = Spec R\nR : CommRingCat\ng : Spec R ⟶ Y\nhg : IsOpenImmersion g\ny : ↥(Spec R)\nhy : g y = f x\n⊢ 1... | obtain ⟨z, hzl, hzr⟩ := Scheme.Pullback.exists_preimage_pullback (f := f) (g := g) x y hy.symm | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 454,
"column": 77
} | {
"line": 456,
"column": 50
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : B\nl : a ⟶ b\nr : b ⟶ a\nadj : l ⊣ r\n⊢ (mateEquiv adj adj) ((λ_ l).hom ≫ (ρ_ l).inv) = (ρ_ r).hom ≫ (λ_ r).inv",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.Iso.inv_hom_id",
"Equiv.instEquivLike",
"CategoryTheo... | by
simp [← cancel_mono (λ_ r).hom,
← conjugateEquiv_id adj, conjugateEquiv_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 96
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : SetLike σ A\ninst✝³ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsHomogeneous 𝒜 I\nx : A\nhx : x ∈ I\n⊢ ∑ i ∈ DFinsupp.support ((decompose 𝒜) x), ↑(((... | exact Ideal.sum_mem _ fun j _ => Ideal.subset_span ⟨⟨_, isHomogeneousElem_coe _⟩, h _ hx, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 37
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : DecidableEq ι\ninst✝³ : AddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ni : ι\nr : A\nhr : r = 0\n⊢ ↑(((decompose 𝒜) r) i) ∈ ⊥",
"usedConstants": [
"Eq.mpr",
"S... | rw [hr, decompose_zero, zero_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal | {
"line": 576,
"column": 91
} | {
"line": 580,
"column": 54
} | [
{
"pp": "ι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : DecidableEq ι\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : PartialOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\ni : ι\nr : A\nhr : ↑(((decompose 𝒜) r) 0)... | by
change (decompose 𝒜 (decompose 𝒜 r _ : A) 0 : A) = 0
by_cases h : i = 0
· rw [h, hr, decompose_zero, zero_apply, ZeroMemClass.coe_zero]
· rw [decompose_of_mem_ne 𝒜 (SetLike.coe_mem _) h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 60
} | [
{
"pp": "case homogeneous_mem_or_mem.refine_2\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\nh : I.IsPrime... | · exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf | {
"line": 123,
"column": 86
} | {
"line": 127,
"column": 68
} | [
{
"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\nU : (Opens ↑(ProjectiveSpectrum.top 𝒜))ᵒᵖ\na : (x : ↥(unop U)) → at ↑x\nha : (isLocallyFraction 𝒜).pred a\nx : ↥(unop U)\n⊢ ∃ V, ∃ (_ : ↑x ∈ V), ∃ i, (isFractionPre... | by
rcases ha x with ⟨V, m, i, j, ⟨r, r_mem⟩, ⟨s, s_mem⟩, nin, hy⟩
refine ⟨V, m, i, j, ⟨-r, neg_mem r_mem⟩, ⟨s, s_mem⟩, nin, fun y => ?_⟩
simp only [ext_iff_val, val_mk] at hy
simp only [Pi.neg_apply, ext_iff_val, val_neg, hy, val_mk, neg_mk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 124,
"column": 4
} | {
"line": 125,
"column": 25
} | [
{
"pp": "case hf.convert_4.add\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\nι : Type u_2\nf : ι → A\nhfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n\nhf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤\nx✝ x y : A\nhx : x ∈ Algebra.adjoin (... | rw [map_add, add_sub_add_comm]
exact add_mem ‹_› ‹_› | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic | {
"line": 124,
"column": 4
} | {
"line": 125,
"column": 25
} | [
{
"pp": "case hf.convert_4.add\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\nι : Type u_2\nf : ι → A\nhfn : ∀ (i : ι), ∃ n, f i ∈ 𝒜 n\nhf : Algebra.adjoin (↥(𝒜 0)) (Set.range f) = ⊤\nx✝ x y : A\nhx : x ∈ Algebra.adjoin (... | rw [map_add, add_sub_add_comm]
exact add_mem ‹_› ‹_› | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 300,
"column": 78
} | {
"line": 308,
"column": 54
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\na : A\nn : ℕ\nhn : a ∈ 𝒜 n\n⊢ a ∈ carrier f_deg q ↔\n HomogeneousLocalization.mk... | by
trans (HomogeneousLocalization.mk ⟨m * n, ⟨proj 𝒜 n a ^ m, by rw [← smul_eq_mul]; mem_tac⟩,
⟨f ^ n, by rw [mul_comm]; mem_tac⟩, ⟨_, rfl⟩⟩ : A⁰_ f) ∈ q.asIdeal
· refine ⟨fun h ↦ h n, fun h i ↦ if hi : i = n then hi ▸ h else ?_⟩
