module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 80
} | {
"line": 590,
"column": 2
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝¹ : RelCWComplex C D\ninst✝ : T2Space X\nA : Set X\nhAC : A ⊆ C\nhDA : IsClosed[t] (A ∩ D)\nh : ∀ (n : ℕ), 0 < n → ∀ (j : cell C n), Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)\n⊢ IsClosed[t] A",
"ppTerm": "?m.42",
"assign... | [
"X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝¹ : RelCWComplex C D\ninst✝ : T2Space X\nA : Set X\nhAC : A ⊆ C\nhDA : IsClosed[t] (A ∩ D)\nh : ∀ (n : ℕ), 0 < n → ∀ (j : cell C n), Disjoint A (openCell n j) ∨ IsClosed[t] (A ∩ closedCell n j)\n⊢ ∀ (n : ℕ), 0 < n → ∀ (j : cell C n), IsClosed[t] (A ∩ openCell... | apply isClosed_of_isClosed_inter_openCell_or_isClosed_inter_closedCell hAC hDA | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 39
} | {
"line": 160,
"column": 40
} | [
{
"pp": "case h\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nx✝ : ↑(smaller C o)\na : LocallyConstant ↑C ℤ\nb : LocallyConstant ↑(π C fun x ↦ ord I x < o) ℤ\nhb : b ∈ range (π C fun x ↦ ord I x < o) ∧ (πs C o) b = a\n⊢ (fun x ↦ ⟨(πs C o) ↑x, ⋯⟩) ⟨b, ⋯⟩ = ⟨a,... | [
"case h\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nx✝ : ↑(smaller C o)\na : LocallyConstant ↑C ℤ\nb : LocallyConstant ↑(π C fun x ↦ ord I x < o) ℤ\nhb : b ∈ range (π C fun x ↦ ord I x < o) ∧ (πs C o) b = a\n⊢ (πs C o) b = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 13
} | {
"line": 651,
"column": 14
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\nn : ℕ\ni : cell C n\n⊢ ∃ I, cellFrontier n i ⊆ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, openCell m j",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nt : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\nn : ℕ\ni : cell C n\n⊢ ∃ I, cellFrontier n i ⊆ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, openCell m j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 681,
"column": 4
} | {
"line": 681,
"column": 15
} | {
"line": 681,
"column": 16
} | [
{
"pp": "case e_I\nX : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nE : Set X\nI✝¹ : (n : ℕ) → Set (cell C n)\nclosed'✝¹ : IsClosed[t] E\nhE : D ∪ ⋃ n, ⋃ j, openCell n ↑j = E\nF : Set X\nI✝ : (n : ℕ) → Set (cell C n)\nclosed'✝ : IsClosed[t] F\nhF : D ∪ ⋃ n, ⋃ j, openCell n ↑j = F\nh ... | [
"case e_I\nX : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nE : Set X\nI✝¹ : (n : ℕ) → Set (cell C n)\nclosed'✝¹ : IsClosed[t] E\nhE : D ∪ ⋃ n, ⋃ j, openCell n ↑j = E\nF : Set X\nI✝ : (n : ℕ) → Set (cell C n)\nclosed'✝ : IsClosed[t] F\nhF : D ∪ ⋃ n, ⋃ j, openCell n ↑j = F\nh :\n (fun E ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 143,
"column": 19
} | {
"line": 143,
"column": 35
} | {
"line": 143,
"column": 36
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nthis : (∀ (n : ℕ), dissipate C n ∈ S ∨ dissipate C n =... | [
"α : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nthis : (∀ (n : ℕ), dissipate C n ∈ S ∨ dissipate C n = ∅) ∧ ⋂ n, d... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 521,
"column": 4
} | {
"line": 521,
"column": 84
} | {
"line": 523,
"column": 0
} | [
{
"pp": "case neg\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nq : ↑(GoodProducts (π C fun x ↦ ord I x < o))\nl : ↑(MaxProducts C ho)\nthis : Inhabited I\nh : ¬↑↑q = []\n⊢ (Ordinal.... | [] | exact Products.prop_of_isGood C _ q.prop q.val.val.head! (List.head!_mem_self h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 25
} | {
"line": 146,
"column": 26
} | [
{
"pp": "case pos\nα : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nn : ℕ\ng : (dissipate C n).Nonempty\n⊢ dissi... | [
"case pos\nα : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nn : ℕ\ng : (dissipate C n).Nonempty\n⊢ dissipate C n = ∅... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 15
} | {
"line": 170,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nC : ℕ → Set α\nhC_cc : ∀ (i : ℕ), C i ∈ {s | IsCompact s ∧ IsClosed s}\nh_nonempty : ∀ (n : ℕ), (dissipate C n).Nonempty\n⊢ IsCompact (dissipate C 0)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Set.dissipate",... | [
"case refine_2\nα : Type u_2\ninst✝ : TopologicalSpace α\nC : ℕ → Set α\nhC_cc : ∀ (i : ℕ), C i ∈ {s | IsCompact s ∧ IsClosed s}\nh_nonempty : ∀ (n : ℕ), (dissipate C n).Nonempty\n⊢ IsCompact (C 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 13
} | {
"line": 176,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\ns : Set α\n⊢ IsCompact s ↔ IsCompact s ∧ IsClosed s",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"IsClosed",
"And",
"Iff",
"iff_self_and._simp_1",
"Eq",
... | [
"α : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\ns : Set α\n⊢ IsCompact s → IsClosed s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.Bounded.ArzelaAscoli | {
"line": 62,
"column": 61
} | {
"line": 62,
"column": 72
} | {
"line": 62,
"column": 73
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : CompactSpace β\nA : Set (α →ᵇ β)\nclosed : IsClosed A\nH : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ nhds x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 <... | [
"α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : CompactSpace α\ninst✝¹ : PseudoMetricSpace β\ninst✝ : CompactSpace β\nA : Set (α →ᵇ β)\nclosed : IsClosed A\nH : ∀ (x₀ : α), ∀ ε > 0, ∃ U ∈ nhds x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ↑A), dist (↑i x) (↑i x') < ε\nε : ℝ\nε0 : ε > 0\nε₁ : ℝ\nε₁0 : 0 < ε₁\nεε₁ : ε... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 98,
"column": 36
} | {
"line": 98,
"column": 47
} | {
"line": 98,
"column": 48
} | [
{
"pp": "E : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : ∀ (x : ℕ → E), (∀ᶠ (n : ℕ) in atTop, x n ∈ A) → ∃ a ∈ A, MapClusterPt a atTop x\nf : Filter E\nx✝¹ : f.NeBot\nx✝ : f.IsCountablyGenerated\nhle : f ≤ 𝓟 A\nx : ℕ → E\nhx : Tendsto x atTop f\n⊢ ∀ᶠ (n : ℕ) in atTop, x n ∈ A",
"ppTerm": "?m.57",... | [
"E : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : ∀ (x : ℕ → E), (∀ᶠ (n : ℕ) in atTop, x n ∈ A) → ∃ a ∈ A, MapClusterPt a atTop x\nf : Filter E\nx✝¹ : f.NeBot\nx✝ : f.IsCountablyGenerated\nhle : f ≤ 𝓟 A\nx : ℕ → E\nhx : Tendsto x atTop f\n⊢ ∃ a, ∀ (b : ℕ), a ≤ b → x b ∈ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.ContinuousMap.BoundedCompactlySupported | {
"line": 78,
"column": 2
} | {
"line": 79,
"column": 9
} | {
"line": 79,
"column": 10
