module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | {
"line": 85,
"column": 14
} | [
{
"pp": "case mk.mk\n𝕜 : Type u\nA : Type v\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ntoProd✝¹ : (A →L[𝕜] A) × (A →L[𝕜] A)\ncentral✝¹ : ∀ (x y : A), toProd✝¹.2 x * y = x * toProd✝¹.1 y\ntoProd✝... | [
"case mk.mk\n𝕜 : Type u\nA : Type v\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\ntoProd✝¹ : (A →L[𝕜] A) × (A →L[𝕜] A)\ncentral✝¹ : ∀ (x y : A), toProd✝¹.2 x * y = x * toProd✝¹.1 y\ntoProd✝ : (A →L[𝕜]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 633,
"column": 24
} | {
"line": 633,
"column": 35
} | {
"line": 633,
"column": 36
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nB' : Type u_5\ninst✝ : TopologicalSpace B'\nh : B ≃ₜ B'\np : Z\n... | [
"B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nB' : Type u_5\ninst✝ : TopologicalSpace B'\nh : B ≃ₜ B'\np : Z\nhp : p ∈ (e.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 781,
"column": 73
} | {
"line": 781,
"column": 84
} | {
"line": 781,
"column": 84
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne₁ e₂ e₃ : Trivialization F proj\nb : B\nh₁ : b ∈ e₁.baseSet\nh₂ : b ∈ e₂.baseSet\nx : F\n⊢ (↑e₃ (↑e₁.symm (b, x))).2 = e₁.coordChange e₃ b x",
"ppTerm": "?m... | [
"B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ninst✝ : TopologicalSpace Z\ne₁ e₂ e₃ : Trivialization F proj\nb : B\nh₁ : b ∈ e₁.baseSet\nh₂ : b ∈ e₂.baseSet\nx : F\n⊢ (↑e₃ (↑e₁.symm (b, x))).2 = (↑e₃ (↑e₁.symm (b, x))).2",
"B : Type u_1\nF : Type... | coordChange | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 538,
"column": 6
} | {
"line": 538,
"column": 42
} | {
"line": 538,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nf : A →L[𝕜] A\nC : ℝ≥0\nh : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nf : A →L[𝕜] A\nC : ℝ≥0\nh : ∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ * ‖b‖₊\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 547,
"column": 8
} | {
"line": 547,
"column": 31
} | {
"line": 547,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nh0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nh0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 555,
"column": 8
} | {
"line": 555,
"column": 31
} | {
"line": 555,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nh0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤... | [
"𝕜 : Type u_1\nA : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : SMulCommClass 𝕜 A A\ninst✝² : IsScalarTower 𝕜 A A\ninst✝¹ : StarRing A\ninst✝ : CStarRing A\na : 𝓜(𝕜, A)\nh0 : ∀ (f : A →L[𝕜] A) (C : ℝ≥0), (∀ (b : A), ‖f b‖₊ ^ 2 ≤ C * ‖f b‖₊ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 994,
"column": 2
} | {
"line": 994,
"column": 13
} | {
"line": 994,
"column": 14
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace F\nproj : Z → B\ninst✝⁴ : TopologicalSpace Z\ninst✝³ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nb✝ : B\ny : E b✝\nT✝ : T... | [
"B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace F\nproj : Z → B\ninst✝⁴ : TopologicalSpace Z\ninst✝³ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nb✝ : B\ny : E b✝\nT✝ : Trivializatio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.SeparatedMap | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 78
} | {
"line": 140,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nf : X → Y\n⊢ IsLocallyInjective f ↔ IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Topo... | [
"X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nf : X → Y\n⊢ (∀ (x : X), ∃ U ∈ 𝓝 x, Set.InjOn f U) ↔\n ∀ (a b : X) (b_1 : f a = f b),\n ⟨(a, b), b_1⟩ ∈ Function.pullbackDiagonal f → ∃ t ∈ 𝓝 a ×ˢ 𝓝 b, Subtype.val ⁻¹' t ⊆ Function.pullbackDiagonal f"
] | simp_rw [isLocallyInjective_iff_nhds, isOpen_iff_mem_nhds,
Subtype.forall, Prod.forall, nhds_induced, nhds_prod_eq, Filter.mem_comap] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 634,
"column": 10
} | {
"line": 635,
"column": 17
} | {
"line": 635,
"column": 18
} | [
{
"pp": "case refine_2.refine_2\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : DenselyNormedField 𝕜\ninst✝⁷ : StarRing 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : CStarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : StarModule 𝕜 A\na : 𝓜(... | [
"case refine_2.refine_2\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : DenselyNormedField 𝕜\ninst✝⁷ : StarRing 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : CStarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : StarModule 𝕜 A\na : 𝓜(𝕜, A)\nhbal... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 56,
"column": 49
} | {
"line": 56,
"column": 65
} | {
"line": 56,
"column": 66
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝ : TopologicalSpace I\nx : X\nh✝ : DiscreteTopology I\nU : Set X\nhxU : x ∈ U\nhU : IsOpen[inst✝¹] U\nhfU : IsOpen[inst✝²] (f ⁻¹' U)\nH : ↑(f ⁻¹' U) ≃ₜ ↑U × I\nhH : ∀ (x : ↑(f ... | [
"E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝ : TopologicalSpace I\nx : X\nh✝ : DiscreteTopology I\nU : Set X\nhxU : x ∈ U\nhU : IsOpen[inst✝¹] U\nhfU : IsOpen[inst✝²] (f ⁻¹' U)\nH : ↑(f ⁻¹' U) ≃ₜ ↑U × I\nhH : ∀ (x : ↑(f ⁻¹' U)), ↑(H... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 17
} | {
"line": 73,
"column": 18
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns : Set X\nx : ↑s\ne : OpenPartialHomeomorph X Y\nhx : ↑x ∈ e.source\nh : IsLocalHomeomorphOn (↑e) s\ninst✝ : DiscreteTopology ↑(↑e '' s)\nU : Set Y\nhU : IsOpen[inst✝¹] U\neq : Subtype.val ⁻¹' U = {⟨↑e... | [
"case refine_1\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns : Set X\nx : ↑s\ne : OpenPartialHomeomorph X Y\nhx : ↑x ∈ e.source\nh : IsLocalHomeomorphOn (↑e) s\ninst✝ : DiscreteTopology ↑(↑e '' s)\nU : Set Y\nhU : IsOpen[inst✝¹] U\neq : Subtype.val ⁻¹' U = {⟨↑e ↑x, ⋯⟩}\nx'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 48,
