module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Pairing | {
"line": 125,
"column": 4
} | {
"line": 126,
"column": 34
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case inl\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nx : A.N\nh : x ∈ P.I\n⊢ ∃ y, x = ↑y ∨ x = ↑(P.p y)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"SSet.Subcomplex.Pairing.p",
"Membership.mem",
"E... | [] | obtain ⟨y, hy⟩ := P.p.surjective ⟨x, h⟩
exact ⟨y, Or.inr (by rw [hy])⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Pairing | {
"line": 125,
"column": 4
} | {
"line": 126,
"column": 34
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case inl\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nx : A.N\nh : x ∈ P.I\n⊢ ∃ y, x = ↑y ∨ x = ↑(P.p y)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"SSet.Subcomplex.Pairing.p",
"Membership.mem",
"E... | [] | obtain ⟨y, hy⟩ := P.p.surjective ⟨x, h⟩
exact ⟨y, Or.inr (by rw [hy])⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Nonsingular | {
"line": 162,
"column": 6
} | {
"line": 162,
"column": 39
} | {
"line": 162,
"column": 40
} | [
{
"pp": "X : SSet\ninst✝ : X.Nonsingular\nx y : X.N\nh : x ≤ y\n⊢ stdSimplex.map (monoOfLE h) ≫ yonedaEquiv.symm y.simplex = yonedaEquiv.symm x.simplex",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"SSet.yonedaEquiv",
"SSet.S.simplex",
"Eq.mpr",
"Opposite",
... | [
"X : SSet\ninst✝ : X.Nonsingular\nx y : X.N\nh : x ≤ y\n⊢ yonedaEquiv.symm ((ConcreteCategory.hom (X.map (monoOfLE h).op)) y.simplex) = yonedaEquiv.symm x.simplex"
] | yonedaEquiv_symm_naturality_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteProd | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 66
} | {
"line": 49,
"column": 0
} | [
{
"pp": "X₁ X₂ : SSet\np q : ℕ\nx₁ : X₁ _⦋p⦌\nx₂ : X₂ _⦋q⦌\n⊢ (ofSimplex x₁).prod (ofSimplex x₂) = range (yonedaEquiv.symm x₁ ⊗ₘ yonedaEquiv.symm x₂)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"SSet.yonedaEquiv",
"SSet.Subcomplex.range",
"SSet.Subcomplex.ofSimplex",
... | [] | simp [Subcomplex.range_tensorHom, Subcomplex.range_eq_ofSimplex] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteProd | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 66
} | {
"line": 49,
"column": 0
} | [
{
"pp": "X₁ X₂ : SSet\np q : ℕ\nx₁ : X₁ _⦋p⦌\nx₂ : X₂ _⦋q⦌\n⊢ (ofSimplex x₁).prod (ofSimplex x₂) = range (yonedaEquiv.symm x₁ ⊗ₘ yonedaEquiv.symm x₂)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"SSet.yonedaEquiv",
"SSet.Subcomplex.range",
"SSet.Subcomplex.ofSimplex",
... | [] | simp [Subcomplex.range_tensorHom, Subcomplex.range_eq_ofSimplex] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.FiniteProd | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 66
} | {
"line": 49,
"column": 0
} | [
{
"pp": "X₁ X₂ : SSet\np q : ℕ\nx₁ : X₁ _⦋p⦌\nx₂ : X₂ _⦋q⦌\n⊢ (ofSimplex x₁).prod (ofSimplex x₂) = range (yonedaEquiv.symm x₁ ⊗ₘ yonedaEquiv.symm x₂)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"SSet.yonedaEquiv",
"SSet.Subcomplex.range",
"SSet.Subcomplex.ofSimplex",
... | [] | simp [Subcomplex.range_tensorHom, Subcomplex.range_eq_ofSimplex] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 67
} | {
"line": 162,
"column": 4
} | [
{
"pp": "case a\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ninst✝ : SuccOrder ι\ni : ι\nhi : ¬IsMax i\n⊢ A ≤ f.filtration i ⊔ ⨆ c, (↑(P.p c.s)).subcomplex ∧\n ∀ i_1 < Order.succ i,\n ∀ (i_3 : f.Cell i_1), (↑(P.p i_3.s)).subcomplex ≤ f.filtration ... | [
"case a\nX : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ninst✝ : SuccOrder ι\ni : ι\nhi : ¬IsMax i\nj : ι\nhj : j < Order.succ i\nc : f.Cell j\n⊢ (↑(P.p c.s)).subcomplex ≤ f.filtration i ⊔ ⨆ c, (↑(P.p c.s)).subcomplex"
] | refine ⟨(f.le_filtration _).trans le_sup_left, fun j hj c ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Presentable.CardinalFilteredPresentation | {
"line": 176,
"column": 71
} | {
"line": 178,
"column": 41
} | {
"line": 180,
"column": 0
} | [
{
"pp": "C✝ : Type u\ninst✝³ : Category.{v, u} C✝\nC : Type u\ninst✝² : Category.{v, u} C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : HasCardinalFilteredGenerator C κ\n⊢ ObjectProperty.EssentiallySmall.{w, v, u} (isCardinalPresentable C κ)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstan... | [] | by
obtain ⟨P, _, hP⟩ := HasCardinalFilteredGenerator.exists_generator C κ
exact hP.essentiallySmall_isPresentable | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallRepresentatives | {
"line": 132,
"column": 29
} | {
"line": 132,
"column": 77
} | {
"line": 133,
"column": 6
} | [
{
"pp": "Ω : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nh : CoreSmallCategoryOfSet Ω C\nx y : ↑h.smallCategoryOfSet.obj\nf : (𝟭 ↑h.smallCategoryOfSet.obj).obj x ⟶ (𝟭 ↑h.smallCategoryOfSet.obj).obj y\nx' y' : ↑h.smallCategoryOfSet.obj\ng : (𝟭 ↑h.smallCategoryOfSet.obj).obj x' ⟶ (𝟭 ↑h.smallCategoryOfSet.o... | [] | by simpa using! congr_arg Arrow.leftFunc.obj hfg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.SmallRepresentatives | {
"line": 183,
"column": 41
} | {
"line": 185,
"column": 25
} | {
"line": 185,
"column": 25
} | [
{
"pp": "κ : Cardinal.{w}\nC : Type u\ninst✝ : Category.{v, u} C\nhC : HasCardinalLT (Arrow C) κ\nΩ : Type w := κ.ord.ToType\n⊢ Nonempty (Arrow C ↪ Ω)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
... | [] | by
rw [← Cardinal.lift_mk_le']
simpa [Ω] using hC.le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.StrongGenerator | {
"line": 75,
"column": 33
} | {
"line": 77,
"column": 26
} | {
"line": 77,
"column": 26
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : ObjectProperty C\nκ : Cardinal.{w}\ninst✝¹ : Fact κ.IsRegular\ninst✝ : HasColimitsOfSize.{w, w, v, u} C\nX : C\nJ : Type w\nx✝ : SmallCategory J\nK : J ⥤ CostructuredArrow (P.colimitsCardinalClosure κ).ι X\nhJ : HasCardinalLT (Arrow J) κ\n⊢ Cocone K",
"pp... | [] | by
have := ObjectProperty.isClosedUnderColimitsOfShape_colimitsCardinalClosure P κ J hJ
exact colimit.cocone K | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic | {
"line": 146,
"column": 69
} | {
"line": 148,
"column": 42
} | {
"line": 149,
"column": 4
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nhA : strongAnodyneExtensions A.ι\n⊢ ∃ B P, P.IsRegular ∧ B = A",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"SSet.Subcomplex.range",
"Opposite",
"CategoryTheory.Mono",
"congrArg",
"Categor... | [] | by
obtain ⟨_, P, _⟩ := hA
exact ⟨_, P, inferInstance, by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.StrongGenerator | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 77
} | {
"line": 183,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : IsCardinalLocallyPresentable C κ\nκ' : Cardinal.{w}\ninst✝ : Fact κ'.IsRegular\nh : κ ≤ κ'\n⊢ ∃ P, ∃ (_ : ObjectProperty.Small.{w, v, u} P), P.IsStrongGenerator ∧ P ≤ isCardinalPresentable C κ'",
