module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicGeometry.Sites.EtalePoint | {
"line": 40,
"column": 59
} | {
"line": 54,
"column": 36
} | {
"line": 56,
"column": 0
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝² : Etale f\nΩ : Type u\ninst✝¹ : Field Ω\ninst✝ : IsSepClosed Ω\ns : Spec (CommRingCat.of Ω) ⟶ S\nx : ↥X\nhx : f x = s default\n⊢ ∃ l, l ≫ f = s ∧ l default = x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Cat... | [] | by
obtain ⟨⟨s, a⟩, rfl⟩ := (SpecToEquivOfField Ω S).symm.surjective s
obtain rfl : f x = s := by simp [hx, SpecToEquivOfField]
let m := (f.residueFieldMap x).hom
dsimp at m
algebraize [m, a.hom]
let b : X.residueField x →ₐ[S.residueField (f x)] Ω :=
IsSepClosed.lift
have : f.residueFieldMap x ≫ CommRi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 357,
"column": 4
} | {
"line": 359,
"column": 33
} | {
"line": 361,
"column": 0
} | [
{
"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.RespectsIso\nf : X ⟶ Y\nh₀ : ∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : IsPullback fst (F.map snd) f g),... | [] | refine (P.arrow_mk_iso_iff ?_).2 H
exact Arrow.isoMk (F.preimageIso (h.isoIsPullback X (F.obj a) BC)) (Iso.refl _)
(F.map_injective (by simp)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Representable | {
"line": 357,
"column": 4
} | {
"line": 359,
"column": 33
} | {
"line": 361,
"column": 0
} | [
{
"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.RespectsIso\nf : X ⟶ Y\nh₀ : ∀ ⦃a : C⦄ (g : F.obj a ⟶ Y), ∃ b fst snd, ∃ (_ : IsPullback fst (F.map snd) f g),... | [] | refine (P.arrow_mk_iso_iff ?_).2 H
exact Arrow.isoMk (F.preimageIso (h.isoIsPullback X (F.obj a) BC)) (Iso.refl _)
(F.map_injective (by simp)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 729,
"column": 4
} | {
"line": 729,
"column": 23
} | {
"line": 730,
"column": 2
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ_[p] ⇑padicNormE\nε : ℚ\nhε : ε > 0\nhε3 : 0 < ε / 3\nN : ℕ\nhN : ∀ i ≥ N, padicNormE (↑f i - ↑(limSeq f i)) < ε / 3\nN2 : ℕ\nhN2 : ∀ j ≥ N2, ∀ k ≥ N2, padicNormE (↑f j - ↑f k) < ε / 3\nj : ℕ\nhj : j ≥ max N N2\nthis : padicNormE (↑(limSeq f j) - ↑(limSeq ... | [] | exact mod_cast this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 963,
"column": 2
} | {
"line": 963,
"column": 59
} | {
"line": 964,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℤ\nn : ℕ\n⊢ ‖↑k‖ ≤ ↑p ^ (-↑n) ↔ ↑p ^ n ∣ k",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Real",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"Real.inst... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℤ\nn : ℕ\nthis : ↑p ^ (-↑n) = ↑↑p ^ (-↑n)\n⊢ ‖↑k‖ ≤ ↑p ^ (-↑n) ↔ ↑p ^ n ∣ k"
] | have : (p : ℝ) ^ (-n : ℤ) = (p : ℚ) ^ (-n : ℤ) := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1066,
"column": 2
} | {
"line": 1066,
"column": 35
} | {
"line": 1067,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\n⊢ (↑q).valuation = padicValRat p q",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"DivisionRing.toRatCast",
"Rat",
"Field.toDivisionRing",
"Padic.valuation",
"Rat.cast",
"Ne",
"... | [
"case inl\np : ℕ\nhp : Fact (Nat.Prime p)\n⊢ (↑0).valuation = padicValRat p 0",
"case inr\np : ℕ\nhp : Fact (Nat.Prime p)\nq : ℚ\nhq : q ≠ 0\n⊢ (↑q).valuation = padicValRat p q"
] | rcases eq_or_ne q 0 with rfl | hq | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1155,
"column": 8
} | {
"line": 1155,
"column": 11
} | {
"line": 1155,
"column": 12
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x = 0\n⊢ (if x * y = 0 then ⊤ else ↑(x * y).valuation) =\n (if x = 0 then ⊤ else ↑x.valuation) + if y = 0 then ⊤ else ↑y.valuation",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
... | [
"case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhx : x = 0\n⊢ (if 0 * y = 0 then ⊤ else ↑(0 * y).valuation) =\n (if 0 = 0 then ⊤ else ↑(valuation 0)) + if y = 0 then ⊤ else ↑y.valuation"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.PadicNumbers | {
"line": 1168,
"column": 10
} | {
"line": 1168,
"column": 13
} | {
"line": 1168,
"column": 14
} | [
{
"pp": "case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : x = 0\n⊢ min (if x = 0 then ⊤ else ↑x.valuation) (if y = 0 then ⊤ else ↑y.valuation) ≤\n if x + y = 0 then ⊤ else ↑(x + y).valuation",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case pos\np : ℕ\nhp : Fact (Nat.Prime p)\nx y : ℚ_[p]\nhxy : ¬x + y = 0\nhx : x = 0\n⊢ min (if 0 = 0 then ⊤ else ↑(valuation 0)) (if y = 0 then ⊤ else ↑y.valuation) ≤\n if 0 + y = 0 then ⊤ else ↑(0 + y).valuation"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 117,
"column": 4
} | {
"line": 120,
"column": 21
} | {
"line": 121,
"column": 2
} | [
{
"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 → ... | [] | rw [X.δ_comp_σ_self' (Fin.castSucc_mk _ _ _).symm,
X.δ_comp_σ_succ' (Fin.succ_mk _ _ _).symm]
simp only [comp_id, pow_add _ (a + 1) 1, pow_one, mul_neg, mul_one, neg_mul, neg_smul,
add_neg_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 117,
"column": 4
} | {
"line": 120,
"column": 21
} | {
"line": 121,
"column": 2
} | [
{
"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 → ... | [] | rw [X.δ_comp_σ_self' (Fin.castSucc_mk _ _ _).symm,
X.δ_comp_σ_succ' (Fin.succ_mk _ _ _).symm]
simp only [comp_id, pow_add _ (a + 1) 1, pow_one, mul_neg, mul_one, neg_mul, neg_smul,
add_neg_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 144,
"column": 4
} | {
"line": 157,
"column": 52
} | {
"line": 159,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫ hσ' q (n + 1) (n + 2) ⋯ ≫ (AlternatingFaceMapComplex.obj X).d (n + 2) (n + 1) = 0",
"ppTerm": "... | [] | simp only [hσ'_eq (show n + 1 = 0 + q by lia) (c_mk (n + 2) (n + 1) rfl), pow_zero,
Fin.mk_zero, one_zsmul, eqToHom_refl, comp_id, comp_sum,
AlternatingFaceMapComplex.obj_d_eq]
-- All terms of the sum but the first two are zeros
rw [Fin.sum_univ_succ, Fin.sum_univ_succ, Fintype.sum_eq_zero, add_zero... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.DoldKan.Faces | {
"line": 144,
"column": 4
} | {
"line": 157,
"column": 52
} | {
"line": 159,
"column": 0
} | [
{
"pp": "case neg\nC : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X _⦋n + 1⦌\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫ hσ' q (n + 1) (n + 2) ⋯ ≫ (AlternatingFaceMapComplex.obj X).d (n + 2) (n + 1) = 0",
"ppTerm": "... | [] | simp only [hσ'_eq (show n + 1 = 0 + q by lia) (c_mk (n + 2) (n + 1) rfl), pow_zero,
Fin.mk_zero, one_zsmul, eqToHom_refl, comp_id, comp_sum,
AlternatingFaceMapComplex.obj_d_eq]
-- All terms of the sum but the first two are zeros
rw [Fin.sum_univ_succ, Fin.sum_univ_succ, Fintype.sum_eq_zero, add_zero... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.DoldKan.FunctorGamma | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 16
} | {
"line": 68,
"column": 2
} | [
{