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : NonUnitalNormedRing γ\ninst✝ : Nontrivial γ\nh : C_cb(α, γ) = ⊤\nx : γ\nhx : x ≠ 0\n⊢ IsCompact univ",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nγ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : NonUnitalNormedRing γ\ninst✝ : Nontrivial γ\nh : C_cb(α, γ) = ⊤\nx : γ\nhx : x ≠ 0\n⊢ IsCompact univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 156,
"column": 77
} | {
"line": 156,
"column": 88
} | {
"line": 156,
"column": 89
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : IsCountablyCompact A\nb : Set ι\nhb : b.Countable\nU : ι → Set E\nhUo : ∀ i ∈ b, IsOpen[inst✝] (U i)\nhAU : A ⊆ ⋃ i ∈ b, U i\nthis : Countable ↑b\n⊢ A ⊆ ⋃ i, U ↑i",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": ... | [
"ι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nA : Set E\nhA : IsCountablyCompact A\nb : Set ι\nhb : b.Countable\nU : ι → Set E\nhUo : ∀ i ∈ b, IsOpen[inst✝] (U i)\nhAU : A ⊆ ⋃ i ∈ b, U i\nthis : Countable ↑b\n⊢ A ⊆ ⋃ i ∈ b, U i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 590,
"column": 4
} | {
"line": 590,
"column": 38
} | {
"line": 591,
"column": 4
} | [
{
"pp": "case h₂\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : ↑(MaxProducts C ho)\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nthis ... | [
"case h₂\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : ↑(MaxProducts C ho)\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nthis : Inhabited ... | rw [max_eq_o_cons_tail C hsC ho l] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 15
} | {
"line": 251,
"column": 16
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nA✝ B : Set E\ninst✝¹ : SequentialSpace E\ninst✝ : CountablyCompactSpace E\nx : ℕ → E\nhx : ∀ (x_1 : E) (x_2 : ℕ → ℕ), StrictMono x_2 → ¬Tendsto (x ∘ x_2) atTop (𝓝 x_1)\nA : Set E := ⋃ i, closure[inst✝³]... | [
"ι : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nA✝ B : Set E\ninst✝¹ : SequentialSpace E\ninst✝ : CountablyCompactSpace E\nx : ℕ → E\nhx : ∀ (x_1 : E) (x_2 : ℕ → ℕ), StrictMono x_2 → ¬Tendsto (x ∘ x_2) atTop (𝓝 x_1)\nA : Set E := ⋃ i, closure[inst✝³] {x i}\nthis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 617,
"column": 2
} | {
"line": 617,
"column": 35
} | {
"line": 618,
"column": 2
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nh : LinearIndependent ℤ (eval (C' C ho)... | [
"I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nh : LinearIndependent ℤ (eval (C' C ho))\nf : ↑(Max... | let f := MaxToGood C hC hsC ho h₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.ContinuousMap.Sigma | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 15
} | {
"line": 57,
"column": 16
} | [
{
"pp": "X : Type u_1\nι : Type u_2\nY : ι → Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : Nonempty X\ni : ι\ng g' : C(X, Y i)\nh : (fun g ↦ (sigmaMk g.fst).comp g.snd) ⟨i, g⟩ = (fun g ↦ (sigmaMk g.fst).comp g.snd) ⟨i, g'⟩\nhg : ⇑g ≍ ⇑g'\n⊢ ⟨i, g⟩ = ⟨i, g'⟩",
"ppT... | [
"X : Type u_1\nι : Type u_2\nY : ι → Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : Nonempty X\ni : ι\ng g' : C(X, Y i)\nh : (fun g ↦ (sigmaMk g.fst).comp g.snd) ⟨i, g⟩ = (fun g ↦ (sigmaMk g.fst).comp g.snd) ⟨i, g'⟩\nhg : ⇑g ≍ ⇑g'\n⊢ g = g'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 391,
"column": 13
} | {
"line": 391,
"column": 24
} | {
"line": 391,
"column": 25
} | [
{
"pp": "case empty\nι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nf : ι → Set E\nhf : ∀ i ∈ ∅, IsCountablyCompact (f i)\n⊢ IsCountablyCompact (⋃ i ∈ ∅, f i)",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Iff.of_eq",
"congrArg",
... | [
"case empty\nι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nf : ι → Set E\nhf : ∀ i ∈ ∅, IsCountablyCompact (f i)\n⊢ IsCountablyCompact ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 392,
"column": 25
} | {
"line": 392,
"column": 60
} | {
"line": 393,
"column": 4
} | [
{
"pp": "case insert\nι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nf : ι → Set E\na : ι\ns : Finset ι\nha : a ∉ s\nih : (∀ i ∈ s, IsCountablyCompact (f i)) → IsCountablyCompact (⋃ i ∈ s, f i)\nhf : ∀ i ∈ insert a s, IsCountablyCompact (f i)\n⊢ IsCountablyCompact (⋃ i ∈ insert a s, f i)",
"ppTerm"... | [
"case insert\nι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nf : ι → Set E\na : ι\ns : Finset ι\nha : a ∉ s\nih : (∀ i ∈ s, IsCountablyCompact (f i)) → IsCountablyCompact (⋃ i ∈ s, f i)\nhf : ∀ i ∈ insert a s, IsCountablyCompact (f i)\n⊢ IsCountablyCompact (f a ∪ ⋃ x ∈ s, f x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DerivedSet | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 13
} | {
"line": 83,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nhA : IsClosed[inst✝] A\n⊢ relDerivedSet A = derivedSet A",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"CompleteLattice.toConditionallyCompleteLattice",
"id",
"... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nhA : IsClosed[inst✝] A\n⊢ derivedSet A ⊆ A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DerivedSet | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 18
} | {
"line": 116,
"column": 19
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\n⊢ Perfect U ↔ U = derivedSet U",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Perfect",
"Preperfect",
"id",
"derivedSet",
"IsClosed",
"perfect_def",
"A... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\n⊢ IsClosed[inst✝] U ∧ Preperfect U ↔ U = derivedSet U"
] | perfect_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Filter | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 34
} | {
"line": 57,
"column": 35
} | [
{
"pp": "α : Type u_2\ns : Set α\n⊢ IsOpen {l | s ∈ l}",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ns : Set α\n⊢ IsOpen {l | s ∈ l}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Filter | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 53
} | {
"line": 78,
"column": 54
} | [
{
"pp": "α : Type u_2\nl : Filter α\n⊢ 𝓝 l = l.lift' fun s ↦ {l' | s ∈ l'}",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nl : Filter α\n⊢ 𝓝 l = l.lift' fun s ↦ {l' | s ∈ l'}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Filter | {
"line": 100,
"column": 54
} | {
"line": 100,
"column": 86
} | {
"line": 100,
"column": 87
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\n⊢ (𝓝 l).HasBasis p fun i ↦ {l' | s i ∈ l'}",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Sort u_1\nα : Type u_2\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\n⊢ (𝓝 l).HasBasis p fun i ↦ {l' | s i ∈ l'}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Convenient.OpenClosed | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 67
