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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 75
} | {
"line": 170,
"column": 0
} | [
{
"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⊢ f.filtration i ⊔ ⨆ c, (↑(P.p c.s)).subcomplex ≤ f.filtration (Order.succ i)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
... | [] | · simp only [sup_le_iff, iSup_le_iff]
exact ⟨f.filtration_monotone (Order.le_succ i),
fun c ↦ f.subcomplex_le_filtration _ (Order.lt_succ_of_not_isMax hi)⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.RelativeCellComplex | {
"line": 313,
"column": 61
} | {
"line": 318,
"column": 48
} | {
"line": 320,
"column": 0
} | [
{
"pp": "X : SSet\nA : X.Subcomplex\nP : A.Pairing\nι : Type v\ninst✝¹ : LinearOrder ι\nf : P.RankFunction ι\ninst✝ : P.IsProper\nj : ι\nc : f.Cell j\n⊢ Mono c.ιSigmaStdSimplex",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
... | [] | by
rw [NatTrans.mono_iff_mono_app]
rintro ⟨⟨d⟩⟩
rw [mono_iff_injective]
intro x y h
simpa [f.ιSigmaStdSimplex_eq_iff] using h.symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Presentable.Limits | {
"line": 87,
"column": 6
} | {
"line": 88,
"column": 51
} | {
"line": 89,
"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... | let ψ (f : Arrow K) : j₀ ⟶ IsCardinalFiltered.max j'' hK :=
g f.hom ≫ IsCardinalFiltered.toMax j'' hK f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Functor.KanExtension.Dense | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 41
} | {
"line": 128,
"column": 0
} | [
{
"pp": "C : Type u₁\nD : Type u₂\ninst✝³ : Category.{v₁, u₁} C\ninst✝² : Category.{v₂, u₂} D\nC' : Type u₃\ninst✝¹ : Category.{v₃, u₃} C'\nF : C ⥤ D\ninst✝ : F.IsDense\nY Z : D\nf : (restrictedULiftYoneda F).obj Y ⟶ (restrictedULiftYoneda F).obj Z\nc : Cocone (CostructuredArrow.proj F Y ⋙ F) :=\n { pt := Z,\n... | [] | simpa using ULift.down_injective this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Basic | {
"line": 167,
"column": 4
} | {
"line": 168,
"column": 27
} | {
"line": 169,
"column": 2
} | [
{
"pp": "X✝ Y✝ Z✝ : SSet\ne : X✝ ⟶ Y✝\nx✝ : isomorphisms SSet e\nf : Y✝ ⟶ Z✝\nleft✝ : Mono f\nP : (Subcomplex.range f).Pairing\nhP : P.IsRegular\n⊢ (Subcomplex.range f).preimage (Iso.refl Z✝).hom = Subcomplex.range (e ≫ f)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"CategoryTheo... | [] | simp [Subcomplex.range_comp, Subcomplex.range_eq_top e,
Subcomplex.image_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 360,
"column": 8
} | {
"line": 362,
"column": 13
} | {
"line": 363,
"column": 6
} | [
{
"pp": "case inl.inr.refine_1\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nhx : IsType₂ x\nd : ℕ\nhd : x.dim = d\nhx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)\ni : Fin d\nhi✝ : i.succ ≤ min x hd\nhi : i.succ = min x hd\n⊢ ((objEquiv (x.cast hd).simplex) i.castSucc).1 ≤ (k.c... | [] | dsimp
rw [simplex_fst_le_castSucc_iff]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 360,
"column": 8
} | {
"line": 362,
"column": 13
} | {
"line": 363,
"column": 6
} | [
{
"pp": "case inl.inr.refine_1\nm : ℕ\nk : Fin (m + 1)\nn : ℕ\nx : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nhx : IsType₂ x\nd : ℕ\nhd : x.dim = d\nhx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)\ni : Fin d\nhi✝ : i.succ ≤ min x hd\nhi : i.succ = min x hd\n⊢ ((objEquiv (x.cast hd).simplex) i.castSucc).1 ≤ (k.c... | [] | dsimp
rw [simplex_fst_le_castSucc_iff]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.PiZero | {
"line": 175,
"column": 57
} | {
"line": 175,
"column": 70
} | {
"line": 175,
"column": 70
} | [
{
"pp": "X : SSet\nx✝ : X.IsPreconnected ∧ X.Nonempty\nleft✝ : X.IsPreconnected\nright✝ : X.Nonempty\n⊢ X.Nonempty",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"right✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd | {
"line": 569,
"column": 6
} | {
"line": 569,
"column": 37
} | {
"line": 570,
"column": 6
} | [
{
"pp": "case neg\nm✝ : ℕ\nk✝ : Fin (m✝ + 1)\nn✝ : ℕ\nx✝ : (Λ[m✝ + 1, k✝.castSucc].unionProd ∂Δ[n✝]).N\nd m : ℕ\nk : Fin (m + 1)\nn : ℕ\nx : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nhx : ∃ x_1, ∃ (h : x.dim = x_1), ∃ x_2, IsIndex x ⋯ x_2\n⊢ ∃ s,\n x.toS = { dim := s.d + 1, simplex := (s.x.cast ⋯).simplex } ... | [
"case neg.zero\nm✝ : ℕ\nk✝ : Fin (m✝ + 1)\nn✝ : ℕ\nx✝ : (Λ[m✝ + 1, k✝.castSucc].unionProd ∂Δ[n✝]).N\nd m : ℕ\nk : Fin (m + 1)\nn : ℕ\nx : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N\nhd : x.dim = 0\ni : Fin (0 + 1)\nhx : IsIndex x ⋯ i\n⊢ ∃ s,\n x.toS = { dim := s.d + 1, simplex := (s.x.cast ⋯).simplex } ∨\n x.... | obtain ⟨_ | d, hd, i, hx⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplexOne | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 79
} | {
"line": 78,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nh : j.castSucc < i\n⊢ (ConcreteCategory.hom (SimplicialObject.σ Δ[1] j)) (objMk₁ i) = objMk₁ i.succ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Opposite",
"Fin.succ",
"SimplexCategory.toMk₁",
"CategoryTheory.Con... | [] | ext k : 1
exact ConcreteCategory.congr_hom (SimplexCategory.σ_comp_toMk₁_of_lt _ _ h) k | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicTopology.SimplicialSet.StdSimplexOne | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 79
} | {
"line": 78,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 2)\nj : Fin (n + 1)\nh : j.castSucc < i\n⊢ (ConcreteCategory.hom (SimplicialObject.σ Δ[1] j)) (objMk₁ i) = objMk₁ i.succ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Opposite",
"Fin.succ",
"SimplexCategory.toMk₁",
"CategoryTheory.Con... | [] | ext k : 1
exact ConcreteCategory.congr_hom (SimplexCategory.σ_comp_toMk₁_of_lt _ _ h) k | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicTopology.SimplicialSet.Monomorphisms | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 65
} | {
"line": 40,
"column": 4
} | [
{
"pp": "J : Type u\nx✝¹ : Category.{u, u} J\nx✝ : IsFiltered J\n⊢ (monomorphisms SSet).IsStableUnderColimitsOfShape J",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.Functor.category",
"CategoryTheory.MorphismProperty.monomorphisms",
"id... | [
"J : Type u\nx✝¹ : Category.{u, u} J\nx✝ : IsFiltered J\n⊢ (monomorphisms (SimplexCategoryᵒᵖ ⥤ Type u)).IsStableUnderColimitsOfShape J"
] | change (monomorphisms (_ ⥤ _)).IsStableUnderColimitsOfShape J | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 110,
"column": 4
} | {
"line": 113,
"column": 43
} | {
"line": 114,
"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 ... | [] | rw [← Functor.map_comp_apply, ← op_comp, δ₂_two_comp_σ₂_zero, op_comp,
