module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.Normalization | {
"line": 59,
"column": 8
} | {
"line": 59,
"column": 72
} | {
"line": 60,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU V : Y.Opens\nx : ↑Γ(X, f ⁻¹ᵁ Opposite.unop (Opposite.op V))\nhx : x ∈ integralClosure ↑Γ(Y, Opposite.unop (Opposite.op V)) ↑Γ(X, f ⁻¹ᵁ Opposite.unop (Opposite.op V))\ni : Opposite.op V ⟶ Opposite.op U\nalgInst✝⁴ : Algebra ↑Γ(Y, U) ↑Γ(X, f ⁻¹ᵁ U) := (CommRingCat.Hom.hom (app f... | [] | simp [RingHom.algebraMap_toAlgebra, ← CommRingCat.hom_comp]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 22
} | {
"line": 309,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\n⊢ (-1) ^ (2 * m + (n + 1)) = (-1) ^ (n + 1)",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
"one_pow",
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNon... | [] | simp [pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 22
} | {
"line": 309,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\n⊢ (-1) ^ (2 * m + (n + 1)) = (-1) ^ (n + 1)",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
"one_pow",
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNon... | [] | simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 22
} | {
"line": 309,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm : ℕ\nhf : f.natDegree ≤ m\nn : ℕ\n⊢ (-1) ^ (2 * m + (n + 1)) = (-1) ^ (n + 1)",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
"one_pow",
"NegZeroClass.toNeg",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNon... | [] | simp [pow_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 342,
"column": 17
} | {
"line": 342,
"column": 74
} | {
"line": 342,
"column": 74
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm n k✝ : ℕ\nhf : f.natDegree ≤ m\nk : ℕ\nIH : f.resultant g (m + k) n = (-1) ^ (n * k) * g.coeff n ^ k * f.resultant g m n\n⊢ f.resultant g (m + (k + 1)) n = (-1) ^ (n * (k + 1)) * g.coeff n ^ (k + 1) * f.resultant g m n",
"ppTerm": "?succ",
... | [
"case succ\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nm n k✝ : ℕ\nhf : f.natDegree ≤ m\nk : ℕ\nIH : f.resultant g (m + k) n = (-1) ^ (n * k) * g.coeff n ^ k * f.resultant g m n\n⊢ (-1) ^ n * g.coeff n * ((-1) ^ (n * k) * g.coeff n ^ k * f.resultant g m n) =\n (-1) ^ (n * (k + 1)) * g.coeff n ^ (k + 1) * f.re... | simp [← add_assoc, resultant_succ_left_deg, hf.trans, IH] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 211,
"column": 6
} | {
"line": 211,
"column": 32
} | {
"line": 211,
"column": 32
} | [
{
"pp": "case e'_3\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\ne : ↑(fromNormalization f ⁻¹ᵁ ↑U) ≅ (normalizationOpenCover f).X U := ⋯\nthis : (CommRingCat.Hom.hom ((normalizationDiagramMap f).app (Opposite.op ↑U))).IsIntegral\ne_1✝ : (normalizationOpenCover ... | [
"case e'_3\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU : ↑Y.affineOpens\ne : ↑(fromNormalization f ⁻¹ᵁ ↑U) ≅ (normalizationOpenCover f).X U :=\n IsOpenImmersion.isoOfRangeEq (fromNormalization f ⁻¹ᵁ ↑U).ι ((normalizationOpenCover f).f U) ⋯\nthis : (CommRingCat.Hom.hom ((normaliza... | ← cancel_mono U.2.fromSpec | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 412,
"column": 6
} | {
"line": 413,
"column": 48
} | {
"line": 414,
"column": 6
} | [
{
"pp": "K : Type u_3\ninst✝ : Field K\nf g : K[X]\nhf : f.Monic\nhg : g.Monic\nhf' : f.Splits\nhg' : g.Splits\nthis :\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree → f.resultant g = (Multiset.m... | [
"K : Type u_3\ninst✝ : Field K\nf g : K[X]\nhf : f.Monic\nhg : g.Monic\nhf' : f.Splits\nhg' : g.Splits\nthis :\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree → f.resultant g = (Multiset.map (fun ij ↦... | rw [resultant_comm, this g f hg hf hg' hf' (le_of_not_ge hfg), ← Multiset.map_swap_product,
Multiset.map_map, Multiset.prod_map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 674,
"column": 4
} | {
"line": 674,
"column": 36
} | {
"line": 675,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FiniteType R S\np : Ideal S\nH : ZariskisMainProperty R p\ns : Finset S\nhs : adjoin R ↑s = ⊤\nr : S\nhrp : r ∉ p\nhr : IsIntegral R r\nm : S → ℕ\nhm : ∀ (x : S), IsIntegral R (r ^ m x * x... | [
"case refine_1\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : FiniteType R S\np : Ideal S\nH : ZariskisMainProperty R p\ns : Finset S\nhs : adjoin R ↑s = ⊤\nr : S\nhrp : r ∉ p\nhr : IsIntegral R r\nm : S → ℕ\nhm : ∀ (x : S), IsIntegral R (r ^ m x * x)\nt : Set S... | simp only [t, Set.finite_insert] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Quotient.Over | {
"line": 32,
"column": 4
} | {
"line": 32,
"column": 62
} | {
"line": 34,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\nf : S →ₐ[R] T\nHf : Function.Surjective ⇑f\nP : Ideal R\nP' : Ideal S\ninst✝¹ : P'.IsPrime\ninst✝ : P'.LiesOver P\nhkP' : ¬RingHom.ker f.to... | [] | rintro ⟨q, ⟨_, _⟩, rfl⟩; exact hkP' (Ideal.ker_le_comap _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Quotient.Over | {
"line": 32,
"column": 4
} | {
"line": 32,
"column": 62
} | {
"line": 34,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\nf : S →ₐ[R] T\nHf : Function.Surjective ⇑f\nP : Ideal R\nP' : Ideal S\ninst✝¹ : P'.IsPrime\ninst✝ : P'.LiesOver P\nhkP' : ¬RingHom.ker f.to... | [] | rintro ⟨q, ⟨_, _⟩, rfl⟩; exact hkP' (Ideal.ker_le_comap _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 68
} | {
"line": 421,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ : X ⟶ T\nf₂ : T ⟶ Y\ninst✝ : IsIntegralHom f₂\nH : f = f₁ ≫ f₂\nU : (directedCover Y).I₀\n⊢ Spec.map\n (appLE f₂ (↑U) (f₂ ⁻¹ᵁ ↑U) ⋯ ≫\n CommRingCat.ofHom\n ((CommRingCat.Hom.hom (appLE... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝² : QuasiCompact f\ninst✝¹ : QuasiSeparated f\nT : Scheme\nf₁ : X ⟶ T\nf₂ : T ⟶ Y\ninst✝ : IsIntegralHom f₂\nH : f = f₁ ≫ f₂\nU : (directedCover Y).I₀\n⊢ Spec.map\n (appLE f₂ (↑U) (f₂ ⁻¹ᵁ ↑U) ⋯ ≫\n CommRingCat.ofHom\n ((CommRingCat.Hom.hom (appLE f₁ (f₂ ⁻¹ᵁ ... | dsimp [normalizationGlueData, relativeGluingData, restrictIsoSpec] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 627,
"column": 4
} | {
"line": 627,
"column": 58
} | {
"line": 628,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH✝ : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\na b : R[X]\ne : a * f + b * g = 1\nha : a ≠ 0\nhb : b ≠ 0\nthis : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\nH : b.natDegree + g.natDegree < a... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nH✝ : IsCoprime f g\nhf : f ≠ 0\nhg : g ≠ 0\na b : R[X]\ne : a * f + b * g = 1\nha : a ≠ 0\nhb : b ≠ 0\nthis : f.resultant (b * g) f.natDegree (b.natDegree + g.natDegree) = f.resultant b * f.resultant g\nH : (b * g).natDegree < (a * f).natDegree\n⊢ ... | rw [← natDegree_mul ha hf, ← natDegree_mul hb hg] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Finiteness.Descent | {
"line": 43,
"column": 34
} | {
"line": 43,
"column": 45
} | {
"line": 43,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nT : Type u_3\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : FaithfullyFlat R T\ninst✝ : Module.Finite T (T ⊗[R] M)\nn : ℕ\ns : Fin n → T ⊗[R] M\nhs : Submodule.span T (Set.range s) = ⊤\nk : Fin n → ℕ\nt... | [
