module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.Circle | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 36
} | [
{
"pp": "case h\n⊢ probChar π ≠ 1 π",
"usedConstants": [
"Real",
"Real.pi",
"Real.instAddMonoid",
"AddChar.instOne",
"AddChar",
"id",
"DivInvMonoid.toMonoid",
"Ne",
"Real.probChar",
"Group.toDivInvMonoid",
"Circle",
"One.toOfNat1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Bundle | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 23
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nb : B\n⊢ range (mk b) = proj ⁻¹' {b}",
"usedConstants": [
"Set.Subset.antisymm",
"Bundle.TotalSpace.mk",
"Set.instSingletonSet",
"Bundle.TotalSpace.proj",
"Set.preimage",
"Bundle.TotalSpace",
"Set.range",
... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.OpenPartialHomeomorph.Composition | {
"line": 184,
"column": 4
} | {
"line": 185,
"column": 68
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\ne' : OpenPartialHomeomorph X Y\nhs : IsOpen[inst✝¹] s\nht : IsOpen[inst✝] (e'.target ∩ ↑e'.symm ⁻¹' s)\n⊢ e'.target ∩ (↑e'.symm ⁻¹' e.source ∩ ↑e'.symm ⁻¹' inter... | rw [interior_eq_iff_isOpen.mpr hs,
← inter_assoc, inter_comm e'.target, inter_assoc, inter_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 240,
"column": 75
} | {
"line": 241,
"column": 51
} | [
{
"pp": "X : Type u_1\nY : Type u_3\nZ : Type u_5\nZ' : Type u_6\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne : OpenPartialHomeomorph X Y\ne' : OpenPartialHomeomorph Y Z\nf'' : Z ≃ₜ Z'\n⊢ (e.trans e').transHomeomorph f'' = e.trans (e'.tra... | by
simp only [transHomeomorph_eq_trans, trans_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 363,
"column": 24
} | {
"line": 363,
"column": 82
} | [
{
"pp": "X✝ : Type u_1\nX'✝ : Type u_2\nY : Type u_3\nY' : Type u_4\nZ✝ : Type u_5\nZ' : Type u_6\ninst✝⁹ : TopologicalSpace X✝\ninst✝⁸ : TopologicalSpace X'✝\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : TopologicalSpace Y'\ninst✝⁵ : TopologicalSpace Z✝\ninst✝⁴ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomorph X✝ Y\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 107,
"column": 2
} | {
"line": 108,
"column": 9
} | [
{
"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✝ : Nonempty F\ne e' : Pretrivialization F proj\nh : e.toPartialEquiv = e'.toPartialEquiv\n⊢ e.baseSet = e'.baseSet",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 106,
"column": 2
} | {
"line": 108,
"column": 57
} | [
{
"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✝ : Nonempty F\n⊢ Injective toPartialEquiv",
"usedConstants": [
"Set.instSProd",
"SProd.sprod",
"congrArg",
"PartialEquiv.target",
"PartialEquiv",
... | refine fun e e' h ↦ ext' _ _ h ?_
simpa only [fst_image_prod, univ_nonempty, target_eq]
using congr_arg (Prod.fst '' PartialEquiv.target ·) h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 106,
"column": 2
} | {
"line": 108,
"column": 57
} | [
{
"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✝ : Nonempty F\n⊢ Injective toPartialEquiv",
"usedConstants": [
"Set.instSProd",
"SProd.sprod",
"congrArg",
"PartialEquiv.target",
"PartialEquiv",
... | refine fun e e' h ↦ ext' _ _ h ?_
simpa only [fst_image_prod, univ_nonempty, target_eq]
using congr_arg (Prod.fst '' PartialEquiv.target ·) h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 277,
"column": 6
} | {
"line": 277,
"column": 89
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 58
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 319,
"column": 4
} | {
"line": 319,
"column": 36
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\ne : Pretrivialization F fun z ↦ proj ↑z\ninst✝ : Nonempty (Z → F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 320,
"column": 25
} | {
"line": 320,
"column": 36
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\ne : Pretrivialization F fun z ↦ proj ↑z\ninst✝ : Nonempty (Z → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 326,
"column": 25
} | {
"line": 326,
"column": 50
} | [
{
"pp": "case h\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\ne : Pretrivialization F fun z ↦ proj ↑z\ninst✝ : Nonempt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 341,
"column": 25
} | {
"line": 341,
"column": 36
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj✝ : Z → B\ne✝ : Pretrivialization F proj✝\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\nhs : IsOpen[inst✝²] s\nproj : Z → ↑s\ne : Pretrivialization F ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 342,
"column": 43
} | {
"line": 342,
"column": 54
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj✝ : Z → B\ne✝ : Pretrivialization F proj✝\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\nhs : IsOpen[inst✝²] s\nproj : Z → ↑s\ne : Pretrivialization F... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 343,
"column": 23
} | {
"line": 343,
"column": 34
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj✝ : Z → B\ne✝ : Pretrivialization F proj✝\nx : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ns : Set B\nhs : IsOpen[inst✝²] s\nproj : Z → ↑s\ne : Pretrivialization F ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Basic | {
"line": 467,
"column": 25
} | {
"line": 467,
"column": 36
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni j : ι\np : B × F\nhp : p ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ\n⊢ (p.1, Z.coordChange i j p.1 p.2) ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Basic | {
"line": 468,
"column": 25
} | {
"line": 468,
"column": 36