"pp": "n : ℕ\nj : Fin (n + 2)\ninst✝ : Mono (SimplexCategory.δ j)\nhi : Isδ₀ (SimplexCategory.δ j)\n⊢ SimplexCategory.δ j = SimplexCategory.δ 0",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"congrArg",
"AlgebraicTopology.DoldKan.Isδ₀.iff",
"SimplexCategory.δ",
"... | [
"n : ℕ\nj : Fin (n + 2)\ninst✝ : Mono (SimplexCategory.δ j)\nhi : j = 0\n⊢ SimplexCategory.δ j = SimplexCategory.δ 0"
] | rw [iff] at hi | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Idempotents.HomologicalComplex | {
"line": 169,
"column": 8
} | {
"line": 169,
"column": 76
} | {
"line": 169,
"column": 77
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nι : Type u_2\nc : ComplexShape ι\nP Q : Karoubi (HomologicalComplex C c)\nφ : P ⟶ Q\ni✝ : ι\n⊢ φ.f.f i✝ ≫ Q.p.f i✝ = P.p.f i✝ ≫ φ.f.f i✝",
"ppTerm": "?m.188",
"assigned": true,
"usedConstants": [
"CategoryTheory.Idem... | [] | simp only [HomologicalComplex.comp_p_d, HomologicalComplex.p_comp_d] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.DoldKan.GammaCompN | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 19
} | {
"line": 108,
"column": 2
} | [
{
"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⊢ (N₁Γ₀.hom.app K).f.f n = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv.f.f n",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"AlgebraicTopol... | [
"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⊢ ((Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv ≫ (toKaroubi (ChainComplex C ℕ)).map (Γ₀NondegComplexIso K).hom).f.f\n n =\n (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.inv... | rw [N₁Γ₀_hom_app] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Quiver.SingleObj | {
"line": 108,
"column": 2
} | {
"line": 110,
"column": 51
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nx : SingleObj α\np : Path (star α) x\n⊢ listToPath (pathToList p) = Path.cast ⋯ ⋯ p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Quiver.SingleObj.pathToList",
"Quiver.Hom",
"Quiver.Path.nil",
"congrArg",
"Quiver.Singl... | [] | induction p with
| nil => rfl
| cons _ _ ih => dsimp [pathToList] at *; rw [ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.Quiver.SingleObj | {
"line": 108,
"column": 2
} | {
"line": 110,
"column": 51
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nx : SingleObj α\np : Path (star α) x\n⊢ listToPath (pathToList p) = Path.cast ⋯ ⋯ p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Quiver.SingleObj.pathToList",
"Quiver.Hom",
"Quiver.Path.nil",
"congrArg",
"Quiver.Singl... | [] | induction p with
| nil => rfl
| cons _ _ ih => dsimp [pathToList] at *; rw [ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.SingleObj | {
"line": 108,
"column": 2
} | {
"line": 110,
"column": 51
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nx : SingleObj α\np : Path (star α) x\n⊢ listToPath (pathToList p) = Path.cast ⋯ ⋯ p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Quiver.SingleObj.pathToList",
"Quiver.Hom",
"Quiver.Path.nil",
"congrArg",
"Quiver.Singl... | [] | induction p with
| nil => rfl
| cons _ _ ih => dsimp [pathToList] at *; rw [ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.Basic | {
"line": 275,
"column": 19
} | {
"line": 275,
"column": 33
} | {
"line": 276,
"column": 2
} | [
{
"pp": "F✝ : Type u_1\nX : Type u\nY : Type v\nZ : Type w\nZ' : Type x\nι : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\nf₀ f₁ g₀ g₁ : C(X, Y)\nF : f₀.Homotopy f₁\nh₀ : f₀ = g₀\nh₁ : f₁ = g₁\n⊢ ∀ (x : X), F (0, x) = g₀ x",
"pp... | [] | by simp [← h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.ModelCategory.BifibrantObjectHomotopy | {
"line": 100,
"column": 45
} | {
"line": 102,
"column": 49
} | {
"line": 102,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : ModelCategory C\nD : Type u_1\ninst✝ : Category.{v_1, u_1} D\nF : BifibrantObject C ⥤ D\nH : (weakEquivalences (BifibrantObject C)).IsInvertedBy F\nK L : BifibrantObject C\nf g : K ⟶ L\nh✝ : homRel C f g\nP : PathObject L.obj\nleft✝ : P.IsVeryGood\nh : P... | [] | by
rw [← weakEquivalence_iff, weakEquivalence_iff_of_objectProperty]
exact inferInstanceAs (WeakEquivalence P.ι) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 54,
"column": 20
} | {
"line": 54,
"column": 43
} | {
"line": 54,
"column": 43
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\n⊢ (⨆ j, stdSimplex.face {↑j}ᶜ).toSSet.HasDimensionLT n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
"Opposite",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"congrArg",
... | [
"n : ℕ\ni : Fin (n + 1)\n⊢ ∀ (i_1 : ↑{i}ᶜ), (stdSimplex.face {↑i_1}ᶜ).toSSet.HasDimensionLT n"
] | hasDimensionLT_iSup_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.Horn | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 36
} | {
"line": 109,
"column": 2
} | [
{
"pp": "n : ℕ\nA : Δ[n + 1].Subcomplex\ni : Fin (n + 2)\n⊢ A ≤ Λ[n + 1, i] ↔ ¬stdSimplex.face {i}ᶜ ≤ A",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Opposite",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"Compl.compl",
"Finset",
"Partial... | [
"case refine_1\nn : ℕ\nA : Δ[n + 1].Subcomplex\ni : Fin (n + 2)\nhA : A ≤ Λ[n + 1, i]\nh : stdSimplex.face {i}ᶜ ≤ A\n⊢ False",
"case refine_2\nn : ℕ\nA : Δ[n + 1].Subcomplex\ni : Fin (n + 2)\nh : ¬stdSimplex.face {i}ᶜ ≤ A\n⊢ A ≤ Λ[n + 1, i]"
] | refine ⟨fun hA h ↦ ?_, fun h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicTopology.SimplicialSet.Boundary | {
"line": 51,
"column": 24
} | {
"line": 51,
"column": 47
} | {
"line": 51,
"column": 47
} | [
{
"pp": "n : ℕ\n⊢ (⨆ i, stdSimplex.face {i}ᶜ).toSSet.HasDimensionLT n",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
"Opposite",
"SimplexCategory.instFintypeToTypeOrderHomFinHAddNatLenOfNat",
"congrArg",
"iSup",
... | [
"n : ℕ\n⊢ ∀ (i : Fin (n + 1)), (stdSimplex.face {i}ᶜ).toSSet.HasDimensionLT n"
] | hasDimensionLT_iSup_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Types.Monomorphisms | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 60
} | {
"line": 66,
"column": 4
} | [
{
"pp": "T : Type u'\nX₁ X₂ : T → Type u\ninst✝¹ : HasCoproduct X₁\ninst✝ : HasCoproduct X₂\nf : (j : T) → X₁ j ⟶ X₂ j\nh : ∀ (j : T), monomorphisms (Type u) (f j)\n⊢ monomorphisms (Type u) (Limits.Sigma.map f)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Category... | [
"T : Type u'\nX₁ X₂ : T → Type u\ninst✝¹ : HasCoproduct X₁\ninst✝ : HasCoproduct X₂\nf : (j : T) → X₁ j ⟶ X₂ j\nh : ∀ (j : T), Function.Injective ⇑(hom (f j))\n⊢ Function.Injective ⇑(hom (Limits.Sigma.map f))"
] | simp only [monomorphisms.iff, mono_iff_injective] at h ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Limits.Types.Pushouts | {
"line": 277,
"column": 6
} | {
"line": 277,
"column": 28
} | {
"line": 278,
"column": 6
} | [
{