} | {
"line": 39,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type u_3\ninst✝² : TopologicalSpace Y\ninst✝¹ : ∀ (i : ι) (U : TopologicalSpace.Opens (X i)), IsGeneratedBy X ↥U\ninst✝ : IsGeneratedBy X Y\nU : Set Y\nhU : IsOpen[inst✝²] U\nW : (a : (i : ι) × C(X i, Y)) → TopologicalSpace.... | [
"ι : Type u_1\nX : ι → Type u_2\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type u_3\ninst✝² : TopologicalSpace Y\ninst✝¹ : ∀ (i : ι) (U : TopologicalSpace.Opens (X i)), IsGeneratedBy X ↥U\ninst✝ : IsGeneratedBy X Y\nU : Set Y\nhU : IsOpen[inst✝²] U\nW : (a : (i : ι) × C(X i, Y)) → TopologicalSpace.Opens (X a.f... | have hg (a) : Continuous (g a) := a.2.continuous.restrictPreimage | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Convenient.OpenClosed | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 67
} | {
"line": 72,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type u_3\ninst✝² : TopologicalSpace Y\ninst✝¹ : ∀ (i : ι) (F : TopologicalSpace.Closeds (X i)), IsGeneratedBy X ↥F\ninst✝ : IsGeneratedBy X Y\nF : Set Y\nhF : IsClosed[inst✝²] F\nW : (a : (i : ι) × C(X i, Y)) → TopologicalSp... | [
"ι : Type u_1\nX : ι → Type u_2\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type u_3\ninst✝² : TopologicalSpace Y\ninst✝¹ : ∀ (i : ι) (F : TopologicalSpace.Closeds (X i)), IsGeneratedBy X ↥F\ninst✝ : IsGeneratedBy X Y\nF : Set Y\nhF : IsClosed[inst✝²] F\nW : (a : (i : ι) × C(X i, Y)) → TopologicalSpace.Closeds ... | have hg (a) : Continuous (g a) := a.2.continuous.restrictPreimage | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Homotopy.HSpaces | {
"line": 197,
"column": 13
} | {
"line": 198,
"column": 35
} | {
"line": 199,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nx y : X\nθ : ↑I\nγ : Path x y\n⊢ γ (qRight (0, θ)) = x",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Icc.instZero",
"congrArg",
... | [] | by
rw [qRight_zero_left, γ.source] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.ENatENNReal | {
"line": 26,
"column": 4
} | {
"line": 26,
"column": 15
} | {
"line": 26,
"column": 16
} | [
{
"pp": "case refine_1\na : ENNReal\n⊢ IsOpen (toENNReal ⁻¹' Set.Ioi a)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
"PartialOrder.toPreorder",
"instPreorderENat",
"id",
"ENat.toENNReal",
"ENat.preimage... | [
"case refine_1\na : ENNReal\n⊢ IsOpen (Set.Ioi ⌊a⌋ₑ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.ENatENNReal | {
"line": 27,
"column": 4
} | {
"line": 27,
"column": 15
} | {
"line": 27,
"column": 16
} | [
{
"pp": "case refine_2\na : ENNReal\n⊢ IsOpen (toENNReal ⁻¹' Set.Iio a)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"instPreorderENat",
"id",
"ENat.ceil",
"ENat.toENNReal",
"ENat.preima... | [
"case refine_2\na : ENNReal\n⊢ IsOpen (Set.Iio ⌈a⌉ₑ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.RatLemmas | {
"line": 57,
"column": 2
} | {
"line": 62,
"column": 28
} | {
"line": 64,
"column": 0
} | [
{
"pp": "⊢ ¬(cocompact ℚ).IsCountablyGenerated",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"False",
"Filter.tendsto_inf",
"congrArg",
"Filter.Inf.isCountablyGenerated",
"TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology",
... | [] | intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.RatLemmas | {
"line": 57,
"column": 2
} | {
"line": 62,
"column": 28
} | {
"line": 64,
"column": 0
} | [
{
"pp": "⊢ ¬(cocompact ℚ).IsCountablyGenerated",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"False",
"Filter.tendsto_inf",
"congrArg",
"Filter.Inf.isCountablyGenerated",
"TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology",
... | [] | intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 38
} | {
"line": 219,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset.instGeneralizedBoolea... | [] | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 38
} | {
"line": 219,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset.instGeneralizedBoolea... | [] | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 38
} | {
"line": 219,
"column": 0
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset.instGeneralizedBoolea... | [] | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 42
} | {
"line": 272,
"column": 4
} | [
{
"pp": "case hab\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\nh : (logSizeBallSeq J hJ a c i).finset.Nonempty\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"ppTerm": "?hab",
"assigned": true,
... | [] | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 42
} | {
"line": 272,
"column": 4
} | [
{
"pp": "case hab\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\nh : (logSizeBallSeq J hJ a c i).finset.Nonempty\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"ppTerm": "?hab",
"assigned": true,
... | [] | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 42
} | {
"line": 272,
"column": 4
} | [
{
"pp": "case hab\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\nh : (logSizeBallSeq J hJ a c i).finset.Nonempty\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"ppTerm": "?hab",
"assigned": true,
... | [] | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.CantorSet | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 37
} | {
"line": 125,
"column": 38
} | [
{
"pp": "case succ.refine_1\nf : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯\ng : ℝ ≃ₜ ℝ := (Homeomorph.addLeft 2).trans f\nn : ℕ\nih : IsClosed (preCantorSet n)\n⊢ IsClosed ((fun x ↦ x / 3) '' preCantorSet n)",
"ppTerm": "?succ.refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real... | [
"case succ.refine_1\nf : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯\ng : ℝ ≃ₜ ℝ := (Homeomorph.addLeft 2).trans f\nn : ℕ\nih : IsClosed (preCantorSet n)\n⊢ IsClosed ((fun a ↦ 3⁻¹ * a) '' preCantorSet n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 40
} | {
"line": 126,
"column": 41
} | [
{
"pp": "case succ.refine_2\nf : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯\ng : ℝ ≃ₜ ℝ := (Homeomorph.addLeft 2).trans f\nn : ℕ\nih : IsClosed (preCantorSet n)\n⊢ IsClosed ((fun x ↦ (2 + x) / 3) '' preCantorSet n)",
"ppTerm": "?succ.refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case succ.refine_2\nf : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯\ng : ℝ ≃ₜ ℝ := (Homeomorph.addLeft 2).trans f\nn : ℕ\nih : IsClosed (preCantorSet n)\n⊢ IsClosed ((fun a ↦ 3⁻¹ * (2 + a)) '' preCantorSet n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 173,
