module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.ContinuousOn | {
"line": 972,
"column": 2
} | {
"line": 972,
"column": 17
} | {
"line": 972,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : Continuous[inst✝¹, inst✝] f\ns u : Set β\nh : u ∈ 𝓝ˢ s\n⊢ f ⁻¹' u ∈ 𝓝ˢ (f ⁻¹' s)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nhf : Continuous[inst✝¹, inst✝] f\ns u : Set β\nh : u ∈ 𝓝ˢ s\n⊢ f ⁻¹' u ∈ 𝓝ˢ (f ⁻¹' s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructions | {
"line": 1095,
"column": 31
} | {
"line": 1095,
"column": 42
} | {
"line": 1095,
"column": 43
} | [
{
"pp": "ι : Type u_5\nA : ι → Type u_6\nB : ι → Type u_7\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : (i : ι) → TopologicalSpace (B i)\nf : (i : ι) → A i → B i\nhf : ∀ (i : ι), IsClosedEmbedding (f i)\n⊢ IsClosed[Pi.topologicalSpace] (range (Pi.map f))",
"ppTerm": "?m.33",
"assigned": true,
"used... | [
"ι : Type u_5\nA : ι → Type u_6\nB : ι → Type u_7\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : (i : ι) → TopologicalSpace (B i)\nf : (i : ι) → A i → B i\nhf : ∀ (i : ι), IsClosedEmbedding (f i)\n⊢ IsClosed[Pi.topologicalSpace] (univ.pi fun i ↦ range (f i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructions | {
"line": 1099,
"column": 31
} | {
"line": 1099,
"column": 42
} | {
"line": 1099,
"column": 43
} | [
{
"pp": "ι : Type u_5\nA : ι → Type u_6\nB : ι → Type u_7\nT : (i : ι) → TopologicalSpace (A i)\ninst✝¹ : (i : ι) → TopologicalSpace (B i)\ninst✝ : Finite ι\nf : (i : ι) → A i → B i\nhf : ∀ (i : ι), IsOpenEmbedding (f i)\n⊢ IsOpen[Pi.topologicalSpace] (range (Pi.map f))",
"ppTerm": "?m.33",
"assigned": ... | [
"ι : Type u_5\nA : ι → Type u_6\nB : ι → Type u_7\nT : (i : ι) → TopologicalSpace (A i)\ninst✝¹ : (i : ι) → TopologicalSpace (B i)\ninst✝ : Finite ι\nf : (i : ι) → A i → B i\nhf : ∀ (i : ι), IsOpenEmbedding (f i)\n⊢ IsOpen[Pi.topologicalSpace] (univ.pi fun i ↦ range (f i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Ultrafilter.Basic | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 13
} | {
"line": 50,
"column": 14
} | [
{
"pp": "α : Type u\nβ : Type v\nf : Ultrafilter α\ninst✝ : Finite β\nP : β → α → Prop\n⊢ (∀ᶠ (i : α) in ↑f, ∃ a, P a i) ↔ ∃ a, ∀ᶠ (i : α) in ↑f, P a i",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\nf : Ultrafilter α\ninst✝ : Finite β\nP : β → α → Prop\n⊢ (∀ᶠ (i : α) in ↑f, ∃ a, P a i) ↔ ∃ a, ∀ᶠ (i : α) in ↑f, P a i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Ultrafilter.Basic | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 38
} | {
"line": 88,
"column": 39
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nl₁ : Filter α\nl₂ : Filter β\n⊢ Tendsto f l₁ l₂ ↔ ∀ (g : Ultrafilter α), ↑g ≤ l₁ → Tendsto f (↑g) l₂",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Order.Filter.Ultrafilter.Basic.0.Filter.tendsto_iff_ultraf... | [
"α : Type u\nβ : Type v\nf : α → β\nl₁ : Filter α\nl₂ : Filter β\n⊢ l₁ ≤ comap f l₂ ↔ ∀ (g : Ultrafilter α), ↑g ≤ l₁ → ↑g ≤ comap f l₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructions | {
"line": 1124,
"column": 2
} | {
"line": 1124,
"column": 35
} | {
"line": 1124,
"column": 36
} | [
{
"pp": "ι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → A a)\nx y : (a : ι) → A a\nI : Finset ι\nt : (i : ι) → Set (A i)\nhtx : ∀ (i : ι), t i ∈ 𝓝 (x i)\nhts : (↑I).pi t ⊆ s\ni : ι\nhi : i ∈ ↑I\n⊢ I.piecewise x y i ∈ t i",
"ppTerm": "?m.50",
... | [
"ι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : DecidableEq ι\ns : Set ((a : ι) → A a)\nx y : (a : ι) → A a\nI : Finset ι\nt : (i : ι) → Set (A i)\nhtx : ∀ (i : ι), t i ∈ 𝓝 (x i)\nhts : (↑I).pi t ⊆ s\ni : ι\nhi : i ∈ ↑I\n⊢ x i ∈ t i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 58
} | {
"line": 274,
"column": 59
} | [
{
"pp": "α : Type u\nβ : Type u_1\nt : TopologicalSpace α\nγ : Type u_2\ns : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\nf₁ : γ → α\nf₂ : γ → β\n⊢ IsTopologicalBasis (image2 (fun x1 x2 ↦ f₁ ⁻¹' x1 ∩ f₂ ⁻¹' x2) B₁ B₂)",
"ppTerm": "?m.24",
... | [
"α : Type u\nβ : Type u_1\nt : TopologicalSpace α\nγ : Type u_2\ns : TopologicalSpace β\nB₁ : Set (Set α)\nB₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\nf₁ : γ → α\nf₂ : γ → β\n⊢ IsTopologicalBasis (image2 (fun x1 x2 ↦ f₁ ⁻¹' x1 ∩ f₂ ⁻¹' x2) B₁ B₂)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 311,
"column": 13
} | {
"line": 311,
"column": 24
} | {
"line": 311,
"column": 25
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsTopologicalBasis ∅\n⊢ IsEmpty α",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nt : TopologicalSpace α\nh : IsTopologicalBasis ∅\n⊢ IsEmpty α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 316,
"column": 13
} | {
"line": 316,
"column": 24
} | {
"line": 316,
"column": 25
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsTopologicalBasis {∅}\n⊢ IsEmpty α",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nt : TopologicalSpace α\nh : IsTopologicalBasis {∅}\n⊢ IsEmpty α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.Ultrafilter.Defs | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 13
} | {
"line": 344,
"column": 14
} | [
{
"pp": "α : Type u\nf : Filter α\ns : Set α\nH : ∀ (g : Ultrafilter α), ↑g ≤ f → s ∈ g\nhf : s ∉ f\ng : Filter ↑sᶜ := comap Subtype.val f\nthis : g.NeBot\n⊢ False",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nf : Filter α\ns : Set α\nH : ∀ (g : Ultrafilter α), ↑g ≤ f → s ∈ g\nhf : s ∉ f\ng : Filter ↑sᶜ := comap Subtype.val f\nthis : g.NeBot\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructions | {
"line": 1285,
"column": 4
} | {
"line": 1285,
"column": 29
} | {
"line": 1285,
"column": 30
} | [
{
"pp": "case refine_1\nX : Type u\nι : Type u_5\nσ : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (σ i)\ninst✝ : TopologicalSpace X\nf : Sigma σ → X\nh : IsInducing f\ni : ι\nU : Set X\nhUo : IsOpen[inst✝] U\nhU : f ⁻¹' U = range (Sigma.mk i)\n⊢ ∀ (x : Sigma σ), f x ∈ U ↔ x.fst = i",
"ppTerm": "?refin... | [
"case refine_1\nX : Type u\nι : Type u_5\nσ : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (σ i)\ninst✝ : TopologicalSpace X\nf : Sigma σ → X\nh : IsInducing f\ni : ι\nU : Set X\nhUo : IsOpen[inst✝] U\nhU : f ⁻¹' U = range (Sigma.mk i)\n⊢ ∀ (x : Sigma σ), f x ∈ U ↔ x.fst = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Constructions | {
"line": 1290,
"column": 4
} | {
"line": 1290,
"column": 20
} | {
"line": 1290,
"column": 21
} | [
{
"pp": "X : Type u\nι : Type u_5\nσ : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (σ i)\ninst✝ : TopologicalSpace X\nf : Sigma σ → X\nx✝¹ : (∀ (i : ι), IsInducing (f ∘ Sigma.mk i)) ∧ ∀ (i : ι), ∃ U, IsOpen[inst✝] U ∧ ∀ (x : Sigma σ), f x ∈ U ↔ x.fst = i\nh₁ : ∀ (i : ι), IsInducing (f ∘ Sigma.mk i)\nh₂ : ... | [
