module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicTopology.SimplicialNerve | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 65
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case inl\nJ : Type u_1\ninst✝ : LinearOrder J\ni j k : SimplicialThickening J\nf : Path i.as j.as\ng : Path j.as k.as\nl : J\nh : l ∈ f.I\n⊢ i.as ≤ l",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"CategoryTheory.SimplicialThickening.Path.left_le",
"CategoryTheory.Simp... | [] | exacts [(f.left_le l h), (Path.le f).trans (g.left_le l h)] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.CategoryTheory.Presentable.Limits | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 58
} | {
"line": 124,
"column": 6
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\nK : Type u'\ninst✝³ : Category.{v', u'} K\nF : K ⥤ C ⥤ Type w'\nc : Cone F\nhc : (Y : C) → IsLimit (((evaluation C (Type w')).obj Y).mapCone c)\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\nhK : HasCardinalLT (Arrow K) κ\nJ : Type w\ninst✝¹ : SmallCategory J\nins... | [
"C : Type u\ninst✝⁴ : Category.{v, u} C\nK : Type u'\ninst✝³ : Category.{v', u'} K\nF : K ⥤ C ⥤ Type w'\nc : Cone F\nhc : (Y : C) → IsLimit (((evaluation C (Type w')).obj Y).mapCone c)\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\nhK : HasCardinalLT (Arrow K) κ\nJ : Type w\ninst✝¹ : SmallCategory J\ninst✝ : IsCardi... | simp only [y₁, y₂, Types.isLimitEquivSections_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.SmallRepresentatives | {
"line": 154,
"column": 71
} | {
"line": 163,
"column": 47
} | {
"line": 165,
"column": 0
} | [
{
"pp": "Ω : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nh₁ : Cardinal.lift.{w, u} (Cardinal.mk C) ≤ Cardinal.lift.{u, w} (Cardinal.mk Ω)\nh₂ : ∀ (X Y : C), Cardinal.lift.{w, v} (Cardinal.mk (X ⟶ Y)) ≤ Cardinal.lift.{v, w} (Cardinal.mk Ω)\n⊢ ∃ h, Nonempty (categoryFamily Ω h ≌ C)",
"ppTerm": "?m.10",
... | [] | by
let f₁ := (Cardinal.lift_mk_le'.1 h₁).some
let f₂ (X Y) := (Cardinal.lift_mk_le'.1 (h₂ X Y)).some
let e := Equiv.ofInjective _ f₁.injective
let h : CoreSmallCategoryOfSet Ω C :=
{ obj := Set.range f₁
hom X Y := Set.range (f₂ (e.symm X) (e.symm Y))
objEquiv := e.symm
homEquiv {_ _} := by... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 493,
"column": 6
} | {
"line": 493,
"column": 45
} | {
"line": 494,
"column": 6
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝³ : LinearOrder ι\nf : P.RankFunction ι\ninst✝² : P.IsProper\ninst✝¹ : SuccOrder ι\ninst✝ : NoMaxOrder ι\nj : ι\nx✝ : SimplexCategoryᵒᵖ\nd : ℕ\nx : f.Cell j\nb : Δ[x.dim + 1] _⦋d⦌\ny : X _⦋d⦌\nhy : y ∈ (f.filtration j).obj (op ⦋d⦌)\nh :\n ⟨↑⟨... | [
"X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝³ : LinearOrder ι\nf : P.RankFunction ι\ninst✝² : P.IsProper\ninst✝¹ : SuccOrder ι\ninst✝ : NoMaxOrder ι\nj : ι\nx✝ : SimplexCategoryᵒᵖ\nd : ℕ\nx : f.Cell j\nb : Δ[x.dim + 1] _⦋d⦌\ny : X _⦋d⦌\nhy : y ∈ (f.filtration j).obj (op ⦋d⦌)\nh :\n ⟨↑⟨y, hy⟩, ⋯⟩ =... | simp only [← x.preimage_filtration_map] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.Basic | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 45
} | {
"line": 67,
"column": 0
} | [
{
"pp": "⊢ modelCategoryQuillen.J.rlp ≤ innerHornInclusions.rlp",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Opposite",
"SSet.modelCategoryQuillen.J",
"CategoryTheory.Functor.category",
"SSet.innerHornInclusions_le_J",
"SSet",
"CategoryTheory.Morphis... | [] | exact antitone_rlp innerHornInclusions_le_J | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialObject.ChainHomotopy | {
"line": 169,
"column": 6
} | {
"line": 169,
"column": 40
} | {
"line": 171,
"column": 0
} | [
{
"pp": "case succ\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH✝ H : Homotopy f g\nn : ℕ\n⊢ ((alternatingFaceMapComplex C).map f).f (n + 1) =\n ((alternatingFaceMapComplex C).obj X).d (n + 1) n ≫ ToChainHomotopy.hom H n (n + 1) +\n ToChainHom... | [] | simp [ToChainHomotopy.comm_succ H] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.LiftingProperties.ParametrizedAdjunction | {
"line": 72,
"column": 10
} | {
"line": 74,
"column": 48
} | {
"line": 74,
"column": 48
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝² : Category.{v₁, u₁} C₁\ninst✝¹ : Category.{v₂, u₂} C₂\ninst✝ : Category.{v₃, u₃} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj₂ : F ⊣₂ G\nX₁ Y₁ : C₁\nf₁ : X₁ ⟶ Y₁\nX₂ Y₂ : C₂\nf₂ : X₂ ⟶ Y₂\nX₃ Y₃ : C₃\nf₃ : X₃ ⟶ Y₃\nsq₁₂ : F.PushoutObjObj f₁ f₂\nsq₁₃ : G... | [] | simp only [← adj₂.homEquiv_symm_naturality_one,
← adj₂.homEquiv_symm_naturality_two,
Arrow.mk_left, Arrow.mk_right, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 246,
"column": 8
} | {
"line": 246,
"column": 63
} | {
"line": 247,
"column": 6
} | [
{
"pp": "case refine_2.inl\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nd : ℕ\nhd : x.dim = d + 1\nl : Fin (d + 1)\nhl : IsIndex x hd l.succ\nj : Fin (m + 1 + 1)\nhj : ¬j = k.castSucc\nhi : (x.cast hd).simplex.1 l.castSucc = j\n⊢ ∃ x_1, (x.cast hd).simplex.1 (l.castSucc.succAbov... | [] | exact (hj (by rw [← hi, hl.simplex_fst_castSucc])).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 246,
"column": 8
} | {
"line": 246,
"column": 63
} | {
"line": 247,
"column": 6
} | [
{
"pp": "case refine_2.inl\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nd : ℕ\nhd : x.dim = d + 1\nl : Fin (d + 1)\nhl : IsIndex x hd l.succ\nj : Fin (m + 1 + 1)\nhj : ¬j = k.castSucc\nhi : (x.cast hd).simplex.1 l.castSucc = j\n⊢ ∃ x_1, (x.cast hd).simplex.1 (l.castSucc.succAbov... | [] | exact (hj (by rw [← hi, hl.simplex_fst_castSucc])).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 246,
"column": 8
} | {
"line": 246,
"column": 63
} | {
"line": 247,
"column": 6
} | [
{
"pp": "case refine_2.inl\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nd : ℕ\nhd : x.dim = d + 1\nl : Fin (d + 1)\nhl : IsIndex x hd l.succ\nj : Fin (m + 1 + 1)\nhj : ¬j = k.castSucc\nhi : (x.cast hd).simplex.1 l.castSucc = j\n⊢ ∃ x_1, (x.cast hd).simplex.1 (l.castSucc.succAbov... | [] | exact (hj (by rw [← hi, hl.simplex_fst_castSucc])).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Adjunction.ParametrizedLimits | {
"line": 49,
"column": 4
} | {
"line": 60,
"column": 76
} | {
"line": 60,
"column": 76
} | [
{