"column": 46
} | {
"line": 173,
"column": 57
} | {
"line": 173,
"column": 58
} | [
{
"pp": "a b : ℕ → Fin 3\nha : ∀ (n : ℕ), a n ≠ 1\nhb : ∀ (n : ℕ), b n ≠ 1\nh✝ : ofDigits a = ofDigits b\nh : ∃ a_1, a a_1 ≠ b a_1\nn0 : ℕ := Nat.find h\nn : ℕ\nhn : n < n0\n⊢ a n = b n",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ → Fin 3\nha : ∀ (n : ℕ), a n ≠ 1\nhb : ∀ (n : ℕ), b n ≠ 1\nh✝ : ofDigits a = ofDigits b\nh : ∃ a_1, a a_1 ≠ b a_1\nn0 : ℕ := Nat.find h\nn : ℕ\nhn : n < n0\n⊢ a n = b n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 174,
"column": 30
} | {
"line": 174,
"column": 41
} | {
"line": 174,
"column": 42
} | [
{
"pp": "a b : ℕ → Fin 3\nha : ∀ (n : ℕ), a n ≠ 1\nhb : ∀ (n : ℕ), b n ≠ 1\nh✝ : ofDigits a = ofDigits b\nh : ∃ a_1, a a_1 ≠ b a_1\nn0 : ℕ := Nat.find h\nh1 : ∀ n < n0, a n = b n\n⊢ a n0 ≠ b n0",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"instOfNatNat",
... | [
"a b : ℕ → Fin 3\nha : ∀ (n : ℕ), a n ≠ 1\nhb : ∀ (n : ℕ), b n ≠ 1\nh✝ : ofDigits a = ofDigits b\nh : ∃ a_1, a a_1 ≠ b a_1\nn0 : ℕ := Nat.find h\nh1 : ∀ n < n0, a n = b n\n⊢ ¬a n0 = b n0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 380,
"column": 68
} | {
"line": 380,
"column": 89
} | {
"line": 380,
"column": 90
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nn : ℕ\nJ : Finset T\ninst✝ : DecidableEq T\nha : 1 < a\nhJ_card : ↑(#J) ≤ a ^ n\ns t : T\nh : (s, t) ∈ pairSet J a c\n⊢ ∃ i < #J, (s, t) ∈ pairSetSeq J a c i",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nn : ℕ\nJ : Finset T\ninst✝ : DecidableEq T\nha : 1 < a\nhJ_card : ↑(#J) ≤ a ^ n\ns t : T\nh : (s, t) ∈ pairSet J a c\n⊢ ∃ i < #J, (s, t) ∈ pairSetSeq J a c i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 160,
"column": 7
} | {
"line": 160,
"column": 18
} | {
"line": 160,
"column": 19
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ M X x)\ny : N → ↑I\nx✝ : y ∈ Cube.boundary N\nn : N\nhn : y n = 0 ∨ y n = 1\n⊢ ((↑p).comp { toFun := fun t m ↦ t (e m), continuous_toFun := ⋯ }) y = x",
"ppTerm": "?m.112",
"assigned": tr... | [
"N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ M X x)\ny : N → ↑I\nx✝ : y ∈ Cube.boundary N\nn : N\nhn : y n = 0 ∨ y n = 1\n⊢ (p fun m ↦ y (e m)) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 160,
"column": 39
} | {
"line": 160,
"column": 50
} | {
"line": 160,
"column": 51
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ M X x)\ny : N → ↑I\nx✝ : y ∈ Cube.boundary N\nn : N\nhn : y n = 0 ∨ y n = 1\n⊢ y (e (e.symm n)) = 0 ∨ y (e (e.symm n)) = 1",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [
... | [
"N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ M X x)\ny : N → ↑I\nx✝ : y ∈ Cube.boundary N\nn : N\nhn : y n = 0 ∨ y n = 1\n⊢ y n = 0 ∨ y n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 15
} | {
"line": 162,
"column": 16
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ N X x)\ny : M → ↑I\nx✝ : y ∈ Cube.boundary M\nm : M\nhm : y m = 0 ∨ y m = 1\n⊢ ((↑p).comp { toFun := fun t n ↦ t (e.symm n), continuous_toFun := ⋯ }) y = x",
"ppTerm": "?m.147",
"assigned... | [
"N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ N X x)\ny : M → ↑I\nx✝ : y ∈ Cube.boundary M\nm : M\nhm : y m = 0 ∨ y m = 1\n⊢ (p fun n ↦ y (e.symm n)) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 162,
"column": 31
} | {
"line": 162,
"column": 42
} | {
"line": 162,
"column": 43
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ N X x)\ny : M → ↑I\nx✝ : y ∈ Cube.boundary M\nm : M\nhm : y m = 0 ∨ y m = 1\n⊢ y (e.symm (e m)) = 0 ∨ y (e.symm (e m)) = 1",
"ppTerm": "?m.155",
"assigned": true,
"usedConstants": [
... | [
"N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.15\nx : X\ne : M ≃ N\np : ↑(Ω^ N X x)\ny : M → ↑I\nx✝ : y ∈ Cube.boundary M\nm : M\nhm : y m = 0 ∨ y m = 1\n⊢ y m = 0 ∨ y m = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 47
} | {
"line": 403,
"column": 48
} | [
{
"pp": "T : Type u_1\ninst✝² : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝¹ : DecidableEq T\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha : 1 < a\nf : T → E\ns : T\nhs : s ∈ J\nt : T\nht : t ∈ J\nhst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c\nhJ : J.Nonempty\nP : ℕ → Prop := ⋯\nl : ℕ := ⋯\n⊢ P 0",
"ppTerm": ... | [
"T : Type u_1\ninst✝² : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝¹ : DecidableEq T\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha : 1 < a\nf : T → E\ns : T\nhs : s ∈ J\nt : T\nht : t ∈ J\nhst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c\nhJ : J.Nonempty\nP : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 269,
"column": 68
} | {
"line": 274,
"column": 25
} | {
"line": 276,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : x ∈ cantorSet\nn : ℕ\n⊢ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i ≤ x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real.... | [] | by
have h_mem := cantorSequence_mem_cantorSet hx n
rw [cantorSequence_eq_self_sub_sum_cantorToTernary x n] at h_mem
apply cantorSet_subset_unitInterval at h_mem
simp only [Set.mem_Icc] at h_mem
simpa using! h_mem.left | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 15
} | {
"line": 183,
"column": 16
} | [
{
"pp": "case inr\nN : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nM : Type u_3\nx : X\np : ↑(Ω^ M (↑(Ω^ N X x)) const)\ny : M ⊕ N → ↑I\nhy : y ∈ Cube.boundary (M ⊕ N)\nhN : y ∘ Sum.inr ∈ Cube.boundary N\n⊢ (p (y ∘ Sum.inl)) (y ∘ Sum.inr) = x",
"ppTerm": "?inr",
"assigned": false,
"usedConst... | [
"case inr\nN : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nM : Type u_3\nx : X\np : ↑(Ω^ M (↑(Ω^ N X x)) const)\ny : M ⊕ N → ↑I\nhy : y ∈ Cube.boundary (M ⊕ N)\nhN : y ∘ Sum.inr ∈ Cube.boundary N\n⊢ (p (y ∘ Sum.inl)) (y ∘ Sum.inr) = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 57
} | {
"line": 292,
"column": 58
} | [
{
"pp": "x : ℝ\nhx : x ∈ cantorSet\n⊢ Summable fun i ↦ ‖ofDigitsTerm (cantorToTernary x).get i‖",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"cantorToTernary",
"congrArg",
"Su... | [
"x : ℝ\nhx : x ∈ cantorSet\n⊢ Summable fun i ↦ ofDigitsTerm (cantorToTernary x).get i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 24
} | {
"line": 421,
"column": 25
} | [
{