"X : Type u\nι : Type u_5\nσ : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (σ i)\ninst✝ : TopologicalSpace X\nf : Sigma σ → X\nx✝¹ : (∀ (i : ι), IsInducing (f ∘ Sigma.mk i)) ∧ ∀ (i : ι), ∃ U, IsOpen[inst✝] U ∧ ∀ (x : Sigma σ), f x ∈ U ↔ x.fst = i\nh₁ : ∀ (i : ι), IsInducing (f ∘ Sigma.mk i)\nh₂ : ∀ (i : ι), ∃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 541,
"column": 6
} | {
"line": 541,
"column": 38
} | {
"line": 541,
"column": 39
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Countable ι\nX : ι → Type u_3\ns : (i : ι) → Set (X i)\ninst✝ : (i : ι) → TopologicalSpace (X i)\nf₀ : (i : ι) → X i\nc : (i : ι) → Set (X i)\nc_count : ∀ (i : ι), (c i).Countable\nhc : ∀ (i : ι), s i ⊆ closure[inst✝ i] (c i)\nthis : ∀ (i : ι), Countable ↑(c i)\ng : (I : Finset ι... | [
"ι : Type u_2\ninst✝¹ : Countable ι\nX : ι → Type u_3\ns : (i : ι) → Set (X i)\ninst✝ : (i : ι) → TopologicalSpace (X i)\nf₀ : (i : ι) → X i\nc : (i : ι) → Set (X i)\nc_count : ∀ (i : ι), (c i).Countable\nhc : ∀ (i : ι), s i ⊆ closure[inst✝ i] (c i)\nthis : ∀ (i : ι), Countable ↑(c i)\ng : (I : Finset ι) × ((i : ↥I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 579,
"column": 2
} | {
"line": 579,
"column": 13
} | {
"line": 579,
"column": 14
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\ns : Set α\ninst✝ : SeparableSpace ↑s\n⊢ IsSeparable s",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nt : TopologicalSpace α\ns : Set α\ninst✝ : SeparableSpace ↑s\n⊢ IsSeparable s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.LocallyCompact | {
"line": 74,
"column": 24
} | {
"line": 74,
"column": 51
} | {
"line": 74,
"column": 52
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : LocallyCompactSpace X\nx : X\n⊢ ∀ t ∈ 𝓝 x, ∃ r ∈ 𝓝 x, IsCompact r ∧ r ⊆ t",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Membership.mem",
"Exists",
... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : LocallyCompactSpace X\nx : X\n⊢ ∀ t ∈ 𝓝 x, ∃ r, (IsCompact r ∧ r ⊆ t) ∧ r ∈ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.SmallSets | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 47
} | {
"line": 198,
"column": 48
} | [
{
"pp": "α : Type u_1\nl : Filter α\np : α → Prop\n⊢ (∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, p x) ↔ ∀ᶠ (x : α) in l, p x",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nl : Filter α\np : α → Prop\n⊢ (∀ᶠ (s : Set α) in l.smallSets, ∀ x ∈ s, p x) ↔ ∀ᶠ (x : α) in l, p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinite | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 44
} | {
"line": 57,
"column": 45
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\ng : ι' → ι\nhg : Surjective g\nhfg : LocallyFinite (f ∘ g)\n⊢ LocallyFinite f",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nι' : Type u_2\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\ng : ι' → ι\nhg : Surjective g\nhfg : LocallyFinite (f ∘ g)\n⊢ LocallyFinite f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 685,
"column": 2
} | {
"line": 685,
"column": 13
} | {
"line": 685,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : PartialOrder α\n⊢ ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∈ s) ∧ ∀ (x : α), IsTop x → x ∈ s",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : SeparableSpace α\ninst✝ : PartialOrder α\n⊢ ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∈ s) ∧ ∀ (x : α), IsTop x → x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinite | {
"line": 197,
"column": 15
} | {
"line": 197,
"column": 62
} | {
"line": 197,
"column": 63
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\ne : ι' ≃ ι\nh : LocallyFinite (f ∘ ⇑e)\n⊢ LocallyFinite f",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nι' : Type u_2\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\ne : ι' ≃ ι\nh : LocallyFinite (f ∘ ⇑e)\n⊢ LocallyFinite f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.LocallyFinite | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 47
} | {
"line": 205,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι ⊕ ι' → Set X\n⊢ LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.smallSets",
"congrArg",
"Filter.Eventu... | [] | simp only [locallyFinite_iff_smallSets, ← forall_and, ← finite_preimage_inl_and_inr,
preimage_setOf_eq, (· ∘ ·), eventually_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.LocallyFinite | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 47
} | {
"line": 205,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι ⊕ ι' → Set X\n⊢ LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.smallSets",
"congrArg",
"Filter.Eventu... | [] | simp only [locallyFinite_iff_smallSets, ← forall_and, ← finite_preimage_inl_and_inr,
preimage_setOf_eq, (· ∘ ·), eventually_and] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinite | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 47
} | {
"line": 205,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι ⊕ ι' → Set X\n⊢ LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.smallSets",
"congrArg",
"Filter.Eventu... | [] | simp only [locallyFinite_iff_smallSets, ← forall_and, ← finite_preimage_inl_and_inr,
preimage_setOf_eq, (· ∘ ·), eventually_and] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.LocallyFinite | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 31
} | {
"line": 38,
"column": 32
} | [
{
"pp": "X : Type u_1\nι : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nf : ι → Set X\nhf : LocallyFinite f\n⊢ {i | (f i).Nonempty}.Finite",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nι : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nf : ι → Set X\nhf : LocallyFinite f\n⊢ {i | (f i).Nonempty}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 897,
"column": 24
} | {
"line": 900,
"column": 52
} | {
"line": 902,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\n⊢ SeparableSpace α",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Set.mem_range_self",
"Iff.mpr",
"TopologicalSpace.countableBasis",
"Membership.mem",
"Dense",
"Set.Elem",
... | [] | by
choose p hp using fun s : countableBasis α => nonempty_of_mem_countableBasis s.2
exact ⟨⟨range p, countable_range _, (isBasis_countableBasis α).dense_iff.2 fun o ho _ =>
⟨p ⟨o, ho⟩, hp ⟨o, _⟩, mem_range_self _⟩⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Bases | {
"line": 971,
"column": 2
} | {
"line": 971,
"column": 23
} | {
"line": 972,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : IsTopologicalBasis t\nu : Set α\nhu : IsOpen[t✝] u\na : α → Set α\nhat : ∀ x ∈ u, a x ∈ t\nxa : ∀ x ∈ u, x ∈ a x\nau : ∀ x ∈ u, a x ⊆ u\nT : Set ↑u\nT_count : T.Countable\nhT : ⋃ i ∈ T, a ↑i = ⋃... | [
"case refine_2.h₁\nα : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : IsTopologicalBasis t\nu : Set α\nhu : IsOpen[t✝] u\na : α → Set α\nhat : ∀ x ∈ u, a x ∈ t\nxa : ∀ x ∈ u, x ∈ a x\nau : ∀ x ∈ u, a x ⊆ u\nT : Set ↑u\nT_count : T.Countable\nhT : ⋃ i ∈ T, a ↑i = ⋃ i, a ↑i\... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Bases | {