"R : Type u_1\ninst✝⁶ : CommRing R\nT : Type u_3\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : FaithfullyFlat R T\ninst✝ : Module.Finite T (T ⊗[R] M)\nn : ℕ\ns : Fin n → T ⊗[R] M\nhs : Submodule.span T (Set.range s) = ⊤\nk : Fin n → ℕ\nt : (i : Fin ... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.Descent | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 57
} | {
"line": 93,
"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✝ : FiniteType T (T ⊗[R] S)\ns : Finset (T ⊗[R] S)\nhs : adjoin T ↑s = ⊤\nk : ↥s → ℕ\nt : (x : ↥s) → Fin (k x) → T... | [
"case h\nR : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u_3\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Module.FaithfullyFlat R T\ninst✝ : FiniteType T (T ⊗[R] S)\ns : Finset (T ⊗[R] S)\nhs : adjoin T ↑s = ⊤\nk : ↥s → ℕ\nt : (x : ↥s) → Fin (k x) → T\nm ... | use ∑ (j : Fin (k i)), t i j ⊗ₜ MvPolynomial.X ⟨i, j⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 89
} | {
"line": 198,
"column": 4
} | [
{
"pp": "R : Type u_2\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : FiniteType R S\ninst✝ : QuasiFiniteAt R q\ns₁ : S\nhs₁q : s₁ ∉ q\nhs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q... | [
"R : Type u_2\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\ninst✝⁴ : p.IsPrime\nq : Ideal S\ninst✝³ : q.IsPrime\ninst✝² : q.LiesOver p\ninst✝¹ : FiniteType R S\ninst✝ : QuasiFiniteAt R q\ns₁ : S\nhs₁q : s₁ ∉ q\nhs₁ : ∀ (q' : Ideal S), q'.IsPrime → q' ≠ q → q'.LiesOver p... | replace this : s₃ ∈ q' := by simpa [← Ideal.mem_comap, ← q's.over_def q'] using! this | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.RingTheory.Extension.Cotangent.BaseChange | {
"line": 98,
"column": 16
} | {
"line": 98,
"column": 26
} | {
"line": 98,
"column": 27
} | [
{
"pp": "case zero\nR : 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\n⊢ (P.tensorCotangentSpace T) (t ⊗ₜ[R] 0) = t • (CotangentSpace.map (toBaseChange T)) 0",
"ppTerm": "?zero",
"a... | [
"case zero\nR : 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\n⊢ (P.tensorCotangentSpace T) 0 = t • (CotangentSpace.map (toBaseChange T)) 0"
] | tmul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 291,
"column": 40
} | {
"line": 311,
"column": 39
} | {
"line": 313,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : LocallyOfFiniteType f\n⊢ IsOpen {x | QuasiFiniteAt f x}",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.locallyOfFinitePresentation_of_isOpenImmersion",
"AlgebraicGeometry.Scheme.Hom.exists_isIso_morph... | [] | by
wlog H : IsAffineHom f
· rw [isOpen_iff_forall_mem_open]
intro x hx
obtain ⟨_, ⟨U : Y.Opens, hU, rfl⟩, hxU, -⟩ := Y.isBasis_affineOpens.exists_subset_of_mem_open
(Set.mem_univ (f x)) isOpen_univ
obtain ⟨_, ⟨V : X.Opens, hV, rfl⟩, hxV, hVU⟩ := X.isBasis_affineOpens.exists_subset_of_mem_open
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 323,
"column": 6
} | {
"line": 325,
"column": 34
} | {
"line": 326,
"column": 4
} | [
{
"pp": "case tmul\nR : 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.IsIntegra... | [] | obtain ⟨b', m, e : _ = _⟩ := Localization.awayMap_surjective_iff.mp hg b
refine ⟨e₀ ^ m * a ⊗ₜ b', m, ?_⟩
simp [e, mul_pow, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 323,
"column": 6
} | {
"line": 325,
"column": 34
} | {
"line": 326,
"column": 4
} | [
{
"pp": "case tmul\nR : 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.IsIntegra... | [] | obtain ⟨b', m, e : _ = _⟩ := Localization.awayMap_surjective_iff.mp hg b
refine ⟨e₀ ^ m * a ⊗ₜ b', m, ?_⟩
simp [e, mul_pow, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 25
} | {
"line": 89,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\np : a ⟶ b\nf g : b ⟶ c\npf : a ⟶ c\nη : f ≅ g\nη_f : p ≫ f ≅ pf\nη_g : p ≫ g ≅ pf\nih : p ◁ η ≪≫ η_g = η_f\n⊢ p ◁ η.symm ≪≫ η_f = η_g",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.t... | [] | rw [← ih]
apply Iso.ext (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence | {
"line": 86,
"column": 2
} | {
"line": 87,
"column": 25
} | {
"line": 89,
"column": 0
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c : B\np : a ⟶ b\nf g : b ⟶ c\npf : a ⟶ c\nη : f ≅ g\nη_f : p ≫ f ≅ pf\nη_g : p ≫ g ≅ pf\nih : p ◁ η ≪≫ η_g = η_f\n⊢ p ◁ η.symm ≪≫ η_f = η_g",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.t... | [] | rw [← ih]
apply Iso.ext (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 112,
"column": 18
} | {
"line": 112,
"column": 58
} | {
"line": 112,
"column": 58
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : Adj B\nα β : a ⟶ b\nx y : α ⟶ β\nhl : x.τl = y.τl\n⊢ x.τr = y.τr",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.Bicategory.Adj.Hom₂.τr",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | simp only [← Hom₂.conjugateEquiv_τl, hl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 112,
"column": 18
} | {
"line": 112,
"column": 58
} | {
"line": 112,
"column": 58
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : Adj B\nα β : a ⟶ b\nx y : α ⟶ β\nhl : x.τl = y.τl\n⊢ x.τr = y.τr",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.Bicategory.Adj.Hom₂.τr",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | simp only [← Hom₂.conjugateEquiv_τl, hl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Bicategory.Adjunction.Adj | {
"line": 112,
"column": 18
} | {
"line": 112,
"column": 58
} | {
"line": 112,
"column": 58
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b : Adj B\nα β : a ⟶ b\nx y : α ⟶ β\nhl : x.τl = y.τl\n⊢ x.τr = y.τr",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.Bicategory.Adj.Hom₂.τr",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | simp only [← Hom₂.conjugateEquiv_τl, hl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 6
} | {
"line": 221,
"column": 2
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d e f : B\ng₁ : a ⟶ c\ng₂ : c ⟶ e\nh₁ : b ⟶ d\nh₂ : d ⟶ f\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : c ⟶ d\nr₂ : d ⟶ c\nl₃ : e ⟶ f\nr₃ : f ⟶ e\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : g₁ ≫ l₂ ⟶ l₁ ≫ h₁\nβ : g₂ ≫ l₃ ⟶ l₂ ≫ h₂\n⊢ 𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫\n r₁ ◁ (g₁... | [
"B : Type u\ninst✝ : Bicategory B\na b c d e f : B\ng₁ : a ⟶ c\ng₂ : c ⟶ e\nh₁ : b ⟶ d\nh₂ : d ⟶ f\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : c ⟶ d\nr₂ : d ⟶ c\nl₃ : e ⟶ f\nr₃ : f ⟶ e\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : g₁ ≫ l₂ ⟶ l₁ ≫ h₁\nβ : g₂ ≫ l₃ ⟶ l₂ ≫ h₂\n⊢ (α_ r₁ g₁ g₂).inv ≫\n (𝟙 (r₁ ≫ g₁) ⊗≫ r₁ ◁... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Etale.QuasiFinite | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 41
} | {
"line": 487,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type (max u v)\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\np : Ideal R\ninst✝¹⁶ : p.IsPrime\nq : Ideal S\nR' : Type u\ninst✝¹⁵ : CommRing R'\ninst✝¹⁴ : Algebra R R'\ninst✝¹³ : Etale R R'\nP : Ideal R'\ninst✝¹² : P.IsPrime\ninst✝¹¹ : P.LiesOver p\ne : R' ⊗[R] S\nP... | [