"pp": "T : Type u_1\ninst✝² : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝¹ : DecidableEq T\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha : 1 < a\nf : T → E\ns : T\nhs : s ∈ J\nt : T\nht : t ∈ J\nhst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c\nhJ : J.Nonempty\nP : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c ... | [
"T : Type u_1\ninst✝² : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝¹ : DecidableEq T\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha : 1 < a\nf : T → E\ns : T\nhs : s ∈ J\nt : T\nht : t ∈ J\nhst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c\nhJ : J.Nonempty\nP : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 348,
"column": 26
} | {
"line": 348,
"column": 35
} | {
"line": 348,
"column": 36
} | [
{
"pp": "case refine_1\nN : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\ni✝ : N\np q : ↑(Ω^ N X x)\nH : (↑p).HomotopyRel (↑q) (Cube.boundary N)\nt : ↑I × ↑I\ny : { j // j ≠ i✝ } → ↑I\ni : { j // j ≠ i✝ }\niH : y i = 0 ∨ y i = 1\n⊢ H (t.1, (Cube.insertAt i✝) (t.2, y)) = x",
... | [
"case refine_1\nN : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\ni✝ : N\np q : ↑(Ω^ N X x)\nH : (↑p).HomotopyRel (↑q) (Cube.boundary N)\nt : ↑I × ↑I\ny : { j // j ≠ i✝ } → ↑I\ni : { j // j ≠ i✝ }\niH : y i = 0 ∨ y i = 1\n⊢ ↑p ((Cube.insertAt i✝) (t.2, y)) = x",
"case refine_1... | H.eq_fst, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.List | {
"line": 148,
"column": 41
} | {
"line": 148,
"column": 58
} | {
"line": 149,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\na : α\nl : List α\n⊢ Tendsto (fun p ↦ (p.1 :: p.2).eraseIdx 0) (𝓝 a ×ˢ 𝓝 l) (𝓝 ((a :: l).eraseIdx 0))",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"nhds",
"List",
"Filter.tendsto_snd",
"instTopologicalSpaceLis... | [] | exact tendsto_snd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 494,
"column": 8
} | {
"line": 494,
"column": 60
} | {
"line": 495,
"column": 4
} | [
{
"pp": "case refine_2\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na : ℝ≥0∞\nn : ℕ\nJ : Finset T\nhJ_card : ↑(#J) ≤ a ^ n\nc : ℝ≥0∞\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha1 : a ≤ 1\nhJ : Nonempty ↥J\n⊢ 1 ≤ #J",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rwa [Finset.one_le_card, ← Finset.nonempty_coe_sort] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 494,
"column": 8
} | {
"line": 494,
"column": 60
} | {
"line": 495,
"column": 4
} | [
{
"pp": "case refine_2\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na : ℝ≥0∞\nn : ℕ\nJ : Finset T\nhJ_card : ↑(#J) ≤ a ^ n\nc : ℝ≥0∞\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha1 : a ≤ 1\nhJ : Nonempty ↥J\n⊢ 1 ≤ #J",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rwa [Finset.one_le_card, ← Finset.nonempty_coe_sort] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 494,
"column": 8
} | {
"line": 494,
"column": 60
} | {
"line": 495,
"column": 4
} | [
{
"pp": "case refine_2\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na : ℝ≥0∞\nn : ℕ\nJ : Finset T\nhJ_card : ↑(#J) ≤ a ^ n\nc : ℝ≥0∞\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha1 : a ≤ 1\nhJ : Nonempty ↥J\n⊢ 1 ≤ #J",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rwa [Finset.one_le_card, ← Finset.nonempty_coe_sort] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.BundledFun | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 13
} | {
"line": 145,
"column": 14
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : AddCommMonoid R\ninst✝² : LinearOrder R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : IsOrderedAddMonoid R\nY : Type u_3\nf : Y → PseudoMetric X R\ns : Finset Y\nhs : s.Nonempty\n⊢ ⇑(s.sup f) = ⇑(s.sup' hs fun x ↦ f x)",
"ppTerm": "?m.20",
"assigned": true,
"us... | [
"X : Type u_1\nR : Type u_2\ninst✝³ : AddCommMonoid R\ninst✝² : LinearOrder R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : IsOrderedAddMonoid R\nY : Type u_3\nf : Y → PseudoMetric X R\ns : Finset Y\nhs : s.Nonempty\n⊢ s.sup f = s.sup' hs fun x ↦ f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 624,
"column": 4
} | {
"line": 626,
"column": 58
} | {
"line": 628,
"column": 0
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\na b : π_ 1 X x\np q : ↑(Ω^ (Fin 1) X x)\n⊢ pi1EquivFundamentalGroup.toFun (⟦p⟧ * ⟦q⟧) = pi1EquivFundamentalGroup.toFun ⟦p⟧ * pi1EquivFundamentalGroup.toFun ⟦q⟧",
"ppTerm": "?m.26",
"assigned": true,
"used... | [] | simp only [HomotopyGroup.mul_spec (i := (0 : Fin 1))]
apply Quotient.sound
rw [Unique.eq_default 0, genLoopEquivOfUnique_transAt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 624,
"column": 4
} | {
"line": 626,
"column": 58
} | {
"line": 628,
"column": 0
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\na b : π_ 1 X x\np q : ↑(Ω^ (Fin 1) X x)\n⊢ pi1EquivFundamentalGroup.toFun (⟦p⟧ * ⟦q⟧) = pi1EquivFundamentalGroup.toFun ⟦p⟧ * pi1EquivFundamentalGroup.toFun ⟦q⟧",
"ppTerm": "?m.26",
"assigned": true,
"used... | [] | simp only [HomotopyGroup.mul_spec (i := (0 : Fin 1))]
apply Quotient.sound
rw [Unique.eq_default 0, genLoopEquivOfUnique_transAt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | {
"line": 121,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : ∀ i ⊆ A, IsSeparated (↑ε) i → i = ∅\nx : X\nhx : x ∈ A\n⊢ False",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : ∀ i ⊆ A, IsSeparated (↑ε) i → i = ∅\nx : X\nhx : x ∈ A\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 314,
"column": 54
} | {
"line": 314,
"column": 65
} | {
"line": 314,
"column": 66
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : packingNumber ε A ≠ ⊤\nx : X\nhxA : x ∈ A\nh_dist : ∀ y ∈ maximalSeparatedSet ε A, (x, y) ∉ {x | edist x.1 x.2 ≤ ↑ε}\nC : Set X := {x} ∪ maximalSeparatedSet ε A\n⊢ x ∉ maximalSeparatedSet ε A",
"ppTerm": "?m.42",
"assigned": fa... | [
"X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : packingNumber ε A ≠ ⊤\nx : X\nhxA : x ∈ A\nh_dist : ∀ y ∈ maximalSeparatedSet ε A, (x, y) ∉ {x | edist x.1 x.2 ≤ ↑ε}\nC : Set X := {x} ∪ maximalSeparatedSet ε A\n⊢ x ∉ maximalSeparatedSet ε A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 322,
"column": 43
} | {
"line": 322,
"column": 54
} | {
"line": 322,
"column": 55
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : packingNumber ε A ≠ ⊤\nx : X\nhxA : x ∈ A\nh_dist : ∀ y ∈ maximalSeparatedSet ε A, (x, y) ∉ {x | edist x.1 x.2 ≤ ↑ε}\nC : Set X := {x} ∪ maximalSeparatedSet ε A\nhx_not_mem : x ∉ maximalSeparatedSet ε A\n⊢ ∀ y ∈ maximalSeparatedSet ε A... | [