Functor.map_comp_apply, hσ, SimplexCategory.mkOfSucc_zero_eq_δ,
← Functor.map_comp_apply, ← op_comp, δ₂_two_comp_σ₂_zero,
op_comp, Functor.map_comp_apply, hδ₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Order | {
"line": 224,
"column": 86
} | {
"line": 227,
"column": 63
} | {
"line": 229,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf g : 𝕜 → E\nz₀ : 𝕜\nhfg : analyticOrderAt f z₀ ≠ analyticOrderAt g z₀\n⊢ analyticOrderAt (f + g) z₀ = min (analyticOrderAt f z₀) (analyticOrderAt g z₀)",
"ppTerm": "?m.36",
... | [] | by
obtain hfg | hgf := hfg.lt_or_gt
· simpa [hfg.le] using analyticOrderAt_add_eq_left_of_lt hfg
· simpa [hgf.le] using analyticOrderAt_add_eq_right_of_lt hgf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction | {
"line": 296,
"column": 6
} | {
"line": 297,
"column": 35
} | {
"line": 298,
"column": 6
} | [
{
"pp": "X : Truncated 2\nC D : Type u\ninst✝¹ : SmallCategory C\ninst✝ : SmallCategory D\nφ : X ⟶ (truncation 2).obj (nerve C)\nx₀ x₁ : X.obj (op { obj := ⦋0⦌, property := Edge._proof_1 })\nf : Edge x₀ x₁\n⊢ (ConcreteCategory.hom ((homToNerveMk (descOfTruncation φ)).app (op { obj := ⦋1⦌, property := ⋯ }))) f.e... | [
"X : Truncated 2\nC D : Type u\ninst✝¹ : SmallCategory C\ninst✝ : SmallCategory D\nφ : X ⟶ (truncation 2).obj (nerve C)\nx₀ x₁ : X.obj (op { obj := ⦋0⦌, property := Edge._proof_1 })\nf : Edge x₀ x₁\n⊢ ComposableArrows.mk₁ (nerve.homEquiv (f.map φ)) =\n (ConcreteCategory.hom (φ.app (op { obj := ⦋1⦌, property := ⋯... | simp only [homToNerveMk_app_edge, descOfTruncation_obj_mk,
descOfTruncation_map_homMk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 117,
"column": 8
} | {
"line": 117,
"column": 29
} | {
"line": 117,
"column": 29
} | [
{
"pp": "case insert\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nι : Type u_4\ns✝ : Finset ι\nF : ι → 𝕜 → 𝕜'\nx : 𝕜\na : ι\ns : Finset ι\nha : a ∉ s\nhs : (∀ σ ∈ s, MeromorphicAt (F σ) x) → MeromorphicAt (∏ i ∈ s, F ... | [
"case insert\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nι : Type u_4\ns✝ : Finset ι\nF : ι → 𝕜 → 𝕜'\nx : 𝕜\na : ι\ns : Finset ι\nha : a ∉ s\nhs : (∀ σ ∈ s, MeromorphicAt (F σ) x) → MeromorphicAt (∏ i ∈ s, F i) x\nhf : ∀... | Finset.prod_insert ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 32
} | {
"line": 144,
"column": 4
} | [
{
"pp": "case empty\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nι : Type u_4\ns : Finset ι\nG : ι → 𝕜 → E\nx : 𝕜\nh : ∀ σ ∈ ∅, MeromorphicAt (G σ) x\n⊢ MeromorphicAt (∑ n ∈ ∅, G n) x",
"ppTerm": "?empty",
"assigned": true,... | [
"case empty\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nι : Type u_4\ns : Finset ι\nG : ι → 𝕜 → E\nx : 𝕜\nh : ∀ σ ∈ ∅, MeromorphicAt (G σ) x\n⊢ MeromorphicAt 0 x"
] | simp only [Finset.sum_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 341,
"column": 4
} | {
"line": 341,
"column": 30
} | {
"line": 343,
"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\nf' : E ≃L[𝕜] F\nc : ℝ≥0\ninst✝ : CompleteSpace E\nhf : ApproximatesLinearOn f (↑f') univ c\nhc ... | [] | exact fun R h y hy => h hy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 45
} | {
"line": 285,
"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⊢ MeromorphicAt f⁻¹ x",
"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"
] | refine (MeromorphicAt.const 0 x).congr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 302,
"column": 6
} | {
"line": 302,
"column": 37
} | {
"line": 304,
"column": 0
} | [
{
"pp": "case inr\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\nn : ℕ\ng : 𝕜 → 𝕜'\nhg_an : A... | [] | simp [pow_succ', mul_smul, hfg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn | {
"line": 370,
"column": 4
} | {
"line": 382,
"column": 52
} | {
"line": 384,
"column": 0
} | [] | [] | ‖x' - y' - f'.symm (A x' - A y')‖ ≤ N * ‖f' (x' - y' - f'.symm (A x' - A y'))‖ :=
(f' : E →L[𝕜] F).bound_of_antilipschitz f'.antilipschitz _
_ = N * ‖A y' - A x' - f' (y' - x')‖ := by
congr 2
simp only [ContinuousLinearEquiv.apply_symm_apply, map_sub]
abel
_ ≤ N * (c * ‖y' - x'‖) := by ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 323,
"column": 15
} | {
"line": 323,
"column": 77
} | {
"line": 324,
"column": 2
} | [
{
"pp": "case ofNat\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'\nhf : MeromorphicAt f x\nm : ℕ\n⊢ MeromorphicAt (f ^ Int.ofNat m) x",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants... | [] | simpa only [Int.ofNat_eq_natCast, zpow_natCast] using hf.pow m | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 323,
"column": 15
} | {
"line": 323,
"column": 77
} | {
"line": 324,
"column": 2
} | [
{
"pp": "case ofNat\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'\nhf : MeromorphicAt f x\nm : ℕ\n⊢ MeromorphicAt (f ^ Int.ofNat m) x",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants... | [] | simpa only [Int.ofNat_eq_natCast, zpow_natCast] using hf.pow m | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 323,
"column": 15
} | {
"line": 323,
"column": 77
} | {
"line": 324,
"column": 2
} | [
{
"pp": "case ofNat\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'\nhf : MeromorphicAt f x\nm : ℕ\n⊢ MeromorphicAt (f ^ Int.ofNat m) x",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants... | [] | simpa only [Int.ofNat_eq_natCast, zpow_natCast] using hf.pow m | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Basic | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 58
} | {
"line": 333,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nn : ℕ\nh : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) x\nthis : ∀ᶠ (y : 𝕜) in 𝓝[≠] x, ContinuousAt (fun z ↦ (z - x) ^ n • f z) y\n⊢ ∀ᶠ (y : 𝕜) in 𝓝[≠] x, Con... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nn : ℕ\nh : AnalyticAt 𝕜 (fun z ↦ (z - x) ^ n • f z) x\nthis : ∀ᶠ (y : 𝕜) in 𝓝[≠] x, ContinuousAt (fun z ↦ (z - x) ^ n • f z) y\ny : 𝕜\nhy : ContinuousAt (fun z ↦ (z - x... | filter_upwards [this, self_mem_nhdsWithin] with y hy h'y | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Calculus.FDeriv.Extend | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 73
} | {
"line": 65,
"column": 6
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\ns : Set E\nx : E\nf' : E →L[ℝ] F\nf_diff : DifferentiableOn ℝ f s\ns_conv : Convex ℝ s\ns_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalS... | [