"line": 986,
"column": 42
} | {
"line": 986,
"column": 53
} | {
"line": 986,
"column": 54
} | [
{
"pp": "α : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : IsTopologicalBasis t\ns : Set α → Set (Set α)\nhst : ∀ u ∈ countableBasis α, s u ⊆ t\ns_count : ∀ u ∈ countableBasis α, (s u).Countable\nhs : ∀ u ∈ countableBasis α, u = ⋃ a ∈ s u, a\n⊢ ⋃ u ∈ countableBasis α,... | [
"α : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : IsTopologicalBasis t\ns : Set α → Set (Set α)\nhst : ∀ u ∈ countableBasis α, s u ⊆ t\ns_count : ∀ u ∈ countableBasis α, (s u).Countable\nhs : ∀ u ∈ countableBasis α, u = ⋃ a ∈ s u, a\n⊢ ∀ i ∈ countableBasis α, s i ⊆ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 82,
"column": 55
} | {
"line": 82,
"column": 70
} | {
"line": 82,
"column": 71
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsCompact s\np : Set X → Prop\nhe : p ∅\nhmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s\nhunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)\nhnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t\nf : Filter X := comk p he ⋯ ⋯\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f",
"ppTer... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsCompact s\np : Set X → Prop\nhe : p ∅\nhmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s\nhunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)\nhnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t\nf : Filter X := comk p he ⋯ ⋯\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 999,
"column": 48
} | {
"line": 999,
"column": 59
} | {
"line": 999,
"column": 60
} | [
{
"pp": "α : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : IsTopologicalBasis t\ns : Set α → Set (Set α)\nhst : ∀ u ∈ countableBasis α, s u ⊆ t\ns_count : ∀ u ∈ countableBasis α, (s u).Countable\nhs : ∀ u ∈ countableBasis α, u = ⋃ a ∈ s u, a\nx : α\nv : Set α\nhx : x ... | [
"α : Type u\nt✝ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : IsTopologicalBasis t\ns : Set α → Set (Set α)\nhst : ∀ u ∈ countableBasis α, s u ⊆ t\ns_count : ∀ u ∈ countableBasis α, (s u).Countable\nhs : ∀ u ∈ countableBasis α, u = ⋃ a ∈ s u, a\nx : α\nv : Set α\nhx : x ∈ v\nhv : Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 1015,
"column": 18
} | {
"line": 1015,
"column": 45
} | {
"line": 1015,
"column": 46
} | [
{
"pp": "α : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\nt' : Set (Set α) := (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ t}\nthis : IsTopologicalBasis t'\ns' : Set (Set α)\ns't' : s' ⊆ t'\ns'_count : s'.Countable\nhs' : IsTopologicalBasis s'\nu : Se... | [
"α : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\nt' : Set (Set α) := (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ t}\nthis : IsTopologicalBasis t'\ns' : Set (Set α)\ns't' : s' ⊆ t'\ns'_count : s'.Countable\nhs' : IsTopologicalBasis s'\nu : Set α\nhu : u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Bases | {
"line": 1017,
"column": 28
} | {
"line": 1017,
"column": 39
} | {
"line": 1017,
"column": 40
} | [
{
"pp": "α : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\nt' : Set (Set α) := (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ t}\nthis : IsTopologicalBasis t'\ns' : Set (Set α)\ns't' : s' ⊆ t'\ns'_count : s'.Countable\nhs' : IsTopologicalBasis s'\nf : Se... | [
"α : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\nt' : Set (Set α) := (fun f ↦ ⋂₀ f) '' {f | f.Finite ∧ f ⊆ t}\nthis : IsTopologicalBasis t'\ns' : Set (Set α)\ns't' : s' ⊆ t'\ns'_count : s'.Countable\nhs' : IsTopologicalBasis s'\nf : Set α → Set (S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 34
} | {
"line": 261,
"column": 35
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nl : Filter X\nhs : IsCompact s\n⊢ Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x)",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nl : Filter X\nhs : IsCompact s\n⊢ Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 13
} | {
"line": 302,
"column": 14
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsCompact s\nt : Set (Set X)\nht : ∀ a ∈ t, IsClosed[inst✝] a\nh : ∀ a ⊆ t, a.Finite → (s ∩ ⋂₀ a).Nonempty\na : Finset ↑t\n⊢ (s ∩ ⋂ i ∈ a, ↑i).Nonempty",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"E... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsCompact s\nt : Set (Set X)\nht : ∀ a ∈ t, IsClosed[inst✝] a\nh : ∀ a ⊆ t, a.Finite → (s ∩ ⋂₀ a).Nonempty\na : Finset ↑t\n⊢ (s ∩ ⋂ i, ⋂ (x : i ∈ t), ⋂ (_ : ⟨i, ⋯⟩ ∈ a), i).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 13
} | {
"line": 306,
"column": 14
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\ns : Set (Set X)\nhsc : ∀ t ∈ s, IsClosed[inst✝¹] t\nhs : ∀ t ⊆ s, t.Finite → (⋂₀ t).Nonempty\n⊢ (⋂₀ s).Nonempty",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Exists",
... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\ns : Set (Set X)\nhsc : ∀ t ∈ s, IsClosed[inst✝¹] t\nhs : ∀ t ⊆ s, t.Finite → (⋂₀ t).Nonempty\n⊢ ∃ a, ∀ b ∈ s, a ∈ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 306,
"column": 59
} | {
"line": 306,
"column": 70
} | {
"line": 306,
"column": 71
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\ns : Set (Set X)\nhsc : ∀ t ∈ s, IsClosed[inst✝¹] t\nhs : ∀ t ⊆ s, t.Finite → (⋂₀ t).Nonempty\n⊢ ∀ a ⊆ s, a.Finite → (univ ∩ ⋂₀ a).Nonempty",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrA... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\ns : Set (Set X)\nhsc : ∀ t ∈ s, IsClosed[inst✝¹] t\nhs : ∀ t ⊆ s, t.Finite → (⋂₀ t).Nonempty\n⊢ ∀ a ⊆ s, a.Finite → ∃ a_3, ∀ b ∈ a, a_3 ∈ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 413,
"column": 62
} | {
"line": 413,
"column": 73
} | {
"line": 413,
"column": 74
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nK : Set X\nhK : IsCompact K\nY : Type u_2\nl : Filter Y\ns : Set (X × Y)\nhs : s ∈ ⨆ x ∈ K, 𝓝 x ×ˢ l\n⊢ ∀ x ∈ K, s ∈ 𝓝 x ×ˢ l",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝ : TopologicalSpace X\nK : Set X\nhK : IsCompact K\nY : Type u_2\nl : Filter Y\ns : Set (X × Y)\nhs : s ∈ ⨆ x ∈ K, 𝓝 x ×ˢ l\n⊢ ∀ x ∈ K, s ∈ 𝓝 x ×ˢ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 425,
"column": 41
} | {
"line": 425,
"column": 52
} | {
"line": 425,
"column": 53
} | [
{
"pp": "Y : Type v\ninst✝ : TopologicalSpace Y\nK : Set Y\nX : Type u_2\nl : Filter X\ns : Set (X × Y)\nhK : IsCompact K\nhs : ∀ y ∈ K, s ∈ l ×ˢ 𝓝 y\n⊢ s ∈ ⨆ y ∈ K, l ×ˢ 𝓝 y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.inst... | [
"Y : Type v\ninst✝ : TopologicalSpace Y\nK : Set Y\nX : Type u_2\nl : Filter X\ns : Set (X × Y)\nhK : IsCompact K\nhs : ∀ y ∈ K, s ∈ l ×ˢ 𝓝 y\n⊢ ∀ i ∈ K, s ∈ l ×ˢ 𝓝 i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 444,
"column": 40
} | {
"line": 444,
"column": 51
} | {
"line": 444,
"column": 52