"pp": "case pos\nX₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nx₄ : X₄\nx₃ : X₃\nhx₃ : (ConcreteCategory.hom b) x₃ = x₄\nh₂ : x₃ ∈ Set.range ⇑(ConcreteCategory.hom l)\n⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨\n ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ ... | [
"case pos\nX₁ X₂ X₃ X₄ : Type u\nt : X₁ ⟶ X₂\nr : X₂ ⟶ X₄\nl : X₁ ⟶ X₃\nb : X₃ ⟶ X₄\nh : IsPushout t l r b\nx₄ : X₄\nx₁ : X₁\nhx₃ : (ConcreteCategory.hom b) ((ConcreteCategory.hom l) x₁) = x₄\n⊢ (∃ x₂, (ConcreteCategory.hom r) x₂ = x₄) ∨\n ∃ x₃, (ConcreteCategory.hom b) x₃ = x₄ ∧ x₃ ∉ Set.range ⇑(ConcreteCategor... | obtain ⟨x₁, rfl⟩ := h₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.Path | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | {
"line": 90,
"column": 0
} | [
{
"pp": "case hₐ\nn : ℕ\nX : Truncated (n + 1)\nm : ℕ\nf g : X.Path (m + 1)\nh : ∀ (i : Fin (m + 1)), f.arrow i = g.arrow i\nj : Fin (m + 1)\n⊢ f.arrow j = g.arrow j",
"ppTerm": "?hₐ",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h",
"j"
],
"usedGoals": []
}
] | [] | · exact h j | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.StrictSegal | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 45
} | {
"line": 190,
"column": 46
} | [
{
"pp": "n : ℕ\nX Y : Truncated (n + 1)\nsx : X.StrictSegal\nsy : Y.StrictSegal\nm : ℕ\nh : m ≤ n\nf : X.Path (m + 1)\nσ : X ⟶ Y\nk : Fin (m + 1)\n⊢ (ConcreteCategory.hom (Y.map (tr (mkOfSucc k) ⋯ ⋯).op)) (sy.spineToSimplex (m + 1) ⋯ (f.map σ)) =\n (ConcreteCategory.hom (Y.map (tr (mkOfSucc k) ⋯ ⋯).op))\n ... | [
"n : ℕ\nX Y : Truncated (n + 1)\nsx : X.StrictSegal\nsy : Y.StrictSegal\nm : ℕ\nh : m ≤ n\nf : X.Path (m + 1)\nσ : X ⟶ Y\nk : Fin (m + 1)\n⊢ (ConcreteCategory.hom (Y.map (tr (mkOfSucc k) ⋯ ⋯).op)) (sy.spineToSimplex (m + 1) ⋯ (f.map σ)) =\n (ConcreteCategory.hom (σ.app (op { obj := ⦋m + 1⦌, property := ⋯ }) ≫ Y.... | ← types_comp_apply (σ.app _) (Y.map _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.StrictSegal | {
"line": 375,
"column": 6
} | {
"line": 375,
"column": 45
} | {
"line": 375,
"column": 46
} | [
{
"pp": "X Y : SSet\nsx : X.StrictSegal\nsy : Y.StrictSegal\nn : ℕ\nf : X.Path (n + 1)\nσ : X ⟶ Y\nk : Fin (n + 1)\n⊢ (ConcreteCategory.hom (Y.map (mkOfSucc k).op)) (sy.spineToSimplex (f.map σ)) =\n (ConcreteCategory.hom (Y.map (mkOfSucc k).op))\n ((ConcreteCategory.hom (σ.app (Opposite.op ⦋n + 1⦌))) (s... | [
"X Y : SSet\nsx : X.StrictSegal\nsy : Y.StrictSegal\nn : ℕ\nf : X.Path (n + 1)\nσ : X ⟶ Y\nk : Fin (n + 1)\n⊢ (ConcreteCategory.hom (Y.map (mkOfSucc k).op)) (sy.spineToSimplex (f.map σ)) =\n (ConcreteCategory.hom (σ.app (Opposite.op ⦋n + 1⦌) ≫ Y.map (mkOfSucc k).op)) (sx.spineToSimplex f)"
] | ← types_comp_apply (σ.app _) (Y.map _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Category.ReflQuiv | {
"line": 108,
"column": 4
} | {
"line": 109,
"column": 22
} | {
"line": 110,
"column": 2
} | [
{
"pp": "V W : Type u\ninst✝¹ : ReflQuiver V\ninst✝ : ReflQuiver W\ne : Quiv.of V ≅ Quiv.of W\nh_id : ∀ (X : V), e.hom.map (𝟙rq X) = 𝟙rq (e.hom.obj X)\n⊢ { toPrefunctor := e.hom, map_id := h_id } ≫ { toPrefunctor := e.inv, map_id := ⋯ } = 𝟙 (of V)",
"ppTerm": "?m.65",
"assigned": true,
"usedConst... | [] | apply forgetToQuiv.map_injective
exact e.hom_inv_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.ReflQuiv | {
"line": 108,
"column": 4
} | {
"line": 109,
"column": 22
} | {
"line": 110,
"column": 2
} | [
{
"pp": "V W : Type u\ninst✝¹ : ReflQuiver V\ninst✝ : ReflQuiver W\ne : Quiv.of V ≅ Quiv.of W\nh_id : ∀ (X : V), e.hom.map (𝟙rq X) = 𝟙rq (e.hom.obj X)\n⊢ { toPrefunctor := e.hom, map_id := h_id } ≫ { toPrefunctor := e.inv, map_id := ⋯ } = 𝟙 (of V)",
"ppTerm": "?m.65",
"assigned": true,
"usedConst... | [] | apply forgetToQuiv.map_injective
exact e.hom_inv_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.Quasicategory.TwoTruncated | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 53
} | {
"line": 175,
"column": 2
} | [
{
"pp": "A : Truncated 2\ninst✝ : A.Quasicategory₂\nx y z : A.obj (Opposite.op { obj := { len := 0 }, property := Quasicategory₂._proof_1 })\nf f' : Edge x y\ng g' : Edge y z\nh h' : Edge x z\ns : f.CompStruct g h\ns' : f'.CompStruct g' h'\nhg✝ : HomotopicL g g'\nhg : (id y).CompStruct g g'\nhf : f.CompStruct (... | [
"A : Truncated 2\ninst✝ : A.Quasicategory₂\nx y z : A.obj (Opposite.op { obj := { len := 0 }, property := Quasicategory₂._proof_1 })\nf f' : Edge x y\ng g' : Edge y z\nh h' : Edge x z\ns : f.CompStruct g h\ns' : f'.CompStruct g' h'\nhg✝ : HomotopicL g g'\nhg : (id y).CompStruct g g'\nhf : f.CompStruct (id y) f'\ns₁... | let ⟨s₁⟩ := Quasicategory₂.fill32 hf (idComp g') s' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 14
} | {
"line": 62,
"column": 15
} | [
{
"pp": "case of\nx✝¹ y : SimplexCategoryGenRel\ne : x✝¹ ⟶ y\nx✝ y✝ : SimplexCategoryGenRel\nx : x✝ ⟶ y✝\nhx : degeneracies x\n⊢ IsSplitEpi x",
"ppTerm": "?of",
"assigned": true,
"usedConstants": [
"SimplexCategoryGenRel.degeneracies.casesOn",
"CategoryTheory.CategoryStruct.toQuiver",
... | [] | | of x hx => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.EpiMono | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 14
} | {
"line": 70,
"column": 15
} | [
{
"pp": "case of\nx✝¹ y : SimplexCategoryGenRel\nm : x✝¹ ⟶ y\nx✝ y✝ : SimplexCategoryGenRel\nx : x✝ ⟶ y✝\nhx : faces x\n⊢ IsSplitMono x",
"ppTerm": "?of",
"assigned": true,
"usedConstants": [
"SimplexCategoryGenRel.faces",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | [] | | of x hx => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.AlgebraicTopology.SimplexCategory.GeneratorsRelations.NormalForms | {
"line": 260,
"column": 4
} | {
"line": 263,
"column": 95
} | {
"line": 265,
"column": 0
} | [
{
"pp": "case cons\nL✝ : List ℕ\nj a : ℕ\nL : List ℕ\nh_rec :\n ∀ (m₁ m₂ : ℕ),\n IsAdmissible m₂ L →\n ∀ (hk : m₂ + L.length = m₁) (hj : j < m₁ + 1),\n ↑((SimplexCategory.Hom.toOrderHom (toSimplexCategory.map (standardσ L hk))) ⟨j, hj⟩) = simplicialEvalσ L j\nm₂ : ℕ\nhL : IsAdmissible m₂ (a :: L... | [] | simpa only [toSimplexCategory_obj_mk, SimplexCategory.len_mk, standardσ_cons, Functor.map_comp,
toSimplexCategory_map_σ, SimplexCategory.σ, SimplexCategory.mkHom,
SimplexCategory.comp_toOrderHom, SimplexCategory.Hom.toOrderHom_mk, OrderHom.comp_coe,
Function.comp_apply, Fin.predAboveOrderHom_coe, simp... | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne | {
"line": 107,
"column": 4
} | {
"line": 112,
"column": 68
} | {
"line": 113,
"column": 2
} | [
{
"pp": "case pos\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nh : j.castSucc < i\nk : Fin (⦋n + 1⦌.len + 1)\nhk : i < k\n⊢ (ConcreteCategory.hom (toMk₁ i)) (j.predAbove k) = (ConcreteCategory.hom (toMk₁ i.succ)) k",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Fin.succ",
"Simpl... | [] | #adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
(replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this goal.