"R : Type u\nS : Type (max u v)\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\np : Ideal R\ninst✝¹⁶ : p.IsPrime\nq : Ideal S\nR' : Type u\ninst✝¹⁵ : CommRing R'\ninst✝¹⁴ : Algebra R R'\ninst✝¹³ : Etale R R'\nP : Ideal R'\ninst✝¹² : P.IsPrime\ninst✝¹¹ : P.LiesOver p\ne : R' ⊗[R] S\nP' : Ideal (R... | have : P'φ.1.LiesOver P := .trans _ Q _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 69
} | {
"line": 242,
"column": 6
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d e f : B\ng₁ : a ⟶ c\ng₂ : c ⟶ e\nh₁ : b ⟶ d\nh₂ : d ⟶ f\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : c ⟶ d\nr₂ : d ⟶ c\nl₃ : e ⟶ f\nr₃ : f ⟶ e\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : g₁ ≫ l₂ ⟶ l₁ ≫ h₁\nβ : g₂ ≫ l₃ ⟶ l₂ ≫ h₂\n⊢ 𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫\n r₁ ◁ g₁ ... | [
"B : Type u\ninst✝ : Bicategory B\na b c d e f : B\ng₁ : a ⟶ c\ng₂ : c ⟶ e\nh₁ : b ⟶ d\nh₂ : d ⟶ f\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : c ⟶ d\nr₂ : d ⟶ c\nl₃ : e ⟶ f\nr₃ : f ⟶ e\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : g₁ ≫ l₂ ⟶ l₁ ≫ h₁\nβ : g₂ ≫ l₃ ⟶ l₂ ≫ h₂\n⊢ 𝟙 (r₁ ≫ g₁ ≫ g₂) ⊗≫\n r₁ ◁ g₁ ◁ (𝟙 c ◁ g₂... | rw [← whisker_exchange, ← whisker_exchange, ← whisker_exchange] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 688,
"column": 2
} | {
"line": 688,
"column": 6
} | {
"line": 689,
"column": 2
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : B\ng : a ⟶ c\nh : b ⟶ d\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : c ⟶ d\nr₂ : d ⟶ c\nl₃ : c ⟶ d\nr₃ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : g ≫ l₂ ⟶ l₁ ≫ h\nβ : l₃ ⟶ l₂\n⊢ (mateEquiv adj₁ adj₃) (leftAdjointSquareConjugate.vcomp α β) =\n rightAdjoi... | [
"B : Type u\ninst✝ : Bicategory B\na b c d : B\ng : a ⟶ c\nh : b ⟶ d\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : c ⟶ d\nr₂ : d ⟶ c\nl₃ : c ⟶ d\nr₃ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : g ≫ l₂ ⟶ l₁ ≫ h\nβ : l₃ ⟶ l₂\n⊢ rightAdjointSquareConjugate.vcomp ((mateEquiv adj₁ adj₂) α) ((conjugateEquiv adj₂ adj₃) β)... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Bicategory.Adjunction.Mate | {
"line": 726,
"column": 2
} | {
"line": 726,
"column": 6
} | {
"line": 727,
"column": 2
} | [
{
"pp": "B : Type u\ninst✝ : Bicategory B\na b c d : B\ng : a ⟶ c\nh : b ⟶ d\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : a ⟶ b\nr₂ : b ⟶ a\nl₃ : c ⟶ d\nr₃ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : l₂ ⟶ l₁\nβ : g ≫ l₃ ⟶ l₂ ≫ h\n⊢ (mateEquiv adj₁ adj₃) (leftAdjointConjugateSquare.vcomp α β) =\n rightAdjoi... | [
"B : Type u\ninst✝ : Bicategory B\na b c d : B\ng : a ⟶ c\nh : b ⟶ d\nl₁ : a ⟶ b\nr₁ : b ⟶ a\nl₂ : a ⟶ b\nr₂ : b ⟶ a\nl₃ : c ⟶ d\nr₃ : d ⟶ c\nadj₁ : l₁ ⊣ r₁\nadj₂ : l₂ ⊣ r₂\nadj₃ : l₃ ⊣ r₃\nα : l₂ ⟶ l₁\nβ : g ≫ l₃ ⟶ l₂ ≫ h\n⊢ rightAdjointConjugateSquare.vcomp ((conjugateEquiv adj₁ adj₂) α) ((mateEquiv adj₂ adj₃) β)... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicGeometry.Morphisms.FlatRank | {
"line": 274,
"column": 4
} | {
"line": 275,
"column": 88
} | {
"line": 276,
"column": 4
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nh : finrank f = 1\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f], finrank f = 1 → (∃ R, Y = Spec R) → IsIso f\nhY : ¬∃ R, Y = Spec R\n⊢ IsIso f",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : Flat f\ninst✝ : IsFinite f\nh : finrank f = 1\nthis : ∀ {X Y : Scheme} (f : X ⟶ Y) [Flat f] [IsFinite f], finrank f = 1 → (∃ R, Y = Spec R) → IsIso f\nhY : ¬∃ R, Y = Spec R\n⊢ ∀ (i : Y.affineCover.toPreZeroHypercover.1),\n MorphismProperty.isomorphisms Scheme (Cover.p... | rw [← MorphismProperty.isomorphisms.iff,
IsZariskiLocalAtTarget.iff_of_openCover (P := .isomorphisms Scheme) Y.affineCover] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.OrderOfVanishing.Noetherian | {
"line": 155,
"column": 86
} | {
"line": 157,
"column": 52
} | {
"line": 159,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDomain R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : KrullDimLE 1 R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nx : R\nhx : IsUnit x\n⊢ (ordFrac R) ((algebraMap R K) x) = 1",
"ppTerm": "?m.29",
"assigned": true,
... | [] | by
simp [ordFrac_eq_ord R (IsUnit.ne_zero hx), IsUnit.mem_nonZeroDivisors hx,
ordMonoidWithZeroHom_eq_ord, ord_of_isUnit hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.GradedAlgebra.Radical | {
"line": 117,
"column": 10
} | {
"line": 117,
"column": 51
} | {
"line": 118,
"column": 8
} | [
{
"pp": "case inl\nι : Type u_1\nσ : Type u_2\nA : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : AddCommMonoid ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : IsOrderedCancelAddMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nhI : IsHomogeneous 𝒜 I\nI_ne_top : I ≠ ⊤... | [] | exact Ideal.mul_mem_right _ I (notMem H₂) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme | {
"line": 289,
"column": 6
} | {
"line": 289,
"column": 31
} | {
"line": 289,
"column": 32
} | [
{
"pp": "A : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\na : A\n⊢ (∀ (i : ℕ),\n HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m... | [
"case mp\nA : Type u_1\nσ : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nf : A\nm : ℕ\nf_deg : f ∈ 𝒜 m\nq : ↑↑(Spec A⁰_ f).toPresheafedSpace\na : A\nh :\n ∀ (i : ℕ),\n HomogeneousLocalization.mk { deg := m * i, num := ⟨(proj 𝒜 i) a ^ m,... | constructor <;> intro h i | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 899,
"column": 31
} | {
"line": 899,
"column": 57
} | {
"line": 899,
"column": 58
} | [
{
"pp": "case map_units\nA : Type u_2\nσ : Type u_3\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\ne d : ℕ\nf : A\nhf : f ∈ 𝒜 d\ng : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\nhd : d ≠ 0\nthis : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).t... | [
"case map_units\nA : Type u_2\nσ : Type u_3\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\ne d : ℕ\nf : A\nhf : f ∈ 𝒜 d\ng : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\nhd : d ≠ 0\nthis : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).toAlgebra\nn ... | Localization.mk_eq_mk_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Modules.Tilde | {
"line": 696,
"column": 8
} | {
"line": 698,
"column": 86
} | {
"line": 699,
"column": 8
} | [
{
"pp": "R : 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 ((algebr... | [
"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... | have := (h₂ i j).uniqueness (f * (g i * g j)) (basicOpen_mul_le_right _ _)
(M.presheaf.map (homOfLE (basicOpen_mul_le_left (g i) (g j))).op (t i) -
M.presheaf.map (homOfLE (basicOpen_mul_le_right (g i) (g j))).op (t j)) ?_ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization | {
"line": 913,
"column": 8
} | {
"line": 913,
"column": 34
} | {
"line": 913,
"column": 35
} | [
{
"pp": "case surj.succ\nA : Type u_2\nσ : Type u_3\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\ne : ℕ\nf g : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\nthis : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).toAlgebra\nn : ℕ\ns : A\nd : ℕ\nhf ... | [
"case surj.succ\nA : Type u_2\nσ : Type u_3\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\ne : ℕ\nf g : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\nthis : Algebra (Away 𝒜 f) (Away 𝒜 x) := (awayMap 𝒜 hg hx).toAlgebra\nn : ℕ\ns : A\nd : ℕ\nhf : f ∈ 𝒜 (d ... | Localization.mk_eq_mk_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 88