convert! zero_mem q.asIdeal
apply HomogeneousLocalization.val_injective
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.Homogeneous.Maps | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 12
} | [
{
"pp": "case add\nA : Type u_1\nB : Type u_2\nC : Type u_3\nσ : Type u_4\nτ : Type u_5\nω : Type u_6\nι : Type u_7\nF : Type u_8\nG : Type u_9\ninst✝¹³ : Semiring A\ninst✝¹² : Semiring B\ninst✝¹¹ : Semiring C\ninst✝¹⁰ : SetLike σ A\ninst✝⁹ : SetLike τ B\ninst✝⁸ : SetLike ω C\ninst✝⁷ : AddSubmonoidClass σ A\nin... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 821,
"column": 2
} | {
"line": 822,
"column": 55
} | [
{
"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\n⊢ Away 𝒜 f →+* Away 𝒜 x",
"usedConstants":... | let e := RingEquiv.ofLeftInverse (f := algebraMap (Away 𝒜 x) (Localization.Away x))
(h := (val_injective _).hasLeftInverse.choose_spec) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 830,
"column": 2
} | {
"line": 831,
"column": 55
} | [
{
"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 : Away 𝒜 f\n⊢ val ((awayMap 𝒜 hg hx) a) = (a... | let e := RingEquiv.ofLeftInverse (f := algebraMap (Away 𝒜 x) (Localization.Away x))
(h := (val_injective _).hasLeftInverse.choose_spec) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 401,
"column": 37
} | {
"line": 401,
"column": 47
} | [
{
"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\n⊢ (proj 𝒜 (i -... | by mem_tac | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 432,
"column": 13
} | {
"line": 434,
"column": 68
} | [
{
"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\ni : ℕ\na : A\nha : a ∈ asIdeal f_deg hm q\nj : ℕ\nh : ¬i = j\n⊢ HomogeneousLocalizat... | by
simpa only [proj_apply, decompose_of_mem_ne 𝒜 (SetLike.coe_mem (decompose 𝒜 a i)) h,
zero_pow hm.ne', map_zero] using carrier.zero_mem f_deg hm q j | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 32
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nR : LocalSubring K\nhR : IsMax R\nx : K\nhx : x ∉ R.toSubring\nhx0 : x ≠ 0\n⊢ IsIntegral (↥R.toSubring) x⁻¹",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneC... | let := invertibleOfNonzero hx0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 32
} | [
{
"pp": "case inr\nK : Type u_3\ninst✝¹ : Field K\nx : K\nR : Subring K\nhxR : x ∉ R\ninst✝ : IsIntegrallyClosedIn (↥R) K\nhx0 : x ≠ 0\n⊢ ∃ V, R ≤ V.toSubring ∧ x ∉ V",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.toOne",
"DivInv... | let := invertibleOfNonzero hx0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 32
} | [
{
"pp": "case inr\nK : Type u_3\ninst✝¹ : Field K\nx : K\nR : LocalSubring K\nhxR : x ∉ R.toSubring\ninst✝ : IsIntegrallyClosedIn (↥R.toSubring) K\nhx0 : x ≠ 0\n⊢ ∃ V, R ≤ V.toLocalSubring ∧ x ∉ V",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"GroupWithZero.toDivisionMonoid",
"Inv... | let := invertibleOfNonzero hx0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 227,
"column": 39
} | {
"line": 227,
"column": 96
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nK : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Field K\ninst✝⁵ : IsDomain R\ninst✝⁴ : ValuationRing R\ninst✝³ : IsLocalRing S\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nf : R →+* S\ng : S →+* K\nh : g.comp f = algebraMap R K\ninst✝ : IsLocalHom f\n... | by convert! (IsFractionRing.injective R K); rw [← h]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.LocalSubring | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 54
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nK : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Field K\ninst✝⁵ : IsDomain R\ninst✝⁴ : ValuationRing R\ninst✝³ : IsLocalRing S\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\nf : R →+* S\ng : S →+* K\nh : g.comp f = algebraMap R K\ninst✝ : IsLocalHom f\n... | exact ⟨y, Subtype.ext (by simpa [← h] using hy)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 728,
"column": 2
} | {
"line": 775,
"column": 40