It is not yet clear whether this is due to defeq abuse in Mathlib or a problem in the new
canonicalizer; a minimization would help. The orig... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplexCategory.ToMkOne | {
"line": 107,
"column": 4
} | {
"line": 112,
"column": 68
} | {
"line": 113,
"column": 2
} | [
{
"pp": "case pos\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nh : j.castSucc < i\nk : Fin (⦋n + 1⦌.len + 1)\nhk : i < k\n⊢ (ConcreteCategory.hom (toMk₁ i)) (j.predAbove k) = (ConcreteCategory.hom (toMk₁ i.succ)) k",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Fin.succ",
"Simpl... | [] | #adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
(replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this goal.
It is not yet clear whether this is due to defeq abuse in Mathlib or a problem in the new
canonicalizer; a minimization would help. The orig... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.UpperLower.Relative | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 31
} | {
"line": 183,
"column": 0
} | [
{
"pp": "α : Type u_1\na : α\ninst✝ : Preorder α\nc x✝ : α\nb : a ≤ x✝ ∧ x✝ ≤ c\n⊢ ∀ ⦃b : α⦄, x✝ ≤ b → b ≤ c → a ≤ b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Preorder.toLE",
"LE.le",
"And.left",
"LE.le.trans"
],
"usedFVars": [
"α",
"inst✝... | [] | exact fun _ x _ ↦ b.1.trans x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 132,
"column": 10
} | {
"line": 132,
"column": 90
} | {
"line": 133,
"column": 10
} | [
{
"pp": "case inl.inl.inl.left\nn m p : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\ng : Fin (m + 1) →o Fin (p + 1)\nx : Fin (p + 1)\ny : Fin (m + 1)\nhy : x.castSucc ≤ (g y).castSucc ∧ ∀ i < y, (g i).castSucc < x.castSucc\nz : Fin (n + 1)\nhz : y.castSucc ≤ (f z).castSucc ∧ ∀ i < z, (f i).castSucc < y.castSucc\n⊢ (g y).... | [
"case inl.inl.inl.left\nn m p : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\ng : Fin (m + 1) →o Fin (p + 1)\nx : Fin (p + 1)\ny : Fin (m + 1)\nhy : x.castSucc ≤ (g y).castSucc ∧ ∀ i < y, (g i).castSucc < x.castSucc\nz : Fin (n + 1)\nhz : y.castSucc ≤ (f z).castSucc ∧ ∀ i < z, (f i).castSucc < y.castSucc\n⊢ g y ≤ g (f z)"
] | simp only [OrderHom.comp_coe, Function.comp_apply, Fin.castSucc_le_castSucc_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 140,
"column": 19
} | {
"line": 140,
"column": 22
} | {
"line": 140,
"column": 23
} | [
{
"pp": "case inl.inr\nn m p : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\ng : Fin (m + 1) →o Fin (p + 1)\nx : Fin (p + 1)\nhx : map' g x.castSucc = Fin.last (m + 1)\n⊢ map' (g.comp f) x.castSucc = map' f (map' g x.castSucc)",
"ppTerm": "?inl.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case inl.inr\nn m p : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\ng : Fin (m + 1) →o Fin (p + 1)\nx : Fin (p + 1)\nhx : map' g x.castSucc = Fin.last (m + 1)\n⊢ map' (g.comp f) x.castSucc = map' f (Fin.last (m + 1))"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 198,
"column": 8
} | {
"line": 199,
"column": 41
} | {
"line": 200,
"column": 6
} | [
{
"pp": "case neg.left\nn : ℕ\ni x : Fin (n + 1)\nhi : x ≤ i\n⊢ x ≤ i.predAbove x.castSucc",
"ppTerm": "?neg.left✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fin.castSucc_le_castSucc_iff._simp_1",
"PartialOrder.toPreorder",
"id",
"instOfNatNat",... | [] | simp only [i.predAbove_of_le_castSucc x.castSucc (by simpa),
Fin.castPred_castSucc, le_refl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 198,
"column": 8
} | {
"line": 199,
"column": 41
} | {
"line": 200,
"column": 6
} | [
{
"pp": "case neg.left\nn : ℕ\ni x : Fin (n + 1)\nhi : x ≤ i\n⊢ x ≤ i.predAbove x.castSucc",
"ppTerm": "?neg.left✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fin.castSucc_le_castSucc_iff._simp_1",
"PartialOrder.toPreorder",
"id",
"instOfNatNat",... | [] | simp only [i.predAbove_of_le_castSucc x.castSucc (by simpa),
Fin.castPred_castSucc, le_refl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 198,
"column": 8
} | {
"line": 199,
"column": 41
} | {
"line": 200,
"column": 6
} | [
{
"pp": "case neg.left\nn : ℕ\ni x : Fin (n + 1)\nhi : x ≤ i\n⊢ x ≤ i.predAbove x.castSucc",
"ppTerm": "?neg.left✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Fin.castSucc_le_castSucc_iff._simp_1",
"PartialOrder.toPreorder",
"id",
"instOfNatNat",... | [] | simp only [i.predAbove_of_le_castSucc x.castSucc (by simpa),
Fin.castPred_castSucc, le_refl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 211,
"column": 41
} | {
"line": 215,
"column": 10
} | {
"line": 215,
"column": 10
} | [
{
"pp": "n m : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\nx y : Fin (m + 2)\nhxy : x ≤ y\nz : Fin (n + 2)\nhz : z ∈ finset f y\n⊢ z ∈ finset f x",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimplexCategory.II.finset",
"SimplexCategory.II.castSucc_mem_finset_iff._sim... | [] | by
obtain ⟨z, rfl⟩ | rfl := z.eq_castSucc_or_eq_last
· simp only [castSucc_mem_finset_iff] at hz ⊢
exact hxy.trans hz
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 210,
"column": 2
} | {
"line": 215,
"column": 11
} | {
"line": 217,
"column": 0
} | [
{
"pp": "n m : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\n⊢ Monotone (map' f)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimplexCategory.II.finset",
"SimplexCategory.II.castSucc_mem_finset_iff._simp_1",
"Finset",
"PartialOrder.toPreorder",
"Preor... | [] | intro x y hxy
exact Finset.min'_subset _ (fun z hz ↦ by
obtain ⟨z, rfl⟩ | rfl := z.eq_castSucc_or_eq_last
· simp only [castSucc_mem_finset_iff] at hz ⊢
exact hxy.trans hz
· simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialObject.II | {
"line": 210,
"column": 2
} | {
"line": 215,
"column": 11
} | {
"line": 217,
"column": 0
} | [
{
"pp": "n m : ℕ\nf : Fin (n + 1) →o Fin (m + 1)\n⊢ Monotone (map' f)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimplexCategory.II.finset",
"SimplexCategory.II.castSucc_mem_finset_iff._simp_1",
"Finset",
"PartialOrder.toPreorder",
"Preor... | [] | intro x y hxy
exact Finset.min'_subset _ (fun z hz ↦ by
obtain ⟨z, rfl⟩ | rfl := z.eq_castSucc_or_eq_last
· simp only [castSucc_mem_finset_iff] at hz ⊢
exact hxy.trans hz
· simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteColimits | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 27
} | {
"line": 50,
"column": 27
} | [
{
"pp": "J : Type u_1\ninst✝¹ : Category.{u_2, u_1} J\ninst✝ : HasColimitsOfShape J (Type u)\nF : J ⥤ SSet\nc : Cocone F\nhc : IsColimit c\nn : ℕ\nh : ∀ (j : J), (F.obj j).HasDimensionLT n\n⊢ (⨆ j, Subcomplex.range (c.ι.app j)).toSSet.HasDimensionLT n",
"ppTerm": "?m.32",
"assigned": true,
"usedCons... | [