"pp": "C₁ : Type u_1\nC₂ : Type u_2\nC₃ : Type u_3\ninst✝⁴ : Category.{v_1, u_1} C₁\ninst✝³ : Category.{v_2, u_2} C₂\ninst✝² : Category.{v_3, u_3} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj₂ : F ⊣₂ G\nJ : Type u_4\ninst✝¹ : Category.{v_4, u_4} J\nP : J ⥤ C₁ᵒᵖ\ninst✝ : ∀ (X₂ : C₂), PreservesColimit P.leftOp... | [] | exact {
lift s := adj₂.homEquiv ((hc' s).desc (cocone s))
fac s j := by
dsimp
rw [← dsimp% adj₂.homEquiv_naturality_one (c.π.app j).unop,
dsimp% (hc' s).fac (cocone s) (op j)]
simp [cocone]
uniq s m hm := adj₂.homEquiv.symm.injective (by
simp only [op_unop, Eq... | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct | {
"line": 80,
"column": 6
} | {
"line": 82,
"column": 40
} | {
"line": 82,
"column": 40
} | [
{
"pp": "X₁ Y₁ E✝ B✝ : SSet\ni : X₁ ⟶ Y₁\np✝ : E✝ ⟶ B✝\ninst✝¹ : Mono i\ninst✝ : InnerFibration p✝\nsq₁₃ : internalHom.PullbackObjObj i p✝\nX✝ Y✝ : SSet\ng✝ : X✝ ⟶ Y✝\nn✝ : ℕ\nk : Fin (n✝ + 3)\nh0 : 0 < k\nhn : k < Fin.last (n✝ + 2)\nsq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[n✝ + 2, k].ι :=\n Functor.Pusho... | [
"X₁ Y₁ E✝ B✝ : SSet\ni : X₁ ⟶ Y₁\np✝ : E✝ ⟶ B✝\ninst✝¹ : Mono i\ninst✝ : InnerFibration p✝\nsq₁₃ : internalHom.PullbackObjObj i p✝\nX✝ Y✝ : SSet\ng✝ : X✝ ⟶ Y✝\nn✝ : ℕ\nk : Fin (n✝ + 3)\nh0 : 0 < k\nhn : k < Fin.last (n✝ + 2)\nsq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[n✝ + 2, k].ι :=\n Functor.PushoutObjObj.ofH... | HasLiftingProperty.iff_of_arrow_iso_left
(show Arrow.mk sq₁₂.ι ≅ Arrow.mk sq₁₂.flipTensor.ι from
Arrow.isoMk (Iso.refl _) (β_ _ _)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct | {
"line": 82,
"column": 6
} | {
"line": 84,
"column": 40
} | {
"line": 84,
"column": 40
} | [
{
"pp": "X₁ Y₁ E✝ B✝ : SSet\ni : X₁ ⟶ Y₁\np✝ : E✝ ⟶ B✝\ninst✝¹ : Mono i\ninst✝ : Fibration p✝\nsq₁₃ : internalHom.PullbackObjObj i p✝\nm : ℕ\nX Y : SSet\nk : Fin (m + 2)\nsq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=\n Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι\nE B ... | [
"X₁ Y₁ E✝ B✝ : SSet\ni : X₁ ⟶ Y₁\np✝ : E✝ ⟶ B✝\ninst✝¹ : Mono i\ninst✝ : Fibration p✝\nsq₁₃ : internalHom.PullbackObjObj i p✝\nm : ℕ\nX Y : SSet\nk : Fin (m + 2)\nsq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=\n Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι\nE B : SSet\np : ... | HasLiftingProperty.iff_of_arrow_iso_left
(show Arrow.mk sq₁₂.ι ≅ Arrow.mk sq₁₂.flipTensor.ι from
Arrow.isoMk (Iso.refl _) (β_ _ _)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.CoherentIso | {
"line": 126,
"column": 6
} | {
"line": 127,
"column": 17
} | {
"line": 127,
"column": 18
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nF : WalkingIso ⥤ C\nX Y : WalkingIso\nf : X ⟶ Y\n⊢ ((fun p ↦ fromIso p.snd.snd) ((fun F ↦ ⟨F.obj zero, ⟨F.obj one, toIso F⟩⟩) F)).map f = eqToHom ⋯ ≫ F.map f ≫ eqToHom ⋯",
"ppTerm": "?m.101",
"assigned": true,
"usedConstants": [
"CategoryTheory.F... | [
"case zero.zero\nC : Type u\ninst✝ : Category.{v, u} C\nF : WalkingIso ⥤ C\nf : zero ⟶ zero\n⊢ ((fun p ↦ fromIso p.snd.snd) ((fun F ↦ ⟨F.obj zero, ⟨F.obj one, toIso F⟩⟩) F)).map f = eqToHom ⋯ ≫ F.map f ≫ eqToHom ⋯",
"case zero.one\nC : Type u\ninst✝ : Category.{v, u} C\nF : WalkingIso ⥤ C\nf : zero ⟶ one\n⊢ ((fun... | induction X <;>
induction Y | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 487,
"column": 6
} | {
"line": 487,
"column": 22
} | {
"line": 488,
"column": 4
} | [
{
"pp": "case inr.inr.inl\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx u : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nhu : IsType₂ u\nhd : x.dim = u.dim + 1\nl : Fin (u.dim + 1)\nhl : IsIndex x hd l.succ\nhu' : u.simplex = (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) l.succ)) (x.cast hd).simplex\nhi : l.cast... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 487,
"column": 6
} | {
"line": 487,
"column": 22
} | {
"line": 488,
"column": 4
} | [
{
"pp": "case inr.inr.inl\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx u : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nhu : IsType₂ u\nhd : x.dim = u.dim + 1\nl : Fin (u.dim + 1)\nhl : IsIndex x hd l.succ\nhu' : u.simplex = (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) l.succ)) (x.cast hd).simplex\nhi : l.cast... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 487,
"column": 6
} | {
"line": 487,
"column": 22
} | {
"line": 488,
"column": 4
} | [
{
"pp": "case inr.inr.inl\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx u : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nhu : IsType₂ u\nhd : x.dim = u.dim + 1\nl : Fin (u.dim + 1)\nhl : IsIndex x hd l.succ\nhu' : u.simplex = (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) l.succ)) (x.cast hd).simplex\nhi : l.cast... | [] | exact Or.inr rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 519,
"column": 6
} | {
"line": 519,
"column": 55
} | {
"line": 520,
"column": 6
} | [
{
"pp": "case pos.snd\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\ns : Type₁ k n\nht : IsType₂ s.δ\n⊢ (s.x.cast ⋯).simplex.2 (s.index.castSucc.succAbove (min s.δ ⋯)) = ((objEquiv (s.x.cast ⋯).simplex) s.index.castSucc).2",
"ppTerm": "?pos.snd✝",
"assigned": true,
"usedConstants": [
"SSet.S.simplex",
... | [
"case pos.snd\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\ns : Type₁ k n\nht : IsType₂ s.δ\n⊢ (s.x.cast ⋯).simplex.2 s.index.succ = ((objEquiv (s.x.cast ⋯).simplex) s.index.castSucc).2"
] | rw [s.isIndex.min_δ, Fin.succAbove_castSucc_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Homology.Nondegenerate | {
"line": 256,
"column": 31
} | {
"line": 258,
"column": 54
} | {
"line": 260,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasCoproducts C\ninst✝² : Preadditive C\nX : SSet\nR : C\ninst✝¹ : CategoryWithHomology C\nn d : ℕ\ninst✝ : X.HasDimensionLT d\nh : d ≤ n\n⊢ IsZero (X.homology R n)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"HomologicalC... | [] | by
rw [← exactAt_iff_isZero_homology]
exact X.exactAt_chainComplex_of_hasDimensionLT R n d | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 683,
"column": 18
} | {
"line": 683,
"column": 43
} | {
"line": 684,
"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 + 2)\nn : ℕ\nhk : k = Fin.last (m + 1)\n⊢ (Λ[m + 1, 0].unionProd ∂Δ[n]).op.preimage\n (((stdSimplex.opIso ⦋m + 1⦌).inv ⊗ₘ (stdSimplex.opIso ⦋n⦌).inv) ≫ Functor.LaxMonoidal.μ opFunctor Δ[m + 1] Δ... | [