"ppTerm":... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nκ : Cardinal.{w}\ninst✝² : Fact κ.IsRegular\ninst✝¹ : IsCardinalLocallyPresentable C κ\nκ' : Cardinal.{w}\ninst✝ : Fact κ'.IsRegular\nh : κ ≤ κ'\nS : ObjectProperty C\nw✝ : ObjectProperty.Small.{w, v, u} S\nh₁ : S.IsStrongGenerator\nh₂ : S ≤ isCardinalPresentable C κ\n⊢ ∃ P,... | obtain ⟨S, _, h₁, h₂⟩ := (iff_exists_isStrongGenerator C κ).1 inferInstance | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 588,
"column": 8
} | {
"line": 588,
"column": 14
} | {
"line": 588,
"column": 15
} | [
{
"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\nx₁ x₂ : CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaStdSimplex j)\nhx₁ : ... | [
"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\nx₁ x₂ : CategoryTheory.evaluation SimplexCategoryᵒᵖ (Type u) _⦋d⦌.obj (f.sigmaStdSimplex j)\nhx₁ : x₁ ∉ Set.ran... | ← hg₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 224,
"column": 4
} | {
"line": 226,
"column": 73
} | {
"line": 227,
"column": 4
} | [
{
"pp": "m : 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\n⊢ (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) l.castSucc)) (x.cast hd).simplex ∉\n (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).obj (Opposite.o... | [
"m : 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\n⊢ (∀ (x_1 : Fin (n + 1)), ∃ x_2, (x.cast hd).simplex.2 (l.castSucc.succAbove x_2) = x_1) ∧\n ∀ (x_1 : Fin (m + 1 + 1)), ¬x_1 = k.castSucc → ∃ x_2, (x.cast hd).simple... | simp only [Subcomplex.mem_unionProd_iff, prod_δ_snd, mem_boundary_iff_notMem_range,
Set.mem_range, stdSimplex.δ_apply, not_exists, prod_δ_fst, mem_horn_iff_notMem_range,
ne_eq, exists_prop, not_or, not_forall, Decidable.not_not, not_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Monoidal.PushoutProduct | {
"line": 182,
"column": 6
} | {
"line": 183,
"column": 90
} | {
"line": 184,
"column": 4
} | [
{
"pp": "case refine_1.refine_3\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : HasPushouts C\ninst✝⁴ : MonoidalCategory C\nX₁ X₂ X₃ : Arrow C\nW : C\ninst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)\ninst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.... | [] | apply pushout.hom_ext (by simp)
apply ((tensorRight _).map_isPushout (IsPushout.of_hasPushout _ _)).hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.PushoutProduct | {
"line": 182,
"column": 6
} | {
"line": 183,
"column": 90
} | {
"line": 184,
"column": 4
} | [
{
"pp": "case refine_1.refine_3\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : HasPushouts C\ninst✝⁴ : MonoidalCategory C\nX₁ X₂ X₃ : Arrow C\nW : C\ninst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)\ninst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.... | [] | apply pushout.hom_ext (by simp)
apply ((tensorRight _).map_isPushout (IsPushout.of_hasPushout _ _)).hom_ext <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.RelativeMorphism | {
"line": 156,
"column": 9
} | {
"line": 158,
"column": 61
} | {
"line": 160,
"column": 0
} | [
{
"pp": "X Y Z : SSet\nA : X.Subcomplex\nB : Y.Subcomplex\nφ : A.toSSet ⟶ B.toSSet\nf✝ g : RelativeMorphism A B φ\nC : Z.Subcomplex\nψ : B.toSSet ⟶ C.toSSet\nf' g' : RelativeMorphism B C ψ\nh : f'.Homotopy g'\nf : RelativeMorphism A B φ\nφψ : A.toSSet ⟶ C.toSSet\nfac : φ ≫ ψ = φψ\n⊢ A.ι ▷ Δ[1] ≫ f.map ▷ Δ[1] ≫ ... | [] | by
rw [← fac, Category.assoc, ← comp_whiskerRight_assoc, f.comm,
comp_whiskerRight_assoc, h.rel, whiskerRight_fst_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.Splitting | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 58
} | {
"line": 43,
"column": 6
} | [
{
"pp": "case refine_1\nX : SSet\nx✝ : SimplexCategoryᵒᵖ\nn : ℕ\nx :\n (SimplicialObject.Splitting.cofan' (fun n ↦ ↑(X.nonDegenerate n)) X (fun n ↦ ↾Subtype.val)\n (Opposite.op ⦋n⦌)).cofanTypes.pt\n⊢ ∃ i y,\n (SimplicialObject.Splitting.cofan' (fun n ↦ ↑(X.nonDegenerate n)) X (fun n ↦ ↾Subtype.val)\n... | [
"case refine_1\nX : SSet\nx✝ : SimplexCategoryᵒᵖ\nn m : ℕ\nf : ⦋n⦌ ⟶ ⦋m⦌\nw✝ : Epi f\ny : ↑(X.nonDegenerate m)\n⊢ ∃ i y_1,\n (SimplicialObject.Splitting.cofan' (fun n ↦ ↑(X.nonDegenerate n)) X (fun n ↦ ↾Subtype.val)\n (Opposite.op ⦋n⦌)).cofanTypes.inj\n i y_1 =\n (ConcreteCategory.hom (X... | obtain ⟨m, f, _, y, rfl⟩ := X.exists_nonDegenerate x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.KanComplex.MulStruct | {
"line": 101,
"column": 27
} | {
"line": 101,
"column": 42
} | {
"line": 101,
"column": 43
} | [
{
"pp": "X : SSet\nn : ℕ\nx : X _⦋0⦌\nf g : X.PtSimplex n x\ni : Fin (n + 1)\nr : f.RelStruct g i\nf' g' : X.PtSimplex n x\nhf : f = f'\nhg : g = g'\n⊢ stdSimplex.δ i.castSucc ≫ r.map = f'.map",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"SSet.Subcomplex.toSSet",
"Eq.mpr",
... | [
"X : SSet\nn : ℕ\nx : X _⦋0⦌\nf g : X.PtSimplex n x\ni : Fin (n + 1)\nr : f.RelStruct g i\nf' g' : X.PtSimplex n x\nhf : f = f'\nhg : g = g'\n⊢ f.map = f'.map"
] | δ_castSucc_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Homotopy.TopCat.ZerothHomotopy | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 41
} | {
"line": 107,
"column": 0
} | [
{
"pp": "X : TopCat\ninst✝ : PathConnectedSpace ↑X\nthis : Unique (ZerothHomotopy ↑X) := ⋯.some\n⊢ Nonempty (Unique (toSSet.obj X).π₀)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"TopCat.zerothHomotopyEquiv",
"Equiv.unique",
"Opposite",
"ZerothHomotopy",
"... | [] | exact ⟨zerothHomotopyEquiv.symm.unique⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 376,
"column": 31
} | {
"line": 376,
"column": 62
} | {
"line": 377,
"column": 4
} | [
{
"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\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨... | [] | by gcongr; exact hf _ y's _ x's | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 36
} | {
"line": 288,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nx : 𝕜\nf : 𝕜 → 𝕜'\nm : ℕ\nhf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x\nh_eq : (fun z ↦ (z - x) ^ m • f z) =ᶠ[𝓝 x] 0\n⊢ (fun x ↦ 0) =ᶠ[𝓝[≠] x] f⁻¹",... | [
"case pos\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nx : 𝕜\nf : 𝕜 → 𝕜'\nm : ℕ\nhf : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ m • f z) x\nh_eq : (fun z ↦ (z - x) ^ m • f z) =ᶠ[𝓝 x] 0\n⊢ ∀ᶠ (x_1 : 𝕜) in 𝓝 x, x_1 ∈ {x}ᶜ → 0 = f... | rw [eventuallyEq_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric | {
"line": 185,
"column": 32
} | {
"line": 185,
"column": 39
} | {
"line": 186,
"column": 4
} | [
{
"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... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric | {
"line": 185,
"column": 32
} | {
"line": 185,
"column": 39
} | {
"line": 186,
"column": 4
} | [
{
"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... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric | {
"line": 185,
"column": 32
} | {
"line": 185,
"column": 39
} | {