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↥(pbo f)\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\n⊢ IsLocalization ((ConcreteCategory.hom (toSpec 𝒜 f).base) x).asIdeal.primeCompl\n (AtPrime 𝒜 (↑x).asHo... | letI : Algebra (Away 𝒜 f) (AtPrime 𝒜 x.1.asHomogeneousIdeal.toIdeal) :=
(mapId 𝒜 (Submonoid.powers_le.mpr x.2)).toAlgebra
constructor; constructor
· rintro ⟨y, hy⟩
obtain ⟨y, rfl⟩ := HomogeneousLocalization.mk_surjective y
refine .of_mul_eq_one
(.mk ⟨y.deg, y.den, y.num, (mk_mem_toSpec_base_app... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 728,
"column": 2
} | {
"line": 775,
"column": 40
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nx : ↥(pbo f)\nm : ℕ\nf_deg : f ∈ 𝒜 m\nhm : 0 < m\n⊢ IsLocalization ((ConcreteCategory.hom (toSpec 𝒜 f).base) x).asIdeal.primeCompl\n (AtPrime 𝒜 (↑x).asHo... | letI : Algebra (Away 𝒜 f) (AtPrime 𝒜 x.1.asHomogeneousIdeal.toIdeal) :=
(mapId 𝒜 (Submonoid.powers_le.mpr x.2)).toAlgebra
constructor; constructor
· rintro ⟨y, hy⟩
obtain ⟨y, rfl⟩ := HomogeneousLocalization.mk_surjective y
refine .of_mul_eq_one
(.mk ⟨y.deg, y.den, y.num, (mk_mem_toSpec_base_app... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 254,
"column": 10
} | {
"line": 254,
"column": 44
} | [
{
"pp": "P Q : MorphismProperty Scheme\nS : Scheme\ninst✝⁵ : P.IsStableUnderBaseChange\ninst✝⁴ : P.IsMultiplicative\ninst✝³ : P.RespectsIso\ninst✝² : Q.IsStableUnderComposition\ninst✝¹ : Q.IsStableUnderBaseChange\ninst✝ : Q.HasOfPostcompProperty Q\nX : Q.Over ⊤ S\nR : Sieve X\nY : ↥X.left → Q.Over ⊤ S\nf : (x :... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 581,
"column": 14
} | {
"line": 581,
"column": 22
} | [
{
"pp": "case fac\nC₀ : 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✝ : HasLi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 581,
"column": 14
} | {
"line": 581,
"column": 22
} | [
{
"pp": "case fac\nC₀ : 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✝ : HasLi... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 581,
"column": 14
} | {
"line": 581,
"column": 22
} | [
{
"pp": "case fac\nC₀ : 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✝ : HasLi... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 581,
"column": 14
} | {
"line": 581,
"column": 22
} | [
{
"pp": "case fac\nC₀ : 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✝ : HasLi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 581,
"column": 14
} | {
"line": 581,
"column": 22
} | [
{
"pp": "case fac\nC₀ : 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✝ : HasLi... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 581,
"column": 14
} | {
"line": 581,
"column": 22
} | [
{
"pp": "case fac\nC₀ : 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✝ : HasLi... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 622,
"column": 49
} | {
"line": 622,
"column": 57
} | [
{
"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... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 622,
"column": 49
} | {
"line": 622,
"column": 57
} | [
{
"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... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 622,
"column": 49
} | {
"line": 622,
"column": 57
} | [
{
"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... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 38
} | [
{
"pp": "S✝ : Scheme\nP : MorphismProperty Scheme\nS : Scheme\n𝒰 : Cover (precoverage P) S\ninst✝ : QuasiCompactCover 𝒰.toPreZeroHypercover\n⊢ (𝒰.ulift.sum (QuasiCompactCover.ulift 𝒰.toPreZeroHypercover)).presieve₀ ∈ (precoverage P).coverings S",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeomet... | rw [presieve₀_mem_precoverage_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 90
} | [
{
"pp": "case intro\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFract... | have ⟨i₀, hi1⟩ : ∃ a, ψ a = Kmax := by simpa using Finset.max'_mem (Finset.univ.image ψ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 95
} | [
{