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nK : Set X\nl : Filter X\ns : Set X\nhK : IsCompact K\nhs : ∀ x ∈ K, s ∈ 𝓝 x ⊓ l\n⊢ s ∈ ⨆ x ∈ K, 𝓝 x ⊓ l",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.instSupSet",
"iSup",... | [
"X : Type u\ninst✝ : TopologicalSpace X\nK : Set X\nl : Filter X\ns : Set X\nhK : IsCompact K\nhs : ∀ x ∈ K, s ∈ 𝓝 x ⊓ l\n⊢ ∀ i ∈ K, s ∈ 𝓝 i ⊓ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 451,
"column": 40
} | {
"line": 451,
"column": 51
} | {
"line": 451,
"column": 52
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nK : Set X\nl : Filter X\ns : Set X\nhK : IsCompact K\nhs : ∀ y ∈ K, s ∈ l ⊓ 𝓝 y\n⊢ s ∈ ⨆ x ∈ K, l ⊓ 𝓝 x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.instSupSet",
"iSup",... | [
"X : Type u\ninst✝ : TopologicalSpace X\nK : Set X\nl : Filter X\ns : Set X\nhK : IsCompact K\nhs : ∀ y ∈ K, s ∈ l ⊓ 𝓝 y\n⊢ ∀ i ∈ K, s ∈ l ⊓ 𝓝 i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 471,
"column": 21
} | {
"line": 471,
"column": 59
} | {
"line": 471,
"column": 60
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nx : X\nx✝ : Filter X\nhf : x✝.NeBot\nhfa : x✝ ≤ 𝓟 {x}\n⊢ 𝓟 {x} ≤ 𝓝 x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"nhds"... | [
"X : Type u\ninst✝ : TopologicalSpace X\nx : X\nx✝ : Filter X\nhf : x✝.NeBot\nhfa : x✝ ≤ 𝓟 {x}\n⊢ pure x ≤ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 556,
"column": 55
} | {
"line": 556,
"column": 86
} | {
"line": 556,
"column": 87
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\n⊢ ∀ x ∈ s, ∃ t, x ∈ t ∧ t ∈ S ∧ t ∉ F",
"ppTerm": "?m.50",
"assigned": false,
"usedConsta... | [
"X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\n⊢ ∀ x ∈ s, ∃ t, x ∈ t ∧ t ∈ S ∧ t ∉ F"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 563,
"column": 4
} | {
"line": 563,
"column": 84
} | {
"line": 563,
"column": 85
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\nU : X → Set X\nhxU : ∀ x ∈ s, x ∈ U x\nhSU : ∀ x ∈ s, U x ∈ S\nhUF : ∀ x ∈ s, U x ∉ F\nQ : Set (Set X... | [
"X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\nU : X → Set X\nhxU : ∀ x ∈ s, x ∈ U x\nhSU : ∀ x ∈ s, U x ∈ S\nhUF : ∀ x ∈ s, U x ∉ F\nQ : Set (Set X)\nhQU : Q ⊆... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.UpperLower.Closure | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 34
} | {
"line": 182,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\ns t : Set α\nhs : IsLowerSet s\n⊢ Disjoint s ↑(upperClosure t) ↔ Disjoint s t",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : Preorder α\ns t : Set α\nhs : IsLowerSet s\n⊢ Disjoint s ↑(upperClosure t) ↔ Disjoint s t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 609,
"column": 11
} | {
"line": 609,
"column": 49
} | {
"line": 609,
"column": 50
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nf : Filter X\n⊢ Disjoint f (cocompact X) ↔ ∃ K ∈ f, IsCompact K",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Compl.compl",
"Filter.HasBasis.disjoint_iff_right",... | [
"X : Type u\ninst✝ : TopologicalSpace X\nf : Filter X\n⊢ (∃ i, IsCompact i ∧ iᶜᶜ ∈ f) ↔ ∃ K ∈ f, IsCompact K"
] | hasBasis_cocompact.disjoint_iff_right, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.UpperLower.Closure | {
"line": 218,
"column": 18
} | {
"line": 218,
"column": 34
} | {
"line": 218,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : NoMinOrder α\ns : Set α\nh₁ : upperClosure s = ⊥\nh₂ : BddBelow s\n⊢ False",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : NoMinOrder α\ns : Set α\nh₁ : upperClosure s = ⊥\nh₂ : BddBelow s\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 730,
"column": 2
} | {
"line": 730,
"column": 13
} | {
"line": 730,
"column": 14
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : Filter Y\nx : X\nhf : Disjoint f (cocompact Y)\n⊢ 𝓝 x ×ˢ f ≤ 𝓝ˢ ({x} ×ˢ univ)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : Filter Y\nx : X\nhf : Disjoint f (cocompact Y)\n⊢ 𝓝 x ×ˢ f ≤ 𝓝ˢ ({x} ×ˢ univ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 735,
"column": 2
} | {
"line": 735,
"column": 13
} | {
"line": 735,
"column": 14
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : Filter X\ny : Y\nhf : Disjoint f (cocompact X)\n⊢ f ×ˢ 𝓝 y ≤ 𝓝ˢ (univ ×ˢ {y})",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : Filter X\ny : Y\nhf : Disjoint f (cocompact X)\n⊢ f ×ˢ 𝓝 y ≤ 𝓝ˢ (univ ×ˢ {y})"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 792,
"column": 2
} | {
"line": 792,
"column": 13
} | {
"line": 792,
"column": 14
} | [
{
"pp": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nf : Filter X\ninst✝ : f.NeBot\n⊢ ∃ x, ClusterPt x f",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\nf : Filter X\ninst✝ : f.NeBot\n⊢ ∃ x, ClusterPt x f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 807,
"column": 69
} | {
"line": 807,
"column": 80
} | {
"line": 807,
"column": 81
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nh : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed[inst✝] (t i)) → ⋂ i, t i = ∅ → ∃ u, ⋂ i ∈ u, t i = ∅\nι✝ : Type u\nt : ι✝ → Set X\n⊢ (∀ (i : ι✝), IsClosed[inst✝] (t i)) → univ ∩ ⋂ i, t i = ∅ → ∃ u, univ ∩ ⋂ i ∈ u, t i = ∅",
"ppTerm": "?m.30",
"as... | [
"X : Type u\ninst✝ : TopologicalSpace X\nh : ∀ {ι : Type u} (t : ι → Set X), (∀ (i : ι), IsClosed[inst✝] (t i)) → ⋂ i, t i = ∅ → ∃ u, ⋂ i ∈ u, t i = ∅\nι✝ : Type u\nt : ι✝ → Set X\n⊢ (∀ (i : ι✝), IsClosed[inst✝] (t i)) → ⋂ i, t i = ∅ → ∃ u, ⋂ i ∈ u, t i = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 817,
"column": 2
} | {
"line": 817,
"column": 13
} | {
"line": 817,
"column": 14
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nι : Type v\ninst✝ : CompactSpace X\nt : ι → Set X\nhtc : ∀ (i : ι), IsClosed[inst✝¹] (t i)\nhst : ∀ (s : Finset ι), (⋂ i ∈ s, t i).Nonempty\n⊢ (⋂ i, t i).Nonempty",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.iI... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\nι : Type v\ninst✝ : CompactSpace X\nt : ι → Set X\nhtc : ∀ (i : ι), IsClosed[inst✝¹] (t i)\nhst : ∀ (s : Finset ι), (⋂ i ∈ s, t i).Nonempty\n⊢ ∃ x, ∀ (i : ι), x ∈ t i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 817,
"column": 61
} | {
"line": 817,
"column": 72
} | {
"line": 817,
"column": 73
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\nι : Type v\ninst✝ : CompactSpace X\nt : ι → Set X\nhtc : ∀ (i : ι), IsClosed[inst✝¹] (t i)\nhst : ∀ (s : Finset ι), (⋂ i ∈ s, t i).Nonempty\n⊢ ∀ (u : Finset ι), (univ ∩ ⋂ i ∈ u, t i).Nonempty",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\nι : Type v\ninst✝ : CompactSpace X\nt : ι → Set X\nhtc : ∀ (i : ι), IsClosed[inst✝¹] (t i)\nhst : ∀ (s : Finset ι), (⋂ i ∈ s, t i).Nonempty\n⊢ ∀ (u : Finset ι), ∃ x, ∀ i ∈ u, x ∈ t i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Inseparable | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 36
} | {
"line": 220,
"column": 37