"E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\ns : Set E\nx : E\nf' : E →L[ℝ] F\nf_diff : DifferentiableOn ℝ f s\ns_conv : Convex ℝ s\ns_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nf_c... | have : B ∩ closure s ⊆ closure (B ∩ s) := isOpen_ball.inter_closure | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.Deriv.Prod | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 22
} | {
"line": 91,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nι : Type u_1\nE' : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E' i)\nφ : 𝕜 → (i : ι) → E' i\nh : ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x ↦ φ x i) s x\nhsx : UniqueDiff... | [
"case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nι : Type u_1\nE' : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (E' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (E' i)\nφ : 𝕜 → (i : ι) → E' i\nh : ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x ↦ φ x i) s x\nhsx : ¬UniqueDiffWithinAt 𝕜... | · rw [derivWithin, fderivWithin_pi h hsx]
simp [derivWithin] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.Order | {
"line": 591,
"column": 6
} | {
"line": 591,
"column": 28
} | {
"line": 592,
"column": 6
} | [
{
"pp": "case left.inr\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 ∈ ... | [
"case h\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 → f y... | use Subtype.val ⁻¹' t' | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Analytic.Order | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 26
} | {
"line": 609,
"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\nz : ↑U\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', f y = 0\nh₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t'\... | [
"case h\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\nt' : Set 𝕜\nh₁t' : ∀ y ∈ t', f y = 0\nh₂t' : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t'\nh₃t' : ↑z ∈ t'\... | use Subtype.val ⁻¹' t' | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 287,
"column": 10
} | {
"line": 287,
"column": 20
} | {
"line": 287,
"column": 21
} | [
{
"pp": "case e'_9\nx : ℝ\nhx✝ : x ≠ 0\nr : ℂ\nhr : r + 1 ≠ 0\nhx : x < 0\nthis✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1)) / (r + 1)\nthis : HasDerivAt (fun y ↦ ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x)\n⊢ -(r + 1) * ↑(-x) ^ r = -1 • ((r + 1) * ↑(-x) ^ r)",
"ppT... | [
"case e'_9\nx : ℝ\nhx✝ : x ≠ 0\nr : ℂ\nhr : r + 1 ≠ 0\nhx : x < 0\nthis✝ : ∀ᶠ (y : ℝ) in 𝓝 x, ↑y ^ (r + 1) / (r + 1) = (-↑y) ^ (r + 1) * cexp (↑π * I * (r + 1)) / (r + 1)\nthis : HasDerivAt (fun y ↦ ↑y ^ (r + 1)) ((r + 1) * ↑(-x) ^ r) (-x)\n⊢ -(r + 1) * ↑(-x) ^ r = ↑(-1) * ((r + 1) * ↑(-x) ^ r)"
] | real_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Deriv | {
"line": 728,
"column": 2
} | {
"line": 728,
"column": 60
} | {
"line": 729,
"column": 2
} | [
{
"pp": "f : ℝ → ℝ\nf' x : ℝ\ns : Set ℝ\na : ℝ\nha : 0 < a\nhf : HasDerivWithinAt f f' s x\n⊢ HasDerivWithinAt (fun x ↦ a ^ f x) (Real.log a * f' * a ^ f x) s x",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow"... | [
"f : ℝ → ℝ\nf' x : ℝ\ns : Set ℝ\na : ℝ\nha : 0 < a\nhf : HasDerivWithinAt f f' s x\n⊢ Real.log a * f' * a ^ f x = 0 * f x * a ^ (f x - 1) + f' * a ^ f x * Real.log a"
] | convert! (hasDerivWithinAt_const x s a).rpow hf ha using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Analytic.Binomial | {
"line": 264,
"column": 6
} | {
"line": 264,
"column": 34
} | {
"line": 264,
"column": 34
} | [
{
"pp": "a : ℝ\nthis :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)\n (FormalMultilinearSeries.restrictScalars ℝ\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))\n 0 1\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ a)\n (FormalMultilinearSeries.ofScalars ℝ fun n... | [
"a : ℝ\nthis :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ ↑a)\n (FormalMultilinearSeries.restrictScalars ℝ\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ Ring.choose (↑a + ↑n - 1) n))\n (Complex.ofRealCLM 0) 1\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (1 - x) ^ a)\n (FormalMultilinearSeries.ofScalars... | ← Complex.ofRealCLM.map_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Binomial | {
"line": 283,
"column": 6
} | {
"line": 283,
"column": 34
} | {
"line": 283,
"column": 34
} | [
{
"pp": "a : ℕ\nr : ℝ\nhr : r ≠ 0\nthis :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (↑r - x) ^ (a + 1))\n (FormalMultilinearSeries.restrictScalars ℝ\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (↑r ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)))\n 0 ‖↑r‖ₑ\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (r - x) ^ (a + 1... | [
"a : ℕ\nr : ℝ\nhr : r ≠ 0\nthis :\n HasFPowerSeriesOnBall (fun x ↦ 1 / (↑r - x) ^ (a + 1))\n (FormalMultilinearSeries.restrictScalars ℝ\n (FormalMultilinearSeries.ofScalars ℂ fun n ↦ (↑r ^ (n + a + 1))⁻¹ * ↑((a + n).choose a)))\n (Complex.ofRealCLM 0) ‖↑r‖ₑ\n⊢ HasFPowerSeriesOnBall (fun x ↦ 1 / (r - x) ... | ← Complex.ofRealCLM.map_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 271,
"column": 4
} | {
"line": 271,
"column": 36
} | {
"line": 272,
"column": 4
} | [
{
"pp": "u 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 ≠ ⊤",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"EReal",
"instTopEReal"... | [
"u 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 ≠ ⊥"
] | · exact Or.inr EReal.zero_ne_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 36
} | {
"line": 282,
"column": 4
} | [
{
"pp": "u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthSup v ≠ ⊥ ∨ 0 ≠ ⊤",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"EReal.instDivInvMonoid",
"EReal",
"instTopEReal"... | [
"u v : ℕ → EReal\nb : EReal\nhb : b ≠ ⊤\nh : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b\nb_bot : ⊥ < b\n⊢ linearGrowthSup v ≠ ⊤ ∨ 0 ≠ ⊥"
] | · exact Or.inr EReal.zero_ne_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Asymptotics.LinearGrowth | {
"line": 424,
"column": 8
} | {
"line": 424,
"column": 38
} | {
"line": 424,
"column": 38
} | [
{
"pp": "case inr\nu : ℕ → EReal\nh : Monotone u\nm : ℕ\nhm : ¬u m = ⊥ m\nm_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n\nhm' : u m ≠ ⊤\n⊢ 0 ≤ linearGrowthInf u",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMonoid",
... | [
"case inr\nu : ℕ → EReal\nh : Monotone u\nm : ℕ\nhm : ¬u m = ⊥ m\nm_n : ∀ᶠ (n : ℕ) in atTop, u m ≤ u n\nhm' : u m ≠ ⊤\n⊢ (linearGrowthInf fun x ↦ u m) ≤ linearGrowthInf u"
] | ← linearGrowthInf_const hm hm' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 118,
"column": 8
} | {
"line": 118,
"column": 47
} | {
"line": 119,
"column": 8
} | [
{
"pp": "α : 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 ∩ B ... | [