"m✝ : ℕ\nk✝ : Fin (m✝ + 1)\nn✝ : ℕ\nx : (Λ[m✝ + 1, k✝.castSucc].unionProd ∂Δ[n✝]).N\nd m : ℕ\nk : Fin (m + 2)\nn : ℕ\nhk : k = Fin.last (m + 1)\n⊢ ((Λ[m + 1, 0].unionProd ∂Δ[n]).op.preimage (Functor.LaxMonoidal.μ opFunctor Δ[m + 1] Δ[n])).preimage\n ((stdSimplex.opIso ⦋m + 1⦌).inv ⊗ₘ (stdSimplex.opIso ⦋n⦌).inv... | Subcomplex.preimage_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackObjObj | {
"line": 612,
"column": 29
} | {
"line": 615,
"column": 94
} | {
"line": 617,
"column": 0
} | [
{
"pp": "C₁ : Type u₁\nC₂ : Type u₂\nC₃ : Type u₃\ninst✝⁴ : Category.{v₁, u₁} C₁\ninst✝³ : Category.{v₂, u₂} C₂\ninst✝² : Category.{v₃, u₃} C₃\nF : C₁ ⥤ C₂ ⥤ C₃\nG : C₁ᵒᵖ ⥤ C₃ ⥤ C₂\nadj₂ : F ⊣₂ G\ninst✝¹ : HasPullbacks C₂\ninst✝ : HasPushouts C₃\nX₁✝ Y₁✝ : Arrow C₁\nx✝ : X₁✝ ⟶ Y₁✝\n⊢ (LeibnizAdjunction.adj F G ... | [] | by
ext
· simp [← homEquiv_naturality_one, ← homEquiv_naturality_three]
· apply pullback.hom_ext <;> simp [← homEquiv_naturality_one, ← homEquiv_naturality_three] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.CountableCover | {
"line": 65,
"column": 23
} | {
"line": 65,
"column": 41
} | {
"line": 66,
"column": 6
} | [
{
"pp": "α ι : Type u\na : Cardinal.{u}\ninst✝¹ : Countable ι\nf : ι → Set α\nl : Filter ι\ninst✝ : l.NeBot\nt : Set α\nht : ∀ x ∈ t, ∀ᶠ (i : ι) in l, x ∈ f i\nh'f : ∀ (i : ι), #↑(f i) ≤ a\nha : ℵ₀ ≤ a\nthis : t ⊆ ⋃ i, f i\n⊢ #ι * a ≤ ℵ₀ * a",
"ppTerm": "?m.385",
"assigned": true,
"usedConstants": [... | [] | grw [mk_le_aleph0] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.SetTheory.Cardinal.CountableCover | {
"line": 65,
"column": 23
} | {
"line": 65,
"column": 41
} | {
"line": 66,
"column": 6
} | [
{
"pp": "α ι : Type u\na : Cardinal.{u}\ninst✝¹ : Countable ι\nf : ι → Set α\nl : Filter ι\ninst✝ : l.NeBot\nt : Set α\nht : ∀ x ∈ t, ∀ᶠ (i : ι) in l, x ∈ f i\nh'f : ∀ (i : ι), #↑(f i) ≤ a\nha : ℵ₀ ≤ a\nthis : t ⊆ ⋃ i, f i\n⊢ #ι * a ≤ ℵ₀ * a",
"ppTerm": "?m.385",
"assigned": true,
"usedConstants": [... | [] | grw [mk_le_aleph0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.CountableCover | {
"line": 65,
"column": 23
} | {
"line": 65,
"column": 41
} | {
"line": 66,
"column": 6
} | [
{
"pp": "α ι : Type u\na : Cardinal.{u}\ninst✝¹ : Countable ι\nf : ι → Set α\nl : Filter ι\ninst✝ : l.NeBot\nt : Set α\nht : ∀ x ∈ t, ∀ᶠ (i : ι) in l, x ∈ f i\nh'f : ∀ (i : ι), #↑(f i) ≤ a\nha : ℵ₀ ≤ a\nthis : t ⊆ ⋃ i, f i\n⊢ #ι * a ≤ ℵ₀ * a",
"ppTerm": "?m.385",
"assigned": true,
"usedConstants": [... | [] | grw [mk_le_aleph0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 63
} | {
"line": 36,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\nr : ENNReal\nhr_pos : 0 < r\nh : AnalyticOnNhd 𝕜 f (Metric.eball x r)\np : FormalMultilinearSeries 𝕜 𝕜 𝕜 := FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial\nhr : r ≤ p.radius\ng : 𝕜 → 𝕜 := fun t ↦ p.sum (t - x)... | [] | simpa using (p.hasFPowerSeriesOnBall (by order)).comp_sub x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 131,
"column": 4
} | {
"line": 132,
"column": 42
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case «0»\nX Y : Truncated 2\nf₀ : X.obj (op { obj := ⦋0⦌, property := _proof_11 }) → Y.obj (op { obj := ⦋0⦌, property := _proof_11 })\nf₁ : X.obj (op { obj := ⦋1⦌, property := _proof_12 }) → Y.obj (op { obj := ⦋1⦌, property := _proof_12 })\nhδ₁ :\n ∀ (x : X.obj (op { obj := ⦋1⦌, property := _proof_12 ... | [] | simp [StrictSegal.spineEquiv, SimplexCategory.mkOfSucc_zero_eq_δ,
← Functor.map_comp_apply, ← op_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 98
} | {
"line": 150,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E → F\ninst✝ : CompleteSpace E\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearO... | [
"case inr\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : E → F\ninst✝ : CompleteSpace E\ns : Set E\nc : ℝ≥0\nf' : E →L[𝕜] F\nhf : ApproximatesLinearOn f f' s c\n... | have If' : (0 : ℝ) < f'symm.nnnorm := by rw [← inv_pos]; exact (NNReal.coe_nonneg _).trans_lt hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 254,
"column": 10
} | {
"line": 256,
"column": 31
} | {
"line": 257,
"column": 6
} | [
{
"pp": "case «1»\nX : Truncated 2\nC D : Type u\ninst✝¹ : SmallCategory C\ninst✝ : SmallCategory D\nF : X.HomotopyCategory ⥤ C\ny : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_14 })\nh' :\n ∀ {a b : X.obj (op { obj := ⦋0⦌, property := ⋯ })} (e : Edge a b),\n ComposableArrows.mk₁... | [] | dsimp
simp only [SimplexCategory.mkOfSucc_one_eq_δ, ← h₀]
apply nerve.δ₀_mk₂_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 254,
"column": 10
} | {
"line": 256,
"column": 31
} | {
"line": 257,
"column": 6
} | [
{
"pp": "case «1»\nX : Truncated 2\nC D : Type u\ninst✝¹ : SmallCategory C\ninst✝ : SmallCategory D\nF : X.HomotopyCategory ⥤ C\ny : ((truncation 2).obj (nerve C)).obj (op { obj := ⦋2⦌, property := _proof_14 })\nh' :\n ∀ {a b : X.obj (op { obj := ⦋0⦌, property := ⋯ })} (e : Edge a b),\n ComposableArrows.mk₁... | [] | dsimp
simp only [SimplexCategory.mkOfSucc_one_eq_δ, ← h₀]
apply nerve.δ₀_mk₂_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 45
} | {
"line": 398,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nε : ℝ\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f... | [] | rwa [f'.toEquiv.subsingleton_congr] at hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 45
} | {
"line": 398,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nε : ℝ\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f... | [] | rwa [f'.toEquiv.subsingleton_congr] at hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 45
} | {
"line": 398,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nε : ℝ\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f... | [] | rwa [f'.toEquiv.subsingleton_congr] at hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 396,
"column": 64
} | {
"line": 397,
"column": 45
} | {
"line": 398,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nε : ℝ\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f... | [] | by
rwa [f'.toEquiv.subsingleton_congr] at hc | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 39
} | {
"line": 152,
"column": 39
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝² : RCLike 𝕂\ninst✝¹ : NormedDivisionRing 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔸\na c : 𝕂\nk : ℕ\n⊢ (ordinaryHypergeometricSeries 𝔸 a (-↑k) c).radius = ⊤",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | [
"𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝² : RCLike 𝕂\ninst✝¹ : NormedDivisionRing 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔸\na c : 𝕂\nk : ℕ\n⊢ (ordinaryHypergeometricSeries 𝔸 (-↑k) a c).radius = ⊤"
] | ordinaryHypergeometricSeries_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Extend | {
"line": 44,
"column": 34
} | {
"line": 44,
"column": 78
} | {
"line": 45,
"column": 2
} | [
{
"pp": "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.toTo... | [] | exact HasFDerivWithinAt.of_notMem_closure hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric | {
"line": 180,
"column": 2
} | {
"line": 195,
"column": 48
} | {
"line": 197,
"column": 0
} | [
{
"pp": "𝕂 : Type u_1\ninst✝ : RCLike 𝕂\na b c : 𝕂\nn : ℕ\nhabc : ∀ kn < n, ↑kn ≠ -a ∧ ↑kn ≠ -b ∧ ↑kn ≠ -c\n⊢ ‖↑n !‖⁻¹ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ *\n ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ /\n (‖↑n !‖⁻¹ * ‖↑n + 1‖⁻¹ * (‖Polynomial.eva... | [] | calc
_ = ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval a (ascPochhammer 𝕂 n)‖⁻¹ *
‖Polynomial.eval b (ascPochhammer 𝕂 n)‖ * ‖Polynomial.eval b (ascPochhammer 𝕂 n)‖⁻¹ *
‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹⁻¹ * ‖Polynomial.eval c (ascPochhammer 𝕂 n)‖⁻¹ *
‖(n ! : 𝕂)‖⁻¹⁻... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Analytic.Order | {
"line": 605,
"column": 8
} | {
"line": 605,
"column": 18
} | {
"line": 606,
"column": 8
} | [
{
"pp": "case h.left\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nz : ↑U\nhz : z ∈ {u | analyticOrderAt f ↑u = ⊤}ᶜ\nh : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t'... | [
"case h.left\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nz : ↑U\nhz : z ∈ {u | analyticOrderAt f ↑u = ⊤}ᶜ\nh : ∀ᶠ (z : 𝕜) in 𝓝[≠] ↑z, f z ≠ 0\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', y ∈ {↑z}ᶜ ... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.Order | {
"line": 653,
"column": 89
} | {
"line": 653,
"column": 91
} | {
"line": 654,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = 0\na : 𝕜\n⊢ (∀ᶠ (x : 𝕜) in 𝓝 a, f x = 0) → a ∈ (U \\ Subtype.val '' {u | ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = 0\na : 𝕜\nha : ∀ᶠ (x : 𝕜) in 𝓝 a, f x = 0\n⊢ a ∈ (U \\ Subtype.val '' {u | analyticOr... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Order | {
"line": 655,
"column": 32
} | {
"line": 655,
"column": 34
} | {
"line": 656,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z ≠ 0\na : 𝕜\n⊢ f a ≠ 0 → a ∈ (U \\ Subtype.val '' {u | analyticOrderAt f ↑u... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z ≠ 0\na : 𝕜\nha : f a ≠ 0\n⊢ a ∈ (U \\ Subtype.val '' {u | analyticOrderAt f ↑u = 0 ∨ a... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Order | {
"line": 671,
"column": 89
} | {
"line": 671,
"column": 91
} | {
"line": 672,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = 0\na : 𝕜\n⊢ (∀ᶠ (x : 𝕜) in 𝓝 a, f x = 0) → a ∈ (U \\ {u | analyticOrderAt... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝 x, f z = 0\na : 𝕜\nha : ∀ᶠ (x : 𝕜) in 𝓝 a, f x = 0\n⊢ a ∈ (U \\ {u | analyticOrderAt f u = 0 ∨... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Order | {
"line": 673,
"column": 32
} | {
"line": 673,
"column": 34
} | {
"line": 674,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z ≠ 0\na : 𝕜\n⊢ f a ≠ 0 → a ∈ (U \\ {u | analyticOrderAt f u = 0 ∨ analyticO... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nU : Set 𝕜\nf : 𝕜 → E\nhf : AnalyticOnNhd 𝕜 f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z ≠ 0\na : 𝕜\nha : f a ≠ 0\n⊢ a ∈ (U \\ {u | analyticOrderAt f u = 0 ∨ analyticOrderAt f... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 508,
"column": 34
} | {
"line": 508,
"column": 36
} | {
"line": 509,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nh₂ : EqOn f g Uᶜ\nx : 𝕜\nhx : x ∈ U\nh₁ : ∀ x ∈ U, (U \\ {x | f x = g x})ᶜ ∈ 𝓝[≠] x\na : 𝕜\n⊢ a ∈ (U \\ {x | f x = g x})ᶜ → f a... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nh₂ : EqOn f g Uᶜ\nx : 𝕜\nhx : x ∈ U\nh₁ : ∀ x ∈ U, (U \\ {x | f x = g x})ᶜ ∈ 𝓝[≠] x\na : 𝕜\nha : a ∈ (U \\ {x | f x = g x})ᶜ\n⊢ f a = g a"
... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Binomial | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 14
} | {
"line": 187,
"column": 4
} | [
{
"pp": "case convert_2\na : ℕ\nz : ℂ\nhz : z ≠ 0\nthis✝ :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) ((z⁻¹ • 1) 0) 1\nH : 1 / ‖z⁻¹ • 1‖ₑ = ‖z‖ₑ\nthis :\n HasFPowerSeriesOnBall (fun x ↦ ((1 - (z⁻¹ • ⇑1) x) ^ (a + 1))⁻¹)\n (... | [
"case convert_2\na : ℕ\nz : ℂ\nhz : z ≠ 0\nthis✝ :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ (a + 1))\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ ↑((a + n).choose a)) ((z⁻¹ • 1) 0) 1\nH : 1 / ‖z⁻¹ • 1‖ₑ = ‖z‖ₑ\nthis :\n HasFPowerSeriesOnBall (fun x ↦ ((1 - (z⁻¹ • ⇑1) x) ^ (a + 1))⁻¹)\n ((FormalMulti... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.Binomial | {
"line": 175,
"column": 94
} | {
"line": 187,
"column": 68
} | {
"line": 189,
"column": 0
} | [
{
"pp": "a : ℕ\nz : ℂ\nhz : z ≠ 0\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (z - x) ^ (a + 1))\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (z ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)) 0 ‖z‖ₑ",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"HasFPowerSeriesOnBall.congr",
"enorm_s... | [] | by
have := one_div_one_sub_pow_hasFPowerSeriesOnBall_zero a
rw [← map_zero (z⁻¹ • 1 : ℂ →L[ℂ] ℂ)] at this
have := this.compContinuousLinearMap
have H : 1 / ‖(z⁻¹ • 1 : ℂ →L[ℂ] ℂ)‖ₑ = ‖z‖ₑ := by simp [enorm_smul, enorm_inv, hz]
simp only [one_div, FunLike.coe_smul, H, Function.comp_def] at this
convert (this... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Binomial | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 91
} | {
"line": 240,
"column": 2
} | [
{
"pp": "a : ℝ\nH : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)\nthis : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1\n⊢ HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ a) (binomialSeries ℝ a) 0 1",
"ppTerm": "?m.156",
"assign... | [