"J : Type u_1\ninst✝¹ : Category.{u_2, u_1} J\ninst✝ : HasColimitsOfShape J (Type u)\nF : J ⥤ SSet\nc : Cocone F\nhc : IsColimit c\nn : ℕ\nh : ∀ (j : J), (F.obj j).HasDimensionLT n\n⊢ ∀ (i : J), (Subcomplex.range (c.ι.app i)).toSSet.HasDimensionLT n"
] | hasDimensionLT_iSup_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteColimits | {
"line": 55,
"column": 19
} | {
"line": 57,
"column": 16
} | {
"line": 59,
"column": 0
} | [
{
"pp": "J : Type u_1\ninst✝² : Category.{u_2, u_1} J\ninst✝¹ : HasColimitsOfShape J (Type u)\nF : J ⥤ SSet\nc : Cocone F\nhc : IsColimit c\ninst✝ : Finite J\nh : ∀ (j : J), (F.obj j).Finite\n⊢ c.pt.Finite",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
... | [] | by
rw [← finite_subcomplex_top_iff, ← iSup_range_eq_top_of_isColimit hc, finite_iSup_iff]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PairingCore | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 26
} | {
"line": 156,
"column": 4
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nh : A.PairingCore\ns : A.N\n⊢ s ∈ h.I ∪ h.II ↔ s ∈ Set.univ",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.PairingCore.ι",
"SSet.Subcomplex.PairingCore.type₂",
"Exists",
"SSet.Subcomplex.N",
"SSet.Subcompl... | [
"X : SSet\nA : X.Subcomplex\nh : A.PairingCore\ns : A.N\nthis : ∃ s_1, s = h.type₁ s_1 ∨ s = h.type₂ s_1\n⊢ s ∈ h.I ∪ h.II ↔ s ∈ Set.univ"
] | have := h.surjective s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Pullbacks | {
"line": 69,
"column": 96
} | {
"line": 70,
"column": 23
} | {
"line": 72,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF G H : D ⥤ C\ninst✝ : HasPullbacks C\nf : F ⟶ H\ng : G ⟶ H\nd : D\n⊢ (pullbackObjIso f g d).hom ≫ pullback.fst (f.app d) (g.app d) = (pullback.fst f g).app d",
"ppTerm": "?m.60",
"assigned": true,
"usedCo... | [] | by
simp [pullbackObjIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Pullbacks | {
"line": 75,
"column": 96
} | {
"line": 76,
"column": 23
} | {
"line": 78,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF G H : D ⥤ C\ninst✝ : HasPullbacks C\nf : F ⟶ H\ng : G ⟶ H\nd : D\n⊢ (pullbackObjIso f g d).hom ≫ pullback.snd (f.app d) (g.app d) = (pullback.snd f g).app d",
"ppTerm": "?m.60",
"assigned": true,
"usedCo... | [] | by
simp [pullbackObjIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Pullbacks | {
"line": 82,
"column": 96
} | {
"line": 83,
"column": 23
} | {
"line": 85,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF G H : D ⥤ C\ninst✝ : HasPullbacks C\nf : F ⟶ H\ng : G ⟶ H\nd : D\n⊢ (pullbackObjIso f g d).inv ≫ (pullback.fst f g).app d = pullback.fst (f.app d) (g.app d)",
"ppTerm": "?m.60",
"assigned": true,
"usedCo... | [] | by
simp [pullbackObjIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Pullbacks | {
"line": 89,
"column": 96
} | {
"line": 90,
"column": 23
} | {
"line": 92,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nF G H : D ⥤ C\ninst✝ : HasPullbacks C\nf : F ⟶ H\ng : G ⟶ H\nd : D\n⊢ (pullbackObjIso f g d).inv ≫ (pullback.snd f g).app d = pullback.snd (f.app d) (g.app d)",
"ppTerm": "?m.60",
"assigned": true,
"usedCo... | [] | by
simp [pullbackObjIso] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.ExtremalEpi | {
"line": 73,
"column": 18
} | {
"line": 77,
"column": 29
} | {
"line": 78,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasPullbacks C\nx✝¹ : ExtremalEpi f\nA B : C\ni : A ⟶ B\nx✝ : Mono i\nt : X ⟶ A\nb : Y ⟶ B\nsq : CommSq t f i b\nthis : IsIso (pullback.snd i b)\n⊢ f ≫ inv (pullback.snd i b) ≫ pullback.fst i b = t",
"ppTerm": "?m.132",
"assign... | [] | by
rw [← cancel_mono i, sq.w, Category.assoc, Category.assoc]
congr 1
rw [← cancel_epi (pullback.snd i b), IsIso.hom_inv_id_assoc,
pullback.condition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Generator.StrongGenerator | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 33
} | {
"line": 66,
"column": 4
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nP : ObjectProperty C\nx✝¹ : P.IsStrongGenerator\nhS₁ : P.IsSeparating\nhS₂ : ∀ ⦃X : C⦄ (A : Subobject X), (∀ (G : C), P G → ∀ (f : G ⟶ X), A.Factors f) → A = ⊤\nX Y : C\ni : X ⟶ Y\nx✝ : Mono i\nh : ∀ (G : C), P G → Function.Surjective fun f ↦ f ≫ i\... | [
"case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nP : ObjectProperty C\nx✝¹ : P.IsStrongGenerator\nhS₁ : P.IsSeparating\nhS₂ : ∀ ⦃X : C⦄ (A : Subobject X), (∀ (G : C), P G → ∀ (f : G ⟶ X), A.Factors f) → A = ⊤\nX Y : C\ni : X ⟶ Y\nx✝ : Mono i\nh : ∀ (G : C), P G → Function.Surjective fun f ↦ f ≫ i\nG : C\nhG :... | rw [Subobject.mk_factors_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Presentable | {
"line": 44,
"column": 6
} | {
"line": 46,
"column": 20
} | {
"line": 47,
"column": 2
} | [
{
"pp": "case inst\nX : SSet\ninst✝ : X.Finite\n⊢ ∀ (k : Discrete X.N), IsCardinalPresentable ((Discrete.functor fun s ↦ Δ[s.dim]).obj k) Cardinal.aleph0",
"ppTerm": "?inst✝",
"assigned": true,
"usedConstants": [
"Opposite",
"SSet.Finite.instIsFinitelyPresentableObjSimplexCategoryStdSimp... | [] | rintro s
dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.Presentable | {
"line": 44,
"column": 6
} | {
"line": 46,
"column": 20
} | {
"line": 47,
"column": 2
} | [
{
"pp": "case inst\nX : SSet\ninst✝ : X.Finite\n⊢ ∀ (k : Discrete X.N), IsCardinalPresentable ((Discrete.functor fun s ↦ Δ[s.dim]).obj k) Cardinal.aleph0",
"ppTerm": "?inst✝",
"assigned": true,
"usedConstants": [
"Opposite",
"SSet.Finite.instIsFinitelyPresentableObjSimplexCategoryStdSimp... | [] | rintro s
dsimp
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 631,
"column": 12
} | {
"line": 631,
"column": 75
} | {
"line": 632,
"column": 12
} | [
{
"pp": "m✝ : ℕ\nk✝ : Fin (m✝ + 1)\nn✝ : ℕ\nx : (Λ[m✝ + 1, k✝.castSucc].unionProd ∂Δ[n✝]).N\nd✝ m : ℕ\nk : Fin (m + 1)\nn d : ℕ\nis it : Fin (d + 1)\ns : (Δ[m + 1] ⊗ Δ[n]) _⦋d + 1⦌\nhs₁ : s ∈ (Δ[m + 1] ⊗ Δ[n]).nonDegenerate (d + 1)\nhs₂ : s ∉ (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).obj (Opposite.op ⦋d + 1⦌)\nhs ... | [
"m✝ : ℕ\nk✝ : Fin (m✝ + 1)\nn✝ : ℕ\nx : (Λ[m✝ + 1, k✝.castSucc].unionProd ∂Δ[n✝]).N\nd✝ m : ℕ\nk : Fin (m + 1)\nn d : ℕ\nis it : Fin (d + 1)\ns : (Δ[m + 1] ⊗ Δ[n]) _⦋d + 1⦌\nhs₁ : s ∈ (Δ[m + 1] ⊗ Δ[n]).nonDegenerate (d + 1)\nhs₂ : s ∉ (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).obj (Opposite.op ⦋d + 1⦌)\nhs : IsIndex (S... | Finset.card_image_of_injective _ Fin.succAbove_right_injective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.Homotopy | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 51
} | {
"line": 102,
"column": 6
} | [
{
"pp": "case snd\nX Y : SSet\nf g : X ⟶ Y\nH : Homotopy f g\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nhij : i ≤ j.castSucc\nx : X _⦋n + 1⦌\nk : Fin (n + 1 + 1)\n⊢ ((ConcreteCategory.hom\n (X.map\n (stdSimplex.objEquiv\n ((ConcreteCategory.hom (δ Δ[n + 1] i.castSuc... | [