"line": 186,
"column": 4
} | [
{
"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... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 40
} | {
"line": 601,
"column": 4
} | [
{
"pp": "case right\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\n⊢ IsOpen[instTopologicalSpaceSubtype] {u | analyticOrderAt f ↑u = ⊤}",
"ppTerm": "?right✝",
"assigned": true,... | [
"case right\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\n⊢ ∀ x ∈ {u | analyticOrderAt f ↑u = ⊤},\n ∃ t ⊆ {u | analyticOrderAt f ↑u = ⊤}, IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈ t"... | apply isOpen_iff_forall_mem_open.mpr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv | {
"line": 728,
"column": 2
} | {
"line": 728,
"column": 61
} | {
"line": 730,
"column": 0
} | [
{
"pp": "x✝ : ℂ\n⊢ logDeriv sin x✝ = x✝.cot",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"logDeriv",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"Semiring.toModule",
"Complex.cos",
"congrArg",
"Pi.instDiv",
"deriv",
... | [] | rw [logDeriv, Complex.deriv_sin, Pi.div_apply, Complex.cot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Binomial | {
"line": 237,
"column": 56
} | {
"line": 237,
"column": 63
} | {
"line": 237,
"column": 63
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nr : R\nk : ℕ\nih : (ascPochhammer ℕ k).smeval (-r) = (↑k).negOnePow • (descPochhammer ℤ k).smeval r\nh : (X + ↑k).smeval (-r) = -(X + -↑k).smeval r\n⊢ -(↑(↑k).negOnePow • (descPochhammer ℤ k).smeval r) * (X + -↑k... | [
"case succ\nR : Type u_1\ninst✝² : NonAssocRing R\ninst✝¹ : Pow R ℕ\ninst✝ : NatPowAssoc R\nr : R\nk : ℕ\nih : (ascPochhammer ℕ k).smeval (-r) = (↑k).negOnePow • (descPochhammer ℤ k).smeval r\nh : (X + ↑k).smeval (-r) = -(X + -↑k).smeval r\n⊢ -(↑(↑k).negOnePow • (descPochhammer ℤ k).smeval r * (X + -↑k).smeval r) =... | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Binomial | {
"line": 400,
"column": 2
} | {
"line": 402,
"column": 90
} | {
"line": 404,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nn k : ℕ\n⊢ choose (↑n) k = ↑(n.choose k)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
... | [] | rw [← nsmul_right_inj (Nat.factorial_ne_zero k),
← descPochhammer_eq_factorial_smul_choose, nsmul_eq_mul, ← Nat.cast_mul,
← Nat.descFactorial_eq_factorial_mul_choose, ← descPochhammer_smeval_eq_descFactorial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Binomial | {
"line": 400,
"column": 2
} | {
"line": 402,
"column": 90
} | {
"line": 404,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nn k : ℕ\n⊢ choose (↑n) k = ↑(n.choose k)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
... | [] | rw [← nsmul_right_inj (Nat.factorial_ne_zero k),
← descPochhammer_eq_factorial_smul_choose, nsmul_eq_mul, ← Nat.cast_mul,
← Nat.descFactorial_eq_factorial_mul_choose, ← descPochhammer_smeval_eq_descFactorial] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Binomial | {
"line": 400,
"column": 2
} | {
"line": 402,
"column": 90
} | {
"line": 404,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonAssocRing R\ninst✝² : Pow R ℕ\ninst✝¹ : BinomialRing R\ninst✝ : NatPowAssoc R\nn k : ℕ\n⊢ choose (↑n) k = ↑(n.choose k)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
... | [] | rw [← nsmul_right_inj (Nat.factorial_ne_zero k),
← descPochhammer_eq_factorial_smul_choose, nsmul_eq_mul, ← Nat.cast_mul,
← Nat.descFactorial_eq_factorial_mul_choose, ← descPochhammer_smeval_eq_descFactorial] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 634,
"column": 2
} | {
"line": 635,
"column": 29
} | {
"line": 636,
"column": 2
} | [
{
"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\n⊢ {u | analyticOrderAt f ↑u = 0 ∨ analyticOrderAt f ↑u = ⊤} ∈ codiscrete ↑U",
"ppTerm": "?m.32",
"assigned": true,
"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\n⊢ ∀ x ∈ U, (U \\ Subtype.val '' {u | analyticOrderAt f ↑u = 0 ∨ analyticOrderAt f ↑u = ⊤})ᶜ ∈ 𝓝[≠] x"
] | simp_rw [mem_codiscrete_subtype_iff_mem_codiscreteWithin, mem_codiscreteWithin,
disjoint_principal_right] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Analytic.Order | {
"line": 639,
"column": 4
} | {
"line": 639,
"column": 37
} | {
"line": 640,
"column": 2
} | [
{
"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\nha : ∀ᶠ (x : 𝕜) in 𝓝 a, f x = 0\n⊢ a ∈ (U \\ Subtype.val '' {u ... | [] | simp [analyticOrderAt_eq_top, ha] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 237,
"column": 79
} | {
"line": 247,
"column": 31
} | {
"line": 249,
"column": 0
} | [
{
"pp": "f : ℂ → ℂ\ns : Set ℂ\nx : ℂ\nhf : DifferentiableWithinAt ℂ f s x\nc : ℂ\n⊢ derivWithin (fun x ↦ c ^ f x) s x = log c * derivWithin f s x * c ^ f x",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"derivWithin_zero_of_frequently_mem",
"NonUnitalNonAssocCommRing.toNonUnit... | [] | by
by_cases h : UniqueDiffWithinAt ℂ s x; swap
· rw [derivWithin_zero_of_not_uniqueDiffWithinAt h,
derivWithin_zero_of_not_uniqueDiffWithinAt h, mul_zero, zero_mul]
by_cases hc : c = 0; swap
· rw [mul_comm, ← mul_assoc]
exact (hf.hasDerivWithinAt.const_cpow (Or.inl hc)).derivWithin h
rw [uniqueDiffW... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Order | {
"line": 653,
"column": 4
} | {
"line": 653,
"column": 37
} | {
"line": 654,
"column": 2
} | [
{
"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\nha : ∀ᶠ (x : 𝕜) in 𝓝 a, f x = 0\n⊢ a ∈ (U \\ {u | analyticOrder... | [] | simp [analyticOrderAt_eq_top, ha] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Binomial | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 9
} | {
"line": 73,
"column": 2
} | [
{
"pp": "𝕂 : Type u\ninst✝⁵ : Field 𝕂\ninst✝⁴ : CharZero 𝕂\n𝔸 : Type v\ninst✝³ : Ring 𝔸\ninst✝² : Algebra 𝕂 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : IsTopologicalRing 𝔸\na b : 𝕂\nh : ∀ (k : ℕ), ↑k ≠ -b\nn : ℕ\n⊢ (↑n !)⁻¹ • Polynomial.eval (a - ↑n + 1) (ascPochhammer 𝕂 n) =\n (-1) ^ n * ((↑n !)⁻¹ * ... | [
"𝕂 : Type u\ninst✝⁵ : Field 𝕂\ninst✝⁴ : CharZero 𝕂\n𝔸 : Type v\ninst✝³ : Ring 𝔸\ninst✝² : Algebra 𝕂 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : IsTopologicalRing 𝔸\na b : 𝕂\nh : ∀ (k : ℕ), ↑k ≠ -b\nn : ℕ\n⊢ Polynomial.eval (1 + a - ↑n) (ascPochhammer 𝕂 n) * (↑n !)⁻¹ =\n Polynomial.eval (1 + a - ↑n) (ascPo... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 363,
"column": 2
} | {
"line": 371,
"column": 49
} | {
"line": 373,
"column": 0
} | [
{
"pp": "p : ℝ × ℝ\nhp : p.1 < 0\n⊢ HasStrictFDerivAt (fun x ↦ x.1 ^ x.2)\n ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +\n (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ)\n p",
"ppTerm": "?m.111",
"assigned": true,
"usedConst... | [] | have : (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) * cos (x.2 * π) :=
(continuousAt_fst.eventually (gt_mem_nhds hp)).mono fun p hp => rpow_def_of_neg hp _
refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm
convert!