} | {
"line": 184,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nR : Type u\ncommRing✝ : CommRing R\ndomain✝ : IsDomain R\nvaluationRing✝ : ValuationRing R\nK : Type u\nfield✝ : Field K\nalgebra✝ : Algebra R K\nisFractionRing✝ : IsFractionRing R K\ni₁ : Spec (CommRingCat.of K) ⟶ X\ni₂ : Spec (CommRingCat.of R) ⟶ Y\nw : i₁ ≫ f = Spec.map (Com... | [
"case refine_1\nX Y : Scheme\nf : X ⟶ Y\nR : Type u\ncommRing✝ : CommRing R\ndomain✝ : IsDomain R\nvaluationRing✝ : ValuationRing R\nK : Type u\nfield✝ : Field K\nalgebra✝ : Algebra R K\nisFractionRing✝ : IsFractionRing R K\ni₁ : Spec (CommRingCat.of K) ⟶ X\ni₂ : Spec (CommRingCat.of R) ⟶ Y\nw : i₁ ≫ f = Spec.map (... | refine ⟨⟨⟨Spec.map ((pullback.snd i₂ f).stalkMap x ≫ φ) ≫ X.fromSpecStalk _, ?_, ?_⟩⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 33
} | [
{
"pp": "case h\nσ : Type u_1\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nd e : ℕ\nf : A\nhf : f ∈ 𝒜 d\ng : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\nhd : 0 < d\nn : ℕ\na : A\nha : a ∈ 𝒜 n\nj : ℕ\nhb' : (fun x_1 ↦ x ^ x_1) j ∈ 𝒜 n... | [
"case h\nσ : Type u_1\nA : Type u_2\ninst✝³ : CommRing A\ninst✝² : SetLike σ A\ninst✝¹ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝ : GradedRing 𝒜\nd e : ℕ\nf : A\nhf : f ∈ 𝒜 d\ng : A\nhg : g ∈ 𝒜 e\nx : A\nhx : x = f * g\nhd : 0 < d\nn : ℕ\na : A\nha : a ∈ 𝒜 n\nj : ℕ\nhb' : (fun x_1 ↦ x ^ x_1) j ∈ 𝒜 n\nhfg : ¬(f ... | Localization.mk_eq_mk_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ValuativeCriterion | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 19
} | {
"line": 286,
"column": 2
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsSeparated f\nS : ValuativeCommSq f\nl₁ : Spec (CommRingCat.of S.R) ⟶ X\nhl₁ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₁ = S.i₁\nhl₁' : l₁ ≫ f = S.i₂\nl₂ : Spec (CommRingCat.of S.R) ⟶ X\nhl₂ : Spec.map (CommRingCat.ofHom (algebraMap S.R S.K)) ≫ l₂ = S.i₁\n... | [] | exact this.left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 221,
"column": 4
} | {
"line": 221,
"column": 34
} | {
"line": 222,
"column": 4
} | [
{
"pp": "case mpr\nP 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\nhPQ : P ≤ Q\nX : Q.Over ⊤ S\nR : Sieve X\nthis : (Mor... | [
"case mpr\nP 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\nhPQ : P ≤ Q\nX : Q.Over ⊤ S\nR : Sieve X\nthis : (MorphismPropert... | rintro ⟨T, ⟨𝒰, h, p, rfl⟩, le⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 250,
"column": 10
} | {
"line": 251,
"column": 45
} | {
"line": 251,
"column": 46
} | [
{
"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]
refine ⟨fun x ↦ ⟨x, y x, hy x⟩, hP⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Sites.Small | {
"line": 250,
"column": 10
} | {
"line": 251,
"column": 45
} | {
"line": 251,
"column": 46
} | [
{
"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]
refine ⟨fun x ↦ ⟨x, y x, hy x⟩, hP⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 185,
"column": 4
} | {
"line": 206,
"column": 52
} | {
"line": 208,
"column": 0
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝⁵ : Category.{v₀, u₀} C₀\ninst✝⁴ : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\ninst✝³ : IsDenseSubsite J₀ J F\ninst✝² : HasPullbacks C\ninst✝¹ : F.Full\ninst✝ : F.Faithful\nS : C\nι : C → Type w\nU : (S : C) → ι S → C₀\nf : (S ... | [] | exact ⟨{
I₀ := ι S
X := U S
f := f S
I₁ i j := ι (pullback (f _ i) (f _ j))
Y i j := U (pullback (f _ i) (f _ j))
p₁ i j k := F.preimage (f _ k ≫ pullback.fst _ _)
p₂ i j k := F.preimage (f _ k ≫ pullback.snd _ _)
w i j k := by simp [pullback.condition]
mem₀ := hf S... | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.Sites.DenseSubsite.OneHypercoverDense | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 22
} | {
"line": 349,
"column": 4
} | [
{
"pp": "C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presh... | [
"C₀ : Type u₀\nC : Type u\ninst✝³ : Category.{v₀, u₀} C₀\ninst✝² : Category.{v, u} C\nF : C₀ ⥤ C\nJ₀ : GrothendieckTopology C₀\nJ : GrothendieckTopology C\nA : Type u'\ninst✝¹ : Category.{v', u'} A\ninst✝ : IsDenseSubsite J₀ J F\ndata : (X : C) → F.OneHypercoverDenseData J₀ J X\nG : Cᵒᵖ ⥤ A\nhG₀ : Presheaf.IsSheaf ... | dsimp at i₁ i₂ j ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.Sites.Hypercover.ZeroFamily | {
"line": 112,
"column": 41
} | {
"line": 112,
"column": 71
} | {
"line": 112,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : PreZeroHypercoverFamily C\nh₁ : ∀ {X : C}, P.property.IsClosedUnderIsomorphisms\nh₂ :\n ∀ {X Y : C} (f : X ⟶ Y) (E : PreZeroHypercover Y) [inst : ∀ (i : E.I₀), HasPullback f (E.f i)],\n P.property E → P.property (PreZeroHypercover.pullback₁ f E)\nι : Type ... | [
"C : Type u\ninst✝ : Category.{v, u} C\nP : PreZeroHypercoverFamily C\nh₁ : ∀ {X : C}, P.property.IsClosedUnderIsomorphisms\nh₂ :\n ∀ {X Y : C} (f : X ⟶ Y) (E : PreZeroHypercover Y) [inst : ∀ (i : E.I₀), HasPullback f (E.f i)],\n P.property E → P.property (PreZeroHypercover.pullback₁ f E)\nι : Type (max u v)\nS... | (P (X := Y)).prop_iff_of_iso e | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 268,
"column": 8
} | {
"line": 268,
"column": 31
} | {
"line": 268,
"column": 31
} | [
{
"pp": "case h.succ\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 : IsFrac... | [
"case h.succ\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFractionRing O K... | rw [Nat.add_sub_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper | {
"line": 281,
"column": 14
} | {
"line": 281,
"column": 40
} | {
"line": 281,
"column": 41
} | [
{
"pp": "case e_6.e_6\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 : IsFra... | [
"case e_6.e_6\nσ : Type u_1\nA : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : SetLike σ A\ninst✝⁸ : AddSubgroupClass σ A\n𝒜 : ℕ → σ\ninst✝⁷ : GradedRing 𝒜\nO : Type u_3\ninst✝⁶ : CommRing O\ninst✝⁵ : IsDomain O\ninst✝⁴ : ValuationRing O\nK : Type u_4\ninst✝³ : Field K\ninst✝² : Algebra O K\ninst✝¹ : IsFractionRing O ... | Localization.mk_eq_mk_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Sites.QuasiCompact | {
"line": 177,
"column": 6
} | {
"line": 177,
"column": 91
} | {
"line": 178,
"column": 4
} | [
{
"pp": "case refine_1\nS✝ : Scheme\nP : MorphismProperty Scheme\nS : Scheme\n𝒰 : Cover (propQCPrecoverage P) S\n⊢ QuasiCompactCover\n (𝒰.restrictIndex (Sum.elim 𝒰.forgetQc.idx (QuasiCompactCover.uliftHom 𝒰.forgetQc.toPreZeroHypercover).s₀))",
"ppTerm": "?refine_1",
"assigned": true,
"usedCon... | [] | exact .of_hom (𝒱 := QuasiCompactCover.ulift 𝒰.1) ⟨Sum.inr, fun i ↦ 𝟙 _, by cat_disch⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Sites.ConstantSheaf | {
"line": 94,
"column": 17
} | {
"line": 97,
"column": 72
} | {
"line": 99,
"column": 0
} | [
{
"pp": "S : Scheme\nT : Type v\ninst✝¹ : TopologicalSpace T\ninst✝ : TotallyDisconnectedSpace T\nU : Scheme\nf : C(ConnectedComponents ↥U, T)\n⊢ (fun f ↦ { toFun := ⋯.connectedComponentsLift, continuous_toFun := ⋯ })\n ((fun f ↦ f.comp { toFun := ConnectedComponents.mk, continuous_toFun := ⋯ }) f) =\n ... | [] | by