"column": 32
} | {
"line": 48,
"column": 43
} | {
"line": 48,
"column": 44
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : AddGroup G\ninst✝ : AddAction G E\nhf : IsAddQuotientCoveringMap f G\ng : Multiplicative G\n⊢ Continuous[inst✝³, inst✝³] fun x ↦ g • x",
"ppTerm": "?m.33",
"assigned": false,
... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : AddGroup G\ninst✝ : AddAction G E\nhf : IsAddQuotientCoveringMap f G\ng : Multiplicative G\n⊢ Continuous[inst✝³, inst✝³] fun x ↦ g • x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 48,
"column": 32
} | {
"line": 48,
"column": 98
} | {
"line": 49,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : AddGroup G\ninst✝ : AddAction G E\nhf : IsAddQuotientCoveringMap f G\ng : Multiplicative G\n⊢ Continuous[inst✝³, inst✝³] fun x ↦ g • x",
"ppTerm": "?m.33",
"assigned": true,
... | [] | simpa using hf.continuous_const_vadd (Multiplicative.ofAdd.symm g) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Covering.Quotient | {
"line": 48,
"column": 32
} | {
"line": 48,
"column": 98
} | {
"line": 49,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : AddGroup G\ninst✝ : AddAction G E\nhf : IsAddQuotientCoveringMap f G\ng : Multiplicative G\n⊢ Continuous[inst✝³, inst✝³] fun x ↦ g • x",
"ppTerm": "?m.33",
"assigned": true,
... | [] | simpa using hf.continuous_const_vadd (Multiplicative.ofAdd.symm g) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.Quotient | {
"line": 48,
"column": 32
} | {
"line": 48,
"column": 98
} | {
"line": 49,
"column": 2
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : AddGroup G\ninst✝ : AddAction G E\nhf : IsAddQuotientCoveringMap f G\ng : Multiplicative G\n⊢ Continuous[inst✝³, inst✝³] fun x ↦ g • x",
"ppTerm": "?m.33",
"assigned": true,
... | [] | simpa using hf.continuous_const_vadd (Multiplicative.ofAdd.symm g) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Quotient | {
"line": 52,
"column": 29
} | {
"line": 52,
"column": 40
} | {
"line": 52,
"column": 41
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : AddGroup G\ninst✝ : AddAction G E\nhf : IsAddQuotientCoveringMap f G\ne : E\nU : Set E\nhU : U ∈ 𝓝 e\nhU' : ∀ (g : G), ((fun x ↦ g +ᵥ x) '' U ∩ U).Nonempty → g = 0\ng : Multiplicativ... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : AddGroup G\ninst✝ : AddAction G E\nhf : IsAddQuotientCoveringMap f G\ne : E\nU : Set E\nhU : U ∈ 𝓝 e\nhU' : ∀ (g : G), ((fun x ↦ g +ᵥ x) '' U ∩ U).Nonempty → g = 0\ng : Multiplicative G\n⊢ (U ∩ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 58,
"column": 32
} | {
"line": 58,
"column": 43
} | {
"line": 58,
"column": 44
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\ng : Additive G\n⊢ Continuous[inst✝³, inst✝³] fun x ↦ g +ᵥ x",
"ppTerm": "?m.33",
"assigned": false,
"usedCo... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\ng : Additive G\n⊢ Continuous[inst✝³, inst✝³] fun x ↦ g +ᵥ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 76,
"column": 40
} | {
"line": 76,
"column": 56
} | {
"line": 76,
"column": 57
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝¹ : TopologicalSpace I\nx : X\ninst✝ : Nonempty I\nh✝ : DiscreteTopology I\nU : Set X\nhxU : x ∈ U\nhU : IsOpen[inst✝²] U\nhfU : IsOpen[inst✝³] (f ⁻¹' U)\nH : ↑(f ⁻¹' U) ≃ₜ ↑U ... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝¹ : TopologicalSpace I\nx : X\ninst✝ : Nonempty I\nh✝ : DiscreteTopology I\nU : Set X\nhxU : x ∈ U\nhU : IsOpen[inst✝²] U\nhfU : IsOpen[inst✝³] (f ⁻¹' U)\nH : ↑(f ⁻¹' U) ≃ₜ ↑U × I\nhH : ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 40
} | {
"line": 62,
"column": 41
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\ne : E\nU : Set E\nhU : U ∈ 𝓝 e\nhU' : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1\ng : Additive G\n⊢ ((fun ... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\ne : E\nU : Set E\nhU : U ∈ 𝓝 e\nhU' : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1\ng : Additive G\n⊢ (U ∩ (fun x ↦ g +ᵥ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | {
"line": 87,
"column": 16
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nhs : IsOpen[inst✝¹] s\nhX : ∀ (a : ↑s), IsOpen[instTopologicalSpaceSubtype] {a}\nx : X\nhx : x ∈ s\ne : OpenPartialHomeomorph X Y\nhxe : x ∈ e.source\nh : IsLocalHomeomorphOn (↑e) s\n⊢ IsOpen[... | [
"case refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nhs : IsOpen[inst✝¹] s\nhX : ∀ (a : ↑s), IsOpen[instTopologicalSpaceSubtype] {a}\nx : X\nhx : x ∈ s\ne : OpenPartialHomeomorph X Y\nhxe : x ∈ e.source\nh : IsLocalHomeomorphOn (↑e) s\n⊢ IsOpen[inst✝¹] {x}"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 41
} | {
"line": 79,
"column": 42
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\ng g' : G\ne : E\neq : g • e = g' • e\nU : Set E\nheU : U ∈ 𝓝 e\nhU : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → ... | [
"E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhf : IsQuotientCoveringMap f G\ng g' : G\ne : E\neq : g • e = g' • e\nU : Set E\nheU : U ∈ 𝓝 e\nhU : ∀ (g : G), ((fun x ↦ g • x) '' U ∩ U).Nonempty → g = 1\n⊢ g =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 86,
"column": 31
} | {
"line": 86,
"column": 42
} | {
"line": 86,
"column": 43
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhf : IsQuotientCoveringMap f G\nY : Type u_4\ninst✝ : TopologicalSpace Y\nφ : X ≃ₜ Y\n⊢ ∀ {e₁ e₂ : E}, (⇑φ ∘ f) e₁ = (⇑φ ∘ f) e₂ ↔ e₁ ∈ MulAction.orbi... | [
"E : Type u_1\nX : Type u_2\ninst✝⁴ : TopologicalSpace E\ninst✝³ : TopologicalSpace X\nf : E → X\nG : Type u_3\ninst✝² : Group G\ninst✝¹ : MulAction G E\nhf : IsQuotientCoveringMap f G\nY : Type u_4\ninst✝ : TopologicalSpace Y\nφ : X ≃ₜ Y\n⊢ ∀ {e₁ e₂ : E}, f e₁ = f e₂ ↔ e₁ ∈ MulAction.orbit G e₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 165,