((hasStrictFDerivAt_fst.log hp.ne).fun_mul hasStrictFDerivAt_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 363,
"column": 2
} | {
"line": 371,
"column": 49
} | {
"line": 373,
"column": 0
} | [
{
"pp": "p : ℝ × ℝ\nhp : p.1 < 0\n⊢ HasStrictFDerivAt (fun x ↦ x.1 ^ x.2)\n ((p.2 * p.1 ^ (p.2 - 1)) • ContinuousLinearMap.fst ℝ ℝ ℝ +\n (p.1 ^ p.2 * log p.1 - rexp (log p.1 * p.2) * sin (p.2 * π) * π) • ContinuousLinearMap.snd ℝ ℝ ℝ)\n p",
"ppTerm": "?m.111",
"assigned": true,
"usedConst... | [] | have : (fun x : ℝ × ℝ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) * cos (x.2 * π) :=
(continuousAt_fst.eventually (gt_mem_nhds hp)).mono fun p hp => rpow_def_of_neg hp _
refine HasStrictFDerivAt.congr_of_eventuallyEq ?_ this.symm
convert!
((hasStrictFDerivAt_fst.log hp.ne).fun_mul hasStrictFDerivAt_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.IteratedFDeriv | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 87
} | {
"line": 233,
"column": 0
} | [
{
"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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\ninst✝ : CompleteSpace F\nn : ℕ\nv : Fin n →... | [] | exact h.iteratedFDerivWithin_eq_sum_of_completeSpace uniqueDiffOn_univ (mem_univ _) v | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 141,
"column": 6
} | {
"line": 143,
"column": 51
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case inr.inl.inr.inr\ny : ℝ≥0∞\ny_real : 0 < y.toReal\n⊢ (∞ * y).log = ⊤ + y.log",
"ppTerm": "?inr.inl.inr.inr",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog.0.ENNReal.log_mul_add._simp_1_1",
"ENNReal.log_pos_real'",
"Eq.m... | [] | rw [log_pos_real' y_real, ENNReal.top_mul', EReal.top_add_coe, log_eq_top_iff]
simp only [ite_eq_right_iff, zero_ne_top, imp_false]
exact (ENNReal.toReal_pos_iff.1 y_real).1.ne' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 141,
"column": 6
} | {
"line": 143,
"column": 51
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case inr.inl.inr.inr\ny : ℝ≥0∞\ny_real : 0 < y.toReal\n⊢ (∞ * y).log = ⊤ + y.log",
"ppTerm": "?inr.inl.inr.inr",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog.0.ENNReal.log_mul_add._simp_1_1",
"ENNReal.log_pos_real'",
"Eq.m... | [] | rw [log_pos_real' y_real, ENNReal.top_mul', EReal.top_add_coe, log_eq_top_iff]
simp only [ite_eq_right_iff, zero_ne_top, imp_false]
exact (ENNReal.toReal_pos_iff.1 y_real).1.ne' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 33
} | {
"line": 182,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\n⊢ x⁻¹.log = -x.log",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"congrArg",
"ENNReal.instPowReal",
"EReal.instNeg",
"EReal",
"ENNReal.log",
"MulZeroOneClass.toMulOne... | [] | simp [← rpow_neg_one, log_rpow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 33
} | {
"line": 182,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\n⊢ x⁻¹.log = -x.log",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"congrArg",
"ENNReal.instPowReal",
"EReal.instNeg",
"EReal",
"ENNReal.log",
"MulZeroOneClass.toMulOne... | [] | simp [← rpow_neg_one, log_rpow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 33
} | {
"line": 182,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\n⊢ x⁻¹.log = -x.log",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"congrArg",
"ENNReal.instPowReal",
"EReal.instNeg",
"EReal",
"ENNReal.log",
"MulZeroOneClass.toMulOne... | [] | simp [← rpow_neg_one, log_rpow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.ENNRealLogExp | {
"line": 156,
"column": 60
} | {
"line": 161,
"column": 16
} | {
"line": 163,
"column": 0
} | [
{
"pp": "b : ℝ≥0∞\nhb : b < 1\n⊢ Filter.Tendsto (fun x ↦ b ^ x) Filter.atBot (𝓝 ∞)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
"Real",
"ENNReal.log_lt_zero_iff",
"Preorder.toLT",
"HMul.hMul",
"congrArg",
... | [] | by
simp_rw [ENNReal.rpow_eq_exp_mul_log]
refine EReal.tendsto_exp_nhds_top_nhds_top.comp ?_
convert! EReal.Tendsto.mul_const tendsto_coe_atBot _ _
· rw [EReal.bot_mul_of_neg (log_lt_zero_iff.2 hb)]
all_goals simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 75
} | {
"line": 480,
"column": 2
} | [
{
"pp": "case neg.inr\nu : ℕ → EReal\nv : ℕ → ℕ\nh : Monotone u\nhv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0\nhv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤\nu_0 : ¬u = ⊥\nv_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)\na : EReal\nv_a : a > linearGrowthSup fun n ↦ ↑(v n)\nb : EReal\na_0 : 0 < a\nb_0 : 0 < b\na_top : a ≠ ⊤... | [
"case neg.inr\nu : ℕ → EReal\nv : ℕ → ℕ\nh : Monotone u\nhv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0\nhv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤\nu_0 : ¬u = ⊥\nv_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)\na : EReal\nv_a : a > linearGrowthSup fun n ↦ ↑(v n)\nb : EReal\na_0 : 0 < a\nb_0 : 0 < b\na_top : a ≠ ⊤\na' : EReal... | obtain ⟨n, n_M', ⟨un_bn, _⟩, k, an_k, k_an'⟩ := frequently_atTop.1 u_b M' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 128,
"column": 8
} | {
"line": 128,
"column": 25
} | {
"line": 128,
"column": 25
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u_2\nB : ι → Set α\nt : Set ι\nδ : ι → ℝ\nτ : ℝ\nhτ : 1 < τ\nδnonneg : ∀ a ∈ t, 0 ≤ δ a\nR : ℝ\nδle : ∀ a ∈ t, δ a ≤ R\nhne : ∀ a ∈ t, (B a).Nonempty\nT : Set (Set ι) :=\n {u |\n u ⊆ t ∧\n u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c... | [
"case refine_1\nα : Type u_1\nι : Type u_2\nB : ι → Set α\nt : Set ι\nδ : ι → ℝ\nτ : ℝ\nhτ : 1 < τ\nδnonneg : ∀ a ∈ t, 0 ≤ δ a\nR : ℝ\nδle : ∀ a ∈ t, δ a ≤ R\nhne : ∀ a ∈ t, (B a).Nonempty\nT : Set (Set ι) :=\n {u |\n u ⊆ t ∧\n u.PairwiseDisjoint B ∧ ∀ a ∈ t, ∀ b ∈ u, (B a ∩ B b).Nonempty → ∃ c ∈ u, (B a ∩... | insert_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 44
} | {
"line": 53,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), μ s ≠ ∞\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : S... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), μ s ≠ ∞\nhν : ∀ (s : Set α), ν s ≠ ∞\n⊢ ∃ s,\n MeasurableSet s ∧... | have hν : ∀ s, ν s ≠ ∞ := measure_ne_top ν | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 95
} | {
"line": 177,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\nr : ℝ≥0\nhmeas : Measurable (μ.rnDeriv ν)\nhsing : μ.singularPart ν ⟂ₘ ν\nhadd : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)\nhr : ¬r = 0\n⊢ μ = (μ.singularPart ν, r⁻¹ • μ.rnDeriv ν).1 + ... | [