"case snd.h\nX Y : SSet\nf g : X ⟶ Y\nH : Homotopy f g\nn : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nhij : i ≤ j.castSucc\nx : X _⦋n + 1⦌\nk : Fin (n + 1 + 1)\n⊢ i.castSucc.castSucc < j.castSucc.succ.succ"
] | rw [stdSimplex.δ_objMk₁_of_lt, Fin.pred_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplicesColimit | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 20
} | {
"line": 76,
"column": 8
} | [
{
"pp": "X : SSet\ns : Cocone X.functorN\nx✝ : (MultispanShape.prod X.N).L\nx y : X.N\nH :\n ∀ (u t : X.N) (h : t.simplex ∈ u.subcomplex.obj (op ⦋t.dim⦌)),\n (ConcreteCategory.hom ((s.ι.app u).app (op ⦋t.dim⦌))) ⟨t.simplex, h⟩ =\n (ConcreteCategory.hom ((s.ι.app t).app (op ⦋t.dim⦌))) ⟨t.simplex, ⋯⟩\n⊢ ... | [
"X : SSet\ns : Cocone X.functorN\nx✝ : (MultispanShape.prod X.N).L\nx y : X.N\nH :\n ∀ (u t : X.N) (h : t.simplex ∈ u.subcomplex.obj (op ⦋t.dim⦌)),\n (ConcreteCategory.hom ((s.ι.app u).app (op ⦋t.dim⦌))) ⟨t.simplex, h⟩ =\n (ConcreteCategory.hom ((s.ι.app t).app (op ⦋t.dim⦌))) ⟨t.simplex, ⋯⟩\nn : ℕ\nz : ↑(X... | intro n z hz | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.Order | {
"line": 192,
"column": 43
} | {
"line": 192,
"column": 67
} | {
"line": 193,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\nn : ℕ\n⊢ ∀ (a : 𝕜 → E),\n (AnalyticAt 𝕜 a z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (-f) z = (z - z₀) ^ n • a z) ↔\n AnalyticAt 𝕜 ((Equiv.... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Order | {
"line": 192,
"column": 43
} | {
"line": 192,
"column": 67
} | {
"line": 193,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\nn : ℕ\n⊢ ∀ (a : 𝕜 → E),\n (AnalyticAt 𝕜 a z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (-f) z = (z - z₀) ^ n • a z) ↔\n AnalyticAt 𝕜 ((Equiv.... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Order | {
"line": 192,
"column": 43
} | {
"line": 192,
"column": 67
} | {
"line": 193,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f z₀\nn : ℕ\n⊢ ∀ (a : 𝕜 → E),\n (AnalyticAt 𝕜 a z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (-f) z = (z - z₀) ^ n • a z) ↔\n AnalyticAt 𝕜 ((Equiv.... | [] | simp [neg_eq_iff_eq_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Extend | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 32
} | {
"line": 45,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\ns : Set E\nx : E\nf' : E →L[ℝ] F\nf_diff : DifferentiableOn ℝ f s\ns_conv : Convex ℝ s\ns_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalS... | [
"case pos\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\ns : Set E\nx : E\nf' : E →L[ℝ] F\nf_diff : DifferentiableOn ℝ f s\ns_conv : Convex ℝ s\ns_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | by_cases! hx : x ∉ closure s | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.Analysis.Analytic.Order | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 98
} | {
"line": 559,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nz₀ : 𝕜\nhf : AnalyticAt 𝕜 f (g z₀)\nhg : AnalyticAt 𝕜 g z₀\nhg_nc : ¬analyticOrderAt (fun x ↦ g x - g z₀) z₀ = ⊤\nhf' : ¬analyticOrderAt f (g z₀) = ⊤\nr... | [] | simp only [hfz, Function.comp_def, hgz, smul_eq_mul, mul_pow, mul_smul, mul_comm r s, pow_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Binomial | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 48
} | {
"line": 316,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (ascPochhammer ℕ (n + 0 + 1)).smeval (-↑n) = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"ascPochhammer",
"Module.toMulActionWithZero",
"Nat.instAddM... | [] | rw [add_zero, smeval_ascPochhammer_succ_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Binomial | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 48
} | {
"line": 316,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (ascPochhammer ℕ (n + 0 + 1)).smeval (-↑n) = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"ascPochhammer",
"Module.toMulActionWithZero",
"Nat.instAddM... | [] | rw [add_zero, smeval_ascPochhammer_succ_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Binomial | {
"line": 315,
"column": 4
} | {
"line": 315,
"column": 48
} | {
"line": 316,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (ascPochhammer ℕ (n + 0 + 1)).smeval (-↑n) = 0",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"congrArg",
"AddMonoid.toAddZeroClass",
"ascPochhammer",
"Module.toMulActionWithZero",
"Nat.instAddM... | [] | rw [add_zero, smeval_ascPochhammer_succ_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Binomial | {
"line": 356,
"column": 9
} | {
"line": 357,
"column": 46
} | {
"line": 358,
"column": 2
} | [
{
"pp": "R : Type u_2\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nr : R\n⊢ (ascPochhammer ℤ 0).smeval r = (ascPochhammer ℕ 0).smeval r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"instHSMul",
"congrArg",
"NonUnitalNonAssocRing.toAddCommGroup",
... | [] | by
simp only [ascPochhammer_zero, smeval_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Binomial | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 81
} | {
"line": 109,
"column": 6
} | [
{
"pp": "case convert_7\na : ℂ\nthis : binomialSeries ℂ a = FormalMultilinearSeries.ofScalars ℂ fun n ↦ iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 / ↑n !\n⊢ AnalyticOn ℂ (fun x ↦ (1 + x) ^ a) (Metric.eball 0 1)",
"ppTerm": "?convert_7",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNo... | [
"case convert_7\na : ℂ\nthis : binomialSeries ℂ a = FormalMultilinearSeries.ofScalars ℂ fun n ↦ iteratedDeriv n (fun x ↦ (1 + x) ^ a) 0 / ↑n !\n⊢ ∀ z ∈ Metric.eball 0 1, ((fun x ↦ 1) + fun x ↦ x) z ∈ slitPlane"
] | apply AnalyticOn.cpow (analyticOn_const.add analyticOn_id) analyticOn_const | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Analytic.IteratedFDeriv | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 44
} | {
"line": 119,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nf : E [×n]→L[𝕜] F\nx : E\nv : Fin n → E\n⊢ (iteratedFDeriv 𝕜 n (fun x ↦ f fun x_1 ↦ x) x) v = ∑ σ, ... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nf : E [×n]→L[𝕜] F\nx : E\nv : Fin n → E\n⊢ (iteratedFDeriv 𝕜 n (fun x ↦ f fun x_1 ↦ x) x) v = ∑ i, f fun i_1 ↦ ... | rw [← sum_comp (Equiv.inv (Perm (Fin n)))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Polynomial | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 73
} | {
"line": 33,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CommSemiring A\nz : E\ns : Set E\ninst✝² : NormedRing B\ninst✝¹ : NormedAlgebra 𝕜 B\ninst✝ : Algebra A B\nf : E → B\nhf : AnalyticWithinAt 𝕜... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CommSemiring A\nz : E\ns : Set E\ninst✝² : NormedRing B\ninst✝¹ : NormedAlgebra 𝕜 B\ninst✝ : Algebra A B\nf : E → B\nhf : AnalyticWithinAt... | refine p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.IteratedFDeriv | {