"α : 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 ∩ B c).Nonempty ... | rw [div_lt_iff₀ (zero_lt_one.trans hτ)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 23
} | {
"line": 98,
"column": 2
} | [
{
"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\nh_measure_sub_eq : μ - ν = measure_sub\n⊢ (μ - ν) s = μ s - ν s",
... | [
"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\nh_measure_sub_eq : μ - ν = measure_sub\n⊢ measure_sub s = μ s - ν s"
] | rw [h_measure_sub_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Sub | {
"line": 111,
"column": 56
} | {
"line": 139,
"column": 41
} | {
"line": 141,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nh_meas_s : MeasurableSet s\n⊢ (μ - ν).restrict s = μ.restrict s - ν.restrict s",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Iff.mpr",
"MeasurableSet.diff",
... | [] | by
repeat rw [sub_def]
have h_nonempty : { d | μ ≤ d + ν }.Nonempty := ⟨μ, Measure.le_add_right le_rfl⟩
rw [restrict_sInf_eq_sInf_restrict h_nonempty h_meas_s]
apply le_antisymm
· refine sInf_le_sInf_of_isCoinitialFor ?_
intro ν' h_ν'_in
rw [mem_setOf_eq] at h_ν'_in
refine ⟨ν'.restrict s, ?_, rest... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 12
} | {
"line": 272,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\nh_dec : μ = ν.withDensity (μ.rnDeriv ν)\nh : μ.singularPart ν = 0\n⊢ μ ≪ ν",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\nh_dec : μ = ν.withDensity (μ.rnDeriv ν)\nh : μ.singularPart ν = 0\n⊢ ν.withDensity (μ.rnDeriv ν) ≪ ν"
] | rw [h_dec] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 14
} | {
"line": 281,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\nh_dec : μ = μ.singularPart ν\nh : ν.withDensity (μ.rnDeriv ν) = 0\n⊢ μ ⟂ₘ ν",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure"... | [
"case refine_1\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : μ.HaveLebesgueDecomposition ν\nh_dec : μ = μ.singularPart ν\nh : ν.withDensity (μ.rnDeriv ν) = 0\n⊢ μ.singularPart ν ⟂ₘ ν"
] | rw [h_dec] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 95
} | {
"line": 376,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : μ s ≠ ∞\nhl : ¬μ.HaveLebesgueDecomposition ν\n⊢ ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν < ∞",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Mea... | [] | simp only [Measure.rnDeriv, dif_neg hl, Pi.zero_apply, lintegral_zero, ENNReal.zero_lt_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 95
} | {
"line": 376,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : μ s ≠ ∞\nhl : ¬μ.HaveLebesgueDecomposition ν\n⊢ ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν < ∞",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Mea... | [] | simp only [Measure.rnDeriv, dif_neg hl, Pi.zero_apply, lintegral_zero, ENNReal.zero_lt_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 374,
"column": 4
} | {
"line": 374,
"column": 95
} | {
"line": 376,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : μ s ≠ ∞\nhl : ¬μ.HaveLebesgueDecomposition ν\n⊢ ∫⁻ (x : α) in s, μ.rnDeriv ν x ∂ν < ∞",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Mea... | [] | simp only [Measure.rnDeriv, dif_neg hl, Pi.zero_apply, lintegral_zero, ENNReal.zero_lt_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 234,
"column": 2
} | {
"line": 236,
"column": 46
} | {
"line": 237,
"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\n⊢ ∀ᵐ (x : α) ∂μ, ∃ c, Tendsto (fun a ↦ ρ a / ... | [
"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\nw : Set ℝ≥0∞\nw_count : w.Countable\nw_dense : Dense w\nl... | obtain ⟨w, w_count, w_dense, _, w_top⟩ :
∃ w : Set ℝ≥0∞, w.Countable ∧ Dense w ∧ 0 ∉ w ∧ ∞ ∉ w :=
ENNReal.exists_countable_dense_no_zero_top | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 795,
"column": 4
} | {
"line": 796,
"column": 65
} | {
"line": 798,
"column": 0
} | [
{
"pp": "case right\nα : Sort u_2\nf : ℕ → α → ℝ≥0∞\nm : ℕ\na : α\nc : ℝ≥0∞ := ⋯\nhc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a\nd : ℝ≥0∞ := ⋯\nhd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)\n⊢ max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a",
"ppTerm": "?right",
"assigned":... | [] | refine sup_le ?_ (biSup_mono fun n hn ↦ hn.trans m.le_succ)
exact @le_iSup₂ ℝ≥0∞ ℕ (fun i ↦ i ≤ m + 1) _ _ (m + 1) le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 795,
"column": 4
} | {
"line": 796,
"column": 65
} | {
"line": 798,
"column": 0
} | [
{
"pp": "case right\nα : Sort u_2\nf : ℕ → α → ℝ≥0∞\nm : ℕ\na : α\nc : ℝ≥0∞ := ⋯\nhc : c = ⨆ k, ⨆ (_ : k ≤ m + 1), f k a\nd : ℝ≥0∞ := ⋯\nhd : d = max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a)\n⊢ max (f m.succ a) (⨆ k, ⨆ (_ : k ≤ m), f k a) ≤ ⨆ k, ⨆ (_ : k ≤ m + 1), f k a",
"ppTerm": "?right",
"assigned":... | [] | refine sup_le ?_ (biSup_mono fun n hn ↦ hn.trans m.le_succ)
exact @le_iSup₂ ℝ≥0∞ ℕ (fun i ↦ i ≤ m + 1) _ _ (m + 1) le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 42
} | {
"line": 699,
"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\nt : Measure α := μ.withDensity (ρ.rnDeriv μ)\neq_add : ρ ... | [
"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\nt : Measure α := μ.withDensity (ρ.rnDeriv μ)\neq_add : ρ = ρ.singular... | filter_upwards [A, B, C] with _ Ax Bx Cx | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 852,
"column": 2
} | {
"line": 852,
"column": 64
} | {
"line": 853,
"column": 2
} | [
{
"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\n⊢ ∀ᵐ (x : α) ∂μ, Tendst... | [
"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), ... | rcases hf.exists_nat_integrableOn with ⟨u, u_open, u_univ, hu⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 273,
"column": 2
} | {
"line": 276,
"column": 79
} | {
"line": 278,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝²¹ : Semifield R\ninst✝²⁰ : StarRing R\ninst✝¹⁹ : MetricSpace R\ninst✝¹⁸ : IsTopologicalSemiring R\ninst✝¹⁷ : ContinuousStar R\ninst✝¹⁶ : Field S\ninst✝¹⁵ : StarRing S\ninst✝¹⁴ : MetricSpace S\ninst✝¹³ : IsTopologicalRing S\ninst✝¹² : ContinuousStar S\nins... | [] | have : h.homeomorph.symm 0 = 0 := Subtype.ext (map_zero <| algebraMap _ _)
refine hφ.comp <| IsUniformEmbedding.isClosedEmbedding <| .comp
(ContinuousMapZero.isUniformEmbedding_comp _ halg)
(UniformEquiv.arrowCongrLeft₀ h.homeomorph.symm this |>.isUniformEmbedding) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 273,
"column": 2
} | {
"line": 276,
"column": 79
} | {