"X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : packingNumber ε A ≠ ⊤\nx : X\nhxA : x ∈ A\nh_dist : ∀ y ∈ maximalSeparatedSet ε A, (x, y) ∉ {x | edist x.1 x.2 ≤ ↑ε}\nC : Set X := {x} ∪ maximalSeparatedSet ε A\nhx_not_mem : x ∉ maximalSeparatedSet ε A\n⊢ ∀ y ∈ maximalSeparatedSet ε A, ↑ε < edist... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 352,
"column": 8
} | {
"line": 352,
"column": 37
} | {
"line": 352,
"column": 38
} | [
{
"pp": "case h₂\nX : Type u_1\ninst✝ : PseudoEMetricSpace X\nε : ℝ≥0\nA C : Set X\nhC_cover : IsCover ε A C\nD : Set X\nhD_subset : D ⊆ A\nhD_separated : IsSeparated (2 * ↑ε) D\nf : ↑D → ↑C := ⋯\nhf' : ∀ (x : ↑D), edist ↑x ↑(f x) ≤ ↑ε\nx y : ↑D\nhxy : f x = f y\n⊢ edist ↑(f x) ↑y ≤ ↑ε",
"ppTerm": "?h₂",
... | [
"case h₂\nX : Type u_1\ninst✝ : PseudoEMetricSpace X\nε : ℝ≥0\nA C : Set X\nhC_cover : IsCover ε A C\nD : Set X\nhD_subset : D ⊆ A\nhD_separated : IsSeparated (2 * ↑ε) D\nf : ↑D → ↑C := fun x ↦ ⟨⋯.choose, ⋯⟩\nhf' : ∀ (x : ↑D), edist ↑x ↑(f x) ≤ ↑ε\nx y : ↑D\nhxy : f x = f y\n⊢ edist ↑y ↑(f y) ≤ ↑ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 401,
"column": 44
} | {
"line": 401,
"column": 55
} | {
"line": 401,
"column": 56
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nA : Set X\nε : ℝ≥0\nf : X → Y\nhf : Isometry f\nhf_inj : InjOn f A\nC : Set Y\nhC_subset : C ⊆ f '' A\nhC_cover : IsCover ε (f '' A) C\nx : ↑C\n⊢ ∃ y ∈ A, f y = ↑x",
"ppTerm": "?m.115",
"assigned": false,
... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nA : Set X\nε : ℝ≥0\nf : X → Y\nhf : Isometry f\nhf_inj : InjOn f A\nC : Set Y\nhC_subset : C ⊆ f '' A\nhC_cover : IsCover ε (f '' A) C\nx : ↑C\n⊢ ∃ y ∈ A, f y = ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Kuratowski | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 25
} | {
"line": 82,
"column": 26
} | [
{
"pp": "α : Type u\ninst✝ : MetricSpace α\nx : ℕ → α\nH : DenseRange x\na b : α\ne : ℝ\nepos : 0 < e\nn : ℕ\nhn : dist a (x n) < e / 2\nC : dist b (x n) - dist a (x n) = ↑(embeddingOfSubset x b) n - ↑(embeddingOfSubset x a) n\nthis : dist a b ≤ dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e\n⊢ dist a... | [
"α : Type u\ninst✝ : MetricSpace α\nx : ℕ → α\nH : DenseRange x\na b : α\ne : ℝ\nepos : 0 < e\nn : ℕ\nhn : dist a (x n) < e / 2\nC : dist b (x n) - dist a (x n) = ↑(embeddingOfSubset x b) n - ↑(embeddingOfSubset x a) n\nthis : dist a b ≤ dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e\n⊢ dist a b ≤ dist (e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 15
} | {
"line": 177,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝ : UniformSpace α\nF : Set α\nleft✝ : (∅, F).1 ⊆ SetRel.preimage Set.univ (∅, F).2\nhF : (∅, F).2 ⊆ SetRel.image Set.univ (∅, F).1\n⊢ F ∈ {∅}",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Set.instSingletonSet",
... | [
"α : Type u_1\ninst✝ : UniformSpace α\nF : Set α\nleft✝ : (∅, F).1 ⊆ SetRel.preimage Set.univ (∅, F).2\nhF : (∅, F).2 ⊆ SetRel.image Set.univ (∅, F).1\n⊢ F = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 470,
"column": 23
} | {
"line": 471,
"column": 9
} | {
"line": 471,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : CompactSpace α\n⊢ IsCompact Set.univ",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : CompactSpace α\n⊢ IsCompact Set.univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 16
} | {
"line": 142,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nu : Set (Set α)\nhu₁ : u.Finite\nhu₂ : ∀ U ∈ u, IsOpen[inst✝] U\ns : Set α\nhs₁ : s ⊆ ⋃₀ u\nhs₂ : ∀ U ∈ u, (s ∩ U).Nonempty\nf : Set α → Set α\nhfB : ∀ U ∈ u, f U ∈ B\nhfU : ∀ U ∈ u, f U ⊆ U\nhfs : ∀ U ... | [] | exact ht₂ hU | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 220,
"column": 44
} | {
"line": 224,
"column": 75
} | {
"line": 226,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type u_4\ninst✝ : Finite ι\n⊢ Continuous[Pi.topologicalSpace, TopologicalSpace.vietoris α] range",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"Pi.topologicalSpace",
"congrArg",
"... | [] | by
simp_rw [continuous_iff, powerset, preimage_setOf_eq, range_subset_iff, setOf_forall]
exact ⟨
fun U hU => isOpen_iInter_of_finite fun i => hU.preimage <| continuous_apply i,
fun F hF => isClosed_iInter fun i => hF.preimage <| continuous_apply i⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 46
} | {
"line": 74,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\n⊢ eHolderNorm r f ≠ ∞ ↔ MemHolder r f",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"MemHolder",
"congrArg",
"Iff... | [] | rw [← eHolderNorm_lt_top, lt_top_iff_ne_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 46
} | {
"line": 74,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\n⊢ eHolderNorm r f ≠ ∞ ↔ MemHolder r f",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"MemHolder",
"congrArg",
"Iff... | [] | rw [← eHolderNorm_lt_top, lt_top_iff_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 46
} | {
"line": 74,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\n⊢ eHolderNorm r f ≠ ∞ ↔ MemHolder r f",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"MemHolder",
"congrArg",
"Iff... | [] | rw [← eHolderNorm_lt_top, lt_top_iff_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 249,
"column": 28
} | {
"line": 251,
"column": 25
} | {
"line": 253,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\n⊢ nnHolderNorm r f ≤ C",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"MemHolder.coe_nnHolderNorm_eq_eHolderNor... | [] | by
rw [← ENNReal.coe_le_coe, hf.memHolder.coe_nnHolderNorm_eq_eHolderNorm]
exact hf.eHolderNorm_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 37
} | {
"line": 383,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\n⊢ IsOpen {p | Disjoint p.1 p.2}",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\n⊢ IsOpen {p | Disjoint p.1 p.2}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 36
} | {
"line": 394,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\n⊢ Dense {K | (↑K).Finite}",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.univ",
"setOf",
"Set.Finite",
"Dense",
"id",
"_private.Mathlib.Topology.Sets.VietorisTopology.0.TopologicalSp... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\n⊢ closure {K | (↑K).Finite} = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 331,
"column": 23
} | {
"line": 331,
"column": 50
} | {
"line": 331,
"column": 51
} | [
{
"pp": "X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSp... | [
"X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSpace Y\nbound... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 346,