"pp": "case intro\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFract... | have hφ'1 (s) : φ' (awayMap 𝒜 (hxdi i₀) rfl s) = φ s := IsLocalization.Away.lift_eq _ hunit s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.MorphismProperty.CommaSites | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 95
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nP : MorphismProperty C\nS : C\ninst✝ : P.IsStableUnderComposition\nK : Precoverage C\nH : K ≤ P.precoverage\nX : P.Over ⊤ S\n𝒰 : (Precoverage.comap (CategoryTheory.Over.forget S) K).ZeroHypercover ((Over.forget P ⊤ S).obj X)\n⊢ ∃ T, Presieve.map (Over.forg... | let 𝒱 : PreZeroHypercover X :=
⟨𝒰.I₀, fun i ↦ Over.mk _ (𝒰.X i).hom ?_, fun i ↦ Over.homMk (𝒰.f i).left (by simp) trivial⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Abelian.CommSq | {
"line": 160,
"column": 2
} | {
"line": 161,
"column": 33
} | [
{
"pp": "case refine_2\nC : 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\nA₁ : C\nπ₁ : A₁ ⟶ A₀\nw✝ : Epi π₁\nx₁ :\n A₁ ⟶\n { X₁ := X₁, X₂ := X₂ ⊞ X₃, X₃ := ⋯.cokernelCofork.pt, f ... | · ext
simpa using hx₁ =≫ biprod.snd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Subcanonical | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 33
} | [
{
"pp": "case pos\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\ninst✝¹ : J.Subcanonical\nι : Type u_1\nX : ι → C\nc : Cofan X\nH : Sieve.ofArrows X c.inj ∈ J c.pt\ninst✝ : ∀ (i : ι), Mono (c.inj i)\nhempty : ∀ (Y : C) (a : IsInitial Y), ⊥ ∈ J Y\nhdisj : ∀ {i j : ι}, i ≠ j → ∀ {Y : C} (a :... | rw [(cancel_mono _).mp hab] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 74,
"column": 10
} | {
"line": 76,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\ninst✝³ : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝² : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝¹ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf✝ : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ... | dsimp
simpa only [← cancel_mono g, Category.assoc, Subobject.ofMkLEMk_comp,
Category.comp_id] using MonoOver.w (F.map f) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Subobject | {
"line": 74,
"column": 10
} | {
"line": 76,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\ninst✝³ : IsGrothendieckAbelian.{w, v, u} C\nX : C\nJ : Type w\ninst✝² : SmallCategory J\nF : J ⥤ MonoOver X\ninst✝¹ : IsFiltered J\nc : Cocone (F ⋙ MonoOver.forget X ⋙ Over.forget X)\nhc : IsColimit c\nf✝ : c.pt ⟶ X\nhf : ∀ (j : J), c.ι.app j ... | dsimp
simpa only [← cancel_mono g, Category.assoc, Subobject.ofMkLEMk_comp,
Category.comp_id] using MonoOver.w (F.map f) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Preorder | {
"line": 75,
"column": 88
} | {
"line": 76,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝¹ : Preorder C\nJ : Type u'\ninst✝ : Category.{v, u'} J\nF : J ⥤ C\nc : Cocone F\n⊢ c.pt ∈ upperBounds (Set.range F.obj)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"congrArg",
"Quiver.Hom.le",
"CategoryTheory.Functor.category",
"_pr... | by
intro x ⟨i, p⟩; rw [← p]; exact (c.ι.app i).le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Preorder | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 46
} | [
{
"pp": "C : Type u\ninst✝ : SemilatticeSup C\nF : Discrete WalkingPair ⥤ C\nthis : HasColimit (pair (F.obj { as := WalkingPair.left }) (F.obj { as := WalkingPair.right }))\n⊢ HasColimit F",
"usedConstants": [
"PartialOrder.toPreorder",
"CategoryTheory.Limits.hasColimit_of_iso",
"CategoryT... | apply hasColimit_of_iso (diagramIsoPair F) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.SmallObject.Iteration.FunctorOfCocone | {
"line": 74,
"column": 31
} | {
"line": 85,
"column": 24
} | [
{
"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\ni₁ i₂ i₃ : J\nhi : i₁ ≤ i₂\nhi' : i₂ ≤ i₃\nhi₃ : i₃ ≤ j\n⊢ map c i₁ i₃ ⋯ hi₃ = map c i₁ i₂ hi ⋯ ≫ map c i₂ i₃ hi' hi₃",
"usedConstants": [
"Eq.mpr",
"CategoryTheor... | by
obtain hi₁₂ | rfl := hi.lt_or_eq
· obtain hi₂₃ | rfl := hi'.lt_or_eq
· dsimp [map]