apply ContinuousMap.coe_injective
dsimp
exact (Continuous.connectedComponentsLift_unique _ _ (by simp)).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 179,
"column": 31
} | {
"line": 179,
"column": 68
} | {
"line": 180,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\ninst✝¹ : LocallySmall.{t, v, u} C\ninst✝ : Small.{t, w} I\nφ : ... | [] | simpa [φ] using congr_arg Sigma.fst h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 179,
"column": 31
} | {
"line": 179,
"column": 68
} | {
"line": 180,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\ninst✝¹ : LocallySmall.{t, v, u} C\ninst✝ : Small.{t, w} I\nφ : ... | [] | simpa [φ] using congr_arg Sigma.fst h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.SmallObject.Construction | {
"line": 179,
"column": 31
} | {
"line": 179,
"column": 68
} | {
"line": 180,
"column": 4
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nI : Type w\nA B : I → C\nf : (i : I) → A i ⟶ B i\nS X : C\nπX : X ⟶ S\ninst✝³ : HasColimitsOfShape (Discrete (FunctorObjIndex f πX)) C\ninst✝² : HasPushout (functorObjTop f πX) (functorObjLeft f πX)\ninst✝¹ : LocallySmall.{t, v, u} C\ninst✝ : Small.{t, w} I\nφ : ... | [] | simpa [φ] using congr_arg Sigma.fst h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.SmallObject.TransfiniteCompositionLifting | {
"line": 302,
"column": 86
} | {
"line": 304,
"column": 62
} | {
"line": 306,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW : MorphismProperty C\n⊢ (transfiniteCompositions.{w, v, u} (coproducts.{w, v, u} W).pushouts).retracts ≤ W.rlp.llp",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheor... | [] | by
rw [le_llp_iff_le_rlp, rlp_retracts, ← le_llp_iff_le_rlp]
apply transfiniteCompositions_pushouts_coproducts_le_llp_rlp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallObject.WellOrderInductionData | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 10
} | {
"line": 234,
"column": 6
} | [
{
"pp": "case e_x\nJ : Type u\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝¹ : OrderBot J\nval₀ : F.obj (op ⊥)\ninst✝ : WellFoundedLT J\nj : J\nhj : Order.IsSuccLimit j\ne : (i : J) → i < j → d.Extension val₀ i\ni : J\nhi : i < j\n⊢ ↑⟨fun x ↦\n mat... | [
"case e_x\nJ : Type u\ninst✝³ : LinearOrder J\ninst✝² : SuccOrder J\nF : Jᵒᵖ ⥤ Type v\nd : F.WellOrderInductionData\ninst✝¹ : OrderBot J\nval₀ : F.obj (op ⊥)\ninst✝ : WellFoundedLT J\nj : J\nhj : Order.IsSuccLimit j\ne : (i : J) → i < j → d.Extension val₀ i\ni : J\nhi : i < j\n⊢ (ConcreteCategory.hom (F.map (homOfL... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.SmallObject.IsCardinalForSmallObjectArgument | {
"line": 283,
"column": 2
} | {
"line": 284,
"column": 32
} | {
"line": 286,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nI : MorphismProperty C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : OrderBot κ.ord.ToType\ninst✝ : I.IsCardinalForSmallObjectArgument κ\nf : Arrow C\nj : κ.ord.ToType\nthis :\n 𝟙 (((iterationFunctor I κ).obj j).obj f).right ≫ 𝟙 (((iterationFunctor I ... | [] | rw [← cancel_epi (((iterationFunctor I κ).map (homOfLE (Order.le_succ j))).app f).right,
← reassoc_of% this, comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 156,
"column": 14
} | {
"line": 156,
"column": 49
} | {
"line": 156,
"column": 49
} | [
{
"pp": "p : ℕ\nz : ℤ\nhp : p ≠ 1\nhz : z ≠ 0\n⊢ ↑(padicValInt p z) = ↑(multiplicity (↑p) z)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"padicValInt",
"id",
"Int",
"Nat.cast",
"multiplicity",
"Int.instMonoid",
... | [
"p : ℕ\nz : ℤ\nhp : p ≠ 1\nhz : z ≠ 0\n⊢ ↑(multiplicity (↑p) z) = ↑(multiplicity (↑p) z)"
] | padicValInt.of_ne_one_ne_zero hp hz | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 265,
"column": 8
} | {
"line": 265,
"column": 62
} | {
"line": 265,
"column": 62
} | [
{
"pp": "case e'_5\nC : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nk : ↑(Set.Iic j)\nhk : ¬I... | [
"case e'_5\nC : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nk : ↑(Set.Iic j)\nhk : ¬IsMax k\n⊢ la... | transfiniteIterate_succ _ _ _ (Set.not_isMax_coe _ hk) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 265,
"column": 8
} | {
"line": 265,
"column": 62
} | {
"line": 265,
"column": 62
} | [
{
"pp": "case e'_6\nC : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nk : ↑(Set.Iic j)\nhk : ¬I... | [
"case e'_6\nC : Type u\ninst✝⁶ : Category.{v, u} C\nG : C\ninst✝⁵ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁴ : IsGrothendieckAbelian.{w, v, u} C\nA₀ : Subobject X\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nk : ↑(Set.Iic j)\nhk : ¬IsMax k\ne_5✝... | transfiniteIterate_succ _ _ _ (Set.not_isMax_coe _ hk) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 280,
"column": 2
} | {
"line": 287,
"column": 25
} | {
"line": 289,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nG : C\ninst✝⁶ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁵ : IsGrothendieckAbelian.{w, v, u} C\nA : C\nf : A ⟶ X\ninst✝⁴ : Mono f\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nhj : transfiniteIte... | [] | let t := transfiniteIterate (largerSubobject hG) j (Subobject.mk f)
have := (Subobject.isIso_arrow_iff_eq_top t).2 hj
apply (transfiniteCompositionOfShapeMapFromBot hG (Subobject.mk f) j).ofArrowIso
refine Arrow.isoMk ((Subobject.isoOfEq _ _ (transfiniteIterate_bot _ _) ≪≫
Subobject.underlyingIso f)) (asIso t... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.EnoughInjectives | {
"line": 280,
"column": 2
} | {
"line": 287,
"column": 25
} | {
"line": 289,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁷ : Category.{v, u} C\nG : C\ninst✝⁶ : Abelian C\nhG : IsSeparator G\nX : C\ninst✝⁵ : IsGrothendieckAbelian.{w, v, u} C\nA : C\nf : A ⟶ X\ninst✝⁴ : Mono f\nJ : Type w\ninst✝³ : LinearOrder J\ninst✝² : OrderBot J\ninst✝¹ : SuccOrder J\ninst✝ : WellFoundedLT J\nj : J\nhj : transfiniteIte... | [] | let t := transfiniteIterate (largerSubobject hG) j (Subobject.mk f)
have := (Subobject.isIso_arrow_iff_eq_top t).2 hj
apply (transfiniteCompositionOfShapeMapFromBot hG (Subobject.mk f) j).ofArrowIso
refine Arrow.isoMk ((Subobject.isoOfEq _ _ (transfiniteIterate_bot _ _) ≪≫
Subobject.underlyingIso f)) (asIso t... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNorm | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 23
} | {
"line": 233,
"column": 24
} | [
{
"pp": "case mp\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh : padicNorm p ↑m = 1\n⊢ (↑p)⁻¹ < padicNorm p ↑m",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Eq.mpr",
"inv_lt_one₀",
"Preorder.toLT",
"GroupWithZero.toDivisionMonoi... | [
"case mp\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh : padicNorm p ↑m = 1\n⊢ 1 < ↑p",
"case mp\np : ℕ\nhp : Fact (Nat.Prime p)\nm : ℤ\nh : padicNorm p ↑m = 1\n⊢ 0 < ↑p"
] | rw [h, inv_lt_one₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 389,
"column": 16
} | {
"line": 389,
"column": 43
} | {
"line": 389,
"column": 44
} | [
{
"pp": "case inr\np b : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\nha : b * k ≠ 0\nhb : b ≠ 0\nhk : k ≠ 0\n⊢ padicValNat p (k * b / b) = padicValNat p (k * b) - padicValNat p b",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"Nat.mul_div_c... | [