"column": 23
} | {
"line": 346,
"column": 50
} | {
"line": 346,
"column": 51
} | [
{
"pp": "X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSp... | [
"X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSpace Y\nbound... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.NatEmbedding | {
"line": 38,
"column": 6
} | {
"line": 38,
"column": 30
} | {
"line": 38,
"column": 31
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : Infinite X\nU : ℕ → Set X\nhne : ∀ (n : ℕ), (U n).Nonempty\nho : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhd : Pairwise (Disjoint on U)\nn i j : ℕ\nhij : U (Nat.pair n i) = U (Nat.pair n j)\n⊢ ¬(Disjoint on U) (Nat.pair n i) ... | [
"case refine_1\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : Infinite X\nU : ℕ → Set X\nhne : ∀ (n : ℕ), (U n).Nonempty\nho : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhd : Pairwise (Disjoint on U)\nn i j : ℕ\nhij : U (Nat.pair n i) = U (Nat.pair n j)\n⊢ ¬U (Nat.pair n j) = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.NatEmbedding | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 15
} | {
"line": 44,
"column": 16
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : Infinite X\nh✝ : DiscreteTopology X\nx✝¹ x✝ : ℕ\nh : x✝¹ ≠ x✝\n⊢ (Disjoint on fun n ↦ {(Infinite.natEmbedding X) n}) x✝¹ x✝",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Function.instEmbeddin... | [
"case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : Infinite X\nh✝ : DiscreteTopology X\nx✝¹ x✝ : ℕ\nh : x✝¹ ≠ x✝\n⊢ ¬x✝¹ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 594,
"column": 6
} | {
"line": 594,
"column": 17
} | {
"line": 594,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf : α → β\ninst✝ : CompactSpace α\n⊢ IsCompact univ",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\nf : α → β\ninst✝ : CompactSpace α\n⊢ IsCompact univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 725,
"column": 2
} | {
"line": 726,
"column": 32
} | {
"line": 726,
"column": 33
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 778,
"column": 11
} | {
"line": 778,
"column": 49
} | {
"line": 778,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\n⊢ Continuous fun p ↦ p.1 ⊔ p.2",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"TopologicalSpace.None... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → β\n⊢ Continuous (toCompacts ∘ fun p ↦ p.1 ⊔ p.2)"
] | isEmbedding_toCompacts.continuous_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Order.LowerUpperTopology | {
"line": 313,
"column": 24
} | {
"line": 313,
"column": 88
} | {
"line": 313,
"column": 89
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLower α\nx y : α\nh : Inseparable x y\n⊢ Ici x = Ici y",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLower α\nx y : α\nh : Inseparable x y\n⊢ Ici x = Ici y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 784,
"column": 11
} | {
"line": 784,
"column": 49
} | {
"line": 784,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\n⊢ Continuous fun p ↦ p.1 ×ˢ p.2",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"TopologicalSpace.NonemptyCompacts.toCompacts",
"TopologicalSpace.No... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\n⊢ Continuous (toCompacts ∘ fun p ↦ p.1 ×ˢ p.2)"
] | isEmbedding_toCompacts.continuous_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 795,
"column": 11
} | {
"line": 795,
"column": 49
} | {
"line": 795,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → NonemptyCompacts β\ng : α → β → γ\nhf : Continuous[inst✝², _] f\nhg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)\n⊢ Continuous[inst✝², _] fun x ↦... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : α → NonemptyCompacts β\ng : α → β → γ\nhf : Continuous[inst✝², _] f\nhg : Continuous[instTopologicalSpaceProd, inst✝] (Function.uncurry g)\n⊢ Continuous[inst✝², _] (toCompacts ∘ fun x... | isEmbedding_toCompacts.continuous_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Order.UpperLowerSetTopology | {
"line": 243,
"column": 24
} | {
"line": 243,
"column": 64
} | {
"line": 243,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns : Set α\nS : Set (Set α)\n⊢ (∀ s ∈ S, IsOpen[inst✝¹] s) → IsOpen[inst✝¹] (⋂₀ S)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsUppe... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns : Set α\nS : Set (Set α)\n⊢ (∀ s ∈ S, IsUpperSet s) → IsUpperSet (⋂₀ S)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.ScottTopology | {
"line": 219,
"column": 2
} | {
"line": 220,
"column": 18
} | {
"line": 221,
"column": 2
} | [
{
"pp": "case e'_3\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ns : Set α\ninst✝ : IsScott α univ\n⊢ ↑(lowerClosure s) = closure[upperSet α] s",
"ppTerm": "?e'_3✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Topology.IsUpperSet",
"Preorder... | [
"case e'_4.e'_2\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ns : Set α\ninst✝ : IsScott α univ\n⊢ inst✝¹ = scott α univ"
] | · rw [@IsUpperSet.closure_eq_lowerClosure α _ (upperSet α) ?_ s]
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Order.ScottTopology | {
"line": 240,
"column": 2
} | {
"line": 241,
"column": 9
} | {
"line": 241,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nD : Set (Set α)\ninst✝⁵ : Preorder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsScott β univ\nf : α → β\ninst✝ : IsScott α D\nhf : Continuous[inst✝⁴, inst✝²] f\nx✝ b : α\nhab : x✝ ≤ b\nh : ¬f x✝ ≤ f b\n⊢ False",
"ppTerm": "... | [
"α : Type u_1\nβ : Type u_2\nD : Set (Set α)\ninst✝⁵ : Preorder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsScott β univ\nf : α → β\ninst✝ : IsScott α D\nhf : Continuous[inst✝⁴, inst✝²] f\nx✝ b : α\nhab : x✝ ≤ b\nh : ¬f x✝ ≤ f b\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.ScottTopology | {
"line": 246,
"column": 4
} | {
"line": 250,
"column": 83
} | {
"line": 251,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : Preorder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsScott β univ\nf : α → β\nD : Set (Set α)\ninst✝ : IsScott α D\nhD : ∀ (a b : α), a ≤ b → {a, b} ∈ D\nh : ScottContinuousOn D f\nu : Set β\nhu : IsOpe... | [] | rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)]
exact ⟨(isUpperSet_of_isOpen (D := univ) hu).preimage (h.monotone D hD),
fun t h₀ hd₁ hd₂ a hd₃ ha ↦ image_inter_nonempty_iff.mp <|
(isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ).mp hu).2 trivial (Nonempty.image f hd₁)