obtain hi₃' | rfl := hi₃.lt_or_eq
· rw [dif_pos hi₃', dif_pos (hi₂₃.trans hi₃'), dif_pos hi₃', assoc, assoc,
Iso.inv_hom_id_assoc, ← Functor.map_comp_assoc, homOfLE_comp]
· rw [dif_neg (by simp), di... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.IsSmall | {
"line": 41,
"column": 2
} | {
"line": 45,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW : MorphismProperty C\nι : Type t\ninst✝ : Small.{w, t} ι\nA B : ι → C\nf : (i : ι) → A i ⟶ B i\n⊢ IsSmall.{w, v, u} (ofHoms f)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"HEq.refl",
"Subtype.casesOn"... | let φ (i : ι) : (ofHoms f).toSet := ⟨Arrow.mk (f i), ⟨i⟩⟩
have hφ : Function.Surjective φ := by
rintro ⟨⟨_, _, f⟩, ⟨i⟩⟩
exact ⟨i, rfl⟩
exact ⟨small_of_surjective hφ⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.IsSmall | {
"line": 41,
"column": 2
} | {
"line": 45,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nW : MorphismProperty C\nι : Type t\ninst✝ : Small.{w, t} ι\nA B : ι → C\nf : (i : ι) → A i ⟶ B i\n⊢ IsSmall.{w, v, u} (ofHoms f)",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"HEq.refl",
"Subtype.casesOn"... | let φ (i : ι) : (ofHoms f).toSet := ⟨Arrow.mk (f i), ⟨i⟩⟩
have hφ : Function.Surjective φ := by
rintro ⟨⟨_, _, f⟩, ⟨i⟩⟩
exact ⟨i, rfl⟩
exact ⟨small_of_surjective hφ⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 13
} | [
{
"pp": "p : ℕ\nz : ℤ\nhp : p ≠ 1\nhz : z ≠ 0\n⊢ padicValRat p ↑z = ↑(multiplicity (↑p) z)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"congrArg",
"padicValInt",
"Rat",
"Rat.instIntCast",
"id",
"Int",
"padicValRat",
"Nat.cast",
"multiplicity",... | of_int, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 51
} | [
{
"pp": "p n : ℕ\nhp : 1 ≤ 0\nh : ¬p ∣ n\n⊢ False",
"usedConstants": [
"Nat.instMulZeroClass",
"lt_irrefl",
"Nat.instOne",
"PartialOrder.toPreorder",
"Nat.instZeroLEOneClass",
"instOfNatNat",
"zero_lt_one",
"Nat.instNeZeroSucc",
"Nat",
"Nat.instPar... | exact lt_irrefl 0 (lt_of_lt_of_le zero_lt_one hp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 494,
"column": 28
} | {
"line": 494,
"column": 45
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nh1 : 2 ^ padicValNat 2 (n + 1) < n + 1\nh2 : n < 2 ^ (padicValNat 2 (n + 1) + 1)\n⊢ False",
"usedConstants": [
"instPowNat",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"Nat.instMulOneClass",
"Eq.mp",
"MulOne.toMul",
"padicValNat",
... | ← mul_one (2 ^ _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 61
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : ¬n = 0\n⊢ padicValNat p n ! ≤ n",
"usedConstants": [
"le_of_lt",
"padicValNat",
"Nat.factorial",
"Nat.instPreorder",
"Nat",
"padicValNat_factorial_lt_of_ne_zero"
]
}
] | exact le_of_lt (padicValNat_factorial_lt_of_ne_zero p hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 61
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : ¬n = 0\n⊢ padicValNat p n ! ≤ n",
"usedConstants": [
"le_of_lt",
"padicValNat",
"Nat.factorial",
"Nat.instPreorder",
"Nat",
"padicValNat_factorial_lt_of_ne_zero"
]
}
] | exact le_of_lt (padicValNat_factorial_lt_of_ne_zero p hn) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 602,
"column": 4
} | {
"line": 602,
"column": 61
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : ¬n = 0\n⊢ padicValNat p n ! ≤ n",
"usedConstants": [
"le_of_lt",
"padicValNat",
"Nat.factorial",
"Nat.instPreorder",
"Nat",
"padicValNat_factorial_lt_of_ne_zero"
]
}
] | exact le_of_lt (padicValNat_factorial_lt_of_ne_zero p hn) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 255,
"column": 63
} | {
"line": 256,
"column": 68
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nm : ℕ\n⊢ padicNorm p ↑m = 1 ↔ ¬p ∣ m",
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Dvd.dvd",
"padicNorm.int_eq_one_iff",
"congrArg",
"Iff.rfl",
"Rat",
"AddGroupWithOne.toAd... | by
rw [← Int.natCast_dvd_natCast, ← int_eq_one_iff, Int.cast_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 371,
"column": 18
} | {
"line": 371,
"column": 29