"line": 175,
"column": 12
} | {
"line": 175,
"column": 35
} | {
"line": 175,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nh : HasFPowerSeriesWithinOnBall f... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nh : HasFPowerSeriesWithinOnBall f p s x r\nh'... | iteratedFDeriv_comp_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.Completion | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 34
} | {
"line": 34,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹ : Norm E\ninst✝ : SeminormedAddCommGroup F\nf : α → E\ng : α → F\nl : Filter α\n⊢ (fun x ↦ ↑(g x)) =O[l] f ↔ g =O[l] f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"R... | [] | simp only [isBigO_iff, norm_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Completion | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 34
} | {
"line": 34,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹ : Norm E\ninst✝ : SeminormedAddCommGroup F\nf : α → E\ng : α → F\nl : Filter α\n⊢ (fun x ↦ ↑(g x)) =O[l] f ↔ g =O[l] f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"R... | [] | simp only [isBigO_iff, norm_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Completion | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 34
} | {
"line": 34,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹ : Norm E\ninst✝ : SeminormedAddCommGroup F\nf : α → E\ng : α → F\nl : Filter α\n⊢ (fun x ↦ ↑(g x)) =O[l] f ↔ g =O[l] f",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"R... | [] | simp only [isBigO_iff, norm_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.Completion | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 34
} | {
"line": 38,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹ : Norm E\ninst✝ : SeminormedAddCommGroup F\nf : α → E\ng : α → F\nl : Filter α\n⊢ (f =O[l] fun x ↦ ↑(g x)) ↔ f =O[l] g",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"R... | [] | simp only [isBigO_iff, norm_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Completion | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 34
} | {
"line": 38,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹ : Norm E\ninst✝ : SeminormedAddCommGroup F\nf : α → E\ng : α → F\nl : Filter α\n⊢ (f =O[l] fun x ↦ ↑(g x)) ↔ f =O[l] g",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"R... | [] | simp only [isBigO_iff, norm_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Completion | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 34
} | {
"line": 38,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹ : Norm E\ninst✝ : SeminormedAddCommGroup F\nf : α → E\ng : α → F\nl : Filter α\n⊢ (f =O[l] fun x ↦ ↑(g x)) ↔ f =O[l] g",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Norm.norm",
"UniformSpace.Completion.coe'",
"R... | [] | simp only [isBigO_iff, norm_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.ERealExp | {
"line": 48,
"column": 18
} | {
"line": 48,
"column": 37
} | {
"line": 50,
"column": 0
} | [
{
"pp": "case bot\n⊢ ⊥.exp = 0 ↔ ⊥ = ⊥",
"ppTerm": "?bot",
"assigned": true,
"usedConstants": [
"congrArg",
"EReal",
"Bot.bot",
"iff_self",
"Iff",
"EReal.exp",
"congr",
"True",
"eq_self",
"ENNReal",
"of_eq_true",
"Zero.toOfNat0"... | [] | simp [Real.exp_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.ERealExp | {
"line": 48,
"column": 18
} | {
"line": 48,
"column": 37
} | {
"line": 50,
"column": 0
} | [
{
"pp": "case coe\na✝ : ℝ\n⊢ (↑a✝).exp = 0 ↔ ↑a✝ = ⊥",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Log.ERealExp.0.EReal.exp_eq_zero_iff._simp_1_1",
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"iff_false",... | [] | simp [Real.exp_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.ERealExp | {
"line": 48,
"column": 18
} | {
"line": 48,
"column": 37
} | {
"line": 50,
"column": 0
} | [
{
"pp": "case top\n⊢ ⊤.exp = 0 ↔ ⊤ = ⊥",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"top_ne_bot._simp_1",
"False",
"ENNReal.top_ne_zero._simp_1",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"EReal",
"CompleteLattice.toBoundedOrder",... | [] | simp [Real.exp_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.ERealExp | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 25
} | {
"line": 113,
"column": 2
} | [
{
"pp": "case coe.top\na✝ : ℝ\n⊢ ENNReal.ofReal (Real.exp a✝) ≠ 0",
"ppTerm": "?coe.top✝",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"LE.le",
... | [] | simp [Real.exp_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.ERealExp | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 25
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case top.coe\na✝ : ℝ\n⊢ ENNReal.ofReal (Real.exp a✝) ≠ 0",
"ppTerm": "?top.coe✝",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"LE.le",
... | [] | simp [Real.exp_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 95,
"column": 46
} | {
"line": 95,
"column": 65
} | {
"line": 97,
"column": 0
} | [
{
"pp": "case h\ny : ℝ\n⊢ (ENNReal.ofReal (Real.exp y)).log = ↑y",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.log_exp",
"ite_eq_right_iff._simp_1",
"Real.instZero",
"ENNReal.ofReal",
... | [] | simp [Real.exp_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 64
} | {
"line": 163,
"column": 0
} | [
{
"pp": "n : ℕ\nn_pos : 0 < n\n⊢ ⊥ n / ↑n = ⊥",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Iff.mpr",
"Preorder.toLT",
"NeZero.charZero_one",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"EReal",
... | [] | exact bot_div_of_pos_ne_top (by positivity) (natCast_ne_top n) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 80
} | {
"line": 219,
"column": 2
} | [
{
"pp": "u v : ℕ → EReal\n⊢ linearGrowthInf u + linearGrowthInf v ≤ linearGrowthInf (u + v)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"instHDiv",
"LE.le.trans_eq",
"Filter.liminf",
"Partial... | [
"u v : ℕ → EReal\nn : ℕ\n⊢ ((fun n ↦ u n / ↑n) + fun n ↦ v n / ↑n) n = (u + v) n / ↑n"
] | refine le_liminf_add.trans_eq (liminf_congr (Eventually.of_forall fun n ↦ ?_)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 80
} | {
"line": 205,
"column": 2
} | [
{
"pp": "u v : ℕ → ℝ≥0∞\n⊢ expGrowthInf u + expGrowthInf v ≤ expGrowthInf (u * v)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"ExpGrowth.expGrowthInf",
"instHDiv",
"HMul.hMul",
"LE.le.trans_e... | [
"u v : ℕ → ℝ≥0∞\nn : ℕ\n⊢ ((fun n ↦ (u n).log / ↑n) + fun n ↦ (v n).log / ↑n) n = ((u * v) n).log / ↑n"
] | refine le_liminf_add.trans_eq (liminf_congr (Eventually.of_forall fun n ↦ ?_)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 155,
"column": 5
} | {
"line": 155,
"column": 56
} | {
"line": 155,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : CommRing β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\n⊢ (∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)) ↔ SuperpolynomialDecay l (fun a ↦ |k a|) fun a ↦ |f a|",