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns : Set X\nf g : X → Y\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs : IsClosed[inst✝²] s\nhf : Continuous[inst✝², inst✝¹] f\nhg : Continuous[inst✝², inst✝¹] g\nhspec : ∀ (x : X), g x ⤳ f x\n⊢ Continuous[inst✝², inst✝¹] (s... | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns : Set X\nf g : X → Y\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs : IsClosed[inst✝²] s\nhf : Continuous[inst✝², inst✝¹] f\nhg : Continuous[inst✝², inst✝¹] g\nhspec : ∀ (x : X), g x ⤳ f x\n⊢ Continuous[inst✝², inst✝¹] (s.piecewise f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 850,
"column": 6
} | {
"line": 850,
"column": 61
} | {
"line": 850,
"column": 62
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\n𝔅 : Set (Set X)\nhT : T = generateFrom 𝔅\nh𝔅 : ∀ s ∈ 𝔅, sᶜ ∈ 𝔅\nh : ∀ P ⊆ 𝔅, (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) → (⋂₀ P).Nonempty\nP : Set (Set X)\nhP𝔅 : P ⊆ 𝔅\nhP : ∀ Q ⊆ P, Q.Finite → (⋂₀ (compl '' Q)).Nonempty\nQ : Set (Set X)\nhQP : Q ⊆ compl '' P\nhQ ... | [
"X : Type u\nT : TopologicalSpace X\n𝔅 : Set (Set X)\nhT : T = generateFrom 𝔅\nh𝔅 : ∀ s ∈ 𝔅, sᶜ ∈ 𝔅\nh : ∀ P ⊆ 𝔅, (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) → (⋂₀ P).Nonempty\nP : Set (Set X)\nhP𝔅 : P ⊆ 𝔅\nhP : ∀ Q ⊆ P, Q.Finite → (⋂₀ (compl '' Q)).Nonempty\nQ : Set (Set X)\nhQP : Q ⊆ compl '' P\nhQ : Q.Finite\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Inseparable | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 47
} | {
"line": 622,
"column": 48
} | [
{
"pp": "α : Type u_4\ninst✝ : TopologicalSpace α\n⊢ Nontrivial (SeparationQuotient α) ↔ NontrivialTopology α",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\ninst✝ : TopologicalSpace α\n⊢ Nontrivial (SeparationQuotient α) ↔ NontrivialTopology α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 920,
"column": 2
} | {
"line": 921,
"column": 9
} | {
"line": 921,
"column": 10
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : CompactSpace X\nhs : s.Infinite\n⊢ ∃ x, AccPt x (cofinite ⊓ 𝓟 s)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : CompactSpace X\nhs : s.Infinite\n⊢ ∃ x, AccPt x (cofinite ⊓ 𝓟 s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 928,
"column": 2
} | {
"line": 928,
"column": 21
} | {
"line": 928,
"column": 22
} | [
{
"pp": "X : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : Infinite X\n⊢ ∃ z, (𝓝[≠] z).NeBot",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : Infinite X\n⊢ ∃ z, (𝓝[≠] z).NeBot"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 949,
"column": 2
} | {
"line": 949,
"column": 13
} | {
"line": 949,
"column": 14
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : Continuous[inst✝¹, inst✝] f\nt : Set X\nht : IsCompact t\n⊢ f ⁻¹' (f '' t)ᶜ ⊆ tᶜ",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Compl.compl",
"compl_le_co... | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : Continuous[inst✝¹, inst✝] f\nt : Set X\nht : IsCompact t\n⊢ t ⊆ f ⁻¹' f '' t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Piecewise | {
"line": 144,
"column": 29
} | {
"line": 144,
"column": 45
} | {
"line": 144,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns s' t : Set α\nhs : IsOpen[inst✝] s\nhs' : IsOpen[inst✝] s'\nht : s ∩ frontier t = s' ∩ frontier t\nx : α\nhx : x ∈ frontier t\n⊢ x ∈ s ↔ x ∈ s'",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\ns s' t : Set α\nhs : IsOpen[inst✝] s\nhs' : IsOpen[inst✝] s'\nht : s ∩ frontier t = s' ∩ frontier t\nx : α\nhx : x ∈ frontier t\n⊢ x ∈ s ↔ x ∈ s'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 1038,
"column": 4
} | {
"line": 1038,
"column": 44
} | {
"line": 1039,
"column": 6
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nks : IsCompact s\nι : Type u_3\nt : ι → Set X\nI : Set ι\nhtc : ∀ i ∈ I, IsClosed[instTopologicalSpaceSubtype] (s ↓∩ t i)\nhst : s ∩ ⋂ i ∈ I, t i = ∅\nthis : univ ∩ ⋂ i, (fun i ↦ s ↓∩ t ↑i) i = ∅\n⊢ ∃ u, s ∩ ⋂ i ∈ u, t ↑i = ∅",
"ppTerm": "?m.58",... | [
"X : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nks : IsCompact s\nι : Type u_3\nt : ι → Set X\nI : Set ι\nhtc : ∀ i ∈ I, IsClosed[instTopologicalSpaceSubtype] (s ↓∩ t i)\nhst : s ∩ ⋂ i ∈ I, t i = ∅\nthis : univ ∩ ⋂ i, (fun i ↦ s ↓∩ t ↑i) i = ∅\n⊢ ∃ u, ∀ x ∈ s, ∃ x_1, (∃ (x : x_1 ∈ I), ⟨x_1, ⋯⟩ ∈ u) ∧ x ∉ t x_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Compact | {
"line": 1041,
"column": 2
} | {
"line": 1041,
"column": 62
} | {
"line": 1041,
"column": 63
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nks : IsCompact s\nι : Type u_3\nt : ι → Set X\nI : Set ι\nhtc : ∀ i ∈ I, IsClosed[instTopologicalSpaceSubtype] (s ↓∩ t i)\nhst : s ∩ ⋂ i ∈ I, t i = ∅\n⊢ univ ∩ ⋂ i, (fun i ↦ s ↓∩ t ↑i) i = ∅",
"ppTerm": "?m.57",
"assigned": true,
"usedCon... | [
"X : Type u_2\ninst✝ : TopologicalSpace X\ns : Set X\nks : IsCompact s\nι : Type u_3\nt : ι → Set X\nI : Set ι\nhtc : ∀ i ∈ I, IsClosed[instTopologicalSpaceSubtype] (s ↓∩ t i)\nhst : s ∩ ⋂ i ∈ I, t i = ∅\n⊢ ∀ a ∈ s, ∃ x ∈ I, a ∉ t x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 43
} | {
"line": 298,
"column": 44
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : R0Space X\ninst✝ : TopologicalSpace Y\nf : Y → X\nhf : IsInducing f\na b : Y\n⊢ a ⤳ b → b ⤳ a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Topology.IsInducing.specializes_iff",
"Speci... | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : R0Space X\ninst✝ : TopologicalSpace Y\nf : Y → X\nhf : IsInducing f\na b : Y\n⊢ f a ⤳ f b → f b ⤳ f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 13
} | {
"line": 318,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R0Space X\nx : X\nι✝ : Type u_1\nU : ι✝ → Set X\nhUo : ∀ (i : ι✝), IsOpen[inst✝¹] (U i)\nhxU : closure[inst✝¹] {x} ⊆ ⋃ i, U i\ni : ι✝\nhi : x ∈ U i\ny : X\nhy : y ⤳ x\n⊢ y ∈ ⋃ i_1 ∈ {i}, U i_1",
"ppTerm": "?m.73",
"assigned": true,
"usedCon... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R0Space X\nx : X\nι✝ : Type u_1\nU : ι✝ → Set X\nhUo : ∀ (i : ι✝), IsOpen[inst✝¹] (U i)\nhxU : closure[inst✝¹] {x} ⊆ ⋃ i, U i\ni : ι✝\nhi : x ∈ U i\ny : X\nhy : y ⤳ x\n⊢ y ∈ U i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DiscreteSubset | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 51
} | {
"line": 61,
"column": 52
} | [
{
"pp": "X : Type u_3\ninst✝ : TopologicalSpace X\nE : Set X\nh : ∀ x ∈ E, ¬AccPt x (𝓟 E)\n⊢ DiscreteTopology ↑E",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DiscreteTopology",
"Compl.compl",
"nhdsWithin",
"Membership.mem",
"Set.Elem",
... | [
"X : Type u_3\ninst✝ : TopologicalSpace X\nE : Set X\nh : ∀ x ∈ E, ¬AccPt x (𝓟 E)\n⊢ ∀ x ∈ E, 𝓝[≠] x ⊓ 𝓟 E = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Irreducible | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 48
} | {
"line": 196,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ns t : Set X\ninst✝ : PreirreducibleSpace X\n⊢ IsOpen[inst✝¹] s → IsOpen[inst✝¹] t → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ns t : Set X\ninst✝ : PreirreducibleSpace X\n⊢ IsOpen[inst✝¹] s → IsOpen[inst✝¹] t → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 397,