"line": 278,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝²¹ : Semifield R\ninst✝²⁰ : StarRing R\ninst✝¹⁹ : MetricSpace R\ninst✝¹⁸ : IsTopologicalSemiring R\ninst✝¹⁷ : ContinuousStar R\ninst✝¹⁶ : Field S\ninst✝¹⁵ : StarRing S\ninst✝¹⁴ : MetricSpace S\ninst✝¹³ : IsTopologicalRing S\ninst✝¹² : ContinuousStar S\nins... | [] | have : h.homeomorph.symm 0 = 0 := Subtype.ext (map_zero <| algebraMap _ _)
refine hφ.comp <| IsUniformEmbedding.isClosedEmbedding <| .comp
(ContinuousMapZero.isUniformEmbedding_comp _ halg)
(UniformEquiv.arrowCongrLeft₀ h.homeomorph.symm this |>.isUniformEmbedding) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 1061,
"column": 2
} | {
"line": 1061,
"column": 36
} | {
"line": 1063,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁴ : CommRing R\ninst✝¹³ : PartialOrder R\ninst✝¹² : StarRing R\ninst✝¹¹ : MetricSpace R\ninst✝¹⁰ : IsTopologicalRing R\ninst✝⁹ : ContinuousStar R\ninst✝⁸ : ContinuousSqrt R\ninst✝⁷ : StarOrderedRing R\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : Ring A\ninst✝⁴ ... | [] | exact cfc_le_algebraMap_iff id r a | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 185,
"column": 74
} | {
"line": 193,
"column": 70
} | {
"line": 195,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn ν : ℕ\nh : ν ≤ n\n⊢ eval 0 ((⇑derivative)^[ν] (bernsteinPolynomial R n ν)) ≠ 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.eval",
"Nat.instCanonicall... | [] | by
simp only [bernsteinPolynomial.iterate_derivative_at_0, Ne]
simp only [← ascPochhammer_eval_cast]
norm_cast
apply ne_of_gt
obtain rfl | h' := Nat.eq_zero_or_pos ν
· simp
· rw [← Nat.succ_pred_eq_of_pos h'] at h
exact ascPochhammer_pos _ _ (tsub_pos_of_lt (Nat.lt_of_succ_le h)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 392,
"column": 2
} | {
"line": 400,
"column": 8
} | {
"line": 401,
"column": 2
} | [
{
"pp": "case pos\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... | [
"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\ninst✝¹ : I... | · have hsum : s.sum f = fun z => ∑ i ∈ s, f i z := by ext; simp
have hf' : ContinuousOn (∑ i : s, f i) (σₙ R a) := by
rw [sum_coe_sort s, hsum]
exact continuousOn_finsetSum s fun i hi => hf i hi
rw [← sum_coe_sort s, ← sum_coe_sort s]
rw [cfcₙ_apply_pi _ a ha (fun ⟨i, hi⟩ => hf i hi), ← map_sum,... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 435,
"column": 6
} | {
"line": 436,
"column": 59
} | {
"line": 437,
"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... | [] | 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 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 435,
"column": 6
} | {
"line": 436,
"column": 59
} | {
"line": 437,
"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... | [] | 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 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Bernstein | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 11
} | {
"line": 387,
"column": 12
} | [
{
"pp": "case calc_1.e_a\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x ∈ Finset.range (n + 1),\n (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nk : ℕ\nm : k ∈ Finset.range (n + 1... | [
"case calc_1.e_a.zero\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\np :\n ∑ x ∈ Finset.range (n + 1),\n (↑(x * (x - 1)) + (1 - ↑(2 * n) * X) * ↑x + ↑(n ^ 2) * X ^ 2) * bernsteinPolynomial R n x =\n ↑(n * (n - 1)) * X ^ 2 + (1 - (2 * n) • X) * n • X + n ^ 2 • X ^ 2 * 1\nm : 0 ∈ Finset.range (n + 1)\n⊢ (↑n * X -... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 47
} | {
"line": 136,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nf g : ↥A\n⊢ |↑g - ↑f| ∈ A.topologicalClosure",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"ContinuousMap.abs_mem_subalgebra_closure",
"Normed... | [] | exact mod_cast abs_mem_subalgebra_closure A _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 47
} | {
"line": 156,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nA : Subalgebra ℝ C(X, ℝ)\nf g : ↥A\n⊢ |↑g - ↑f| ∈ A.topologicalClosure",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"ContinuousMap.abs_mem_subalgebra_closure",
"Normed... | [] | exact mod_cast abs_mem_subalgebra_closure A _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Algebra.Unitization | {
"line": 87,
"column": 38
} | {
"line": 87,
"column": 56
} | {
"line": 89,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : SMulCommClass 𝕜 A A\nx : Unitization 𝕜 A\n⊢ (x.toProd.1 + 0, (lift (NonUnitalAlgHom.Lmul 𝕜 A)).toRingHom x) =\n (x.toProd.1, (algebra... | [] | rw [add_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Unitization | {
"line": 87,
"column": 38
} | {
"line": 87,
"column": 56
} | {
"line": 89,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : SMulCommClass 𝕜 A A\nx : Unitization 𝕜 A\n⊢ (x.toProd.1 + 0, (lift (NonUnitalAlgHom.Lmul 𝕜 A)).toRingHom x) =\n (x.toProd.1, (algebra... | [] | rw [add_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 300,
"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\n⊢ ∃ g, ‖↑g - f‖ < ε",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Norm.norm",
"Normed... | [] | have w :=
mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f)
rw [Metric.nhds_basis_ball.frequently_iff] at w
obtain ⟨g, H, m⟩ := w ε pos
rw [Metric.mem_ball, dist_eq_norm] at H
exact ⟨⟨g, m⟩, H⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 300,
"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\n⊢ ∃ g, ‖↑g - f‖ < ε",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Norm.norm",
"Normed... | [] | have w :=
mem_closure_iff_frequently.mp (continuousMap_mem_subalgebra_closure_of_separatesPoints A w f)
rw [Metric.nhds_basis_ball.frequently_iff] at w
obtain ⟨g, H, m⟩ := w ε pos
rw [Metric.mem_ball, dist_eq_norm] at H
exact ⟨⟨g, m⟩, H⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 662,
"column": 12
} | {
"line": 662,
"column": 22
} | {
"line": 663,
"column": 2
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝¹ : RCLike 𝕜\ns : Set 𝕜\ninst✝ : Fact (0 ∈ s)\np : C(↑s, 𝕜)₀ → Prop\nzero : p 0\nid : p (ContinuousMapZero.id s)\nstar_id : p (star (ContinuousMapZero.id s))\nadd : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)\nmul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)\nsmul : ∀... | [] | exact zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 662,
"column": 12
} | {
"line": 662,
"column": 22
} | {
"line": 663,
"column": 2
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝¹ : RCLike 𝕜\ns : Set 𝕜\ninst✝ : Fact (0 ∈ s)\np : C(↑s, 𝕜)₀ → Prop\nzero : p 0\nid : p (ContinuousMapZero.id s)\nstar_id : p (star (ContinuousMapZero.id s))\nadd : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)\nmul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)\nsmul : ∀... | [] | exact zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.StoneWeierstrass | {