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ_[p]\nh : IsUnit z\nw : ℤ_[p]\neq : 1 = z * w\nthis : ‖z‖ * ‖w‖ ≤ ‖z‖\n⊢ 1 ≤ ‖z‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"HMul.hMul",
"PadicInt",
"con... | ← norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 385,
"column": 60
} | {
"line": 386,
"column": 26
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nz : ℤ_[p]\n⊢ ¬IsUnit z ↔ ‖z‖ < 1",
"usedConstants": [
"Norm.norm",
"Real",
"mem_nonunits_iff._simp_1",
"PadicInt",
"congrArg",
"CommSemiring.toSemiring",
"Real.instLT",
"IsUnit",
"nonunits",
"Membership.mem"... | by
simpa using mem_nonunits | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 550,
"column": 12
} | {
"line": 550,
"column": 30
} | [
{
"pp": "case neg\np : ℕ\nhp : Fact (Nat.Prime p)\nx✝ y : ℤ_[p]\nx : ℕ → ℤ_[p]\nhx✝ : ∀ {m n : ℕ}, m ≤ n → ‖x m - x n‖ ≤ ↑p ^ (-↑m)\nx' : CauSeq ℤ_[p] norm := ⟨x, ⋯⟩\nn : ℕ\nthis✝ : 0 < ↑p ^ (-↑n)\ni : ℕ\nhin : n < i\nhi : ‖↑x' i - ↑(const norm x'.lim) i‖ < ↑p ^ (-↑n)\nhx : ‖x n - x i‖ ≤ ↑p ^ (-↑n)\nthis : ‖x n... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 12
} | [
{
"pp": "case neg\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℤ\nh✝ : ¬x = 0\n⊢ -↑(padicValInt p x) ≤ log 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Int.instIsStrictOrderedRing",
"instConditionallyCompleteLinearOrder",
"padicValInt",
"Partia... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 12
} | [
{
"pp": "case neg\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℤ\nh✝ : ¬x = 0\n⊢ 1 ≠ 0",
"usedConstants": [
"WithZero.instNontrivial",
"False",
"Int.instInhabited",
"NeZero.one",
"congrArg",
"AddMonoid.toAddZeroClass",
"WithZero.instMulZeroOneClass",
"Multiplicativ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 113,
"column": 34
} | {
"line": 119,
"column": 12
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℤ\n⊢ (padicValuation p) x ≤ 1",
"usedConstants": [
"WithZero.instNontrivial",
"AddGroup.toSubtractionMonoid",
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"Int.instAddCommMonoid",
"NegZeroClass.toNeg",
"Multiplicative.... | by
simp only [← Rat.padicValuation_cast, Rat.padicValuation, Valuation.coe_mk,
MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, Rat.intCast_eq_zero_iff, padicValRat.of_int]
split_ifs
· simp
· rw [← le_log_iff_exp_le] <;>
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 12
} | [
{
"pp": "case pos\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℤ\nh✝ : x = 0\n⊢ 0 = 1 ↔ ¬↑p ∣ x",
"usedConstants": [
"WithZero.instNontrivial",
"False",
"Int.instInhabited",
"Dvd.dvd",
"NeZero.one",
"congrArg",
"AddMonoid.toAddZeroClass",
"not_true_eq_false",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 12
} | [
{
"pp": "case pos\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℤ\nh✝ : x = 0\n⊢ 0 = 1 ↔ ¬↑p ∣ x",
"usedConstants": [
"WithZero.instNontrivial",
"False",
"Int.instInhabited",
"Dvd.dvd",
"NeZero.one",
"congrArg",
"AddMonoid.toAddZeroClass",
"not_true_eq_false",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 12
} | [
{
"pp": "case pos\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℤ\nh✝ : x = 0\n⊢ 0 = 1 ↔ ¬↑p ∣ x",
"usedConstants": [
"WithZero.instNontrivial",
"False",
"Int.instInhabited",
"Dvd.dvd",
"NeZero.one",
"congrArg",
"AddMonoid.toAddZeroClass",
"not_true_eq_false",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 24
} | [
{
"pp": "case inl\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nε : ℚ\nhε : 0 < ε\nh : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm\nN : ℕ\nhN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε\ni : ℕ\nhi : i ≥ N\nhne : ¬f - const (padicNorm p) (↑f i) ≈ 0\nhge : ε ≤ padicNorm p (↑(f - cons... | exact hN _ hgen _ hi | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 24
} | [
{
"pp": "case inl\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nε : ℚ\nhε : 0 < ε\nh : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm\nN : ℕ\nhN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε\ni : ℕ\nhi : i ≥ N\nhne : ¬f - const (padicNorm p) (↑f i) ≈ 0\nhge : ε ≤ padicNorm p (↑(f - cons... | exact hN _ hgen _ hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 24