"column": 75
} | {
"line": 165,
"column": 86
} | {
"line": 165,
"column": 87
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝ : TopologicalSpace I\nhs : IsOpen[inst✝¹] s\nhfs : IsOpen[inst✝²] (f ⁻¹' s)\nx : X\nhx : x ∈ s\nh : IsEvenlyCovered (fun e ↦ f ↑e) x I\ninst : DiscreteTopology I\nU : Set X\nh... | [
"E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nI : Type u_3\ninst✝ : TopologicalSpace I\nhs : IsOpen[inst✝¹] s\nhfs : IsOpen[inst✝²] (f ⁻¹' s)\nx : X\nhx : x ∈ s\nh : IsEvenlyCovered (fun e ↦ f ↑e) x I\ninst : DiscreteTopology I\nU : Set X\nhxU : x ∈ U\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 280,
"column": 51
} | {
"line": 280,
"column": 62
} | {
"line": 281,
"column": 2
} | [
{
"pp": "case e'_5\nE : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nY : Type u_3\ninst✝ : TopologicalSpace Y\ng : X ≃ₜ Y\nh : IsCoveringMapOn (⇑g ∘ f) (⇑g.symm ⁻¹' s)\n⊢ f = ⇑g.symm ∘ ⇑g ∘ f",
"ppTerm": "?e'_5",
"assigned": true,
"usedConst... | [] | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Topology.Covering.Basic | {
"line": 280,
"column": 51
} | {
"line": 280,
"column": 62
} | {
"line": 281,
"column": 2
} | [
{
"pp": "case e'_6\nE : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ns : Set X\nY : Type u_3\ninst✝ : TopologicalSpace Y\ng : X ≃ₜ Y\nh : IsCoveringMapOn (⇑g ∘ f) (⇑g.symm ⁻¹' s)\n⊢ s = ⇑g.symm.symm ⁻¹' ⇑g.symm ⁻¹' s",
"ppTerm": "?e'_6",
"assigned": true,
... | [] | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Topology.Covering.Basic | {
"line": 300,
"column": 51
} | {
"line": 300,
"column": 62
} | {
"line": 300,
"column": 63
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\n⊢ IsCoveringMapOn (s.restrictPreimage f) Set.univ",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\nf : E → X\ns : Set X\nhf : IsCoveringMapOn f s\n⊢ IsCoveringMapOn (s.restrictPreimage f) Set.univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.AddCircle | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 84
} | {
"line": 76,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : Ring 𝕜\ninst✝³ : IsTopologicalRing 𝕜\np : 𝕜\ninst✝² : T0Space (AddCircle p)\ninst✝¹ : Algebra ℚ 𝕜\nn : ℤ\ninst✝ : NeZero n\n⊢ IsUnit ↑n",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubt... | [] | convert! (Int.cast_ne_zero.mpr <| NeZero.ne n).isUnit.map (algebraMap ℚ 𝕜); simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Covering.AddCircle | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 84
} | {
"line": 76,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : Ring 𝕜\ninst✝³ : IsTopologicalRing 𝕜\np : 𝕜\ninst✝² : T0Space (AddCircle p)\ninst✝¹ : Algebra ℚ 𝕜\nn : ℤ\ninst✝ : NeZero n\n⊢ IsUnit ↑n",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubt... | [] | convert! (Int.cast_ne_zero.mpr <| NeZero.ne n).isUnit.map (algebraMap ℚ 𝕜); simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Covering.Basic | {
"line": 412,
"column": 34
} | {
"line": 412,
"column": 73
} | {
"line": 412,
"column": 74
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsOpen[inst✝] s\nf : E → X\nh : ∀ (x : E), f x ∈ s\nhf : IsCoveringMap fun x ↦ ⟨f x, ⋯⟩\n⊢ f ⁻¹' s = Set.univ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsOpen[inst✝] s\nf : E → X\nh : ∀ (x : E), f x ∈ s\nhf : IsCoveringMap fun x ↦ ⟨f x, ⋯⟩\n⊢ ∀ (a : E), f a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 455,
"column": 46
} | {
"line": 455,
"column": 78
} | {
"line": 455,
"column": 78
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝³ : Nonempty (X → E)\nι : Type ?u.20\ninst✝² : Nonempty ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : DiscreteTopology ι\nU : ι → Set E\nV : Set X\nopen_V : IsOpen[inst✝⁴] V\nopen_iff : ∀ (i : ι) ... | [] | apply (f_inv _ hx.1).symm ▸ hx.1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Covering.Basic | {
"line": 486,
"column": 9
} | {
"line": 486,
"column": 24
} | {
"line": 486,
"column": 25
} | [
{
"pp": "case refine_2\nE : Type u_1\nX : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝³ : Nonempty (X → E)\nι : Type ?u.20\ninst✝² : Nonempty ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : DiscreteTopology ι\nU : ι → Set E\nV : Set X\nopen_V : IsOpen[inst✝⁴] V\nopen_i... | [
"case refine_2\nE : Type u_1\nX : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace X\nf : E → X\ns : Set X\ninst✝³ : Nonempty (X → E)\nι : Type ?u.20\ninst✝² : Nonempty ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : DiscreteTopology ι\nU : ι → Set E\nV : Set X\nopen_V : IsOpen[inst✝⁴] V\nopen_iff : ∀ (i : ... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 13
} | {
"line": 39,
"column": 14
} | [
{
"pp": "z : Circle\n⊢ (↑z).arg = 0 ↔ z = 1",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : Circle\n⊢ (↑z).arg = 0 ↔ z = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 528,
"column": 42
} | {
"line": 528,
"column": 53
} | {
"line": 528,
"column": 54
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ninst✝ : T2Space E\nx : X\nhf : IsClosedMap f\nfin : (f ⁻¹' {x}).Finite\nh : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f\nthis✝ : DiscreteTopology ↑(f ⁻¹' {x})\nφ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X\n... | [
"E : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\nf : E → X\ninst✝ : T2Space E\nx : X\nhf : IsClosedMap f\nfin : (f ⁻¹' {x}).Finite\nh : ∀ e ∈ f ⁻¹' {x}, ∃ φ, e ∈ φ.source ∧ ↑φ = f\nthis✝ : DiscreteTopology ↑(f ⁻¹' {x})\nφ : ↑(f ⁻¹' {x}) → OpenPartialHomeomorph E X\nhφ : ∀ (e : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 29
} | {
"line": 88,
"column": 30
} | [
{
"pp": "n : ℤ\n⊢ exp (2 * π * ↑n) = 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"InvOneCla... | [
"n : ℤ\n⊢ exp (↑n * (π * 2)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 15
} | {