"line": 662,
"column": 12
} | {
"line": 662,
"column": 22
} | {
"line": 663,
"column": 2
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝¹ : RCLike 𝕜\ns : Set 𝕜\ninst✝ : Fact (0 ∈ s)\np : C(↑s, 𝕜)₀ → Prop\nzero : p 0\nid : p (ContinuousMapZero.id s)\nstar_id : p (star (ContinuousMapZero.id s))\nadd : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f + g)\nmul : ∀ (f g : C(↑s, 𝕜)₀), p f → p g → p (f * g)\nsmul : ∀... | [] | exact zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 161,
"column": 75
} | {
"line": 161,
"column": 88
} | {
"line": 163,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA✝ : Type u_2\ninst✝⁶ : NormedField 𝕜\ninst✝⁵ : NormedRing A✝\ninst✝⁴ : NormedAlgebra 𝕜 A✝\ninst✝³ : CompleteSpace A✝\nA : Type u_3\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra ℝ A\na : A\ninst✝ : IsCompact (spectrum ℝ a)\n⊢ IsCompact (spectrum ℝ a)",
"ppTerm": "?m.53",
"assi... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 53
} | {
"line": 179,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NormedField 𝕜\nB : Type u_3\ninst✝⁵ : NonUnitalNormedRing B\ninst✝⁴ : NormedSpace 𝕜 B\ninst✝³ : CompleteSpace B\ninst✝² : IsScalarTower 𝕜 B B\ninst✝¹ : SMulCommClass 𝕜 B B\ninst✝ : ProperSpace 𝕜\na : B\n⊢ IsCompact (quasispectrum 𝕜 a)",
"ppTerm": "?m.19",
"assigned... | [
"𝕜 : Type u_1\ninst✝⁶ : NormedField 𝕜\nB : Type u_3\ninst✝⁵ : NonUnitalNormedRing B\ninst✝⁴ : NormedSpace 𝕜 B\ninst✝³ : CompleteSpace B\ninst✝² : IsScalarTower 𝕜 B B\ninst✝¹ : SMulCommClass 𝕜 B B\ninst✝ : ProperSpace 𝕜\na : B\n⊢ IsCompact (σ ↑a)"
] | Unitization.quasispectrum_eq_spectrum_inr' 𝕜 𝕜, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 193,
"column": 75
} | {
"line": 193,
"column": 88
} | {
"line": 195,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝¹³ : NormedField 𝕜\ninst✝¹² : NormedRing A\ninst✝¹¹ : NormedAlgebra 𝕜 A\ninst✝¹⁰ : CompleteSpace A\nB : Type u_3\ninst✝⁹ : NonUnitalNormedRing B\ninst✝⁸ : NormedSpace 𝕜 B\ninst✝⁷ : CompleteSpace B\ninst✝⁶ : IsScalarTower 𝕜 B B\ninst✝⁵ : SMulCommClass 𝕜 B B\ninst✝⁴... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.GelfandFormula | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 86
} | {
"line": 151,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra ℂ A\ninst✝¹ : CompleteSpace A\ninst✝ : Nontrivial A\na : A\nh : spectrum ℂ a = ∅\n⊢ False",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"spectrum.eq_1",
"NormedRing.toRing",
"spectrum",
"congrArg... | [
"A : Type u_2\ninst✝³ : NormedRing A\ninst✝² : NormedAlgebra ℂ A\ninst✝¹ : CompleteSpace A\ninst✝ : Nontrivial A\na : A\nh : spectrum ℂ a = ∅\nH₀ : resolventSet ℂ a = Set.univ\n⊢ False"
] | have H₀ : resolventSet ℂ a = Set.univ := by rwa [spectrum, Set.compl_empty_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.Spectrum | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 17
} | {
"line": 80,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedRing E\ninst✝³ : StarRing E\ninst✝² : CStarRing E\ninst✝¹ : NormedAlgebra 𝕜 E\ninst✝ : CompleteSpace E\nu : ↥(unitary E)\n⊢ σ 𝕜 ↑u ⊆ Metric.sphere 0 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Nont... | [
"𝕜 : Type u_1\ninst✝⁵ : NormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedRing E\ninst✝³ : StarRing E\ninst✝² : CStarRing E\ninst✝¹ : NormedAlgebra 𝕜 E\ninst✝ : CompleteSpace E\nu : ↥(unitary E)\na✝ : Nontrivial E\n⊢ σ 𝕜 ↑u ⊆ Metric.sphere 0 1"
] | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 348,
"column": 12
} | {
"line": 348,
"column": 40
} | {
"line": 348,
"column": 40
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : HasSummableGeomSeries A\na : A\ny : 𝕜\nhy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹\nh : ¬‖a‖₊ = 0\nnnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹\n⊢ ‖y‖₊ * ‖a‖₊ < 1",
"ppTerm": "?neg✝",
"as... | [
"case neg\n𝕜 : Type u_1\nA : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : HasSummableGeomSeries A\na : A\ny : 𝕜\nhy : y ∈ Metric.eball 0 (↑‖a‖₊)⁻¹\nh : ¬‖a‖₊ = 0\nnnnorm_lt : ‖y‖₊ < ‖a‖₊⁻¹\n⊢ ‖y‖₊ < ‖a‖₊⁻¹"
] | ← NNReal.lt_inv_iff_mul_lt h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 101,
"column": 2
} | {
"line": 102,
"column": 55
} | {
"line": 103,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : IsScalarTower 𝕜 A A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\na : A\nha : ... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : IsScalarTower 𝕜 A A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : StarModule 𝕜 A\np : A → Prop\np₁ : Unitization 𝕜 A → Prop\nhp₁ : ∀ {x : A}, p₁ ↑x ↔ p x\na : A\nha : p a\ninst✝ :... | refine ((cfcHom_isClosedEmbedding (hp₁.mpr ha)).comp ?_).comp
ContinuousMapZero.isClosedEmbedding_toContinuousMap | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Instances | {
"line": 387,
"column": 2
} | {
"line": 387,
"column": 30
} | {
"line": 388,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : T2Space A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\nf : ℂ → ℂ\na : A\nhf_real : ∀ x ∈ σₙ ℂ a, star (... | [
"A : Type u_1\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : Module ℂ A\ninst✝³ : IsScalarTower ℂ A A\ninst✝² : SMulCommClass ℂ A A\ninst✝¹ : T2Space A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℂ A IsStarNormal\nf : ℂ → ℂ\na : A\nhf_real : ∀ x ∈ σₙ ℂ a, star (f x) = f x\n... | rw [cfcₙ_real_eq_complex ..] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.Barrelled | {
"line": 141,
"column": 4
} | {
"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... | [
"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\ninst✝⁴ : Modu... | have hxn' : p x ≤ n := by convert! interior_subset hxn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.WeakDual | {
"line": 175,
"column": 44
} | {
"line": 175,
"column": 57
} | {
"line": 175,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\n⊢ NormedSpace ?m.32 ?m.33",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Affine.AddTorsor | {
"line": 212,
"column": 8
} | {
"line": 212,
"column": 29
} | {
"line": 212,
"column": 30
} | [
{
"pp": "W : Type u_3\nQ : Type u_4\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : MetricSpace Q\ninst✝² : NormedAddTorsor W Q\n𝕜 : Type u_5\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 W\np₁ p₂ : Q\nh : p₁ ≠ p₂\nc₁ c₂ : 𝕜\n⊢ dist c₁ c₂ ≤ ↑(nndist p₁ p₂)⁻¹ * dist ((lineMap p₁ p₂) c₁) ((lineMap p₁ p₂) c₂)",
... | [