"case neg\nα : Type u_1\nm : MeasurableSpace α\nμ✝ ν✝ μ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\nr : ℝ≥0\nhmeas : Measurable (μ.rnDeriv ν)\nhsing : μ.singularPart ν ⟂ₘ ν\nhadd : μ = μ.singularPart ν + ν.withDensity (μ.rnDeriv ν)\nhr : ¬r = 0\nthis : r⁻¹ • μ.rnDeriv ν = ↑r⁻¹ • μ.rnDeriv ν\n⊢ μ = (μ.sing... | have : r⁻¹ • rnDeriv μ ν = ((r⁻¹ : ℝ≥0) : ℝ≥0∞) • rnDeriv μ ν := by simp [ENNReal.smul_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Average | {
"line": 728,
"column": 93
} | {
"line": 730,
"column": 63
} | {
"line": 732,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ninst✝ : IsProbabilityMeasure μ\nhf : AEMeasurable f μ\n⊢ 0 < μ {x | f x ≤ ∫⁻ (a : α), f a ∂μ}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"MeasureTheo... | [] | by
simpa only [laverage_eq_lintegral] using
measure_le_laverage_pos (IsProbabilityMeasure.ne_zero μ) hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 599,
"column": 2
} | {
"line": 600,
"column": 64
} | {
"line": 602,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : μ.HaveLebesgueDecomposition ν\ninst✝ : SigmaFinite ν\ns : Set α\nhs : MeasurableSet s\n⊢ μ.restrict s = (μ.restrict s).singularPart ν + ν.withDensity (s.indicator (μ.rnDeriv ν))",
"ppTerm": "?m.52",
"assigned": true,
"usedConsta... | [] | rw [singularPart_restrict _ _ hs, withDensity_indicator hs, ← restrict_withDensity hs,
← Measure.restrict_add, ← μ.haveLebesgueDecomposition_add ν] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 732,
"column": 35
} | {
"line": 736,
"column": 51
} | {
"line": 737,
"column": 6
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ¬μ ⟂ₘ ν\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ) (t : Set α), MeasurableSet t → ((1 / (↑n + 1)) • ν) (t ∩ f n) ≤ μ (t ∩ f n)\nhf₃ : ∀ (n : ℕ) (t : Set α), Measur... | [] | by
by_contra h
have h' := this (μA / 2) (half_pos (zero_lt_iff.2 h))
rw [← @Classical.not_not (μA ≤ μA / 2)] at h'
exact h' (not_le.2 (NNReal.half_lt_self h)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 1004,
"column": 2
} | {
"line": 1005,
"column": 37
} | {
"line": 1006,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhr : r ≠ ∞\n⊢ (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"instHSMul",
"MeasureTheory.Me... | [
"α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhr : r ≠ ∞\nh : (r.toNNReal • ν).rnDeriv μ =ᵐ[μ] r.toNNReal • ν.rnDeriv μ\n⊢ (r • ν).rnDeriv μ =ᵐ[μ] r • ν.rnDeriv μ"
] | have h : (r.toNNReal • ν).rnDeriv μ =ᵐ[μ] r.toNNReal • ν.rnDeriv μ :=
rnDeriv_smul_left' ν μ r.toNNReal | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 271,
"column": 94
} | {
"line": 408,
"column": 72
} | {
"line": 410,
"column": 0
} | [
{
"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\n⊢ ∃ a b,\n Measura... | [] | by
/- Here is a rough sketch, assuming that the measure is finite and the limit is well defined
everywhere. Let `u := {x | v.limRatio ρ x < p}` and `w := {x | q < v.limRatio ρ x}`. They
have measurable supersets `u'` and `w'` of the same measure. We will show that these satisfy
the conclusion of the theor... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 134,
"column": 4
} | {
"line": 135,
"column": 57
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case inr\nα : 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 ∩ uIcc a b → ∀ ⦃y : α⦄, y ∈ s ∩ uIcc a b → edist (f x) (f... | [] | rw [uIcc_of_ge ba]
exact variationOnFromTo.eq_zero_iff_of_ge hf ha hb ba | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 134,
"column": 4
} | {
"line": 135,
"column": 57
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case inr\nα : 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 ∩ uIcc a b → ∀ ⦃y : α⦄, y ∈ s ∩ uIcc a b → edist (f x) (f... | [] | rw [uIcc_of_ge ba]
exact variationOnFromTo.eq_zero_iff_of_ge hf ha hb ba | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.VariationOnFromTo | {
"line": 169,
"column": 4
} | {
"line": 170,
"column": 100
} | {
"line": 172,
"column": 0
} | [] | [] | f b - f c ≤ |f c - f b| := by grw [le_abs_self (f b - f c), abs_sub_comm (f b) (f c)]
_ ≤ variationOnFromTo f s a c - variationOnFromTo f s a b := abs_sub_le_sub_of_le hf as bs cs bc | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 52
} | {
"line": 395,
"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... | [] | simp [cfc_apply f a, cfcHom_map_spectrum (p := p)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 52
} | {
"line": 395,
"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... | [] | simp [cfc_apply f a, cfcHom_map_spectrum (p := p)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 52
} | {
"line": 395,
"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... | [] | simp [cfc_apply f a, cfcHom_map_spectrum (p := p)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 713,
"column": 2
} | {
"line": 713,
"column": 27
} | {
"line": 714,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : BoundedVariationOn f s\nL : Filter α\nhL : ∀ y ∈ s, s ∩ Ici y ∈ L\nx₀ : α\nhx₀ : x₀ ∈ s\nε : ℝ≥0∞\nεpos : ε > 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ < ε\nH : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ ... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nhf : BoundedVariationOn f s\nL : Filter α\nhL : ∀ y ∈ s, s ∩ Ici y ∈ L\nx₀ : α\nhx₀ : x₀ ∈ s\nε : ℝ≥0∞\nεpos : ε > 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ : δ < ε\nH : ∃ᶠ (x : α) in L, ε ≤ eVariationOn f (s ∩ Ici x)\ny : ... | let v (n : ℕ) := y^[n] x₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 798,
"column": 4
} | {
"line": 798,
"column": 15
} | {
"line": 799,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝⁴ : LinearOrder α\nE : Type u_2\ninst✝³ : PseudoEMetricSpace E\ninst✝² : CompleteSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : α → E\ns : Set α\nhf : BoundedVariationOn f s\nx : α\nhs : s ∩ Iio x = ∅\n⊢ Nonempty E",
"ppTerm": "?inl",
"assigned":... | [] | exact ⟨f x⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 60
} | {
"line": 124,
"column": 4
} | [
{
"pp": "case refine_2.e_a.e_a\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ (n - ν) * (n + 1).choose (ν + 1) = (n + 1) * n.choose (ν + 1)",