"case e'_10\na : ℝ\nH : binomialSeries ℂ a = FormalMultilinearSeries.restrictScalars ℝ (binomialSeries ℂ ↑a)\nthis : HasFPowerSeriesOnBall (fun x ↦ (1 + x) ^ ↑a) (binomialSeries ℂ a) (Complex.ofRealCLM 0) 1\n⊢ binomialSeries ℝ a =\n Complex.reCLM.compFormalMultilinearSeries ((binomialSeries ℂ a).compContinuousLi... | convert! (Complex.reCLM.comp_hasFPowerSeriesOnBall this.compContinuousLinearMap).congr ?_ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 40
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case inr\nu v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v + 0 = linearGrowthInf v",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"AddMono... | [] | exact add_zero (linearGrowthInf v) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 40
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case inr\nu v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v + 0 = linearGrowthInf v",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"AddMono... | [] | exact add_zero (linearGrowthInf v) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 40
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case inr\nu v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthInf v + 0 = linearGrowthInf v",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"AddMono... | [] | exact add_zero (linearGrowthInf v) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 167,
"column": 32
} | {
"line": 167,
"column": 55
} | {
"line": 168,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : CommRing β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\nz : ℕ\nx : α\nhx : (abs ∘ ... | [] | gcongr _ * ?_; exact hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.SuperpolynomialDecay | {
"line": 167,
"column": 32
} | {
"line": 167,
"column": 55
} | {
"line": 168,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nl : Filter α\nk f g : α → β\ninst✝⁴ : TopologicalSpace β\ninst✝³ : CommRing β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : OrderTopology β\nhf : ∀ (n : ℕ), Tendsto (fun a ↦ |k a ^ n * f a|) l (𝓝 0)\nhfg : abs ∘ g ≤ᶠ[l] abs ∘ f\nz : ℕ\nx : α\nhx : (abs ∘ ... | [] | gcongr _ * ?_; exact hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.VitaliFamily | {
"line": 200,
"column": 8
} | {
"line": 201,
"column": 72
} | {
"line": 202,
"column": 4
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝ : PseudoMetricSpace X\nm0 : MeasurableSpace X\nμ : Measure X\nv : VitaliFamily μ\nδ : ℝ\nδpos : 0 < δ\ns : Set X\nf : X → Set (Set X)\nfset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}\nffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f ... | [] | refine False.elim (h't.2.2 ?_)
exact ht.trans (closedBall_subset_closedBall (min_le_right _ _)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.VitaliFamily | {
"line": 200,
"column": 8
} | {
"line": 201,
"column": 72
} | {
"line": 202,
"column": 4
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝ : PseudoMetricSpace X\nm0 : MeasurableSpace X\nμ : Measure X\nv : VitaliFamily μ\nδ : ℝ\nδpos : 0 < δ\ns : Set X\nf : X → Set (Set X)\nfset : ∀ x ∈ s, f x ⊆ v.setsAt x ∪ {s | MeasurableSet s ∧ (interior s).Nonempty ∧ ¬s ⊆ closedBall x δ}\nffine : ∀ x ∈ s, ∀ ε > 0, ∃ t ∈ f ... | [] | refine False.elim (h't.2.2 ?_)
exact ht.trans (closedBall_subset_closedBall (min_le_right _ _)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.VitaliFamily | {
"line": 245,
"column": 61
} | {
"line": 245,
"column": 63
} | {
"line": 245,
"column": 64
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoMetricSpace X\nm0 : MeasurableSpace X\nμ : Measure X\nv : VitaliFamily μ\nx : X\na✝ : Set X\n⊢ a✝ ∈ v.setsAt x → MeasurableSet a✝",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Membership.mem",
"VitaliFamily.setsAt",
"Set.instMembers... | [
"X : Type u_1\ninst✝ : PseudoMetricSpace X\nm0 : MeasurableSpace X\nμ : Measure X\nv : VitaliFamily μ\nx : X\na✝ : Set X\nha : a✝ ∈ v.setsAt x\n⊢ MeasurableSet a✝"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 358,
"column": 57
} | {
"line": 358,
"column": 59
} | {
"line": 358,
"column": 59
} | [
{
"pp": "case insert\nα : Type u_1\nu : α → ℕ → ℝ≥0∞\na : α\nt : Finset α\na_t : a ∉ t\nha : expGrowthSup (∑ x ∈ t, u x) = ⨆ x ∈ t, expGrowthSup (u x)\n⊢ max (expGrowthSup (u a)) (expGrowthSup (∑ x ∈ t, u x)) = max (expGrowthSup (u a)) (⨆ x ∈ t, expGrowthSup (u x))",
"ppTerm": "?insert",
"assigned": tru... | [
"case insert\nα : Type u_1\nu : α → ℕ → ℝ≥0∞\na : α\nt : Finset α\na_t : a ∉ t\nha : expGrowthSup (∑ x ∈ t, u x) = ⨆ x ∈ t, expGrowthSup (u x)\n⊢ max (expGrowthSup (u a)) (⨆ x ∈ t, expGrowthSup (u x)) = max (expGrowthSup (u a)) (⨆ x ∈ t, expGrowthSup (u x))"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 176,
"column": 12
} | {
"line": 176,
"column": 14
} | {
"line": 177,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)\nhu... | [
"α : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 ≤ r a\nt' : Set ι := {a | a ∈ t ∧ 0 ≤ r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ closedBall (x a) (r a)\nhu : ∀ a ∈ t',... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 213,
"column": 12
} | {
"line": 213,
"column": 14
} | {
"line": 214,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 < r a\nt' : Set ι := {a | a ∈ t ∧ 0 < r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)\nhu : ∀ a... | [
"α : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 < r a\nt' : Set ι := {a | a ∈ t ∧ 0 < r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (r a)\nhu : ∀ a ∈ t', ∃ b ∈... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.AEEqOfLIntegral | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 52
} | {
"line": 89,
"column": 6
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nh : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) ... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nh : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∫⁻ (x : α) in s, f x ∂μ ≤ ∫⁻ (x : α) in s, g x ∂μ\nA : ∀ (ε N : ℝ≥0) (p : ℕ), 0 < ε → μ ({x | g x + ↑ε ≤ f x ∧ g x ≤ ↑N} ∩ spanningSets μ p) = 0\nu : ℕ →... | simp only [ENNReal.coe_zero, add_zero] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Average | {
"line": 254,
"column": 4
} | {
"line": 255,
"column": 84
} | {
"line": 256,
"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... | [] | rw [laverage, ← setLIntegral_univ]
exact le_of_eq_of_le (setLIntegral_measure_zero univ f <| by simp [hμ0]) zero_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Average | {
"line": 254,
"column": 4
} | {
"line": 255,
"column": 84
} | {
"line": 256,