"W : Type u_3\nQ : Type u_4\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : MetricSpace Q\ninst✝² : NormedAddTorsor W Q\n𝕜 : Type u_5\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedSpace 𝕜 W\np₁ p₂ : Q\nh : p₁ ≠ p₂\nc₁ c₂ : 𝕜\n⊢ dist c₁ c₂ ≤ ↑(nndist p₁ p₂)⁻¹ * (dist c₁ c₂ * dist p₁ p₂)"
] | dist_lineMap_lineMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.GelfandDuality | {
"line": 229,
"column": 36
} | {
"line": 229,
"column": 72
} | {
"line": 230,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : NonUnitalCommCStarAlgebra A\nι : Type u_2\nf : ι → A\nh0 : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)\nj : ι\ns : Finset ι\nhj : j ∉ s\nih : ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊\nthis : f j * ∑ i ∈ s, f i = 0\n⊢ ‖∑ i ∈ insert j s, f i‖₊ = (insert j s).sup fun x ↦ ‖f x‖₊",
"pp... | [] | by simp_all [nnnorm_add_eq_max this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 172,
"column": 2
} | {
"line": 175,
"column": 48
} | {
"line": 177,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsStarNormal a\n⊢ cfcHom ha = (elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"ContinuousMap.instNonUnitalCStarAlgebra",
"St... | [] | refine cfcHom_eq_of_continuous_of_map_id ha _ ?_ ?_
· exact continuous_subtype_val.comp <|
(StarAlgEquiv.isometry (continuousFunctionalCalculus a)).continuous
· simp [continuousFunctionalCalculus_map_id a] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 172,
"column": 2
} | {
"line": 175,
"column": 48
} | {
"line": 177,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\na : A\nha : IsStarNormal a\n⊢ cfcHom ha = (elemental ℂ a).subtype.comp ↑(continuousFunctionalCalculus a)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"ContinuousMap.instNonUnitalCStarAlgebra",
"St... | [] | refine cfcHom_eq_of_continuous_of_map_id ha _ ?_ ?_
· exact continuous_subtype_val.comp <|
(StarAlgEquiv.isometry (continuousFunctionalCalculus a)).continuous
· simp [continuousFunctionalCalculus_map_id a] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.UniformConvergence | {
"line": 263,
"column": 57
} | {
"line": 265,
"column": 62
} | {
"line": 267,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\n𝔖 : Set (Set α)\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : Finite ↑𝔖\nx : α\nhx : x ∈ ⋃₀ 𝔖\n⊢ LipschitzWith 1 fun f ↦ (toFun 𝔖) f x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | [] | by
intro f g
simpa only [ENNReal.coe_one, one_mul] using edist_eval_le hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 299,
"column": 14
} | {
"line": 302,
"column": 14
} | {
"line": 304,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : Module ℝ A\ninst✝⁹ : SMulCommClass ℝ A A\ninst✝⁸ : IsScalarTower ℝ A A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | [] | by
refine cfcₙ_congr fun x hx ↦ ?_
lift x to σₙ ℝ a using hx
simp [f] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 428,
"column": 14
} | {
"line": 428,
"column": 33
} | {
"line": 428,
"column": 33
} | [
{
"pp": "case mpr.add\nA : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nx✝¹ : PartialOrder A := spectralOrder A\nx✝ p x y : A\nhx✝ : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)\nhy✝ : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)\nhx : IsSelfAdjoint ↑x ∧ SpectrumRestricts ↑x ⇑ContinuousMap.r... | [
"case mpr.add\nA : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nx✝¹ : PartialOrder A := spectralOrder A\nx✝ p x y : A\nhx✝ : x ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)\nhy✝ : y ∈ AddSubmonoid.closure (Set.range fun s ↦ star s * s)\nhx : IsSelfAdjoint ↑x ∧ SpectrumRestricts ↑x ⇑ContinuousMap.realToNNReal\... | Unitization.inr_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 35
} | {
"line": 324,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\n⊢ 1⁻ = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"... | [
"A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\n⊢ cfc (fun x ↦ x⁻) 1 = 0"
] | rw [CFC.negPart_def, cfcₙ_eq_cfc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 35
} | {
"line": 334,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\nr : ℝ\n⊢ ((algebraMap ℝ A) r)⁻ = (algebraMap ℝ A) r⁻",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [
"A : Type u_1\ninst✝⁵ : Ring A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\nr : ℝ\n⊢ cfc (fun x ↦ x⁻) ((algebraMap ℝ A) r) = (algebraMap ℝ A) r⁻"
] | rw [CFC.negPart_def, cfcₙ_eq_cfc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 64
} | {
"line": 105,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CStarRing A\ninst✝⁵ : NormedSpace ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : N... | [] | simpa [hb.star_eq] using norm_star_mul_mul_self_of_nonneg b ha | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 64
} | {
"line": 105,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CStarRing A\ninst✝⁵ : NormedSpace ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : N... | [] | simpa [hb.star_eq] using norm_star_mul_mul_self_of_nonneg b ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Isometric | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 64
} | {
"line": 105,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : CStarRing A\ninst✝⁵ : NormedSpace ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : N... | [] | simpa [hb.star_eq] using norm_star_mul_mul_self_of_nonneg b ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.SpecialFunctions.PosPart | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 57
} | {
"line": 77,
"column": 4
} | [
{
"pp": "case h.right\nA : Type u_2\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ a = ∑ i, I ^ ↑i • ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart a))⁻] i",
"ppTerm": "?h.right",
"assigned": true,
"usedConstants": [
... | [
"case h.right\nA : Type u_2\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\n⊢ (↑(realPart a))⁺ - (↑(realPart a))⁻ + (I • (↑(imaginaryPart a))⁺ - I • (↑(imaginaryPart a))⁻) =\n ∑ i, I ^ ↑i • ![(↑(realPart a))⁺, (↑(imaginaryPart a))⁺, (↑(realPart a))⁻, (↑(imaginaryPart... | nth_rw 1 [← CStarAlgebra.linear_combination_nonneg a] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 438,
"column": 50
} | {
"line": 438,
"column": 68
} | {
"line": 440,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : PartialOrder A\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : TopologicalSpace A\ninst✝³ : StarOrderedRing A\ninst✝² : Algebra ℝ A\ninst✝¹ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : NonnegSpectrumClass ℝ A\nx : ℝ\n⊢ 1 ^ x = 1",
"ppTerm": "?m.32",
"assigned... | [] | by simp [rpow_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 73,