(directedOn_ima... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.ScottTopology | {
"line": 246,
"column": 4
} | {
"line": 250,
"column": 83
} | {
"line": 251,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : Preorder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsScott β univ\nf : α → β\nD : Set (Set α)\ninst✝ : IsScott α D\nhD : ∀ (a b : α), a ≤ b → {a, b} ∈ D\nh : ScottContinuousOn D f\nu : Set β\nhu : IsOpe... | [] | rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)]
exact ⟨(isUpperSet_of_isOpen (D := univ) hu).preimage (h.monotone D hD),
fun t h₀ hd₁ hd₂ a hd₃ ha ↦ image_inter_nonempty_iff.mp <|
(isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ).mp hu).2 trivial (Nonempty.image f hd₁)
(directedOn_ima... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.ScottTopology | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 76
} | {
"line": 274,
"column": 77
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsScott α univ\nx y : α\nh : Inseparable x y\n⊢ Iic x = Iic y",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsScott α univ\nx y : α\nh : Inseparable x y\n⊢ Iic x = Iic y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 621,
"column": 4
} | {
"line": 621,
"column": 44
} | {
"line": 622,
"column": 6
} | [
{
"pp": "δ : ℝ\nδpos : δ > 0\nε : ℝ := 2 / 5 * δ\nεpos : 0 < ε\np : GHSpace\n⊢ ∃ s, s.Finite ∧ univ ⊆ ⋃ x ∈ s, ball x ε",
"ppTerm": "?m.54",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"δ : ℝ\nδpos : δ > 0\nε : ℝ := 2 / 5 * δ\nεpos : 0 < ε\np : GHSpace\n⊢ ∃ s, s.Finite ∧ univ ⊆ ⋃ x ∈ s, ball x ε"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Completion | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 62
} | {
"line": 103,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\nq : ℚ\nx✝¹ : (IsSuccPrelimit x → 0 ≤ q) ∧ (IsPredPrelimit x → q ≤ 0)\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\nhq : q < 0\ny : α\nhy : y ⋖ x\nx✝ : α\n⊢ x✝ ∈ ⇑some ⁻¹' Ioi ⟨toLex (x,... | [
"α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\nq : ℚ\nx✝¹ : (IsSuccPrelimit x → 0 ≤ q) ∧ (IsPredPrelimit x → q ≤ 0)\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\nhq : q < 0\ny : α\nhy : y ⋖ x\nx✝ : α\n⊢ x < x✝ ∨ x = x✝ ↔ y < x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Completion | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 62
} | {
"line": 111,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\nq : ℚ\nx✝¹ : (IsSuccPrelimit x → 0 ≤ q) ∧ (IsPredPrelimit x → q ≤ 0)\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\nhq : 0 < q\ny : α\nhy : x ⋖ y\nx✝ : α\n⊢ x✝ ∈ ⇑some ⁻¹' Iio ⟨toLex (x,... | [
"α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\nq : ℚ\nx✝¹ : (IsSuccPrelimit x → 0 ≤ q) ∧ (IsPredPrelimit x → q ≤ 0)\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\nhq : 0 < q\ny : α\nhy : x ⋖ y\nx✝ : α\n⊢ x✝ < x ∨ x✝ = x ↔ x✝ < y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.HullKernel | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 21
} | {
"line": 97,
"column": 22
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝¹ : SemilatticeInf α\nT : Set α\ninst✝ : OrderTop α\nhT : ∀ p ∈ T, InfPrime p\na : α\nF' : Finset α\nh✝ : a ∉ F'\nI4 : hull T (F'.inf id) = T ↓∩ ⋃ a ∈ ↑F', Set.Ici a\n⊢ hull T ((cons a F' h✝).inf id) = T ↓∩ ⋃ a_1 ∈ ↑(cons a F' h✝), Set.Ici a_1",
"ppTerm": "?cons",
... | [] | | cons a F' _ I4 => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Order.NhdsSet | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\na b : α\n⊢ Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"instClosedIicTopology",
"Set.Ioi",
... | [
"α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\na b : α\n⊢ Ici b ⊆ Ioi a ↔ a < b"
] | isOpen_Ioi.mem_nhdsSet, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.NhdsSet | {
"line": 61,
"column": 64
} | {
"line": 62,
"column": 45
} | {
"line": 64,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\na b : α\n⊢ Ioi a ∈ 𝓝ˢ (Ici b) ↔ a < b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"instClosedIicTopology",
"Set.Ioi",
... | [] | by
rw [isOpen_Ioi.mem_nhdsSet, Ici_subset_Ioi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.PartialSups | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 43
} | {
"line": 37,
"column": 44
} | [
{
"pp": "L : Type u_1\ninst✝² : SemilatticeSup L\ninst✝¹ : TopologicalSpace L\ninst✝ : ContinuousSup L\nα : Type u_2\nl : Filter α\nf : ℕ → α → L\ng : ℕ → L\nn : ℕ\nhf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))\n⊢ Tendsto (fun a ↦ (partialSups fun x ↦ f x a) n) l (𝓝 ((partialSups g) n))",
"ppTerm": "?m.29",
... | [
"L : Type u_1\ninst✝² : SemilatticeSup L\ninst✝¹ : TopologicalSpace L\ninst✝ : ContinuousSup L\nα : Type u_2\nl : Filter α\nf : ℕ → α → L\ng : ℕ → L\nn : ℕ\nhf : ∀ k ≤ n, Tendsto (f k) l (𝓝 (g k))\n⊢ Tendsto (fun a ↦ (partialSups f) n a) l (𝓝 ((partialSups g) n))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.PartialSups | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 43
} | {
"line": 49,
"column": 44
} | [
{
"pp": "L : Type u_1\ninst✝³ : SemilatticeSup L\ninst✝² : TopologicalSpace L\ninst✝¹ : ContinuousSup L\nX : Type u_2\ninst✝ : TopologicalSpace X\nf : ℕ → X → L\nn : ℕ\nx : X\nhf : ∀ k ≤ n, ContinuousAt (f k) x\n⊢ ContinuousAt ((partialSups f) n) x",
"ppTerm": "?m.20",
"assigned": false,
"usedConsta... | [
"L : Type u_1\ninst✝³ : SemilatticeSup L\ninst✝² : TopologicalSpace L\ninst✝¹ : ContinuousSup L\nX : Type u_2\ninst✝ : TopologicalSpace X\nf : ℕ → X → L\nn : ℕ\nx : X\nhf : ∀ k ≤ n, ContinuousAt (f k) x\n⊢ ContinuousAt ((partialSups f) n) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.PartialSups | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 43
} | {
"line": 57,
"column": 44
} | [
{
"pp": "L : Type u_1\ninst✝³ : SemilatticeSup L\ninst✝² : TopologicalSpace L\ninst✝¹ : ContinuousSup L\nX : Type u_2\ninst✝ : TopologicalSpace X\nf : ℕ → X → L\nn : ℕ\ns : Set X\nx : X\nhf : ∀ k ≤ n, ContinuousWithinAt (f k) s x\n⊢ ContinuousWithinAt ((partialSups f) n) s x",
"ppTerm": "?m.20",
"assign... | [
"L : Type u_1\ninst✝³ : SemilatticeSup L\ninst✝² : TopologicalSpace L\ninst✝¹ : ContinuousSup L\nX : Type u_2\ninst✝ : TopologicalSpace X\nf : ℕ → X → L\nn : ℕ\ns : Set X\nx : X\nhf : ∀ k ≤ n, ContinuousWithinAt (f k) s x\n⊢ ContinuousWithinAt ((partialSups f) n) s x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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