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni j : ι\np : B × F\nhp : p ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ univ\n⊢ (p.1, Z.coordChange j i p.1 p.2) ∈ (Z.baseSet i ∩ Z.baseSet j) ×ˢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Basic | {
"line": 486,
"column": 4
} | {
"line": 486,
"column": 28
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni j : ι\n⊢ ContinuousOn (fun p ↦ (p.1, Z.coordChange j i p.1 p.2)) ((Z.baseSet i ∩ Z.baseSet j) ×ˢ univ)",
"usedConstants": []
}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Basic | {
"line": 507,
"column": 4
} | {
"line": 507,
"column": 91
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\np : Z.TotalSpace\nhp : p ∈ Z.proj ⁻¹' Z.baseSet i\n⊢ (p.proj, Z.coordChange (Z.indexAt p.proj) i p.proj p.snd) ∈ Z.baseSet i ×ˢ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Basic | {
"line": 509,
"column": 4
} | {
"line": 509,
"column": 77
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\nX : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\np : B × F\nhp : p ∈ Z.baseSet i ×ˢ univ\n⊢ { proj := p.1, snd := Z.coordChange i (Z.indexAt p.1) p.1 p.2 } ∈ Z.proj ⁻¹' Z.baseS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 613,
"column": 35
} | {
"line": 613,
"column": 46
} | [
{
"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\nZ' : Type u_5\ninst✝ : TopologicalSpace Z'\nh : Z' ≃ₜ Z\np : Z'\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 624,
"column": 24
} | {
"line": 624,
"column": 35
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 767,
"column": 73
} | {
"line": 767,
"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",
"usedConstant... | coordChange | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 840,
"column": 4
} | {
"line": 840,
"column": 45
} | [
{
"pp": "case h.e'_5.h\nB : 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\ne' : Trivialization F TotalSpace.proj\nb : B\ny... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 980,
"column": 2
} | {
"line": 980,
"column": 13
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.SeparatedMap | {
"line": 138,
"column": 2
} | {
"line": 139,
"column": 78
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nf : X → Y\n⊢ IsLocallyInjective f ↔ IsOpen[instTopologicalSpaceSubtype] (Function.pullbackDiagonal f)",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Topology.SeparatedMap.0.isLocallyInjective_iff_is... | 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.Topology.IsLocalHomeomorph | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 17
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.IsLocalHomeomorph | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 15
} | [
{
"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[... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 56,
"column": 49
} | {
"line": 56,
"column": 65
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 76,
"column": 40
} | {
"line": 76,
"column": 56
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 146,
"column": 54
} | {
"line": 146,
"column": 75
} | [
{
"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\nhxs : x ∈ s\nh : IsEvenlyCovered f x I\ninst : DiscreteTopology I\nU : Set X\nhxU : x ∈ U\nhU : IsOpen[inst✝¹] U\nhfU : IsOpen[inst✝²] (f ⁻¹' U)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 165,
"column": 75
} | {
"line": 165,
"column": 86
} | [
{
"pp": "case h\nE : 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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | [
{
"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✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 53,
"column": 4
} | {
"line": 53,
"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_3\ninst✝¹ : Group G\ninst✝ : MulAction G E\nh : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Quotient | {
"line": 62,
"column": 31
} | {
"line": 62,
"column": 42
} | [
{
"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\nh : 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.orbit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 280,
"column": 51
} | {
"line": 280,
"column": 62
} | [
{
"pp": "case h.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",
"usedConstants": [
"congrArg",
"Function.... | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Topology.Covering.Basic | {
"line": 280,
"column": 51
} | {
"line": 280,
"column": 62
} | [
{
"pp": "case h.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",
"usedConstants": [
"Set.ext",
... | (ext; simp) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.Topology.Covering.Basic | {
"line": 300,
"column": 51
} | {
"line": 300,
"column": 62
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 395,
"column": 8
} | {
"line": 395,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NonUnitalNormedRing A\ninst✝⁶ : NormedSpace 𝕜 A\ninst✝⁵ : SMulCommClass 𝕜 A A\ninst✝⁴ : IsScalarTower 𝕜 A A\ninst✝³ : StarRing 𝕜\ninst✝² : StarRing A\ninst✝¹ : StarModule 𝕜 A\ninst✝ : NormedStarGroup A\na : 𝓜(𝕜, A)\nx y :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 412,
"column": 34
} | {
"line": 412,
"column": 73
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Set.univ",
"Membership.mem",
"Ex... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 455,
"column": 46
} | {
"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.23878\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 : ... | apply (f_inv _ hx.1).symm ▸ hx.1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 492,
"column": 8
} | {