"column": 2
} | {
"line": 87,
"column": 20
} | {
"line": 89,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\n⊢ MonotoneOn (cfcₙ fun x ↦ 1 - (1 + x)⁻¹) (Set.Ici 0)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"cfcₙ",
"CFC.monotoneOn_one_sub_one_add_inv",
"Eq.mpr",
... | [] | intro a (ha : 0 ≤ a) b (hb : 0 ≤ b) hab
calc _ = cfcₙ (fun x : ℝ≥0 => 1 - (1 + x)⁻¹) a := by
rw [cfcₙ_nnreal_eq_real _ _ ha]
refine cfcₙ_congr ?_
intro x hx
have hx' : 0 ≤ x := by grind
simp [hx']
_ ≤ cfcₙ (fun x : ℝ≥0 => 1 - (1 + x)⁻¹) b :=
CFC.monotone... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 73,
"column": 2
} | {
"line": 87,
"column": 20
} | {
"line": 89,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\n⊢ MonotoneOn (cfcₙ fun x ↦ 1 - (1 + x)⁻¹) (Set.Ici 0)",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"cfcₙ",
"CFC.monotoneOn_one_sub_one_add_inv",
"Eq.mpr",
... | [] | intro a (ha : 0 ≤ a) b (hb : 0 ≤ b) hab
calc _ = cfcₙ (fun x : ℝ≥0 => 1 - (1 + x)⁻¹) a := by
rw [cfcₙ_nnreal_eq_real _ _ ha]
refine cfcₙ_congr ?_
intro x hx
have hx' : 0 ≤ x := by grind
simp [hx']
_ ≤ cfcₙ (fun x : ℝ≥0 => 1 - (1 + x)⁻¹) b :=
CFC.monotone... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 825,
"column": 58
} | {
"line": 837,
"column": 13
} | {
"line": 839,
"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
tfae_have 1 ↔ 8 := IsStrictlyPositive.iff_of_unital
tfae_have 1 ↔ 9 := ⟨fun h => ⟨h.isSelfAdjoint,
StarOrderedRing.isStrictlyPositive_iff_spectrum_pos a |>.mp h⟩,
fun h => (StarOrderedRing.isStrictlyPositive_iff_spectrum_pos a).mpr h.2⟩
tfae_have 1 → 2 := fun h => ⟨h.sqrt, sqrt_mul_sqrt_self a |>.sym... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 75
} | {
"line": 266,
"column": 2
} | [
{
"pp": "𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\ns : Set 𝕜\n⊢ ContinuousOn (fun fa ↦ cfc ((toFun {s}) fa.1) ... | [
"𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : IsometricContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\ns : Set 𝕜\nx✝¹ : (𝕜 →ᵤ[{t | IsCompact t ∧ t ⊆ s}] 𝕜) × A\nf : 𝕜 →ᵤ[{... | refine continuousOn_of_locally_continuousOn fun (f, a) ⟨hf, ha, has⟩ ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 312,
"column": 2
} | {
"line": 313,
"column": 94
} | {
"line": 314,
"column": 2
} | [
{
"pp": "case e_a.e_a\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nm✝¹ m✝ : A\nε : ℝ≥0\nhε : 0 < ε\nx✝ : A\nx m : Unitization ℂ A\nhm₁ : 0 ≤ m\nhm₂ : ‖m‖ < 1\nhx₂ : ‖x‖ ≤ 1\nhx₀ : 0 ≤ x\ng : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹\nhx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x... | [
"case e_a.e_a\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nm✝¹ m✝ : A\nε : ℝ≥0\nhε : 0 < ε\nx✝ : A\nx m : Unitization ℂ A\nhm₁ : 0 ≤ m\nhm₂ : ‖m‖ < 1\nhx₂ : ‖x‖ ≤ 1\nhx₀ : 0 ≤ x\ng : ℝ≥0 → ℝ≥0 := fun y ↦ 1 - (1 + y)⁻¹\nhx₁ : cfc g (ε⁻¹ ^ 2 • m) ≤ x\nhg : Conti... | rw [← cfc_one (R := ℝ≥0) m, ← cfc_comp_smul _ _ _ hg.continuousOn hm₁,
← cfc_tsub _ _ m (by simp [g]) hm₁ (by fun_prop) (Continuous.continuousOn <| by fun_prop)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 75
} | {
"line": 450,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝¹⁰ : NormedRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : NormedAlgebra ℝ A\ninst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : T2Space A\ninst✝¹ : IsSemit... | [
"A : Type u_2\ninst✝¹⁰ : NormedRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : NormedAlgebra ℝ A\ninst✝⁷ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁶ : ContinuousStar A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : T2Space A\ninst✝¹ : IsSemitopologicalRi... | refine continuousOn_of_locally_continuousOn fun (f, a) ⟨hf, ha, has⟩ ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 536,
"column": 2
} | {
"line": 541,
"column": 35
} | {
"line": 542,
"column": 2
} | [
{
"pp": "X : Type u_1\nA : Type u_2\ninst✝¹¹ : NormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedAlgebra ℝ A\ninst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : T2Space A\ni... | [
"X : Type u_1\nA : Type u_2\ninst✝¹¹ : NormedRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : NormedAlgebra ℝ A\ninst✝⁸ : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁷ : ContinuousStar A\ninst✝⁶ : PartialOrder A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : T2Space A\ninst✝² : IsSe... | have (x : t) : ∃ S, IsCompact S ∧ (∀ᶠ (x' : A) in 𝓝 (a x), spectrum ℝ≥0 x' ⊆ S) ∧ S ⊆ s := by
obtain ⟨S, ⟨hS₁, hS₂⟩, hS₃⟩ :=
spectrum.isCompact_nnreal (a x) |>.nhdsSet_basis_isCompact.mem_iff.mp (hs' x x.2)
refine ⟨S, hS₂, ?_, hS₃⟩
exact upperHemicontinuous_spectrum_nnreal A |>.upperHemicontinuousAt ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 31
} | {
"line": 139,
"column": 2
} | [
{
"pp": "A₁ : Type u_1\nA₂ : Type u_2\ninst✝⁶ : NonUnitalCStarAlgebra A₁\ninst✝⁵ : NonUnitalCStarAlgebra A₂\ninst✝⁴ : PartialOrder A₁\ninst✝³ : PartialOrder A₂\ninst✝² : StarOrderedRing A₁\ninst✝¹ : StarOrderedRing A₂\nn : Type u_3\ninst✝ : Fintype n\nφ : A₁ →CP A₂\nM : CStarMatrix n n A₁\nhM : 0 ≤ M\nk : ℕ := ... | [
"A₁ : Type u_1\nA₂ : Type u_2\ninst✝⁶ : NonUnitalCStarAlgebra A₁\ninst✝⁵ : NonUnitalCStarAlgebra A₂\ninst✝⁴ : PartialOrder A₁\ninst✝³ : PartialOrder A₂\ninst✝² : StarOrderedRing A₁\ninst✝¹ : StarOrderedRing A₂\nn : Type u_3\ninst✝ : Fintype n\nφ : A₁ →CP A₂\nM : CStarMatrix n n A₁\nhM : 0 ≤ M\nk : ℕ := Fintype.card... | rw [← mapₗ_reindexₐ] at hmain | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 60,
"column": 21
} | {
"line": 60,
"column": 33
} | {
"line": 60,
"column": 33
} | [
{
"pp": "case mul\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a b\nhb₂... | [
"case mul\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a b\nhb₂ : Commute (... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Commute | {
"line": 62,
"column": 4
} | {
"line": 65,
"column": 43
} | {
"line": 67,
"column": 0
} | [
{
"pp": "case frequently\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : Ring A\ninst✝⁵ : StarRing A\ninst✝⁴ : Algebra 𝕜 A\ninst✝³ : TopologicalSpace A\ninst✝² : ContinuousFunctionalCalculus 𝕜 A p\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na b : A\nha : p a\nhb₁ : Commute a... | [] | rw [commute_iff_eq, ← Set.mem_setOf (p := fun x => x * b = b * x),
← (isClosed_eq (by fun_prop) (by fun_prop)).closure_eq]
apply mem_closure_of_frequently_of_tendsto hf
exact cfcHom_continuous ha |>.tendsto _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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