"ppTerm": "?refine_2.e_a.e_a",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"Nat.choose_mul_succ_eq",
"HMul.hMul",
... | [
"case e'_2\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ (n - ν) * (n + 1).choose (ν + 1) = (n + 1).choose (ν + 1) * (n + 1 - (ν + 1))",
"case e'_3\nR : Type u_1\ninst✝ : CommRing R\nn ν : ℕ\n⊢ (n + 1) * n.choose (ν + 1) = n.choose (ν + 1) * (n + 1)"
] | convert! (Nat.choose_mul_succ_eq n (ν + 1)).symm using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.ContinuousMap.Polynomial | {
"line": 177,
"column": 4
} | {
"line": 201,
"column": 18
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case mp\na b : ℝ\nh : a < b\nf : C(↑(Set.Icc a b), ℝ)\n⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions I) →\n f ∈ polynomialFunctions (Set.Icc a b)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono... | [] | rintro ⟨p, ⟨-, w⟩⟩
rw [DFunLike.ext_iff] at w
dsimp at w
let q := p.comp ((b - a)⁻¹ • Polynomial.X + Polynomial.C (-a * (b - a)⁻¹))
refine ⟨q, ⟨?_, ?_⟩⟩
· simp
· ext x
simp only [q, neg_mul, map_neg, map_mul, AlgHom.coe_toRingHom,
Polynomial.eval_X, Polynomial.eval_neg, Polynomial.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Polynomial | {
"line": 177,
"column": 4
} | {
"line": 201,
"column": 18
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case mp\na b : ℝ\nh : a < b\nf : C(↑(Set.Icc a b), ℝ)\n⊢ f ∈ Subalgebra.comap (compRightAlgHom ℝ ℝ ↑(iccHomeoI a b h).symm) (polynomialFunctions I) →\n f ∈ polynomialFunctions (Set.Icc a b)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono... | [] | rintro ⟨p, ⟨-, w⟩⟩
rw [DFunLike.ext_iff] at w
dsimp at w
let q := p.comp ((b - a)⁻¹ • Polynomial.X + Polynomial.C (-a * (b - a)⁻¹))
refine ⟨q, ⟨?_, ?_⟩⟩
· simp
· ext x
simp only [q, neg_mul, map_neg, map_mul, AlgHom.coe_toRingHom,
Polynomial.eval_X, Polynomial.eval_neg, Polynomial.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 993,
"column": 4
} | {
"line": 993,
"column": 54
} | {
"line": 994,
"column": 4
} | [
{
"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 g : α → E\ns : Set α\nhg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f\nn : ℕ\nu : ℕ → α\nu_mono : StrictMonoOn u (Iic n)\nu_mem : ∀ i ∈ Iic n, u i ∈ s\nthis✝¹ : Nonemp... | [
"α : Type u_1\ninst✝³ : LinearOrder α\nE : Type u_2\ninst✝² : PseudoEMetricSpace E\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf g : α → E\ns : Set α\nhg : ∀ x ∈ s, MapClusterPt (g x) (𝓝[s] x) f\nn : ℕ\nu : ℕ → α\nu_mono : StrictMonoOn u (Iic n)\nu_mem : ∀ i ∈ Iic n, u i ∈ s\nthis✝¹ : Nonempty α\nc : ℝ≥... | let f' i := if i ∈ Iic n then f (v i) else g (u i) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.SpecialFunctions.Bernstein | {
"line": 121,
"column": 8
} | {
"line": 121,
"column": 24
} | {
"line": 121,
"column": 24
} | [
{
"pp": "case e'_2\nn : ℕ\nhn : n ≠ 0\nx : ↑I\n⊢ ↑n ^ 2 * ∑ x_1, (↑x * ↑n - ↑↑x_1) ^ 2 * ↑(n.choose ↑x_1) * ↑x ^ ↑x_1 * (1 - ↑x) ^ (n - ↑x_1) / ↑n ^ 2 =\n ∑ x_1, (↑x * ↑n - ↑↑x_1) ^ 2 * ↑(n.choose ↑x_1) * ↑x ^ ↑x_1 * (1 - ↑x) ^ (n - ↑x_1)",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
... | [
"case e'_2\nn : ℕ\nhn : n ≠ 0\nx : ↑I\n⊢ ↑n ^ 2 * ((∑ i, (↑x * ↑n - ↑↑i) ^ 2 * ↑(n.choose ↑i) * ↑x ^ ↑i * (1 - ↑x) ^ (n - ↑i)) / ↑n ^ 2) =\n ∑ x_1, (↑x * ↑n - ↑↑x_1) ^ 2 * ↑(n.choose ↑x_1) * ↑x ^ ↑x_1 * (1 - ↑x) ^ (n - ↑x_1)"
] | ← Finset.sum_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 850,
"column": 4
} | {
"line": 850,
"column": 62
} | {
"line": 851,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : Semifield R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : Algebra R A\ninst✝ : ContinuousFunctionalCalculus R A p\n... | [
"R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : Semifield R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : Algebra R A\ninst✝ : ContinuousFunctionalCalculus R A p\na : A\nh_cpc... | have h_cpct : CompactSpace (spectrum R a) := inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 19
} | {
"line": 307,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nw✝ : A.SeparatesPoints\nf : C(X, ℝ)\nε : ℝ\npos : 0 < ε\nw : ∀ (i : ℝ), 0 < i → ∃ x ∈ Metric.ball f i, x ∈ ↑A.toSubsemiring\ng : C(X, ℝ)\nH : ‖g - f‖ < ε\nm : g ∈ ↑A.toSubsemiring\n⊢ ∃ g, ‖↑g - f‖ < ε",
"pp... | [] | exact ⟨⟨g, m⟩, H⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 89
} | {
"line": 412,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\nI : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM\n⊢ (↑I).range ≤ (Submodule.restrictScalars ℝ (Subalgebra.toSub... | [
"𝕜 : Type u_1\nX : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : StarSubalgebra 𝕜 C(X, 𝕜)\nhA : A.SeparatesPoints\nI : C(X, ℝ) →L[ℝ] C(X, 𝕜) := ContinuousLinearMap.compLeftContinuous ℝ X ofRealCLM\nA₀ : Submodule ℝ C(X, ℝ) := Submodule.comap (↑I) (Submodule.restrictScala... | let A₀ : Submodule ℝ C(X, ℝ) := (A.toSubmodule.restrictScalars ℝ).comap I.toLinearMap | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unique | {
"line": 307,
"column": 19
} | {
"line": 319,
"column": 35
} | {
"line": 320,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : Zero X\nA : Type u_2\ninst✝⁴ : NonUnitalRing A\ninst✝³ : StarRing A\ninst✝² : Module ℝ A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsSemitopologicalRing A\nφ : C(X, ℝ≥0)₀ →⋆ₙₐ[ℝ≥0] A\nr : ℝ\nf : C(X, ℝ)₀\n⊢ φ (r • f).toNNReal - φ (-(r • f)).toNNReal = (Mo... | [] | by
simp only [MonoidHom.id_apply]
by_cases! hr : 0 ≤ r
· lift r to ℝ≥0 using hr
simp only [← smul_def, toNNReal_smul, map_smul, toNNReal_neg_smul, smul_sub]
· rw [← neg_pos] at hr
rw [← neg_smul]
nth_rw 1 [← neg_neg r]
nth_rw 3 [← neg_neg r]
lift -r to ℝ≥0 using hr.le with ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 588,
"column": 4
} | {
"line": 588,
"column": 50
} | {
"line": 589,
"column": 4
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ns : Set 𝕜\nh0 : 0 ∈ s\nf : C(↑s, 𝕜)\n⊢ f ∈ ↑(adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) →\n f ∈ ↑(StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)}) ∩ ↑(RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩))",
"ppTerm": "?mpr",
"assigned": true,
... | [