"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... | [] | rw [laverage, ← setLIntegral_univ]
exact le_of_eq_of_le (setLIntegral_measure_zero univ f <| by simp [hμ0]) zero_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 276,
"column": 14
} | {
"line": 276,
"column": 16
} | {
"line": 277,
"column": 6
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set ι\nC : ℝ≥0\nr : ι → ℝ\nc : ι → α\nB : ι → Set α\nhB : ∀ a ∈ t, B a ⊆ closedBall ... | [
"α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set ι\nC : ℝ≥0\nr : ι → ℝ\nc : ι → α\nB : ι → Set α\nhB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Average | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 46
} | {
"line": 509,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ\nhμ : μ s ≠ 0\nhμ₁ : μ s ≠ ∞\nhf : IntegrableOn f s μ\nH : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0\nthis : Fact (μ s < ∞)\n⊢ 0 ≤ᵐ[μ.restrict s] fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ",
"ppTerm": "?r... | [
"case refine_1\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nf : α → ℝ\nhμ : μ s ≠ 0\nhμ₁ : μ s ≠ ∞\nhf : IntegrableOn f s μ\nH : (μ.restrict s) {x | f x ≤ ⨍ (a : α) in s, f a ∂μ} = 0\nthis : Fact (μ s < ∞)\nx : α\nhx : x ∈ {x | (fun x ↦ 0 x ≤ (fun x ↦ f x - ⨍ (a : α) in s, f a ∂μ) x) x}ᶜ\n⊢ x ∈ ... | refine measure_mono_null (fun x hx ↦ ?_) H | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 38
} | {
"line": 91,
"column": 4
} | [
{
"pp": "α : 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\n⊢ μ - ν = measure_sub",
"ppTerm": "?m.105",
"assigned": true,... | [
"α : 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\n⊢ sInf {d | μ ≤ d + ν} = measure_sub"
] | rw [MeasureTheory.Measure.sub_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 152,
"column": 70
} | {
"line": 152,
"column": 72
} | {
"line": 153,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝ : NormedAddCommGroup E\nx : α\nf : α → E\nhf : LocallyIntegrable f μ\nw : Set α\nw_nhds : w ∈ 𝓝 x\nhw : IntegrableOn f w μ\na : Set α\n⊢ a ⊆ w → IntegrableOn f a μ",
"ppTerm":... | [
"α : Type u_1\ninst✝¹ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝ : NormedAddCommGroup E\nx : α\nf : α → E\nhf : LocallyIntegrable f μ\nw : Set α\nw_nhds : w ∈ 𝓝 x\nhw : IntegrableOn f w μ\na : Set α\nha : a ⊆ w\n⊢ IntegrableOn f a μ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.DensityTheorem | {
"line": 89,
"column": 4
} | {
"line": 91,
"column": 14
} | {
"line": 92,
"column": 4
} | [
{
"pp": "case pos.inl.left\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\ninst✝³ : IsUnifLocDoublingMeasure μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nK : ℝ\nx y : α\nr : ℝ\nh : dist x y ≤ K * r\nrpos : 0 < r\nR : ℝ :=... | [
"case pos.inl.right\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\ninst✝³ : IsUnifLocDoublingMeasure μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nK : ℝ\nx y : α\nr : ℝ\nh : dist x y ≤ K * r\nrpos : 0 < r\nR : ℝ := ⋯\nH : clo... | · apply closedBall_subset_closedBall'
rw [dist_comm]
linarith | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 202,
"column": 10
} | {
"line": 202,
"column": 12
} | {
"line": 202,
"column": 13
} | [
{
"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 ⟂ₘ μ\nA : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) i... | [
"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 ⟂ₘ μ\nA : ∀ ε > 0, ∀ᵐ (x : α) ∂μ, ∀ᶠ (a : Set α) in v.filterAt... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.Monotone | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 16
} | {
"line": 208,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nhf : Monotone f\nx : ℝ\nhx :\n Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[<] x)\n (𝓝 (hf.stieltjesFunction.measure.rnDeriv volume x).toReal) ∧\n Tendsto (fun b ↦ (↑hf.stieltjesFunction b - f x) / (b - x)) (𝓝[>] x)\n (𝓝 (hf.stieltjesFunction.measure.rnDer... | [] | exact ⟨L2, L1⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 123,
"column": 90
} | {
"line": 125,
"column": 57
} | {
"line": 127,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : LocallyBoundedVariationOn f s\na b : α\nha : a ∈ s\nhb : b ∈ s\nba : b ≤ a\n⊢ variationOnFromTo f s a b = 0 ↔ ∀ ⦃x : α⦄, x ∈ s ∩ Icc b a → ∀ ⦃y : α⦄, y ∈ s ∩ Icc b a → edist (f x) (f y) = 0",
... | [] | by
rw [variationOnFromTo.eq_of_ge _ _ ba, neg_eq_zero, ENNReal.toReal_eq_zero_iff,
or_iff_left (hf b a hb ha), eVariationOn.eq_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 54
} | {
"line": 259,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\ns : Set α\nf : α →ᵤ[{s}] E\nthis : ∀ s_1 ∈ {s}, TendstoUniformlyOn (⇑(UniformOnFun.toFun {s}) ∘ id) ((UniformOnFun.toFun {s}) f) (𝓝 f) s_1\n⊢ ∀ x ∈ s, Tendsto (fun i ↦ id i x) (𝓝 f) (𝓝 (f x))",
"ppTerm": "?m.43",
... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\ns : Set α\nf : α →ᵤ[{s}] E\nthis : ∀ s_1 ∈ {s}, TendstoUniformlyOn (⇑(UniformOnFun.toFun {s}) ∘ id) ((UniformOnFun.toFun {s}) f) (𝓝 f) s_1\n⊢ ∀ x ∈ s, TendstoUniformlyOn id f (𝓝 f) {x}"
] | simp_rw [← tendstoUniformlyOn_singleton_iff_tendsto] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 1061,
"column": 2
} | {
"line": 1061,
"column": 24
} | {
"line": 1063,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν₁ ν₂ μ : Measure α\ninst✝² : SigmaFinite ν₁\ninst✝¹ : SigmaFinite ν₂\ninst✝ : SigmaFinite μ\nh : ν₂ ⟂ₘ μ\nx : α\nhx_add : (ν₁ + ν₂).rnDeriv μ x = (ν₁.rnDeriv μ + ν₂.rnDeriv μ) x\nhx_zero : ν₂.rnDeriv μ x = 0 x\n⊢ (ν₁ + ν₂).rnDeriv μ x = ν₁.rnDeriv μ x",
"ppTerm... | [] | simp [hx_add, hx_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 310,
"column": 4
} | {
"line": 312,
"column": 8
} | {
"line": 314,
"column": 0
} | [
{
"pp": "case inr.refine_4\nα : Type u_1\ninst✝ : LinearOrder α\nf : α → ℝ\ns : Set α\nh : LocallyBoundedVariationOn f s\nc : α\ncs : c ∈ s\n⊢ ∀ x ∈ s,\n ∀ y ∈ s,\n (fun x ↦ (variationOnFromTo f s c x + f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x + f x) / 2) x +\n ((fun x ↦ (variationOnFr... | [] | intro x hx y hy
rw [← variationOnFromTo.add h hx cs hy, variationOnFromTo.eq_neg_swap]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 93
} | {
"line": 526,
"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\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht ... | [