"case inr\np b : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\nha : b * k ≠ 0\nhb : b ≠ 0\nhk : k ≠ 0\n⊢ padicValNat p k = padicValNat p (k * b) - padicValNat p b"
] | k.mul_div_cancel hb.bot_lt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicVal.Basic | {
"line": 500,
"column": 24
} | {
"line": 500,
"column": 36
} | {
"line": 500,
"column": 36
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nh1 : 2 ^ padicValNat 2 (n + 1) * 1 < n + 1\nh2 : n + 1 ≤ 2 ^ (padicValNat 2 (n + 1) + 1)\n⊢ False",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"instPowNat",
"HMul.hMul",
"congrArg",
"Eq.mp",
"Nat.pow_succ",
"padicValNat",
... | [
"n : ℕ\nhn : n ≠ 0\nh1 : 2 ^ padicValNat 2 (n + 1) * 1 < n + 1\nh2 : n + 1 ≤ 2 ^ padicValNat 2 (n + 1) * 2\n⊢ False"
] | Nat.pow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 139,
"column": 67
} | {
"line": 140,
"column": 62
} | {
"line": 142,
"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... | [] | by
simp [← Coe.ringHom_apply, map_sum PadicInt.Coe.ringHom f s] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 158,
"column": 33
} | {
"line": 158,
"column": 46
} | {
"line": 158,
"column": 47
} | [
{
"pp": "case coe.ofAdd\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ\n⊢ ∃ a,\n { toFun := fun x ↦ if x = 0 then 0 else exp (-padicValRat p x), map_zero' := ⋯, map_one' := ⋯, map_mul' := ⋯,\n map_add_le_max' := ⋯ }\n a =\n ↑(Multiplicative.ofAdd x)",
"ppTerm": "?coe.ofAdd",
"assigned": ... | [
"case coe.ofAdd\np : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ\n⊢ ∃ a,\n { toFun := fun x ↦ if x = 0 then 0 else ↑(Multiplicative.ofAdd (-padicValRat p x)), map_zero' := ⋯, map_one' := ⋯,\n map_mul' := ⋯, map_add_le_max' := ⋯ }\n a =\n ↑(Multiplicative.ofAdd x)"
] | WithZero.exp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Padics.PadicIntegers | {
"line": 475,
"column": 51
} | {
"line": 476,
"column": 68
} | {
"line": 478,
"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",
"Div... | [] | by
rw [norm_def]; exact Padic.norm_le_pow_iff_norm_lt_pow_add_one _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 389,
"column": 98
} | {
"line": 391,
"column": 48
} | {
"line": 393,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\na : PadicSeq p\nha : ¬a ≈ 0\n⊢ ∃ z, a.norm = ↑p ^ (-z)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | [] | by
let ⟨k, hk, hk'⟩ := norm_eq_norm_app_of_nonzero ha
simpa [hk] using padicNorm.values_discrete hk' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 152,
"column": 2
} | {
"line": 162,
"column": 7
} | {
"line": 164,
"column": 0
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁴ : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝³ : LocallySmall.{w, v, u} C\ninst✝² : HasSheafify J (Type w)\ninst✝¹ : J.WEqualsLocallyBijective (Type w)\nhP : P.IsConservativeFamilyOfPoints\nX : C\nι : Type u_1\ninst✝ : Small.{w, u_1... | [] | · rw [J.ofArrows_mem_iff_isLocallySurjective_sigmaDesc_shrinkYoneda_map]
refine hP.jointly_reflect_isLocallySurjective _ (fun Φ x ↦ ?_)
obtain ⟨x, rfl⟩ := (Φ.obj.shrinkYonedaCompPresheafFiberIso.app X).toEquiv.symm.surjective x
obtain ⟨i, y, rfl⟩ := hf Φ x
refine ⟨Φ.obj.presheafFiber.map (Sigma.ι (fun i... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Sites.Point.Conservative | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 17
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case inr\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fi... | [
"case inr\nC : Type u\ninst✝² : Category.{v, u} C\nJ : GrothendieckTopology C\nP : ObjectProperty J.Point\ninst✝¹ : LocallySmall.{w, v, u} C\nhP :\n ∀ ⦃X : C⦄ (S : Sieve X),\n (∀ (Φ : P.FullSubcategory) (x : Φ.obj.fiber.obj X),\n ∃ Y g, ∃ (_ : S.arrows g), ∃ y, (ConcreteCategory.hom (Φ.obj.fiber.map g)) ... | dsimp at hv ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 710,
"column": 2
} | {
"line": 717,
"column": 9
} | {
"line": 719,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : 0 < ε\n⊢ ∃ N, ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Rat.instOfNat",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",... | [] | refine (exists_nat_gt (1 / ε)).imp fun N hN i hi ↦ ?_
have h := Classical.choose_spec (rat_dense' (f i) (div_nat_pos i))
refine lt_of_lt_of_le h ((div_le_iff₀' <| mod_cast succ_pos _).mpr ?_)
rw [right_distrib]
apply le_add_of_le_of_nonneg
· exact (div_le_iff₀ hε).mp (le_trans (le_of_lt hN) (mod_cast hi))
·... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 710,
"column": 2
} | {
"line": 717,
"column": 9
} | {
"line": 719,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : 0 < ε\n⊢ ∃ N, ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Rat.instOfNat",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",... | [] | refine (exists_nat_gt (1 / ε)).imp fun N hN i hi ↦ ?_
have h := Classical.choose_spec (rat_dense' (f i) (div_nat_pos i))
refine lt_of_lt_of_le h ((div_le_iff₀' <| mod_cast succ_pos _).mpr ?_)
rw [right_distrib]
apply le_add_of_le_of_nonneg
· exact (div_le_iff₀ hε).mp (le_trans (le_of_lt hN) (mod_cast hi))
·... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 409,
"column": 6
} | {
"line": 409,
"column": 10
} | {
"line": 410,
"column": 6
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF✝ : C ⥤ D\nX✝ Y : D\nP : MorphismProperty C\ninst✝² : F✝.Faithful\ninst✝¹ : F✝.Full\ninst✝ : P.IsStableUnderComposition\nF G H : D\nf : F ⟶ G\ng : G ⟶ H\nhf : relative F✝ P f\nhg : relative F✝ P g\nZ X... | [
"case refine_2\nC : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\nF✝ : C ⥤ D\nX✝ Y : D\nP : MorphismProperty C\ninst✝² : F✝.Faithful\ninst✝¹ : F✝.Full\ninst✝ : P.IsStableUnderComposition\nF G H : D\nf : F ⟶ G\ng : G ⟶ H\nhf : relative F✝ P f\nhg : relative F✝ P g\nZ X : C\np : F✝... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicTopology.DoldKan.Projections | {
"line": 117,
"column": 8
} | {
"line": 117,
"column": 35
} | {
"line": 118,
"column": 8
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nq : ℕ\nhq : HigherFacesVanish q φ → φ ≫ (P q).f (n + 1) = φ\nv : HigherFacesVanish (q + 1) φ\nhqn : q ≤ n\na : ℕ\nha : q + a = n\nhnaq : n = a + q\n⊢ n + 1 ≤ ↑⟨a, ⋯⟩ + (q + 1)"... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nq : ℕ\nhq : HigherFacesVanish q φ → φ ≫ (P q).f (n + 1) = φ\nv : HigherFacesVanish (q + 1) φ\nhqn : q ≤ n\na : ℕ\nha : q + a = n\nhnaq : n = a + q\n⊢ a + (q + 1) ≤ a + (q + 1)"
] | simp only [hnaq, add_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1107,
"column": 53
} | {
"line": 1113,
"column": 67
} | {
"line": 1115,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ (x * y).valuation = x.valuation + y.valuation",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NormedCommRing.toNormedRing",
"Norm.norm",
"GroupWithZero.toMonoidW... | [] | by
have h_norm : ‖x * y‖ = ‖x‖ * ‖y‖ := norm_mul x y
have hp_ne_one : (p : ℝ) ≠ 1 := mod_cast (Fact.out : p.Prime).ne_one
have hp_pos : (0 : ℝ) < p := mod_cast NeZero.pos _
rwa [norm_eq_zpow_neg_valuation hx, norm_eq_zpow_neg_valuation hy,