"line": 492,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NonUnitalNormedRing A\ninst✝² : NormedSpace 𝕜 A\ninst✝¹ : SMulCommClass 𝕜 A A\ninst✝ : IsScalarTower 𝕜 A A\n⊢ Function.Injective ⇑toProdMulOppositeHom",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 535,
"column": 6
} | {
"line": 535,
"column": 42
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.Basic | {
"line": 486,
"column": 9
} | {
"line": 486,
"column": 24
} | [
{
"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.23878\ninst✝² : Nonempty ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : DiscreteTopology ι\nU : ι → Set E\nV : Set X\nopen_V : IsOpen[inst✝⁴] V\nope... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Covering.Basic | {
"line": 528,
"column": 42
} | {
"line": 528,
"column": 53
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 544,
"column": 8
} | {
"line": 544,
"column": 31
} | [
{
"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 ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 552,
"column": 8
} | {
"line": 552,
"column": 31
} | [
{
"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 ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Covering.AddCircle | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 84
} | [
{
"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",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Monoid",
"Int.ca... | 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": 72,
"column": 4
} | {
"line": 72,
"column": 84
} | [
{
"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",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Monoid",
"Int.ca... | convert! (Int.cast_ne_zero.mpr <| NeZero.ne n).isUnit.map (algebraMap ℚ 𝕜); simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 631,
"column": 10
} | {
"line": 632,
"column": 17
} | [
{
"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(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Basic | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 15
} | [
{
"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))",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 29
} | [
{
"pp": "n : ℤ\n⊢ exp (2 * π * ↑n) = 1",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"InvOneClass.toOne",
"HMul.hMul",
"Division... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 174,
"column": 6
} | {
"line": 174,
"column": 17
} | [
{
"pp": "case h.a\nx y : Circle\nt : ↑unitInterval\n⊢ ↑((x.path y) t) = ↑((⇑exp ∘ ⇑(Path.segment (↑x).arg (x.angleDiff y + (↑x).arg))) t)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toM... | path_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Normalize | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 412,
"column": 33
} | {
"line": 412,
"column": 44
} | [
{
"pp": "n : ℤ\ninst✝ : NeZero n\n⊢ IsUnit ↑n",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"Real.instRCLike",
"congrArg",
"IsUnit",
"DivisionSemiring.toGroupWithZero",
"id",
"Ne",
"Real.instRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 13
} | [
{
"pp": "case h.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‖",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 33
} | [
{
"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) (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 33
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 44
} | [
{
"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‖)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing... | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 155,
"column": 6
} | {
"line": 155,
"column": 44
} | [
{
"pp": "case a\nA : 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",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.... | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 158,
"column": 25
} | {
"line": 158,
"column": 63
} | [
{
"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",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.toS... | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 174,
"column": 42
} | {
"line": 174,
"column": 80
} | [
{
"pp": "case a\nA : 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",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | ← CFC.exp_eq_normedSpace_exp (𝕜 := ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 36
} | [
{
"pp": "case a\nA : 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",
"usedConstants": [
"NormedCommRing.toNormedRing",
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 185,
"column": 44
} | {
"line": 185,
"column": 55
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\ny : ℂ\nhy : y ∈ spectrum ℂ ↑u\n⊢ ‖↑y.arg‖ ≤ π",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
"NormedField.toNorm",
"C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 49
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 267,
"column": 4
} | {
"line": 267,
"column": 47
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : ↥(unitary A)\nhu : u ∈ ball 1 2\n⊢ ‖↑u - 1‖ < 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 268,
"column": 53
} | {
"line": 268,
"column": 64
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : x ∈ ball 0 π\n⊢ ‖x‖ < π",
"usedConstants": [
"Norm.norm",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"Real",
"NonUnitalCStarAlgebra.toNonUnitalNormedRing",
"Real.pi",
"NormedRing.toRing",
"Ring.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 72,
"column": 10
} | {
"line": 72,
"column": 70
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nhx : x ≠ 0\n⊢ ‖ℜ (‖x‖⁻¹ • x)‖ ≤ 1",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"GroupWithZero.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 74,
"column": 10
} | {
"line": 74,