"line": 183,
"column": 16
} | [
{
"pp": "A : Type u_2\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra ℝ A\ninst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\n⊢ {0}ᶜ ∈ nhdsSet (⋃ x ∈ {a | IsSelfAdjoint a ∧ IsUnit a}, spectrum ℝ (id x))",
"ppTerm": "?m.71... | [
"A : Type u_2\ninst✝⁵ : NormedRing A\ninst✝⁴ : StarRing A\ninst✝³ : NormedAlgebra ℝ A\ninst✝² : IsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : ContinuousStar A\ninst✝ : CompleteSpace A\n⊢ ∀ (x : A), IsSelfAdjoint x → IsUnit x → 0 ∉ spectrum ℝ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 31
} | {
"line": 164,
"column": 31
} | [
{
"pp": "z : Circle\ns : ℝ\nn : ℕ\nhs : s ≤ π\nh1 : |(↑z).arg| < π / ↑n\nh2 : ↑n * |(↑(↑z).arg).toReal| < s\nhs0 : 0 < s\nhn0 : n ≠ 0\nhn : 1 ≤ ↑n\n⊢ ↑n * |(↑z).arg| < s",
"ppTerm": "?m.185",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
... | [
"z : Circle\ns : ℝ\nn : ℕ\nhs : s ≤ π\nh1 : |(↑z).arg| < π / ↑n\nh2 : ↑n * |(↑z).arg| < s\nhs0 : 0 < s\nhn0 : n ≠ 0\nhn : 1 ≤ ↑n\n⊢ ↑n * |(↑z).arg| < s",
"z : Circle\ns : ℝ\nn : ℕ\nhs : s ≤ π\nh1 : |(↑z).arg| < π / ↑n\nh2 : |(n • ↑(↑z).arg).toReal| < s\nhs0 : 0 < s\nhn0 : n ≠ 0\nhn : 1 ≤ ↑n\n⊢ (↑(↑z).arg).toReal ... | arg_coe_angle_toReal_eq_arg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 165,
"column": 35
} | {
"line": 165,
"column": 62
} | {
"line": 165,
"column": 62
} | [
{
"pp": "z : Circle\ns : ℝ\nn : ℕ\nhs : s ≤ π\nh1 : |(↑z).arg| < π / ↑n\nh2 : |(n • ↑(↑z).arg).toReal| < s\nhs0 : 0 < s\nhn0 : n ≠ 0\nhn : 1 ≤ ↑n\n⊢ |(↑(↑z).arg).toReal| < π / ↑n",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"z : Circle\ns : ℝ\nn : ℕ\nhs : s ≤ π\nh1 : |(↑z).arg| < π / ↑n\nh2 : |(n • ↑(↑z).arg).toReal| < s\nhs0 : 0 < s\nhn0 : n ≠ 0\nhn : 1 ≤ ↑n\n⊢ |(↑z).arg| < π / ↑n"
] | arg_coe_angle_toReal_eq_arg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 226,
"column": 6
} | {
"line": 226,
"column": 17
} | {
"line": 226,
"column": 18
} | [
{
"pp": "x y : Circle\nt : ↑unitInterval\n⊢ ↑((x.path y) t) = ↑((⇑exp ∘ ⇑(Path.segment (↑x).arg (x.angleDiff y + (↑x).arg))) t)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNon... | [
"x y : Circle\nt : ↑unitInterval\n⊢ ↑(exp ((Path.segment (↑x).arg (x.angleDiff y + (↑x).arg)) t)) =\n ↑((⇑exp ∘ ⇑(Path.segment (↑x).arg (x.angleDiff y + (↑x).arg))) t)"
] | path_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | {
"line": 72,
"column": 14
} | [
{
"pp": "case e'_3\nA : Type u_1\ninst✝ : CStarAlgebra A\nu : A\nhu : u ∈ unitary A\nz : ℂ\nhz : z ∈ spectrum ℂ u\nthis : ‖z‖ = 1\n⊢ √(2 * (1 - z.re)) = ‖z - 1‖",
"ppTerm": "?e'_3✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NonAss... | [
"case e'_3\nA : Type u_1\ninst✝ : CStarAlgebra A\nu : A\nhu : u ∈ unitary A\nz : ℂ\nhz : z ∈ spectrum ℂ u\nthis : ‖z‖ = 1\n⊢ √2 * √(1 - z.re) = ‖z - 1‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 33
} | {
"line": 84,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : A\nhu : u ∈ unitary A\nx : ℝ\nhz : IsLeast (re '' spectrum ℂ u) x\nh✝ : Nontrivial A\nh_eqOn : Set.EqOn (fun z ↦ ‖z - 1‖ ^ 2) (fun z ↦ 2 * (1 - z.re)) (spectrum ℂ u)\nthis : Antitone fun y ↦ 2 * (1 - y)\n⊢ IsGreatest ((fun z ↦ 2 * (1 - z.re)) '' spectrum ℂ u) (... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : A\nhu : u ∈ unitary A\nx : ℝ\nhz : IsLeast (re '' spectrum ℂ u) x\nh✝ : Nontrivial A\nh_eqOn : Set.EqOn (fun z ↦ ‖z - 1‖ ^ 2) (fun z ↦ 2 * (1 - z.re)) (spectrum ℂ u)\nthis : Antitone fun y ↦ 2 * (1 - y)\n⊢ IsGreatest ((fun z ↦ 2 * (1 - z.re)) '' spectrum ℂ u) (2 * (1 - x))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 33
} | {
"line": 87,
"column": 34
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : A\nhu : u ∈ unitary A\nx : ℝ\nhz : IsLeast (re '' spectrum ℂ u) x\nh✝ : Nontrivial A\nh_eqOn : Set.EqOn (fun z ↦ ‖z - 1‖ ^ 2) (fun z ↦ 2 * (1 - z.re)) (spectrum ℂ u)\nh₂ : IsGreatest ((fun z ↦ 2 * (1 - z.re)) '' spectrum ℂ u) (2 * (1 - x))\nthis : MonotoneOn (f... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : A\nhu : u ∈ unitary A\nx : ℝ\nhz : IsLeast (re '' spectrum ℂ u) x\nh✝ : Nontrivial A\nh_eqOn : Set.EqOn (fun z ↦ ‖z - 1‖ ^ 2) (fun z ↦ 2 * (1 - z.re)) (spectrum ℂ u)\nh₂ : IsGreatest ((fun z ↦ 2 * (1 - z.re)) '' spectrum ℂ u) (2 * (1 - x))\nthis : MonotoneOn (fun a ↦ a ^ 2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Normalize | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 13
} | {
"line": 66,
"column": 14
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : NormedSpace ℝ V\nr : ℝ\nhr : r < 0\nx : V\n⊢ normalize (r • x) = -normalize x",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : NormedSpace ℝ V\nr : ℝ\nhr : r < 0\nx : V\n⊢ normalize (r • x) = -normalize x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 44
} | {
"line": 131,
"column": 45
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ ≤ π\na✝ : Nontrivial A\n⊢ IsLeast (re '' spectrum ℂ (NormedSpace.exp (I • ↑x))) (Real.cos ‖x‖)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormed... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ ≤ π\na✝ : Nontrivial A\n⊢ IsLeast (re '' spectrum ℂ (cfc NormedSpace.exp (I • ↑x))) (Real.cos ‖x‖)"
] | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 44
} | {
"line": 156,
"column": 45
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ < π\na✝ : Nontrivial A\nthis : spectrum ℂ ↑(expUnitary x) ⊆ slitPlane\n⊢ cfc (fun x ↦ ↑x.arg) (NormedSpace.exp (I • ↑x)) = ↑x",
"ppTerm": "?m.149",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ < π\na✝ : Nontrivial A\nthis : spectrum ℂ ↑(expUnitary x) ⊆ slitPlane\n⊢ cfc (fun x ↦ ↑x.arg) (cfc NormedSpace.exp (I • ↑x)) = ↑x"
] | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 476,
"column": 2
} | {
"line": 476,
"column": 33
} | {
"line": 477,
"column": 4
} | [
{