norm_eq_zpow_neg_valuation (mul_ne_zero hx hy), ← zpow_add₀ hp_pos.n... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1153,
"column": 2
} | {
"line": 1159,
"column": 28
} | {
"line": 1161,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\n⊢ (x * y).addValuationDef = x.addValuationDef + y.addValuationDef",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"HMul.hMul",
"instZeroPadic",
"MulZeroClass.toMul",... | [] | simp only [addValuationDef]
by_cases hx : x = 0
· rw [hx, if_pos rfl, zero_mul, if_pos rfl, WithTop.top_add]
· by_cases hy : y = 0
· rw [hy, if_pos rfl, mul_zero, if_pos rfl, WithTop.add_top]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← WithTop.coe_add, WithTop.coe_eq_coe,
valuation_m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1153,
"column": 2
} | {
"line": 1159,
"column": 28
} | {
"line": 1161,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\n⊢ (x * y).addValuationDef = x.addValuationDef + y.addValuationDef",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"HMul.hMul",
"instZeroPadic",
"MulZeroClass.toMul",... | [] | simp only [addValuationDef]
by_cases hx : x = 0
· rw [hx, if_pos rfl, zero_mul, if_pos rfl, WithTop.top_add]
· by_cases hy : y = 0
· rw [hy, if_pos rfl, mul_zero, if_pos rfl, WithTop.add_top]
· rw [if_neg hx, if_neg hy, if_neg (mul_ne_zero hx hy), ← WithTop.coe_add, WithTop.coe_eq_coe,
valuation_m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 10
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\n⊢ ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev - (... | [
"case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\n⊢ ∑ i with ↑i < q + 1, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev =\n ∑ i ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 133,
"column": 28
} | {
"line": 133,
"column": 51
} | {
"line": 133,
"column": 52
} | [
{
"pp": "case a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn a q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\nhnaq : n = a + q\nhnaq_shift : ∀ (d : ℕ), n + d = a + d + q\nsimplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → ... | [
"case a\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn a q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\nhnaq : n = a + q\nhnaq_shift : ∀ (d : ℕ), n + d = a + d + q\nsimplif : ∀ (a b c d e f : Y ⟶ X _⦋n + 1⦌), b = f → d + e = 0 → c + a = 0 → a + b + (c +... | add_eq_zero_iff_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 74,
"column": 53
} | {
"line": 74,
"column": 62
} | {
"line": 74,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\nq' : Fin (n + 1) := ⟨q, ⋯⟩\n⊢ q' ∈ {i | ↑i < q + 1}",
"ppTerm": "?m.271",
... | [] | simp [q'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 74,
"column": 53
} | {
"line": 74,
"column": 62
} | {
"line": 74,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\nq' : Fin (n + 1) := ⟨q, ⋯⟩\n⊢ q' ∈ {i | ↑i < q + 1}",
"ppTerm": "?m.271",
... | [] | simp [q'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Decomposition | {
"line": 74,
"column": 53
} | {
"line": 74,
"column": 62
} | {
"line": 74,
"column": 62
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nn q : ℕ\nhq : (Q q).f (n + 1) = ∑ i with ↑i < q, (P ↑i).f (n + 1) ≫ X.δ i.rev.succ ≫ X.σ i.rev\nhqn : q ≤ n\na : ℕ\nha : q + a = n\nq' : Fin (n + 1) := ⟨q, ⋯⟩\n⊢ q' ∈ {i | ↑i < q + 1}",
"ppTerm": "?m.271",
... | [] | simp [q'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 80
} | {
"line": 147,
"column": 2
} | [
{
"pp": "case pos\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nK : ChainComplex C ℕ\nΔ Δ' Δ'' : SimplexCategory\ni' : Δ'' ⟶ Δ'\ni : Δ' ⟶ Δ\ninst✝¹ : Mono i'\ninst✝ : Mono i\nh₁ : ¬Δ = Δ'\nh₂ : Δ' = Δ''\n⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)",
"ppTerm": "?pos✝",
"ass... | [
"case neg\nC : Type u_1\ninst✝³ : Category.{v_1, u_1} C\ninst✝² : Preadditive C\nK : ChainComplex C ℕ\nΔ Δ' Δ'' : SimplexCategory\ni' : Δ'' ⟶ Δ'\ni : Δ' ⟶ Δ\ninst✝¹ : Mono i'\ninst✝ : Mono i\nh₁ : ¬Δ = Δ'\nh₂ : ¬Δ' = Δ''\n⊢ mapMono K i ≫ mapMono K i' = mapMono K (i' ≫ i)"
] | · subst h₂
simp only [SimplexCategory.eq_id_of_mono i', comp_id, id_comp, mapMono_id K] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 115,
"column": 4
} | {
"line": 116,
"column": 26
} | {
"line": 117,
"column": 4
} | [
{
"pp": "case mpr\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nZ : C\nn : ℕ\nf : Z ⟶ X _⦋n⦌\nh : f ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = 0\n⊢ f ≫ PInfty.f n = 0",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case mpr.h\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nZ : C\nn : ℕ\nf : Z ⟶ X _⦋n⦌\nh : f ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = 0\n⊢ ∀ (a : IndexSet (op ⦋n⦌)), f ≫ (s.πSummand a ≫ (s.cofan (op ⦋n⦌)).inj a) ≫ PInfty.f n = 0"
] | rw [← comp_id f, assoc, s.decomposition_id, Preadditive.sum_comp, Preadditive.comp_sum,
Fintype.sum_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 43
} | {
"line": 129,
"column": 2
} | [
{
"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⊢ PInfty.f n ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = s.πSummand (IndexSet.id (op ⦋n⦌))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nn : ℕ\n⊢ PInfty.f n ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = 𝟙 (X _⦋n⦌) ≫ s.πSummand (IndexSet.id (op ⦋n⦌))"
] | conv_rhs => rw [← id_comp (s.πSummand _)] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 6
} | {
"line": 130,
"column": 2
} | [
{
"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⊢ PInfty.f n ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = 𝟙 (X _⦋n⦌) ≫ s.πSummand (IndexSet.id (op ⦋n⦌))",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"CategoryTh... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX : SimplicialObject C\ns : X.Splitting\ninst✝ : Preadditive C\nn : ℕ\n⊢ 𝟙 (X _⦋n⦌) ≫ s.πSummand (IndexSet.id (op ⦋n⦌)) = PInfty.f n ≫ s.πSummand (IndexSet.id (op ⦋n⦌))"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicTopology.DoldKan.EquivalenceAdditive | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 8
} | {
"line": 62,
"column": 4
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nα : N.obj ((𝟭 (Karoubi (SimplicialObject C))).obj P) ≅ N.obj ((N₂ ⋙ Γ₂).obj P) := N.mapIso (Γ₂N₂.app P)\nβ : (Γ₂ ⋙ N₂).obj (N.obj P) ≅ (𝟭 (Karoubi (ChainComplex C ℕ))... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nα : N.obj ((𝟭 (Karoubi (SimplicialObject C))).obj P) ≅ N.obj ((N₂ ⋙ Γ₂).obj P) := ⋯\nβ : (Γ₂ ⋙ N₂).obj (N.obj P) ≅ (𝟭 (Karoubi (ChainComplex C ℕ))).obj (N.obj P) := ⋯\n⊢ 𝟙 (N.ob... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.AlgebraicTopology.DoldKan.NCompGamma | {
"line": 236,
"column": 10
} | {
"line": 236,
"column": 24
} | {
"line": 236,
"column": 24
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nh : N₂Γ₂.inv.app (N₂.obj P) ≫ N₂.map (Γ₂N₂.natTrans.app P) = 𝟙 (N₂.obj P)\n⊢ IsIso (N₂.map (Γ₂N₂.natTrans.app P))",
"ppTerm": "?m.86",
"assigned": true,
"u... | [
"C : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Preadditive C\ninst✝ : HasFiniteCoproducts C\nP : Karoubi (SimplicialObject C)\nh : N₂.map (Γ₂N₂.natTrans.app P) = inv (N₂Γ₂.inv.app (N₂.obj P))\n⊢ IsIso (N₂.map (Γ₂N₂.natTrans.app P))"