"column": 70
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nhx : x ≠ 0\nu₁ : ↥(unitary A) := selfAdjoint.unitarySelfAddISMul (ℜ (‖x‖⁻¹ • x)) ⋯\n⊢ ‖ℑ (‖x‖⁻¹ • x)‖ ≤ 1",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 9
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nhx : x ≠ 0\nu₁ : ↥(unitary A) := selfAdjoint.unitarySelfAddISMul (ℜ (‖x‖⁻¹ • x)) ⋯\nu₂ : ↥(unitary A) := selfAdjoint.unitarySelfAddISMul (ℑ (‖x‖⁻¹ • x)) ⋯\n⊢ x = ‖x‖ • (↑(ℜ ↑u₁) + I • ↑(ℜ ↑u₂))",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 323,
"column": 36
} | {
"line": 323,
"column": 79
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 45
} | [
{
"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",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"CStarAlgebra.toNonUnitalCStarAlgebra",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 15
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\n⊢ ∀ (x : ↥(unitary A)) (i : ℝ), 0 < i ∧ i < 2 → IsPathConnected {y | (y, x) ∈ {p | dist p.1 p.2 < i}}",
"usedConstants": [
"IsPathConnected",
"Eq.mpr",
"Real",
"NormedRing.toRing",
"SetRel",
"Ring.toNonAssocRing",
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 347,
"column": 36
} | {
"line": 347,
"column": 79
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu v : ↥(unitary A)\nhuv : dist u v < 2\nl : List ↥(selfAdjoint A)\nhlv : (List.map expUnitary l).prod = v\n⊢ ‖↑u - ↑v‖ < 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 351,
"column": 22
} | {
"line": 351,
"column": 33
} | [
{
"pp": "case mpr.cons\nA : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nxs : List ↥(selfAdjoint A)\nih : (List.map expUnitary xs).prod ∈ pathComponent 1\n⊢ (List.map expUnitary (x :: xs)).prod ∈ pathComponent 1",
"usedConstants": [
"CStarAlgebra.toNonUnitalCStarAlgebra",
"NonUnitalCS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.AbsolutelyMonotone | {
"line": 67,
"column": 75
} | {
"line": 69,
"column": 21
} | [
{
"pp": "f : ℝ → ℝ\ns : Set ℝ\nhf : AbsolutelyMonotoneOn f s\n⊢ ContDiffOn ℝ ∞ f s",
"usedConstants": [
"Real.instLE",
"Real",
"Semiring.toModule",
"AbsolutelyMonotoneOn",
"_private.Mathlib.Analysis.Calculus.AbsolutelyMonotone.0.AbsolutelyMonotoneOn._proof_2",
"Real.dense... | by
obtain ⟨_, hp, _⟩ := hf
exact hp.contDiffOn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 30
} | [
{
"pp": "case h.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 (affineCombinati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 214,
"column": 26
} | {
"line": 214,
"column": 37
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 138,
"column": 32
} | {
"line": 138,
"column": 43
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 19
} | [
{
"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)) =... | obtain ⟨i₁⟩ := hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 38
} | [
{
"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 ≤ #∅",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 13
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 26
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Finset",
"id",
"Finset.instEmpty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 13
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 37
} | [
{
"pp": "case h\n𝕜 : 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)) ... | refine ext_inner_left 𝕜 fun w => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 37
} | [
{
"pp": "case h\n𝕜 : 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\n... | refine ext_inner_left 𝕜 fun w => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 416,
"column": 27
} | {
"line": 416,
"column": 38
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 491,
"column": 6
} | {
"line": 491,
"column": 28
} | [
{
"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",
"usedConstants": [
"instHSMu... | 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
} | [
{
"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",
"usedConstants": [
"instHSMu... | 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
} | [
{
"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",
"usedConstants": [
"instHSMu... | exact fun h => ⟨p₁, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 38
} | [
{
"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",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 13
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 13
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 598,
"column": 4
} | {
"line": 598,
"column": 75
} | [
{
"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\ns : Set P\nh : Collinear k s\np₁ p₂ p₃ : P\nhp₂ : p₂ ∈ s\nhp₃ : p₃ ∈ s\nhp₂p₃ : p₂ ≠ p₃\n⊢ vectorSpan k (insert p₁ s) = vectorSpan k {p₁, p₂, p₃}",
"usedConstant... | conv_rhs => rw [← direction_affineSpan, ← affineSpan_insert_affineSpan] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 542,
"column": 55
} | {
"line": 542,
"column": 66
} | [
{
"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",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 15
} | [
{
"pp": "case hr\nk : 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\nhmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)\nhmem2 : s.medial.poi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 469,
"column": 12
} | {
"line": 469,
"column": 23
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 37
} | [
{
"pp": "case h\n𝕜 : 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 (adjo... | refine ext_inner_left 𝕜 fun w => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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