"pp": "r : ℝ\nhset : {x | |x| < r} = Ioo (-r) r\n⊢ IsOpen (centeredArc r)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.lattice",
"abs",
"congrArg",
"ContinuousMap",
"setOf",
"PseudoMetricSpace.toUniformSpace",
... | [
"r : ℝ\nhset : {x | |x| < r} = Ioo (-r) r\n⊢ IsOpen (⇑exp '' Ioo (-r) r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 483,
"column": 43
} | {
"line": 483,
"column": 54
} | {
"line": 483,
"column": 55
} | [
{
"pp": "z : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\n⊢ z ∈ centeredArc (π / 2)",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\n⊢ z ∈ centeredArc (π / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 159,
"column": 25
} | {
"line": 159,
"column": 63
} | {
"line": 159,
"column": 64
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ < π\na✝ : Nontrivial A\nthis : spectrum ℂ (NormedSpace.exp (I • ↑x)) ⊆ slitPlane\n⊢ (fun x ↦ NormedSpace.exp (I • x)) '' spectrum ℂ ↑x ⊆ slitPlane",
"ppTerm": "?m.347",
"assigned": true,
"usedConstants": [
"NormedCom... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ < π\na✝ : Nontrivial A\nthis : spectrum ℂ (cfc NormedSpace.exp (I • ↑x)) ⊆ slitPlane\n⊢ (fun x ↦ NormedSpace.exp (I • x)) '' spectrum ℂ ↑x ⊆ slitPlane"
] | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 486,
"column": 14
} | {
"line": 486,
"column": 25
} | {
"line": 486,
"column": 26
} | [
{
"pp": "case zero\nz : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\nhz1 : z ∈ centeredArc (π / 2)\n⊢ z ∈ centeredArc (π / 2 ^ (0 + 1))",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"congrArg",
"Real.instD... | [
"case zero\nz : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\nhz1 : z ∈ centeredArc (π / 2)\n⊢ z ∈ centeredArc (π / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 488,
"column": 8
} | {
"line": 488,
"column": 42
} | {
"line": 488,
"column": 43
} | [
{
"pp": "case succ\nz : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\nhz1 : z ∈ centeredArc (π / 2)\nn : ℕ\nih : z ∈ centeredArc (π / 2 ^ (n + 1))\n⊢ z ∈ centeredArc (π / 2 ^ (n + 1 + 1))",
"ppTerm": "?succ",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case succ\nz : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\nhz1 : z ∈ centeredArc (π / 2)\nn : ℕ\nih : z ∈ centeredArc (π / 2 ^ (n + 1))\n⊢ z ∈ centeredArc (π / 2 ^ (n + 1 + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 17
} | {
"line": 490,
"column": 18
} | [
{
"pp": "z : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\nhz1 : z ∈ centeredArc (π / 2)\nh : ∀ (n : ℕ), z ∈ centeredArc (π / 2 ^ (n + 1))\n⊢ z = 1",
"ppTerm": "?m.83",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"z : Circle\nhz : ∀ n > 0, z ^ n ∈ centeredArc (π / 2)\nhz1 : z ∈ centeredArc (π / 2)\nh : ∀ (n : ℕ), z ∈ centeredArc (π / 2 ^ (n + 1))\n⊢ z = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 498,
"column": 33
} | {
"line": 498,
"column": 44
} | {
"line": 498,
"column": 45
} | [
{
"pp": "n : ℤ\ninst✝ : NeZero n\n⊢ IsUnit ↑n",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"congrArg",
"IsUnit",
"Divis... | [
"n : ℤ\ninst✝ : NeZero n\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 175,
"column": 42
} | {
"line": 175,
"column": 80
} | {
"line": 176,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : ‖↑u - 1‖ < 2\nthis : ContinuousOn arg (spectrum ℂ ↑u)\n⊢ NormedSpace.exp (I • cfc (fun x ↦ ↑x.arg) ↑u) = ↑u",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : ‖↑u - 1‖ < 2\nthis : ContinuousOn arg (spectrum ℂ ↑u)\n⊢ cfc NormedSpace.exp (I • cfc (fun x ↦ ↑x.arg) ↑u) = ↑u"
] | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 36
} | {
"line": 182,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : ‖↑u - 1‖ < 2\nthis✝ : ContinuousOn arg (spectrum ℂ ↑u)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\nhy₁ : ‖y‖ = 1\nthis : I * ↑y.arg = log y\n⊢ NormedSpace.exp (I • ↑y.arg) = y",
"ppTerm": "?m.374",
"assigned": true,
"usedConstants": [
"... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : ‖↑u - 1‖ < 2\nthis✝ : ContinuousOn arg (spectrum ℂ ↑u)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\nhy₁ : ‖y‖ = 1\nthis : I * ↑y.arg = log y\n⊢ cexp (log y) = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 186,
"column": 44
} | {
"line": 186,
"column": 55
} | {
"line": 186,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\n⊢ ‖↑y.arg‖ ≤ π",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"co... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\n⊢ |y.arg| ≤ π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 49
} | {
"line": 235,
"column": 50
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : dist u 1 < 2\nε : ℝ\nhuε : dist u 1 ^ 2 < ε\nhε2 : ε < 2 ^ 2\nhε : 0 < ε\nhuε' : dist u 1 < √ε\nv : ↥(unitary A)\nhv : v ∈ closedBall 1 √ε\nz : ℂ\nhz : z ∈ spectrum ℂ ↑v\n⊢ ‖↑v - 1‖ ≤ √ε",
"ppTerm": "?m.266",
"assigned": false,
"u... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : dist u 1 < 2\nε : ℝ\nhuε : dist u 1 ^ 2 < ε\nhε2 : ε < 2 ^ 2\nhε : 0 < ε\nhuε' : dist u 1 < √ε\nv : ↥(unitary A)\nhv : v ∈ closedBall 1 √ε\nz : ℂ\nhz : z ∈ spectrum ℂ ↑v\n⊢ ‖↑v - 1‖ ≤ √ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 47
} | {
"line": 268,
"column": 48
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : u ∈ ball 1 2\n⊢ ‖↑u - 1‖ < 2",
"ppTerm": "?m.270",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : u ∈ ball 1 2\n⊢ ‖↑u - 1‖ < 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 269,
"column": 53
} | {
"line": 269,
"column": 64
} | {
"line": 269,
"column": 65
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : x ∈ ball 0 π\n⊢ ‖x‖ < π",
"ppTerm": "?m.280",
"assigned": true,
"usedConstants": [
"Norm.norm",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"Real",
"NonUnitalCStarAlgebra.toNonUnitalNormedRing",