] | hom_comp_eq_id | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy | {
"line": 64,
"column": 49
} | {
"line": 64,
"column": 80
} | {
"line": 66,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\n⊢ toHoCat.Full",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteColimits",
"HomotopicalAlgebra.ModelCategory.cm1b",
"CategoryTheory.Quotien... | [] | dsimp [toHoCat]; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.CofibrantObjectHomotopy | {
"line": 64,
"column": 49
} | {
"line": 64,
"column": 80
} | {
"line": 66,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : ModelCategory C\n⊢ toHoCat.Full",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteColimits",
"HomotopicalAlgebra.ModelCategory.cm1b",
"CategoryTheory.Quotien... | [] | dsimp [toHoCat]; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 74,
"column": 49
} | {
"line": 74,
"column": 80
} | {
"line": 76,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\n⊢ toHoCat.Full",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.BifibrantObject",
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteColimits",
"HomotopicalAlgebra.ModelCategor... | [] | dsimp [toHoCat]; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 74,
"column": 49
} | {
"line": 74,
"column": 80
} | {
"line": 76,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : ModelCategory C\n⊢ toHoCat.Full",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"HomotopicalAlgebra.BifibrantObject",
"CategoryTheory.Limits.hasFiniteCoproducts_of_hasFiniteColimits",
"HomotopicalAlgebra.ModelCategor... | [] | dsimp [toHoCat]; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.SubcomplexColimits | {
"line": 58,
"column": 18
} | {
"line": 58,
"column": 36
} | {
"line": 59,
"column": 8
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nι : Type u_1\nU : ι → X.Subcomplex\nV : ι → ι → X.Subcomplex\nh : A.MulticoequalizerDiagram U V\nn : SimplexCategoryᵒᵖ\n⊢ ∀ (i j : ι), (V i j).obj n = (U i).obj n ⊓ (U j).obj n",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"CompleteLattice.Multi... | [] | by simp [h.eq_inf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Boundary | {
"line": 144,
"column": 2
} | {
"line": 148,
"column": 51
} | {
"line": 150,
"column": 0
} | [
{
"pp": "n : ℕ\nA : Δ[n].Subcomplex\n⊢ A = ∂Δ[n] ↔ ∂Δ[n] ≤ A ∧ A ≠ ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"congrArg",
"PartialOrder.toPreorder",
... | [] | constructor
· rintro rfl
exact ⟨by rfl, (boundary_lt_top n).ne⟩
· rintro ⟨h₁, h₂⟩
exact le_antisymm (by rwa [le_boundary_iff]) h₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Boundary | {
"line": 144,
"column": 2
} | {
"line": 148,
"column": 51
} | {
"line": 150,
"column": 0
} | [
{
"pp": "n : ℕ\nA : Δ[n].Subcomplex\n⊢ A = ∂Δ[n] ↔ ∂Δ[n] ≤ A ∧ A ≠ ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Lattice.toSemilatticeSup",
"Opposite",
"CompleteLattice.toLattice",
"congrArg",
"PartialOrder.toPreorder",
... | [] | constructor
· rintro rfl
exact ⟨by rfl, (boundary_lt_top n).ne⟩
· rintro ⟨h₁, h₂⟩
exact le_antisymm (by rwa [le_boundary_iff]) h₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 239,
"column": 16
} | {
"line": 239,
"column": 22
} | {
"line": 240,
"column": 2
} | [
{
"pp": "case refine_2.inl\n⊢ ∀ {i : Fin (2 + 1)} (j : Fin 2), 0 < ↑i → ↑i < 2 → #{i, j.castSucc, j.succ} ≤ 2",
"ppTerm": "?refine_2.inl",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Fin.succ",
"Finset",
"instDecidableEqFin",
"id",
"Insert.insert",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 297,
"column": 2
} | {
"line": 303,
"column": 73
} | {
"line": 306,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 2)\nS : SSet\nσ₁ σ₂ : Λ[n + 1, i].toSSet ⟶ S\nh :\n ∀ (j : Fin (n + 2)) (h : j ≠ i),\n (ConcreteCategory.hom (σ₁.app (op ⦋n⦌))) (face i j h) = (ConcreteCategory.hom (σ₂.app (op ⦋n⦌))) (face i j h)\n⊢ σ₁ = σ₂",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [] | rw [← Subfunctor.equalizer_eq_iff]
apply le_antisymm (Subfunctor.equalizer_le σ₁ σ₂)
simp only [horn_eq_iSup, iSup_le_iff,
Subtype.forall, Set.mem_compl_iff, Set.mem_singleton_iff,
← stdSimplex.ofSimplex_yonedaEquiv_δ, Subcomplex.ofSimplex_le_iff]
intro j hj
exact (Subfunctor.mem_equalizer_iff σ₁ σ₂ (fa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 297,
"column": 2
} | {
"line": 303,
"column": 73
} | {
"line": 306,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 2)\nS : SSet\nσ₁ σ₂ : Λ[n + 1, i].toSSet ⟶ S\nh :\n ∀ (j : Fin (n + 2)) (h : j ≠ i),\n (ConcreteCategory.hom (σ₁.app (op ⦋n⦌))) (face i j h) = (ConcreteCategory.hom (σ₂.app (op ⦋n⦌))) (face i j h)\n⊢ σ₁ = σ₂",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [] | rw [← Subfunctor.equalizer_eq_iff]
apply le_antisymm (Subfunctor.equalizer_le σ₁ σ₂)
simp only [horn_eq_iSup, iSup_le_iff,
Subtype.forall, Set.mem_compl_iff, Set.mem_singleton_iff,
← stdSimplex.ofSimplex_yonedaEquiv_δ, Subcomplex.ofSimplex_le_iff]
intro j hj
exact (Subfunctor.mem_equalizer_iff σ₁ σ₂ (fa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 46
} | {
"line": 81,
"column": 6
} | [
{
"pp": "case mk.inl\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ns : PushoutCocone f g\nm : Pushout f g ⟶ s.pt\nh₁ : inl f g ≫ m = s.inl\nh₂ : inr f g ≫ m = s.inr\nx✝ : Pushout f g\nx₁ : X₁\n⊢ (ConcreteCategory.hom m).toFun (Quot.mk (Rel f g) (Sum.inl x₁)) =\n (ConcreteCategory.hom\n (↾Quot.lift\n ... | [
"case mk.inr\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ns : PushoutCocone f g\nm : Pushout f g ⟶ s.pt\nh₁ : inl f g ≫ m = s.inl\nh₂ : inr f g ≫ m = s.inr\nx✝ : Pushout f g\nx₂ : X₂\n⊢ (ConcreteCategory.hom m).toFun (Quot.mk (Rel f g) (Sum.inr x₂)) =\n (ConcreteCategory.hom\n (↾Quot.lift\n (... | · exact ConcreteCategory.congr_hom h₁ x₁ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 30
} | {
"line": 95,
"column": 0
} | [
{
"pp": "case mpr.inr\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\nw✝¹ w✝ : S\nh : (ConcreteCategory.hom g) w✝¹ = (ConcreteCategory.hom g) w✝\n⊢ Rel' f g (Sum.inl ((ConcreteCategory.hom f) w✝¹)) (Sum.inl ((ConcreteCategory.hom f) w✝))",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedConstants": [
... | [] | · exact Rel'.inl_inl _ _ h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 135,
"column": 8
} | {
"line": 135,
"column": 34
} | {
"line": 136,
"column": 8
} | [
{
"pp": "case inl_inl.inl.inl\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ninst✝ : Mono f\nx₀✝ y₀✝ : S\nh : (ConcreteCategory.hom g) x₀✝ = (ConcreteCategory.hom g) y₀✝\n⊢ Rel' f g (Sum.inl ((ConcreteCategory.hom f) x₀✝)) (Sum.inl ((ConcreteCategory.hom f) y₀✝))",
"ppTerm": "?inl_inl.inl.inl",
"assigned": ... | [
"case inl_inl.inl.inr\nS X₁ X₂ : Type u\nf : S ⟶ X₁\ng : S ⟶ X₂\ninst✝ : Mono f\nx₀✝ y₀✝ : S\nh : (ConcreteCategory.hom g) x₀✝ = (ConcreteCategory.hom g) y₀✝\nw✝¹ w✝ : S\nh' : (ConcreteCategory.hom g) w✝¹ = (ConcreteCategory.hom g) w✝\nh'' : (ConcreteCategory.hom f) y₀✝ = (ConcreteCategory.hom f) w✝¹\n⊢ Rel' f g (S... | · exact Rel'.inl_inl _ _ h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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