"case mpr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ns : Set 𝕜\nh0 : 0 ∈ s\nf : C(↑s, 𝕜)\n⊢ f ∈ adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)} →\n f ∈ StarAlgebra.adjoin 𝕜 {restrict s (ContinuousMap.id 𝕜)} ∧ f ∈ RingHom.ker (evalStarAlgHom 𝕜 𝕜 ⟨0, h0⟩)"
] | simp only [Set.mem_inter_iff, SetLike.mem_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.Unitization | {
"line": 174,
"column": 4
} | {
"line": 175,
"column": 94
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : DenselyNormedField 𝕜\ninst✝⁸ : NonUnitalNormedRing E\ninst✝⁷ : StarRing E\ninst✝⁶ : CStarRing E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : IsScalarTower 𝕜 E E\ninst✝³ : SMulCommClass 𝕜 E E\ninst✝² : StarRing 𝕜\ninst✝¹ : StarModule 𝕜 E\ninst✝ : CStarRing 𝕜... | [] | · replace h := h.le
rw [sq, sq, sup_eq_left.mpr h, sup_eq_left.mpr (mul_self_le_mul_self (norm_nonneg _) h)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 19
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf g : α → Set β\nx : α\nhf : UpperHemicontinuousWithinAt f univ x\nhg : UpperHemicontinuousWithinAt g univ x\n⊢ UpperHemicontinuousWithinAt (fun x ↦ f x ∪ g x) univ x",
"ppTerm": "?m.46",
"assigned": true,
... | [] | exact hf.union hg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 111,
"column": 2
} | {
"line": 122,
"column": 9
} | {
"line": 124,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\na : A\nha : ... | [] | have h₁ := (continuous_cfcₙAux hp₁ a ha).range_subset_closure_image_dense
(ContinuousMapZero.adjoin_id_dense (σₙ 𝕜 a)) ⟨f, rfl⟩
rw [← SetLike.mem_coe]
refine closure_minimal ?_ ?_ h₁
· rw [← NonUnitalStarSubalgebra.coe_map, SetLike.coe_subset_coe, NonUnitalStarSubalgebra.map_le]
apply NonUnitalStarAlgebr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 111,
"column": 2
} | {
"line": 122,
"column": 9
} | {
"line": 124,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : StarRing A\ninst✝⁵ : NormedSpace 𝕜 A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\na : A\nha : ... | [] | have h₁ := (continuous_cfcₙAux hp₁ a ha).range_subset_closure_image_dense
(ContinuousMapZero.adjoin_id_dense (σₙ 𝕜 a)) ⟨f, rfl⟩
rw [← SetLike.mem_coe]
refine closure_minimal ?_ ?_ h₁
· rw [← NonUnitalStarSubalgebra.coe_map, SetLike.coe_subset_coe, NonUnitalStarSubalgebra.map_le]
apply NonUnitalStarAlgebr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 533,
"column": 15
} | {
"line": 533,
"column": 65
} | {
"line": 534,
"column": 2
} | [
{
"pp": "𝕜 : Type u_3\nA : Type u_4\nSA : Type u_5\ninst✝⁵ : NormedRing A\ninst✝⁴ : CompleteSpace A\ninst✝³ : SetLike SA A\ninst✝² : SubringClass SA A\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedAlgebra 𝕜 A\ninstSMulMem : SMulMemClass SA 𝕜 A\nS : SA\nhS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSp... | [] | by rw [← Filter.Tendsto, ← ContinuousAt]; fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 253,
"column": 44
} | {
"line": 254,
"column": 66
} | {
"line": 256,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\na : A\nf : C(↑(σₙ 𝕜 a), 𝕜)₀\nha :... | [] | by
refine isometry_cfcₙHom a |>.norm_map_of_map_zero (map_zero _) f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Isometric | {
"line": 348,
"column": 42
} | {
"line": 349,
"column": 71
} | {
"line": 351,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\na : A\nha : p a\n⊢ IsGreatest ((fun... | [] | by
simpa only [cfcₙ_id 𝕜 a] using! IsGreatest.nnnorm_cfcₙ (id : 𝕜 → 𝕜) a | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.AbsConvex | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 9
} | {
"line": 335,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nr : ℝ\nhr : ‖r‖ ≤ 1\ny : E\nhy : y ∈ s\nha : (fun x ↦ r • x) y ∈ balancedHull ℝ s\nthis✝ : 0 ≤ 1 + r\nthis : 0 ≤ 1 - r\n⊢ ((1 + r) / 2 - (1 - r) / 2) • y = (fun x ↦ r • x) y",
"ppTerm": "?m.206",
"assigned": true,
"usedCo... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Normed.Operator.Completeness | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 57
} | {
"line": 53,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nF : Type u_4\nFₗ : Type u_5\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup Fₗ\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSp... | [
"case refine_2\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nF : Type u_4\nFₗ : Type u_5\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedAddCommGroup Fₗ\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NontriviallyNormedField 𝕜₂\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedSpace 𝕜₂ F\ni... | rcases isBounded_iff_forall_norm_le.1 hs with ⟨C, hC⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.LocallyConvex.Barrelled | {
"line": 141,
"column": 30
} | {
"line": 141,
"column": 58
} | {
"line": 144,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nκ : Type u_3\n𝕜₁ : Type u_4\n𝕜₂ : Type u_5\nE : Type u_6\nF : Type u_7\ninst✝¹⁰ : NontriviallyNormedField 𝕜₁\ninst✝⁹ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜₁ →+* 𝕜₂\ninst✝⁸ : RingHomIsometric σ₁₂\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : Module 𝕜₁ E\ni... | [] | convert! interior_subset hxn | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.UrysohnsLemma | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 49
} | {
"line": 337,
"column": 50
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : NormalSpace X\ns t : Set X\nhs : IsClosed[inst✝¹] s\nht : IsClosed[inst✝¹] t\nhd : Disjoint s t\nP : Set X → Set X → Prop := fun x x_1 ↦ True\nc u : Set X\nc_closed : IsClosed[inst✝¹] c\nu_open : IsOpen[inst✝¹] u\ncu : c ⊆ u\nv : Set X\nv_open : IsOpen... | [] | exact ⟨v, v_open, cv, hv, trivial, trivial⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 60
} | {
"line": 130,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Ring k\ninst✝⁶ : LinearOrder k\ninst✝⁵ : IsStrictOrderedRing k\ninst✝⁴ : AddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module k E\ninst✝ : IsStrictOrderedModule k E\na b : E\nr : k\nh : a < b\n⊢ a < (lineMap a b) r ↔ 0 < r",
"p... | [] | rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 60
} | {
"line": 130,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Ring k\ninst✝⁶ : LinearOrder k\ninst✝⁵ : IsStrictOrderedRing k\ninst✝⁴ : AddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module k E\ninst✝ : IsStrictOrderedModule k E\na b : E\nr : k\nh : a < b\n⊢ a < (lineMap a b) r ↔ 0 < r",