"Real... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : x ∈ ball 0 π\n⊢ ‖↑x‖ < π"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 324,
"column": 36
} | {
"line": 324,
"column": 79
} | {
"line": 324,
"column": 80
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu✝ : ↥(unitary A)\nδ : ℝ\nhδ₀ : 0 < δ\nhδ₂ : δ < 2\nu : ↥(unitary A)\nhu : u ∈ ball 1 δ\n⊢ ‖↑u - 1‖ < δ",
"ppTerm": "?m.134",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu✝ : ↥(unitary A)\nδ : ℝ\nhδ₀ : 0 < δ\nhδ₂ : δ < 2\nu : ↥(unitary A)\nhu : u ∈ ball 1 δ\n⊢ ‖↑u - 1‖ < δ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 45
} | {
"line": 327,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu✝ : ↥(unitary A)\nδ : ℝ\nhδ₀ : 0 < δ\nhδ₂ : δ < 2\nu : ↥(unitary A)\nhu✝ : u ∈ ball 1 δ\nhu : ‖↑u - 1‖ < δ\nt : ↑unitInterval\n⊢ (path 1 u ⋯) t ∈ ball 1 δ",
"ppTerm": "?m.158",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | [
"A : Type u_1\ninst✝ : CStarAlgebra A\nu✝ : ↥(unitary A)\nδ : ℝ\nhδ₀ : 0 < δ\nhδ₂ : δ < 2\nu : ↥(unitary A)\nhu✝ : u ∈ ball 1 δ\nhu : ‖↑u - 1‖ < δ\nt : ↑unitInterval\n⊢ ‖NormedSpace.exp (I • ↑t • cfc (fun x ↦ ↑x.arg) ↑u) - 1‖ < δ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 37
} | {
"line": 97,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nv : E\n⊢ (adjointAux (adjointAux A)) v = A v"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nv : E\nw : F\n⊢ ⟪w, (adjointAux (adjointAux A)) v⟫_𝕜 = ⟪... | refine ext_inner_left 𝕜 fun w => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 37
} | {
"line": 141,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ :... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : CompleteSpace E\ninst✝¹ : CompleteSpa... | refine ext_inner_left 𝕜 fun w => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.AbsolutelyMonotone | {
"line": 67,
"column": 75
} | {
"line": 69,
"column": 21
} | {
"line": 71,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\ns : Set ℝ\nhf : AbsolutelyMonotoneOn f s\n⊢ ContDiffOn ℝ ∞ f s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real.instLE",
"Real",
"Semiring.toModule",
"AbsolutelyMonotoneOn",
"_private.Mathlib.Analysis.Calculus.AbsolutelyMonotone.0.Abso... | [] | by
obtain ⟨_, hp, _⟩ := hf
exact hp.contDiffOn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 38
} | {
"line": 208,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT U : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nhU : (↑U).IsSymmetric\nh : (↑U).ker ≤ (↑T).ker\nthis : CompleteSpace E\n⊢ (↑T).range ≤ (↑U).range",
"ppTerm": "?m... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT U : E →L[𝕜] E\nhT : (↑T).IsSymmetric\nhU : (↑U).IsSymmetric\nh : (↑U).ker ≤ (↑T).ker\nthis : CompleteSpace E\n⊢ (↑T).range ≤ (↑U).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 13
} | {
"line": 217,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] F\n⊢ (↑(T ∘SL adjoint T)).ker = (↑(adjoint T)).k... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] F\n⊢ (↑T ∘ₗ ↑(adjoint T)).ker = (↑(adjoint T)).ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 13
} | {
"line": 236,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] F\n⊢ Function.Injective (⇑T ∘ ⇑(adjoint T)) ↔ Fu... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT : E →L[𝕜] F\n⊢ Function.Injective (⇑T ∘ ⇑(adjoint T)) ↔ Function.Injec... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 13
} | {
"line": 457,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\ninst✝ : CompleteSpace E\nS : E →L[𝕜] E\nhS : IsStarProjection S\nhT : IsStarProjection T\n⊢ S = T ↔ (↑S).range = (↑T).range",
"ppTerm": "?m.62",
"assigned": false,
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\ninst✝ : CompleteSpace E\nS : E →L[𝕜] E\nhS : IsStarProjection S\nhT : IsStarProjection T\n⊢ S = T ↔ (↑S).range = (↑T).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 476,
"column": 12
} | {
"line": 476,
"column": 23
} | {
"line": 476,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nT : E →L[𝕜] E\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\n⊢ Uᗮ ∈ invtSubmodule ↑T → U ∈ invtSubmodule ↑(adjoint T)",
"ppTerm": "?m.106",
"assig... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nT : E →L[𝕜] E\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\n⊢ Uᗮ ∈ invtSubmodule ↑T → U ∈ invtSubmodule ↑(adjoint T)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 37
} | {
"line": 578,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\nv : E\n⊢ (adjoint (adjoint A)) ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\nv : E\nw : F\n⊢ ⟪w, (adjoint (adjoint A)) v... | refine ext_inner_left 𝕜 fun w => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 37
} | {
"line": 587,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : FiniteDimensional 𝕜 E\ni... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : InnerProductSpace 𝕜 E\ninst✝⁴ : InnerProductSpace 𝕜 F\ninst✝³ : InnerProductSpace 𝕜 G\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : Fini... | refine ext_inner_left 𝕜 fun w => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 626,
"column": 2
} | {
"line": 626,
"column": 13
} | {
"line": 626,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ (A ∘ₗ adjoint A).ker = (adjoi... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ (A ∘ₗ adjoint A).ker = (adjoint A).ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 13
} | {
"line": 644,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ Function.Injective (⇑A ∘ ⇑(ad... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ Function.Injective (⇑A ∘ ⇑(adjoint A)) ↔ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 653,
"column": 2
} | {
"line": 653,
"column": 13
} | {
"line": 653,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ (A ∘ₗ adjoint A).range = A.ra... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\n⊢ (A ∘ₗ adjoint A).range = A.range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 727,