"column": 82
} | {
"line": 399,
"column": 38
} | {
"line": 401,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nb : Set (Set X)\nhb : IsTopologicalBasis b\nx y : X\nh : x ≠ y\n⊢ ∃ a ∈ b, x ∈ a ∧ y ∉ a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"TopologicalSpace.IsTopologicalBasis.isOpen_iff",
"setOf",
"Memb... | [] | by
rcases hb.isOpen_iff.1 isOpen_ne x h with ⟨a, ab, xa, ha⟩
exact ⟨a, ab, xa, fun h => ha h rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.DiscreteSubset | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | {
"line": 139,
"column": 14
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : DiscreteTopology X\nhf : IsOpenMap f\n⊢ IsDiscrete (range f)",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : DiscreteTopology X\nhf : IsOpenMap f\n⊢ IsDiscrete (range f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DiscreteSubset | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 13
} | {
"line": 146,
"column": 14
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : DiscreteTopology X\nhf : IsInducing f\n⊢ IsDiscrete (range f)",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : DiscreteTopology X\nhf : IsInducing f\n⊢ IsDiscrete (range f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Irreducible | {
"line": 209,
"column": 33
} | {
"line": 223,
"column": 12
} | {
"line": 225,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nH : IsPreirreducible s\nf : X → Y\nhf : ContinuousOn f s\n⊢ IsPreirreducible (f '' s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",... | [] | by
rintro u v hu hv ⟨_, ⟨⟨x, hx, rfl⟩, hxu⟩⟩ ⟨_, ⟨⟨y, hy, rfl⟩, hyv⟩⟩
rw [← mem_preimage] at hxu hyv
rcases continuousOn_iff'.1 hf u hu with ⟨u', hu', u'_eq⟩
rcases continuousOn_iff'.1 hf v hv with ⟨v', hv', v'_eq⟩
have := H u' v' hu' hv'
rw [inter_comm s u', ← u'_eq] at this
rw [inter_comm s v', ← v'_eq]... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Irreducible | {
"line": 262,
"column": 4
} | {
"line": 263,
"column": 31
} | {
"line": 265,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\ninst✝ : IndiscreteTopology X\nu v : Set X\n⊢ IsOpen[inst✝²] u → IsOpen[inst✝²] v → (univ ∩ u).Nonempty → (univ ∩ v).Nonempty → (univ ∩ (u ∩ v)).Nonempty",
"ppTerm": "?m.7",
"assigned": true,
"... | [] | simp only [IndiscreteTopology.isOpen_iff, univ_inter]
rintro ⟨h | h⟩ <;> simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Irreducible | {
"line": 262,
"column": 4
} | {
"line": 263,
"column": 31
} | {
"line": 265,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\ninst✝ : IndiscreteTopology X\nu v : Set X\n⊢ IsOpen[inst✝²] u → IsOpen[inst✝²] v → (univ ∩ u).Nonempty → (univ ∩ v).Nonempty → (univ ∩ (u ∩ v)).Nonempty",
"ppTerm": "?m.7",
"assigned": true,
"... | [] | simp only [IndiscreteTopology.isOpen_iff, univ_inter]
rintro ⟨h | h⟩ <;> simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Irreducible | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 12
} | {
"line": 269,
"column": 13
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\ns t : Set X✝\nX : Type u_3\ninst✝ : Infinite X\nu v : Set (CofiniteTopology X)\n⊢ (u.Nonempty → uᶜ.Finite) → (v.Nonempty → vᶜ.Finite) → u.Nonempty → v.Nonempty → (u ∩ v).Nonempty",
"ppTerm": "?m.11",
"assign... | [
"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\ns t : Set X✝\nX : Type u_3\ninst✝ : Infinite X\nu v : Set (CofiniteTopology X)\nhu : u.Nonempty → uᶜ.Finite\n⊢ (v.Nonempty → vᶜ.Finite) → u.Nonempty → v.Nonempty → (u ∩ v).Nonempty"
] | intro hu | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Topology.Irreducible | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 47
} | {
"line": 270,
"column": 48
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\ns t : Set X✝\nX : Type u_3\ninst✝ : Infinite X\nu v : Set (CofiniteTopology X)\nhu : u.Nonempty → uᶜ.Finite\nhv : v.Nonempty → vᶜ.Finite\nhu' : u.Nonempty\nhv' : v.Nonempty\n⊢ (u ∩ v).Nonempty",
"ppTerm": "?m.18... | [
"X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\ns t : Set X✝\nX : Type u_3\ninst✝ : Infinite X\nu v : Set (CofiniteTopology X)\nhu : u.Nonempty → uᶜ.Finite\nhv : v.Nonempty → vᶜ.Finite\nhu' : u.Nonempty\nhv' : v.Nonempty\n⊢ (u ∩ v).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Irreducible | {
"line": 289,
"column": 15
} | {
"line": 289,
"column": 26
} | {
"line": 289,
"column": 27
} | [
{
"pp": "case refine_1.empty\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsIrreducible s\nhu : ∀ u ∈ ∅, IsOpen[inst✝] u\nhU : ∀ u ∈ ∅, (s ∩ u).Nonempty\n⊢ (s ∩ ⋂₀ ↑∅).Nonempty",
"ppTerm": "?refine_1.empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.coe_empty",
... | [
"case refine_1.empty\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsIrreducible s\nhu : ∀ u ∈ ∅, IsOpen[inst✝] u\nhU : ∀ u ∈ ∅, (s ∩ u).Nonempty\n⊢ s.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DiscreteSubset | {
"line": 171,
"column": 2
} | {
"line": 172,
"column": 19
} | {
"line": 172,
"column": 20
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsDiscrete s\na b : X\nhab : a ⤳ b\nha : a ∈ s\nhb : b ∈ s\nthis : DiscreteTopology ↑s := hs.to_subtype\n⊢ a = b",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsDiscrete s\na b : X\nhab : a ⤳ b\nha : a ∈ s\nhb : b ∈ s\nthis : DiscreteTopology ↑s := hs.to_subtype\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Irreducible | {
"line": 294,
"column": 4
} | {
"line": 294,
"column": 15
} | {
"line": 294,
"column": 16
} | [
{
"pp": "case refine_2\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty\n⊢ s.Nonempty",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"case refine_2\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty\n⊢ s.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Irreducible | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 19
} | {
"line": 296,
"column": 20
} | [
{
"pp": "case refine_3\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty\nu v : Set X\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhu' : (s ∩ u).Nonempty\nhv' : (s ∩ v).Nonempty\n⊢ (s ∩ (u ∩ v)).Nonem... | [
"case refine_3\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : ∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen[inst✝] u) → (∀ u ∈ U, (s ∩ u).Nonempty) → (s ∩ ⋂₀ ↑U).Nonempty\nu v : Set X\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhu' : (s ∩ u).Nonempty\nhv' : (s ∩ v).Nonempty\n⊢ (s ∩ (u ∩ v)).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DiscreteSubset | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 57
} | {
"line": 226,
"column": 58
} | [
{
"pp": "case mpr\nX : Type u_1\ninst✝ : TopologicalSpace X\nS : Set X\nx : X\nH : Disjoint (𝓝[≠] x) (𝓟 S)\nhx : x ∉ S\n⊢ Disjoint (𝓝 x) (𝓟 S)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Topology.DiscreteSubset.0.isClosed_and_discrete_iff._sim... | [