"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\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_n... | have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne] using t_ne_zero' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 310,
"column": 4
} | {
"line": 312,
"column": 8
} | {
"line": 314,
"column": 0
} | [
{
"pp": "case inr.refine_4\nα : Type u_1\ninst✝ : LinearOrder α\nf : α → ℝ\ns : Set α\nh : LocallyBoundedVariationOn f s\nc : α\ncs : c ∈ s\n⊢ ∀ x ∈ s,\n ∀ y ∈ s,\n (fun x ↦ (variationOnFromTo f s c x + f x) / 2) y - (fun x ↦ (variationOnFromTo f s c x + f x) / 2) x +\n ((fun x ↦ (variationOnFr... | [] | intro x hx y hy
rw [← variationOnFromTo.add h hx cs hy, variationOnFromTo.eq_neg_swap]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoundedVariation | {
"line": 111,
"column": 9
} | {
"line": 111,
"column": 46
} | {
"line": 111,
"column": 47
} | [
{
"pp": "α : Type u_2\ninst✝⁶ : LinearOrder α\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\ns : Set α\nf : α → E\ng : α → F\nhf : BoundedVariatio... | [
"α : Type u_2\ninst✝⁶ : LinearOrder α\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\ns : Set α\nf : α → E\ng : α → F\nhf : BoundedVariationOn f s\nhg ... | show g y = g x + (g y - g x) by abel, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.BoundedVariation | {
"line": 160,
"column": 6
} | {
"line": 160,
"column": 9
} | {
"line": 160,
"column": 10
} | [
{
"pp": "s : Set ℝ\np q : ℝ → ℝ\nhp : MonotoneOn p s\nhq : MonotoneOn q s\nh : LocallyBoundedVariationOn (p - q) s\nx : ℝ\n⊢ (x ∈ s → DifferentiableWithinAt ℝ p s x) →\n (x ∈ s → DifferentiableWithinAt ℝ q s x) → x ∈ s → DifferentiableWithinAt ℝ (p - q) s x",
"ppTerm": "?m.127",
"assigned": true,
... | [
"s : Set ℝ\np q : ℝ → ℝ\nhp : MonotoneOn p s\nhq : MonotoneOn q s\nh : LocallyBoundedVariationOn (p - q) s\nx : ℝ\nhxp : x ∈ s → DifferentiableWithinAt ℝ p s x\n⊢ (x ∈ s → DifferentiableWithinAt ℝ q s x) → x ∈ s → DifferentiableWithinAt ℝ (p - q) s x"
] | hxp | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 594,
"column": 2
} | {
"line": 594,
"column": 93
} | {
"line": 595,
"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\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht ... | [
"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\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_n... | have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne] using t_ne_zero' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 614,
"column": 6
} | {
"line": 625,
"column": 73
} | {
"line": 626,
"column": 2
} | [] | [] | ρ (s ∩ f ⁻¹' I) ≤ (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by
rw [← ENNReal.coe_zpow t_ne_zero']
apply v.measure_le_mul_of_subset_limRatioMeas_lt hρ
intro x hx
apply hx.2.2.trans_le (le_of_eq _)
rw [ENNReal.coe_zpow t_ne_zero']
_ = ∫⁻ _ in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1)... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 737,
"column": 64
} | {
"line": 737,
"column": 66
} | {
"line": 738,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (t ∩ a) / μ a) (... | [
"α : Type u_1\ninst✝³ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set α := toMeasurable μ s\nA : ∀ᵐ (x : α) ∂μ.restrict s, Tendsto (fun a ↦ μ (t ∩ a) / μ a) (v.filterAt x... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 750,
"column": 64
} | {
"line": 750,
"column": 66
} | {
"line": 751,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → ℝ≥0∞\nhf : Measurable f\nh'f : ∫⁻ (y : α), f y ∂μ ≠ ∞\nρ : Measure α := μ.withDensity f\nthis : IsF... | [
"α : Type u_1\ninst✝³ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → ℝ≥0∞\nhf : Measurable f\nh'f : ∫⁻ (y : α), f y ∂μ ≠ ∞\nρ : Measure α := μ.withDensity f\nthis : IsFiniteMeasure... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 808,
"column": 52
} | {
"line": 808,
"column": 54
} | {
"line": 808,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : Integrable f μ\nh'f : StronglyMeasurable f\nA ... | [
"α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : Integrable f μ\nh'f : StronglyMeasurable f\nA : μ.FiniteSp... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 818,
"column": 6
} | {
"line": 818,
"column": 8
} | {
"line": 818,
"column": 9
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : Integrable f μ\nh'f : StronglyMeasurable f\nA ... | [
"α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : Integrable f μ\nh'f : StronglyMeasurable f\nA : μ.FiniteSp... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 860,
"column": 50
} | {
"line": 860,
"column": 52
} | {
"line": 860,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : LocallyIntegrable f μ\nu : ℕ → Set α\nu_open :... | [
"α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : LocallyIntegrable f μ\nu : ℕ → Set α\nu_open : ∀ (n : ℕ), ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 746,
"column": 12
} | {
"line": 746,
"column": 14
} | {
"line": 746,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\nε : ℝ≥0∞\ns : Set α\nh : ε < eVariationOn f s\nn : ℕ\nu : ℕ → α\nhu : ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x))\nu_mono : Monotone u\nu_mem : ∀ (i : ℕ), u i ∈ s\nA : ε < eVariationOn f (s ∩ Icc ... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\nε : ℝ≥0∞\ns : Set α\nh : ε < eVariationOn f s\nn : ℕ\nu : ℕ → α\nhu : ε < ∑ x ∈ Finset.range n, edist (f (u (x + 1))) (f (u x))\nu_mono : Monotone u\nu_mem : ∀ (i : ℕ), u i ∈ s\nA : ε < eVariationOn f (s ∩ Icc (u 0) (u n))... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 892,
"column": 11
} | {
"line": 892,
"column": 13
} | {
"line": 892,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\... | [
"α : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : α → E\nhf : Locall... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 840,
"column": 48
} | {
"line": 840,
"column": 50
} | {
"line": 840,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\nhf : BoundedVariationOn f s\nx : α\nA : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0)\na : α\n⊢ a ∈ s ∩ Ici x → 0 = eV... | [
"α : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\nhf : BoundedVariationOn f s\nx : α\nA : Tendsto (fun y ↦ eVariationOn f (s ∩ Ico y x)) (𝓝[s ∩ Iio x] x) (𝓝 0)\na : α\nha : a ∈ s ∩ Ici x\n⊢ 0 = eVariation... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 541,
"column": 2
} | {
"line": 544,
"column": 59