... | [] | by simp_rw [SuperpolynomialDecay, abs_mul, abs_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 274,
"column": 39
} | {
"line": 279,
"column": 49
} | {
"line": 281,
"column": 0
} | [
{
"pp": "u v : ℕ → ℝ≥0∞\nb : ℝ≥0∞\nhb : b ≠ 0\nh : ∀ᶠ (n : ℕ) in atTop, b * u n ≤ v n\n⊢ expGrowthInf u ≤ expGrowthInf v",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"ExpGrowth.expGrowthInf",
"Preorder.toLT",
"HMul.hMul",
"ExpGrow... | [] | by
apply (expGrowthInf_eventually_monotone h).trans' (le_expGrowthInf_mul.trans' _)
rcases eq_top_or_lt_top b with rfl | b_top
· rw [← Pi.top_def, expGrowthInf_top]
exact le_add_of_nonneg_left le_top
· rw [expGrowthInf_const hb b_top.ne, zero_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.AEMeasurableOrder | {
"line": 56,
"column": 2
} | {
"line": 58,
"column": 87
} | {
"line": 59,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns_count : s.Countable\ns_d... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns_count : s.Countable\ns_dense : Dense... | have u'_meas : ∀ i, MeasurableSet (u' i) := by
intro i
exact MeasurableSet.biInter (s_count.mono inter_subset_left) fun b _ => (huv i b).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 535,
"column": 4
} | {
"line": 535,
"column": 29
} | {
"line": 536,
"column": 4
} | [
{
"pp": "u : ℕ → EReal\nv : ℕ → ℕ\na : EReal\nh : Monotone u\nhv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)\nha : a ≠ 0\nha' : a ≠ ⊤\n⊢ 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMon... | [
"u : ℕ → EReal\nv : ℕ → ℕ\na : EReal\nh : Monotone u\nhv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)\nha : liminf (fun n ↦ ↑(v n) / ↑n) atTop ≠ 0\nha' : a ≠ ⊤\n⊢ 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop"
] | rw [← hv.liminf_eq] at ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.AEMeasurableOrder | {
"line": 42,
"column": 2
} | {
"line": 101,
"column": 26
} | {
"line": 103,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns_count : s.Countable\ns_d... | [] | classical
haveI : Encodable s := s_count.toEncodable
have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
intro p q
by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
· rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv,... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.Function.AEMeasurableOrder | {
"line": 42,
"column": 2
} | {
"line": 101,
"column": 26
} | {
"line": 103,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns_count : s.Countable\ns_d... | [] | classical
haveI : Encodable s := s_count.toEncodable
have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
intro p q
by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
· rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AEMeasurableOrder | {
"line": 42,
"column": 2
} | {
"line": 101,
"column": 26
} | {
"line": 103,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_2\ninst✝⁶ : CompleteLinearOrder β\ninst✝⁵ : DenselyOrdered β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : OrderTopology β\ninst✝² : SecondCountableTopology β\ninst✝¹ : MeasurableSpace β\ninst✝ : BorelSpace β\ns : Set β\ns_count : s.Countable\ns_d... | [] | classical
haveI : Encodable s := s_count.toEncodable
have h' : ∀ p q, ∃ u v, MeasurableSet u ∧ MeasurableSet v ∧
{ x | f x < p } ⊆ u ∧ { x | q < f x } ⊆ v ∧ (p ∈ s → q ∈ s → p < q → μ (u ∩ v) = 0) := by
intro p q
by_cases H : p ∈ s ∧ q ∈ s ∧ p < q
· rcases h p H.1 q H.2.1 H.2.2 with ⟨u, v, hu, hv,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 29
} | {
"line": 565,
"column": 4
} | [
{
"pp": "u : ℕ → EReal\nv : ℕ → ℕ\na : EReal\nh : Monotone u\nhv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)\nha : a ≠ 0\nha' : a ≠ ⊤\n⊢ 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMon... | [
"u : ℕ → EReal\nv : ℕ → ℕ\na : EReal\nh : Monotone u\nhv : Tendsto (fun n ↦ ↑(v n) / ↑n) atTop (𝓝 a)\nha : liminf (fun n ↦ ↑(v n) / ↑n) atTop ≠ 0\nha' : a ≠ ⊤\n⊢ 0 < liminf (fun n ↦ ↑(v n) / ↑n) atTop"
] | rw [← hv.liminf_eq] at ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Average | {
"line": 253,
"column": 2
} | {
"line": 254,
"column": 84
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhμ : IsFiniteMeasure (μ.restrict s)\nhμ0 : μ s = 0\n⊢ ⨍⁻ (x : α) in s, f x ∂μ ≤ essSup f μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"zero... | [
"case neg\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ≥0∞\nhμ : IsFiniteMeasure (μ.restrict s)\nhμ0 : ¬μ s = 0\n⊢ ⨍⁻ (x : α) in s, f x ∂μ ≤ essSup f μ"
] | · rw [laverage, ← setLIntegral_univ]
exact le_of_eq_of_le (setLIntegral_measure_zero univ f <| by simp [hμ0]) zero_le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 61
} | {
"line": 96,
"column": 4
} | [
{
"pp": "case a\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x ↦ μ t - ν t) ⋯ ⋯\nh_measure_sub_add : ν + measure_sub = μ\nd : Measure α\nh_d : d ∈ {d | μ ≤ d + ν}\n⊢ measure_sub ≤ d",... | [
"case a\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\ninst✝ : IsFiniteMeasure ν\nh₁ : MeasurableSet s\nh₂ : ν ≤ μ\nmeasure_sub : Measure α := ofMeasurable (fun t x ↦ μ t - ν t) ⋯ ⋯\nh_measure_sub_add : ν + measure_sub = μ\nd : Measure α\nh_d : ν + measure_sub ≤ ν + d\n⊢ measure_sub ≤ d"
] | rw [← h_measure_sub_add, mem_setOf_eq, add_comm d] at h_d | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Average | {
"line": 782,
"column": 66
} | {
"line": 782,
"column": 72
} | {
"line": 782,
"column": 73
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure α\ninst✝ : CompleteSpace E\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int :... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure α\ninst✝ : CompleteSpace E\nι : Type u_4\na : ι → Set α\nl : Filter ι\nf : α → E\nc : E\ng : ι → α → ℝ\nK : ℝ\nhf : Tendsto (fun i ↦ ⨍ (y : α) in a i, ‖f y - c‖ ∂μ) l (𝓝 0)\nf_int : ∀ᶠ (i : ι) ... | hisupp | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 294,
"column": 2
} | {
"line": 296,
"column": 78
} | {
"line": 298,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\np q : ℝ≥0\nhpq : p < q\ns : Set α := {x | ∃ c... | [
"case refine_1\nα : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\np q : ℝ≥0\nhpq : p < q\ns : Set α := {x | ... | refine
⟨toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (u n),
toMeasurable μ sᶜ ∪ ⋃ n, toMeasurable (ρ + μ) (w n), ?_, ?_, ?_, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 889,
"column": 6
} | {
"line": 889,
"column": 21
} | {
"line": 890,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f ↦ ∫⁻ (... | [] | exact hξle B hB | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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