"case mpr\nX : Type u_1\ninst✝ : TopologicalSpace X\nS : Set X\nx : X\nH : Disjoint (𝓝[≠] x) (𝓟 S)\nhx : x ∉ S\n⊢ 𝓝 x ⊓ 𝓟 S = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DiscreteSubset | {
"line": 385,
"column": 6
} | {
"line": 385,
"column": 47
} | {
"line": 385,
"column": 48
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\np : X\ns : Set X\nhs : Disjoint (𝓝[s] p) cofinite\nt : Set X\nh₁t : t ∈ 𝓝[s] p\nh₂t : t.Finite\nS : Set X := {y | y ∈ t ∩ s ∧ ¬y ⤳ p}\ny : X\nhy : y ∈ S\n⊢ p ∈ (closure[inst✝] {y})ᶜ",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\np : X\ns : Set X\nhs : Disjoint (𝓝[s] p) cofinite\nt : Set X\nh₁t : t ∈ 𝓝[s] p\nh₂t : t.Finite\nS : Set X := {y | y ∈ t ∩ s ∧ ¬y ⤳ p}\ny : X\nhy : y ∈ S\n⊢ p ∉ closure[inst✝] {y}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Irreducible | {
"line": 397,
"column": 44
} | {
"line": 397,
"column": 55
} | {
"line": 397,
"column": 56
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nt S U : Set X\nhU' : IsOpen[inst✝] U\nh₁ : U ⊆ S\nh₂ : S ⊆ t\nz : X\nhz : z ∈ U\nht : IsIrreducible t\nu v : Set X\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nx✝ : X\nhx : x✝ ∈ S\nhx'✝ : x✝ ∈ u\ny : X\nhy : y ∈ S\nhy' : y ∈ v\nx : X\nhx' : x ∈ ⋂₀ ↑{U, u, v}\n⊢... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nt S U : Set X\nhU' : IsOpen[inst✝] U\nh₁ : U ⊆ S\nh₂ : S ⊆ t\nz : X\nhz : z ∈ U\nht : IsIrreducible t\nu v : Set X\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nx✝ : X\nhx : x✝ ∈ S\nhx'✝ : x✝ ∈ u\ny : X\nhy : y ∈ S\nhy' : y ∈ v\nx : X\nhx' : x ∈ ⋂₀ ↑{U, u, v}\n⊢ x ∈ U ∧ x ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DiscreteSubset | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 15
} | {
"line": 396,
"column": 16
} | [
{
"pp": "case a\nX : Type u_1\ninst✝ : TopologicalSpace X\np : X\ns : Set X\nhs : Disjoint (𝓝[s] p) cofinite\n⊢ 𝓝[s] p ≤ 𝓟 ({x | x ⤳ p} ∩ s)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Specializes",
"nhdsWithin",
"Parti... | [
"case a\nX : Type u_1\ninst✝ : TopologicalSpace X\np : X\ns : Set X\nhs : Disjoint (𝓝[s] p) cofinite\n⊢ {x | x ⤳ p} ∈ 𝓝[s] p ∧ s ∈ 𝓝[s] p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DiscreteSubset | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 49
} | {
"line": 402,
"column": 0
} | [
{
"pp": "case h₁\nX : Type u_1\ninst✝ : TopologicalSpace X\np : X\ns : Set X\nhs : Disjoint (𝓝[s] p) cofinite\n⊢ ∀ V ∈ 𝓝 p, {x | x ⤳ p} ⊆ V",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Specializes",
"setOf",
"Membership.mem",
"nhds... | [] | exact fun s hs x hx ↦ mem_of_mem_nhds (hx hs) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Separation.Hausdorff | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 87
} | {
"line": 157,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : T2Space X\nx y : X\nh : x ≠ y\n⊢ ∃ u v, u ∈ 𝓝 x ∧ v ∈ 𝓝 y ∧ IsCompact u ∧ IsCompact v ∧ Disjoint u v",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : LocallyCompactSpace X\ninst✝ : T2Space X\nx y : X\nh : x ≠ y\n⊢ ∃ u v, IsCompact u ∧ IsCompact v ∧ Disjoint u v ∧ u ∈ 𝓝 x ∧ v ∈ 𝓝 y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Hausdorff | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 29
} | {
"line": 214,
"column": 30
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nx : X\nt : Set X\nH1 : IsCompact t\nH2 : x ∉ t\n⊢ ∃ U V, IsOpen[inst✝¹] U ∧ IsOpen[inst✝¹] V ∧ t ⊆ U ∧ x ∈ V ∧ Disjoint U V",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.t... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nx : X\nt : Set X\nH1 : IsCompact t\nH2 : x ∉ t\n⊢ ∃ U, IsOpen[inst✝¹] U ∧ ∃ x_1, IsOpen[inst✝¹] x_1 ∧ t ⊆ U ∧ x ∈ x_1 ∧ Disjoint U x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Hausdorff | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 13
} | {
"line": 222,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nx : X\nt : Set X\nH1 : IsCompact t\nH2 : x ∉ t\n⊢ Disjoint (𝓝ˢ t) (𝓝 x)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\nx : X\nt : Set X\nH1 : IsCompact t\nH2 : x ∉ t\n⊢ Disjoint (𝓝ˢ t) (𝓝 x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 675,
"column": 13
} | {
"line": 675,
"column": 24
} | {
"line": 675,
"column": 25
} | [
{
"pp": "case empty\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T1Space X\ninst✝ : ∀ (x : X), (𝓝[≠] x).NeBot\ns : Set X\nhs : Dense s\n⊢ Dense (s \\ ↑∅)",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.coe_empty",
"congrArg",
"Finset",
... | [
"case empty\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T1Space X\ninst✝ : ∀ (x : X), (𝓝[≠] x).NeBot\ns : Set X\nhs : Dense s\n⊢ Dense s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Hausdorff | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 47
} | {
"line": 449,
"column": 48
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\ni✝ : T2Quotient X\nx : X\nj✝ : T2Quotient X\ny : X\nh : ¬mk x = mk y\n⊢ ∃ s, T2Space (Quotient s) ∧ ⟦x⟧ ≠ ⟦y⟧",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Exists",
"id",
... | [
"X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\ni✝ : T2Quotient X\nx : X\nj✝ : T2Quotient X\ny : X\nh : ¬mk x = mk y\n⊢ ∃ s, T2Space (Quotient s) ∧ ¬s x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 785,
"column": 27
} | {
"line": 785,
"column": 83
} | {
"line": 785,
"column": 84
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\ns : Set X\nhs : s ∈ 𝓝 x\nhsf : s.Finite\nA : {x} ⊆ s\nB : IsClosed[inst✝¹] (s \\ {x})\nC : (s \\ {x})ᶜ ∈ 𝓝 x\n⊢ {x} ∈ 𝓝 x",
"ppTerm": "?m.77",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\ns : Set X\nhs : s ∈ 𝓝 x\nhsf : s.Finite\nA : {x} ⊆ s\nB : IsClosed[inst✝¹] (s \\ {x})\nC : (s \\ {x})ᶜ ∈ 𝓝 x\n⊢ {x} ∈ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 834,
"column": 2
} | {
"line": 834,
"column": 70
} | {
"line": 835,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : DiscreteTopology ↑s\nx : X\nhx : x ∈ s\nthis : {⟨x, hx⟩} ∈ 𝓝 ⟨x, hx⟩\n⊢ {x} ∈ 𝓝[s] x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Filter.map",
... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : DiscreteTopology ↑s\nx : X\nhx : x ∈ s\nthis : {⟨x, hx⟩} ∈ 𝓝 ⟨x, hx⟩\n⊢ {x} ∈ map Subtype.val (𝓝 ⟨x, hx⟩)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 854,
"column": 2
} | {
"line": 854,
"column": 13
} | {
"line": 854,
"column": 14
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsDiscrete s\nx : X\nhx : x ∈ s\n⊢ ∃ U ∈ 𝓝 x, U ∩ s = {x}",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsDiscrete s\nx : X\nhx : x ∈ s\n⊢ ∃ U ∈ 𝓝 x, U ∩ s = {x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 1030,
"column": 4
} | {
"line": 1030,
"column": 88
} | {
"line": 1030,
"column": 89
} | [
{