} | {
"line": 546,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCa... | [] | · obtain (ha | hf) := not_and_or.mp h
· simp [cfc_apply_of_not_predicate a ha]
· rw [cfc_apply_of_not_continuousOn a hf, cfc_apply_of_not_continuousOn, star_zero]
exact fun hf_star ↦ hf <| by simpa using hf_star.star | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 594,
"column": 2
} | {
"line": 594,
"column": 40
} | {
"line": 596,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R A... | [] | rw [cfc_map_polynomial .., cfc_id' ..] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 594,
"column": 2
} | {
"line": 594,
"column": 40
} | {
"line": 596,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R A... | [] | rw [cfc_map_polynomial .., cfc_id' ..] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 594,
"column": 2
} | {
"line": 594,
"column": 40
} | {
"line": 596,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\ninst✝⁶ : MetricSpace R\ninst✝⁵ : IsTopologicalSemiring R\ninst✝⁴ : ContinuousStar R\ninst✝³ : TopologicalSpace A\ninst✝² : Ring A\ninst✝¹ : StarRing A\ninst✝ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R A... | [] | rw [cfc_map_polynomial .., cfc_id' ..] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 260,
"column": 2
} | {
"line": 262,
"column": 20
} | {
"line": 263,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ :... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\ninst✝¹ : IsScalarTow... | have hg0 : (Function.extend Subtype.val f g) 0 = 0 := by
rw [← quasispectrum.coe_zero (R := R) a, Subtype.val_injective.extend_apply]
exact map_zero f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 42
} | {
"line": 298,
"column": 4
} | [
{
"pp": "case neg.inr.inl\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Mo... | [] | rwa [cfcₙ_apply_of_not_map_zero _ h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 42
} | {
"line": 298,
"column": 4
} | [
{
"pp": "case neg.inr.inl\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Mo... | [] | rwa [cfcₙ_apply_of_not_map_zero _ h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 42
} | {
"line": 298,
"column": 4
} | [
{
"pp": "case neg.inr.inl\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Mo... | [] | rwa [cfcₙ_apply_of_not_map_zero _ h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 436,
"column": 2
} | {
"line": 442,
"column": 61
} | {
"line": 444,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A... | [] | · simp only [not_and_or] at h
obtain (ha | hf | h0) := h
· simp [cfcₙ_apply_of_not_predicate a ha]
· rw [cfcₙ_apply_of_not_continuousOn a hf, cfcₙ_apply_of_not_continuousOn, star_zero]
exact fun hf_star ↦ hf <| by simpa using hf_star.star
· rw [cfcₙ_apply_of_not_map_zero a h0, cfcₙ_apply_of_not_ma... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 70
} | {
"line": 195,
"column": 0
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\nh' : ν > 0\n⊢ 0 < eval (n - (ν - 1)) (ascPochhammer ℕ ν)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",
"Nat.succ_pred_eq_o... | [] | · rw [← Nat.succ_pred_eq_of_pos h'] at h
exact ascPochhammer_pos _ _ (tsub_pos_of_lt (Nat.lt_of_succ_le h)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 223,
"column": 46
} | {
"line": 223,
"column": 62
} | {
"line": 223,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬↑(eval ν.succ (ascPochhammer ℕ (n - ν))) = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ ¬↑(eval ν.succ (ascPochhammer ℕ (n - ν))) = ↑0"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 846,
"column": 22
} | {
"line": 846,
"column": 76
} | {
"line": 846,
"column": 77
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Algebra R A\ninst✝¹ : ClosedEmbeddingContinuousFunctiona... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁰ : Semifield R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : Ring A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Algebra R A\ninst✝¹ : ClosedEmbeddingContinuousFunctionalCalculus R ... | ← isUniformEmbedding_toContinuousMap.comap_uniformity, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 32
} | {
"line": 199,
"column": 6
} | [
{
"pp": "case refine_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nL : Set C(X, ℝ)\nnA : L.Nonempty\ninf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L\nsup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L\nsep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nnX : Nonempty... | [
"case refine_2\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nL : Set C(X, ℝ)\nnA : L.Nonempty\ninf_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊓ g ∈ L\nsup_mem : ∀ f ∈ L, ∀ g ∈ L, f ⊔ g ∈ L\nsep : ∀ (v : X → ℝ) (x y : X), ∃ f ∈ L, f x = v x ∧ f y = v y\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nnX : Nonempty X\ng : X → ... | rw [Set.mem_ofPred_eq, w₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 28
} | {
"line": 290,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\ns : Set α\nhf : ∀ x ∈ s, UpperHemicontinuousWithinAt f s x\nu : Set β\nhu : IsClosed[inst✝] u\n⊢ ∀ x ∈ s, UpperHemicontinuousWithinAt (fun x ↦ f x ∩ u) s x",
"ppTerm": "?m.34",
"assigned": true,
... | [] | exact (hf · · |>.inter hu) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 128,
"column": 42
} | {
"line": 128,
"column": 62
} | {
"line": 128,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ ∀ s ∈ 𝓤 α, SetRel.id ⊆ hausdorffEntourage s",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"SetRel.id",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ ∀ s ∈ 𝓤 α, (hausdorffEntourage s).IsRefl"
] | SetRel.id_subset_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 199,
"column": 2
} | {
"line": 202,
"column": 96
} | {
"line": 204,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\n⊢ UniformContinuous fun x ↦ x.1 ×ˢ x.2",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Set.instSProd",
"entourageProd",
"instUniformSpace... | [] | refine (𝓤 α).basis_sets.uniformity_prod (𝓤 β).basis_sets |>.lift' monotone_hausdorffEntourage
|>.tendsto_right_iff.mpr fun ⟨U, V⟩ ⟨hU, hV⟩ => ?_
filter_upwards [entourageProd_mem_uniformity (Filter.mem_lift' hU) (Filter.mem_lift' hV)]
with ⟨⟨s₁, s₂⟩, ⟨t₁, t₂⟩⟩ ⟨h₁, h₂⟩ using prod_mem_hausdorffEntourage_ento... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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