"column": 2
} | {
"line": 727,
"column": 13
} | {
"line": 727,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\n⊢ (T ∘ₗ adjoint T).IsSymmetric"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\n⊢ (T ∘ₗ adjoint T).IsSymmetric"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 732,
"column": 2
} | {
"line": 732,
"column": 13
} | {
"line": 732,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nS : F →ₗ[𝕜... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nS : F →ₗ[𝕜] E\n⊢ (adjo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 736,
"column": 2
} | {
"line": 736,
"column": 13
} | {
"line": 736,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\n⊢ (adjoint T ∘ₗ T).IsSymmetric"... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\n⊢ (adjoint T ∘ₗ T).IsSymmetric"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 788,
"column": 2
} | {
"line": 788,
"column": 13
} | {
"line": 788,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : IsStarProjection S\nhT : IsStarProjection T\nthis : CompleteSpace E\n⊢ S = T ↔ S.range = T.range",
"ppTerm": "?m.77",
"assigned... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : IsStarProjection S\nhT : IsStarProjection T\nthis : CompleteSpace E\n⊢ S = T ↔ S.range = T.range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 826,
"column": 4
} | {
"line": 826,
"column": 53
} | {
"line": 826,
"column": 54
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nu : H →L[𝕜] K\nh : ∀ (x y : H), ⟪u x, u y⟫_𝕜... | [
"case refine_1\n𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nu : H →L[𝕜] K\nh : ∀ (x y : H), ⟪u x, u y⟫_𝕜 = ⟪x, y⟫_𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 915,
"column": 34
} | {
"line": 915,
"column": 50
} | {
"line": 915,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny : ... | [
"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny : 𝕜\nh : y • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 30
} | {
"line": 169,
"column": 31
} | [
{
"pp": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : CharZero k\nn : ℕ\ns : Simplex k P n\nfs₁ fs₂ : Finset (Fin (n + 1))\nm₁ m₂ : ℕ\nh₁ : #fs₁ = m₁ + 1\nh₂ : #fs₂ = m₂ + 1\nh :\n (affineCombination... | [
"case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : CharZero k\nn : ℕ\ns : Simplex k P n\nfs₁ fs₂ : Finset (Fin (n + 1))\nm₁ m₂ : ℕ\nh₁ : #fs₁ = m₁ + 1\nh₂ : #fs₂ = m₂ + 1\nh :\n (affineCombination k univ s.po... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 916,
"column": 48
} | {
"line": 916,
"column": 59
} | {
"line": 916,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝¹ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny :... | [
"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝¹ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny : 𝕜\nh : y •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 919,
"column": 15
} | {
"line": 920,
"column": 59
} | {
"line": 920,
"column": 60
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny : ... | [
"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny : 𝕜\nh : y • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 928,
"column": 34
} | {
"line": 928,
"column": 45
} | {
"line": 928,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny : ... | [
"𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nH : Type u_5\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\nK : Type u_6\ninst✝² : NormedAddCommGroup K\ninst✝¹ : InnerProductSpace 𝕜 K\ninst✝ : CompleteSpace K\nf g : H ≃ₗᵢ[𝕜] K\nx✝ : ∃ y, y • 1 = ↑(↑g).symm ∘SL ↑↑f\ny : 𝕜\nh : y • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 214,
"column": 26
} | {
"line": 214,
"column": 37
} | {
"line": 214,
"column": 38
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nm n : ℕ\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ m = n",
"ppTerm": "?m.71",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nm n : ℕ\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 138,
"column": 32
} | {
"line": 138,
"column": 43
} | {
"line": 138,
"column": 44
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : DecidableEq P\np : ι → P\nhi : AffineIndependent k p\ns : Finset ι\nn : ℕ\nhc : #s = n + 1\nhi' : AffineIndependent k fun x ↦ ↑x\nhc' : #(Fins... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : DecidableEq P\np : ι → P\nhi : AffineIndependent k p\ns : Finset ι\nn : ℕ\nhc : #s = n + 1\nhi' : AffineIndependent k fun x ↦ ↑x\nhc' : #(Finset.image p s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 19
} | {
"line": 211,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\nhn : Nonempty ι\n⊢ AffineIndependent k p ↔ finrank k ↥(vectorSpan k (Set.range p)) =... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\ni₁ : ι\n⊢ AffineIndependent k p ↔ finrank k ↥(vectorSpan k (Set.range p)) = n"
] | obtain ⟨i₁⟩ := hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Calculus.BumpFunction.Basic | {
"line": 139,
"column": 2
} | {
"line": 140,
"column": 31
} | {
"line": 140,
"column": 32
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : HasContDiffBump E\nc : E\nf : ContDiffBump c\nx : E\nhx : x ∈ closedBall c f.rIn\n⊢ ‖(fun x ↦ f.rIn⁻¹ • (x - c)) x‖ ≤ 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : HasContDiffBump E\nc : E\nf : ContDiffBump c\nx : E\nhx : x ∈ closedBall c f.rIn\n⊢ ‖x - c‖ ≤ f.rIn"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 416,
"column": 27
} | {
"line": 416,
"column": 38
} | {
"line": 416,
"column": 39
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\ni : Fin (n + 1)\n⊢ m = n",
"ppTerm": "?m.83",
"assigned": fa... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\ni : Fin (n + 1)\n⊢ m = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 38
} | {
"line": 262,