"pp": "case empty\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nι : Type u_3\ns : Set X\nhs : IsCompact s\nU : ι → Set X\nhU : ∀ i ∈ ∅, IsOpen[inst✝¹] (U i)\nhsC : s ⊆ ⋃ i ∈ ∅, U i\n⊢ s = ⋃ i ∈ ∅, (fun x ↦ ∅) i",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case empty\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nι : Type u_3\ns : Set X\nhs : IsCompact s\nU : ι → Set X\nhU : ∀ i ∈ ∅, IsOpen[inst✝¹] (U i)\nhsC : s ⊆ ⋃ i ∈ ∅, U i\n⊢ s = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Clopen | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 23
} | {
"line": 61,
"column": 24
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsClopen s\nht : IsClopen t\n⊢ IsClopen (s ⇨ t)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"himp_eq",
"BooleanAlgebra.toCompl",
"Set.instUnion",
... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsClopen s\nht : IsClopen t\n⊢ IsClopen (t ∪ sᶜ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Clopen | {
"line": 107,
"column": 17
} | {
"line": 107,
"column": 54
} | {
"line": 107,
"column": 55
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns a b : Set X\nh : IsClopen s\ncover : s ⊆ a ∪ b\nha : IsOpen[inst✝] a\nhb : IsOpen[inst✝] b\nhab : Disjoint a b\nthis : IsClosed[inst✝] (s ∩ bᶜ)\nx : X\nhx₁ : x ∈ s\nhx₂ : x ∈ bᶜ\n⊢ x ∈ a",
"ppTerm": "?m.126",
"assigned": false,
"usedConstants": [],
... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns a b : Set X\nh : IsClopen s\ncover : s ⊆ a ∪ b\nha : IsOpen[inst✝] a\nhb : IsOpen[inst✝] b\nhab : Disjoint a b\nthis : IsClosed[inst✝] (s ∩ bᶜ)\nx : X\nhx₁ : x ∈ s\nhx₂ : x ∈ bᶜ\n⊢ x ∈ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 1068,
"column": 2
} | {
"line": 1069,
"column": 60
} | {
"line": 1070,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\nhTd : ∃ t ∈ T, Disjoint (𝓝 x) (𝓝 y)\n⊢ ⨅ t ∈ T, 𝓝 x ≤ ⨅ t ∈ T, 𝓝 y ∨ Disjoint (⨅ t ∈ T, 𝓝 x) (⨅ t ∈ T, 𝓝 y)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants"... | [
"case neg\nX : Type u_3\nT : Set (TopologicalSpace X)\nhT : ∀ t ∈ T, R1Space X\nx✝ : TopologicalSpace X := sInf T\nx y : X\nhTd : ∀ t ∈ T, ¬Disjoint (𝓝 x) (𝓝 y)\n⊢ ⨅ t ∈ T, 𝓝 x ≤ ⨅ t ∈ T, 𝓝 y ∨ Disjoint (⨅ t ∈ T, 𝓝 x) (⨅ t ∈ T, 𝓝 y)"
] | · rcases hTd with ⟨t, htT, htd⟩
exact .inr <| htd.mono (iInf₂_le t htT) (iInf₂_le t htT) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Separation.Basic | {
"line": 1100,
"column": 4
} | {
"line": 1100,
"column": 35
} | {
"line": 1101,
"column": 2
} | [
{
"pp": "X : Type u_3\nY : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : R1Space Y\nf : X → Y\nx : X\nK : Set X\ns : Set Y\nhf : Continuous[inst✝², inst✝¹] f\nhs : s ∈ 𝓝 (f x)\nhKc : IsCompact K\nhKx : K ∈ 𝓝 x\nhc : IsCompact (f '' K \\ interior s)\ny : Y\nhys : y ∉ interior s\nh... | [] | rwa [mem_interior_iff_mem_nhds] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.Order.OrderClosed | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 36
} | {
"line": 283,
"column": 37
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : ClosedIicTopology α\na b : α\nH : a < b\n⊢ Ioo a b ∈ 𝓝[<] b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Set.Ioi",
"congrArg",
"nhdsWithin... | [
"α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : ClosedIicTopology α\na b : α\nH : a < b\n⊢ Iio b ∩ Ioi a ∈ 𝓝[<] b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Basic | {
"line": 1172,
"column": 2
} | {
"line": 1173,
"column": 9
} | {
"line": 1173,
"column": 10
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : R1Space X\ninst✝ : WeaklyLocallyCompactSpace X\nx : X\n⊢ ∃ U, IsOpen[inst✝²] U ∧ x ∈ U ∧ IsCompact (closure[inst✝²] U)",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : R1Space X\ninst✝ : WeaklyLocallyCompactSpace X\nx : X\n⊢ ∃ U, IsOpen[inst✝²] U ∧ x ∈ U ∧ IsCompact (closure[inst✝²] U)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.OrderClosed | {
"line": 664,
"column": 2
} | {
"line": 664,
"column": 69
} | {
"line": 664,
"column": 70
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderClosedTopology α\nf g : β → α\ninst✝ : TopologicalSpace β\nhf : Continuous[inst✝, inst✝³] f\nhg : Continuous[inst✝, inst✝³] g\n⊢ frontier {b | f b < g b} ⊆ {b | f b = g b}",
"ppTerm": "?m.22",
"assigned":... | [
"α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderClosedTopology α\nf g : β → α\ninst✝ : TopologicalSpace β\nhf : Continuous[inst✝, inst✝³] f\nhg : Continuous[inst✝, inst✝³] g\n⊢ frontier {b | f b < g b} ⊆ {b | f b = g b}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.OrderClosed | {
"line": 672,
"column": 2
} | {
"line": 672,
"column": 40
} | {
"line": 673,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderClosedTopology α\nf g : β → α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : (x : β) → Decidable (f x ≤ g x)\nf' g' : β → γ\nhf : Continuous[inst✝², inst✝⁵] f\nhg : ... | [
"case refine_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderClosedTopology α\nf g : β → α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : (x : β) → Decidable (f x ≤ g x)\nf' g' : β → γ\nhf : Continuous[inst✝², inst✝⁵] f\nhg : Continuous[i... | · rwa [(isClosed_le hf hg).closure_eq] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Order.OrderClosed | {
"line": 673,
"column": 4
} | {
"line": 673,
"column": 22
} | {
"line": 674,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderClosedTopology α\nf g : β → α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : (x : β) → Decidable (f x ≤ g x)\nf' g' : β → γ\nhf : Continuous[inst✝², inst✝⁵] f\nhg : ... | [
"case refine_2\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderClosedTopology α\nf g : β → α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : (x : β) → Decidable (f x ≤ g x)\nf' g' : β → γ\nhf : Continuous[inst✝², inst✝⁵] f\nhg : Continuous[i... | simp only [not_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order.OrderClosed | {
"line": 714,
"column": 2
} | {
"line": 714,
"column": 29
} | {
"line": 714,
"column": 30
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf : β → α\nl : Filter β\na : α\nh : Tendsto f l (𝓝 a)\n⊢ Tendsto (fun i ↦ max a (f i)) l (𝓝 a)",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf : β → α\nl : Filter β\na : α\nh : Tendsto f l (𝓝 a)\n⊢ Tendsto (fun i ↦ max a (f i)) l (𝓝 a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 20
} | {
"line": 193,
"column": 20
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nι : Type u_3\ns : ι → Set α\nH : ∀ (i : ι), IsPreconnected (s i)\nK : ∀ (i j : ι), ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty) i j\n⊢ IsPreconnected (⋃ n, s n)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg... | [
"α : Type u\ninst✝ : TopologicalSpace α\nι : Type u_3\ns : ι → Set α\nH : ∀ (i : ι), IsPreconnected (s i)\nK : ∀ (i j : ι), ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty) i j\n⊢ IsPreconnected (⋃ x ∈ univ, s x)"
] | ← biUnion_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.