} | [
{
"pp": "case inl\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : PadicSeq p\nε : ℚ\nhε : 0 < ε\nh : ∀ (N : ℕ), ∃ i ≥ N, ε ≤ (f - const (padicNorm p) (↑f i)).norm\nN : ℕ\nhN : ∀ j ≥ N, ∀ k ≥ N, padicNorm p (↑f j - ↑f k) < ε\ni : ℕ\nhi : i ≥ N\nhne : ¬f - const (padicNorm p) (↑f i) ≈ 0\nhge : ε ≤ padicNorm p (↑(f - cons... | exact hN _ hgen _ hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 885,
"column": 2
} | {
"line": 888,
"column": 11
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ_[p]\n⊢ ∃ q', ‖q‖ = ↑q'",
"usedConstants": [
"Rat.instOfNat",
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instZeroPadic",
"Real.instZero",
"congrArg",
"Real.instRatCast",
"Rat",
"norm_z... | exact if h : q = 0 then ⟨0, by simp [h]⟩
else
let ⟨n, hn⟩ := padicNormE.image h
⟨_, hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 885,
"column": 2
} | {
"line": 888,
"column": 11
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ_[p]\n⊢ ∃ q', ‖q‖ = ↑q'",
"usedConstants": [
"Rat.instOfNat",
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instZeroPadic",
"Real.instZero",
"congrArg",
"Real.instRatCast",
"Rat",
"norm_z... | exact if h : q = 0 then ⟨0, by simp [h]⟩
else
let ⟨n, hn⟩ := padicNormE.image h
⟨_, hn⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 885,
"column": 2
} | {
"line": 888,
"column": 11
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ_[p]\n⊢ ∃ q', ‖q‖ = ↑q'",
"usedConstants": [
"Rat.instOfNat",
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instZeroPadic",
"Real.instZero",
"congrArg",
"Real.instRatCast",
"Rat",
"norm_z... | exact if h : q = 0 then ⟨0, by simp [h]⟩
else
let ⟨n, hn⟩ := padicNormE.image h
⟨_, hn⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 938,
"column": 8
} | {
"line": 938,
"column": 28
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ\n⊢ ‖↑p‖ * 1 < 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real",
"HMul.hMul",
"congrArg",
"Real.instInv",
"AddGroupWithOne.toAddMonoidWithOne",
"Real.instLT",
"Real.semiring",
... | rw [mul_one, norm_p] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1004,
"column": 4
} | {
"line": 1004,
"column": 20
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] norm\ncau_seq_norm_e : IsCauSeq ⇑padicNormE ↑f\nq : ℚ_[p]\nhq : ∀ ε > 0, ∃ N, ∀ i ≥ N, padicNormE (↑⟨↑f, cau_seq_norm_e⟩ i - q) < ε\nε : ℝ\nhε : ε > 0\nε' : ℚ\nhε' : 0 < ↑ε' ∧ ↑ε' < ε\n⊢ ∃ i, ∀ j ≥ i, ‖↑(f - const norm q) j‖ < ε",
"usedConstants": [
... | norm_cast at hε' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1211,
"column": 2
} | {
"line": 1211,
"column": 83
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nhx : x ≠ 0\n⊢ addValuation x = ↑x.valuation",
"usedConstants": [
"Int.instAddCommGroup",
"instZeroPadic",
"congrArg",
"Int.instLinearOrder",
"Decidable",
"AddCommGroup.toAddCancelCommMonoid",
"Classical.propDecidab... | simp only [Padic.addValuation, AddValuation.of_apply, addValuationDef, if_neg hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1211,
"column": 2
} | {
"line": 1211,
"column": 83
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nhx : x ≠ 0\n⊢ addValuation x = ↑x.valuation",
"usedConstants": [
"Int.instAddCommGroup",
"instZeroPadic",
"congrArg",
"Int.instLinearOrder",
"Decidable",
"AddCommGroup.toAddCancelCommMonoid",
"Classical.propDecidab... | simp only [Padic.addValuation, AddValuation.of_apply, addValuationDef, if_neg hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1211,
"column": 2
} | {
"line": 1211,
"column": 83
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nhx : x ≠ 0\n⊢ addValuation x = ↑x.valuation",
"usedConstants": [
"Int.instAddCommGroup",
"instZeroPadic",
"congrArg",
"Int.instLinearOrder",
"Decidable",
"AddCommGroup.toAddCancelCommMonoid",