"column": 39
} | [
{
"pp": "case inr.inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : Finset V\nhs : AffineIndependent k Subtype.val\nhs' : s.Nonempty\nhst : ↑s ⊆ ↑(affineSpan k ↑∅)\n⊢ #s ≤ #∅",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"... | [
"case inr.inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : Finset V\nhs : AffineIndependent k Subtype.val\nhs' : s.Nonempty\nhst : ↑s ⊆ ↑(affineSpan k ↑∅)\n⊢ s = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Integral | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 13
} | {
"line": 64,
"column": 14
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Ici c⁻¹) μ\n⊢ IntegrableOn (fun... | [
"G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Ici c⁻¹) μ\n⊢ IntegrableOn (fun x ↦ f x⁻¹) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Integral | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | {
"line": 69,
"column": 14
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Iic c⁻¹) μ\n⊢ IntegrableOn (fun... | [
"G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Iic c⁻¹) μ\n⊢ IntegrableOn (fun x ↦ f x⁻¹) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Integral | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 13
} | {
"line": 74,
"column": 14
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Ioi c⁻¹) μ\n⊢ IntegrableOn (fun... | [
"G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Ioi c⁻¹) μ\n⊢ IntegrableOn (fun x ↦ f x⁻¹) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Integral | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Iio c⁻¹) μ\n⊢ IntegrableOn (fun... | [
"G : Type u_4\nF : Type u_6\ninst✝⁶ : MeasurableSpace G\ninst✝⁵ : NormedAddCommGroup F\nμ : Measure G\ninst✝⁴ : PartialOrder G\ninst✝³ : CommGroup G\ninst✝² : IsOrderedMonoid G\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsInvInvariant\nc : G\nf : G → F\nhf : IntegrableOn f (Set.Iio c⁻¹) μ\n⊢ IntegrableOn (fun x ↦ f x⁻¹) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 13
} | {
"line": 271,
"column": 14
} | [
{
"pp": "case inr.inr\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns t : Finset V\nhs : AffineIndependent k Subtype.val\nhst : ↑s ⊆ ↑(affineSpan k ↑t)\nhs' : s.Nonempty\nht' : t.Nonempty\nthis✝ : Nonempty ↥s\nthis : Nonempty ↑↑t\ndirection_le : vectorSpan k ... | [
"case inr.inr\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns t : Finset V\nhs : AffineIndependent k Subtype.val\nhst : ↑s ⊆ ↑(affineSpan k ↑t)\nhs' : s.Nonempty\nht' : t.Nonempty\nthis✝ : Nonempty ↥s\nthis : Nonempty ↑↑t\ndirection_le : vectorSpan k (Set.range S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 26
} | {
"line": 280,
"column": 27
} | [
{
"pp": "case inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nt : Finset V\nhs : AffineIndependent k Subtype.val\nhst : affineSpan k ↑∅ < affineSpan k ↑t\n⊢ #∅ < #t",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nt : Finset V\nhs : AffineIndependent k Subtype.val\nhst : affineSpan k ↑∅ < affineSpan k ↑t\n⊢ t.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 13
} | {
"line": 291,
"column": 14
} | [
{
"pp": "case inr.inr\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns t : Finset V\nhs : AffineIndependent k Subtype.val\nhst : affineSpan k ↑s < affineSpan k ↑t\nhs' : s.Nonempty\nht' : t.Nonempty\nthis✝ : Nonempty ↥s\nthis : Nonempty ↑↑t\ndir_lt : vectorSpa... | [
"case inr.inr\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns t : Finset V\nhs : AffineIndependent k Subtype.val\nhst : affineSpan k ↑s < affineSpan k ↑t\nhs' : s.Nonempty\nht' : t.Nonempty\nthis✝ : Nonempty ↥s\nthis : Nonempty ↑↑t\ndir_lt : vectorSpan k (Set.ran... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Group.Integral | {
"line": 111,
"column": 27
} | {
"line": 111,
"column": 59
} | {
"line": 111,
"column": 59
} | [
{
"pp": "G : Type u_4\nE : Type u_5\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : μ.IsMulRightInvariant\nf : G → E\ng : G\n⊢ ∫ (x : G), f (x * g⁻¹) ∂μ = ∫ (x : G), f x ∂μ",
"ppTerm": "?m.27",
"assi... | [] | integral_mul_right_eq_self f g⁻¹ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 491,
"column": 6
} | {
"line": 491,
"column": 28
} | {
"line": 492,
"column": 4
} | [
{
"pp": "case inr.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\n⊢ (∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁) → ∃ p₀ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀",
"ppTerm": "?inr.mp",
"assigned... | [] | exact fun h => ⟨p₁, h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 491,
"column": 6
} | {
"line": 491,
"column": 28
} | {
"line": 492,
"column": 4
} | [
{
"pp": "case inr.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\n⊢ (∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁) → ∃ p₀ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀",
"ppTerm": "?inr.mp",
"assigned... | [] | exact fun h => ⟨p₁, h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 491,
"column": 6
} | {
"line": 491,
"column": 28
} | {
"line": 492,
"column": 4
} | [
{
"pp": "case inr.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nhp₁ : p₁ ∈ s\n⊢ (∃ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₁) → ∃ p₀ v, ∀ p ∈ s, ∃ r, p = r • v +ᵥ p₀",
"ppTerm": "?inr.mp",
"assigned... | [] | exact fun h => ⟨p₁, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 542,
"column": 55
} | {
"line": 542,
"column": 66
} | {
"line": 542,
"column": 67
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\nh : LinearIndependent k fun i ↦ s.points ↑i -ᵥ s.points 0\nx✝ : { x // x ≠ 0 }\n⊢ (-↑n)⁻¹ ≠ 0",
... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\nh : LinearIndependent k fun i ↦ s.points ↑i -ᵥ s.points 0\nx✝ : { x // x ≠ 0 }\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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