"p... | [] | rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 60
} | {
"line": 130,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Ring k\ninst✝⁶ : LinearOrder k\ninst✝⁵ : IsStrictOrderedRing k\ninst✝⁴ : AddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module k E\ninst✝ : IsStrictOrderedModule k E\na b : E\nr : k\nh : a < b\n⊢ a < (lineMap a b) r ↔ 0 < r",
"p... | [] | rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 60
} | {
"line": 133,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Ring k\ninst✝⁶ : LinearOrder k\ninst✝⁵ : IsStrictOrderedRing k\ninst✝⁴ : AddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module k E\ninst✝ : IsStrictOrderedModule k E\na b : E\nr : k\nh : a < b\n⊢ (lineMap a b) r < a ↔ r < 0",
"p... | [] | rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 60
} | {
"line": 133,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Ring k\ninst✝⁶ : LinearOrder k\ninst✝⁵ : IsStrictOrderedRing k\ninst✝⁴ : AddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module k E\ninst✝ : IsStrictOrderedModule k E\na b : E\nr : k\nh : a < b\n⊢ (lineMap a b) r < a ↔ r < 0",
"p... | [] | rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 60
} | {
"line": 133,
"column": 0
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁷ : Ring k\ninst✝⁶ : LinearOrder k\ninst✝⁵ : IsStrictOrderedRing k\ninst✝⁴ : AddCommGroup E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module k E\ninst✝ : IsStrictOrderedModule k E\na b : E\nr : k\nh : a < b\n⊢ (lineMap a b) r < a ↔ r < 0",
"p... | [] | rw [← lineMap_lt_lineMap_iff_of_lt' h, lineMap_apply_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 49
} | {
"line": 311,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : PseudoEMetricSpace γ\n𝔖 𝔗 : Set (Set α)\ninst✝² : Finite ↑𝔖\ninst✝¹ : PseudoMetricSpace β\ninst✝ : BoundedSpace β\nf : α →ᵤ[𝔖] β\n⊢ ∃ y, (ofFun 𝔖) (UniformFun.toFun y) = f",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
... | [] | exact ⟨UniformFun.ofFun (toFun 𝔖 f), by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 371,
"column": 73
} | {
"line": 377,
"column": 63
} | {
"line": 379,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : NonUnitalRing A\ninst✝⁸ : Module ℂ A\ninst✝⁷ : SMulCommClass ℂ A A\ninst✝⁶ : IsScalarTower ℂ A A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarModule ℂ A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : PartialOrder A\ninst✝ : StarOrde... | [] | by
refine eq_top_iff.mpr fun x _ => ?_
rw [← CStarAlgebra.linear_combination_nonneg x]
apply_rules [sub_mem, Submodule.smul_mem, add_mem]
all_goals
refine subset_span ?_
first | apply CFC.negPart_nonneg | apply CFC.posPart_nonneg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 82,
"column": 59
} | {
"line": 85,
"column": 66
} | {
"line": 87,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedAlgebra ℝ A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\nr : ℝ\n⊢ Con... | [] | by
refine continuousOn_id.cfc_nnreal_of_mem_nhdsSet _ (s := {0}ᶜ) ?_
simp_rw [nhdsSet_iUnion, Filter.mem_iSup, isOpen_compl_singleton.mem_nhdsSet]
exact fun a ha ↦ by simpa using spectrum.zero_notMem _ ha.isUnit | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 439,
"column": 41
} | {
"line": 450,
"column": 22
} | {
"line": 452,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : 0 ≤ a\n⊢ ∀ x ∈ σₙ ℝ a, 0 ≤ x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.range_comp",
"Eq.mpr",
"NonUnitalCStarAlgebra.toStarModule",
"U... | [] | by
rw [Unitization.quasispectrum_eq_spectrum_inr' _ ℂ]
-- should this actually be an instance on the `Unitization`? (probably scoped)
let _ := CStarAlgebra.spectralOrder A⁺¹
have := CStarAlgebra.spectralOrderedRing A⁺¹
apply spectrum_nonneg_of_nonneg
rw [StarOrderedRing.nonneg_iff] at ha ⊢
h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 419,
"column": 71
} | {
"line": 424,
"column": 7
} | {
"line": 426,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : Ring A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na :... | [] | by
rw [CFC.rpow_def, cfc_nnreal_eq_real ..]
refine cfc_congr ?_
intro x hx
simp only [NNReal.coe_rpow, Real.coe_toNNReal']
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 485,
"column": 25
} | {
"line": 486,
"column": 50
} | {
"line": 488,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : Ring A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na :... | [] | by
simpa using rpow_rpow_inv a x⁻¹ (inv_ne_zero hx) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 183,
"column": 30
} | {
"line": 188,
"column": 8
} | {
"line": 190,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsSelfAdjoint a\n⊢ a ≤ (algebraMap ℝ A) ‖a‖",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Norm.norm",
"spectrum.norm_le_norm_of_mem",
"Re... | [] | by
by_cases! nontriv : Nontrivial A
· refine le_algebraMap_of_spectrum_le fun r hr => ?_
calc r ≤ ‖r‖ := Real.le_norm_self r
_ ≤ ‖a‖ := spectrum.norm_le_norm_of_mem hr
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 90
} | {
"line": 710,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : Ring A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na :... | [] | rw [sqrt_eq_rpow, rpow_rpow_of_exponent_nonneg _ _ _ (by simp) h₁, one_div_mul_eq_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Matrix.Normed | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 49
} | {
"line": 96,
"column": 0
} | [
{
"pp": "m : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : SeminormedAddCommGroup α\nr : ℝ\nhr : 0 ≤ r\nA : Matrix m n α\n⊢ ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedA... | [] | simp_rw [norm_def, pi_norm_le_iff_of_nonneg hr] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Matrix.Normed | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 49
} | {
"line": 96,
"column": 0
} | [
{
"pp": "m : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : SeminormedAddCommGroup α\nr : ℝ\nhr : 0 ≤ r\nA : Matrix m n α\n⊢ ‖A‖ ≤ r ↔ ∀ (i : m) (j : n), ‖A i j‖ ≤ r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedA... | [] | simp_rw [norm_def, pi_norm_le_iff_of_nonneg hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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