"Classical.propDecidab... | simp only [Padic.addValuation, AddValuation.of_apply, addValuationDef, if_neg hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.Normalized | {
"line": 71,
"column": 36
} | {
"line": 71,
"column": 59
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Category.{v_1, u_1} A\ninst✝ : Abelian A\nX✝ X : SimplicialObject A\nn : ℕ\n⊢ (fun n ↦ (NormalizedMooreComplex.objX X n).factorThru (PInfty.f n) ⋯) (n + 1) ≫\n (NormalizedMooreComplex.objX X (n + 1)).arrow ≫ ((alternatingFaceMapComplex A).obj X).d (n + 1) n =\n K[X].d (n ... | factorThru_arrow_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 16
} | [
{
"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\nx✝ : Discrete (Splitting.IndexSet Δ)\nA : Splitting.IndexSet Δ\nfac : A.e ≫ 𝟙 (unop A.fs... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 53,
"column": 8
} | {
"line": 53,
"column": 20
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nK : ChainComplex C ℕ\nn : ℕ\n⊢ K.d (n + 1) n = K.d (n + 1) n ≫ 𝟙 (K.X n)",
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom"... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 58
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nn : ℕ\n⊢ s.πSummand (IndexSet.id (op ⦋n⦌)) ≫ (s.cofan (op ⦋n⦌)).inj (IndexSet.id (op ⦋n⦌)) ≫ PInfty.f n =\n ∑ j, (s.πSummand j ≫ (s.cofan (op ⦋n⦌)).inj j) ≫ PInfty.f n",
"usedConstants":... | rw [Fintype.sum_eq_single (IndexSet.id (op ⦋n⦌)), assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected | {
"line": 113,
"column": 4
} | {
"line": 124,
"column": 26
} | [
{
"pp": "case mpr\nY : Type u_2\ninst✝ : TopologicalSpace Y\n⊢ (PathConnectedSpace Y ∧ ∀ (x : Y) (γ : Path x x), γ.Homotopic (Path.refl x)) →\n PathConnectedSpace Y ∧ ∀ {x y : Y} (p₁ p₂ : Path x y), p₁.Homotopic p₂",
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyC... | intro ⟨hpc, hloops⟩
refine ⟨hpc, fun {x y} p₁ p₂ => ?_⟩
-- Work in the quotient where structural steps can be done by simp
rw [← eq]
replace hloops : ∀ (x : Y) (γ : Path x x),
(⟦γ⟧ : Path.Homotopic.Quotient x x) = ⟦Path.refl x⟧ :=
fun x γ => Quotient.sound (hloops x γ)
have h : trans ⟦... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected | {
"line": 113,
"column": 4
} | {
"line": 124,
"column": 26
} | [
{
"pp": "case mpr\nY : Type u_2\ninst✝ : TopologicalSpace Y\n⊢ (PathConnectedSpace Y ∧ ∀ (x : Y) (γ : Path x x), γ.Homotopic (Path.refl x)) →\n PathConnectedSpace Y ∧ ∀ {x y : Y} (p₁ p₂ : Path x y), p₁.Homotopic p₂",
"usedConstants": [
"_private.Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyC... | intro ⟨hpc, hloops⟩
refine ⟨hpc, fun {x y} p₁ p₂ => ?_⟩
-- Work in the quotient where structural steps can be done by simp
rw [← eq]
replace hloops : ∀ (x : Y) (γ : Path x x),
(⟦γ⟧ : Path.Homotopic.Quotient x x) = ⟦Path.refl x⟧ :=
fun x γ => Quotient.sound (hloops x γ)
have h : trans ⟦... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 55,
"column": 6
} | {
"line": 57,
"column": 17
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ModelCategory C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : IsCofibrant X\nh✝ : LeftHomotopyRel f g\nQ : PathObject Y\ninst✝ : Q.IsGood\nP : Cylinder X := ⋯.choose\nh : ⋯.choose.LeftHomotopy f g\nh' : ⋯.choose.IsGood\nsq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π... | have := sq.fac_right =≫ prod.fst
rw [Category.assoc, Q.p_fst, prod.lift_fst] at this
simp [this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.Homotopy | {
"line": 55,
"column": 6
} | {
"line": 57,
"column": 17
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : ModelCategory C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : IsCofibrant X\nh✝ : LeftHomotopyRel f g\nQ : PathObject Y\ninst✝ : Q.IsGood\nP : Cylinder X := ⋯.choose\nh : ⋯.choose.LeftHomotopy f g\nh' : ⋯.choose.IsGood\nsq : CommSq (f ≫ Q.ι) P.i₀ Q.p (prod.lift (P.π... | have := sq.fac_right =≫ prod.fst
rw [Category.assoc, Q.p_fst, prod.lift_fst] at this
simp [this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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