module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.NhdsSet | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 45
} | {
"line": 214,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Sort u_4\ns : ι → Set X\nP : X → Prop\n⊢ (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, s i), P x) ↔ ∀ (i : ι), ∀ᶠ (x : X) in 𝓝ˢ (s i), P x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instSupSet",
"nhdsSet_iUnion",
"congrArg",... | [] | simp only [nhdsSet_iUnion, eventually_iSup] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.NhdsSet | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 45
} | {
"line": 214,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Sort u_4\ns : ι → Set X\nP : X → Prop\n⊢ (∀ᶠ (x : X) in 𝓝ˢ (⋃ i, s i), P x) ↔ ∀ (i : ι), ∀ᶠ (x : X) in 𝓝ˢ (s i), P x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Filter.instSupSet",
"nhdsSet_iUnion",
"congrArg",... | [] | simp only [nhdsSet_iUnion, eventually_iSup] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 46
} | {
"line": 234,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\nm : ℤ\n⊢ toIocDiv hp (a + m • p) b = toIocDiv hp a b - m",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"instHSMul",
"toIocDiv_... | [
"α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\nm : ℤ\n⊢ b - (toIocDiv hp a b - m) • p ∈ Set.Ioc (a + m • p) (a + m • p + p)"
] | refine toIocDiv_eq_of_sub_zsmul_mem_Ioc _ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Maps.Basic | {
"line": 83,
"column": 73
} | {
"line": 83,
"column": 88
} | {
"line": 83,
"column": 88
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nf : X → Y\ng : Y → Z\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Z\nhf : Continuous[inst✝¹, inst✝²] f\nhg : Continuous[inst✝², inst✝] g\nhgf : IsInducing (g ∘ f)\n⊢ induced f inst✝² ≤ induced f (induced g inst✝)",
"pp... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\nf : X → Y\ng : Y → Z\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Z\nhf : Continuous[inst✝¹, inst✝²] f\nhg : Continuous[inst✝², inst✝] g\nhgf : IsInducing (g ∘ f)\n⊢ induced f inst✝² ≤ induced f inst✝²"
] | ← hg.le_induced | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Topology.Maps.Basic | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 41
} | {
"line": 494,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\n⊢ IsOpenMap f ↔ ∀ {u : Set X}, IsClosed[inst✝¹] u → IsClosed[inst✝] (kernImage f u)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.Surjective.forall",
... | [
"X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\n⊢ (∀ (x : Set X), IsOpen[inst✝¹] xᶜ → IsOpen[inst✝] (f '' xᶜ)) ↔\n ∀ {u : Set X}, IsClosed[inst✝¹] u → IsClosed[inst✝] (kernImage f u)"
] | rw [IsOpenMap, compl_surjective.forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order | {
"line": 932,
"column": 2
} | {
"line": 932,
"column": 30
} | {
"line": 933,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nt : TopologicalSpace β\nf : α → β\ns : Set β\nh : IsClosed[t] s\n⊢ IsClosed[induced f t] (f ⁻¹' s)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Topology.Order.0.isClosed_induced._simp_1_1",
"Compl.compl",
... | [
"α : Type u_1\nβ : Type u_2\nt : TopologicalSpace β\nf : α → β\ns : Set β\nh : IsClosed[t] s\n⊢ IsOpen[induced f t] (f ⁻¹' s)ᶜ"
] | simp_rw [← isOpen_compl_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Bornology.Basic | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 23
} | {
"line": 298,
"column": 24
} | [
{
"pp": "α : Type u_2\ninst✝ : Bornology α\n⊢ cobounded α = ⊥ ↔ BoundedSpace α",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"id",
"Bornology.IsBounded",
"Bot.bot",
"BoundedSpace",
"Iff",
"Bornology.is... | [
"α : Type u_2\ninst✝ : Bornology α\n⊢ cobounded α = ⊥ ↔ IsBounded univ"
] | ← isBounded_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Bases | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 21
} | {
"line": 140,
"column": 0
} | [
{
"pp": "case ne\nα : Type u\nt : TopologicalSpace α\na : α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\ni : Set α\nh1 : i ∈ b\nh2 : a ∈ i\n⊢ {s | a ∈ s ∧ s ∈ b}.Nonempty",
"ppTerm": "?ne",
"assigned": true,
"usedConstants": [
"setOf",
"Membership.mem",
"And",
"And... | [] | exact ⟨i, h2, h1⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Bases | {
"line": 130,
"column": 2
} | {
"line": 138,
"column": 21
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\na : α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\n⊢ s ∈ 𝓝 a ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Eq.mpr",
"Filter.mem_sets._simp_1",
... | [] | change s ∈ (𝓝 a).sets ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s
rw [hb.eq_generateFrom, nhds_generateFrom, biInf_sets_eq]
· simp [and_assoc, and_left_comm]
· rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩
let ⟨u, hu₁, hu₂, hu₃⟩ := hb.1 _ hs₂ _ ht₂ _ ⟨hs₁, ht₁⟩
exact ⟨u, ⟨hu₂, hu₁⟩, le_principal_iff.2 (hu₃.trans inter_subset_left),
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 130,
"column": 2
} | {
"line": 138,
"column": 21
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\na : α\ns : Set α\nb : Set (Set α)\nhb : IsTopologicalBasis b\n⊢ s ∈ 𝓝 a ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Eq.mpr",
"Filter.mem_sets._simp_1",
... | [] | change s ∈ (𝓝 a).sets ↔ ∃ t ∈ b, a ∈ t ∧ t ⊆ s
rw [hb.eq_generateFrom, nhds_generateFrom, biInf_sets_eq]
· simp [and_assoc, and_left_comm]
· rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩
let ⟨u, hu₁, hu₂, hu₃⟩ := hb.1 _ hs₂ _ ht₂ _ ⟨hs₁, ht₁⟩
exact ⟨u, ⟨hu₂, hu₁⟩, le_principal_iff.2 (hu₃.trans inter_subset_left),
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousOn | {
"line": 957,
"column": 4
} | {
"line": 957,
"column": 77
} | {
"line": 958,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nhf : ContinuousOn f s\nt u t' : Set β\nh : u ∈ 𝓝ˢ[t'] t\nv : Set β\nhv : IsOpen[inst✝] v ∧ t ⊆ v ∧ v ∩ t' ⊆ u\nw : Set α\nhw : IsOpen[inst✝¹] w ∧ f ⁻¹' v ∩ s = w ∩ s\n⊢ w ∩ (s ∩ f ... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set α\nhf : ContinuousOn f s\nt u t' : Set β\nh : u ∈ 𝓝ˢ[t'] t\nv : Set β\nhv : IsOpen[inst✝] v ∧ t ⊆ v ∧ v ∩ t' ⊆ u\nw : Set α\nhw : IsOpen[inst✝¹] w ∧ f ⁻¹' v ∩ s = w ∩ s\n⊢ s ∩ f ⁻¹' (v ∩ t') ⊆ f ... | rw [← inter_assoc, ← hw.2, inter_comm _ s, inter_assoc, ← preimage_inter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Constructions | {
"line": 1086,
"column": 2
} | {
"line": 1087,
"column": 86
} | {
"line": 1089,
"column": 0
} | [
{
"pp": "case intro.refine_2\nι : Type u_5\nA : ι → Type u_6\nT : (i : ι) → TopologicalSpace (A i)\ninst✝ : Finite ι\ns : Set ((a : ι) → A a)\nval✝ : Fintype ι\na : (a : ι) → A a\nx✝ : a ∈ s\n⊢ (∃ u, (∀ (a_1 : ι), IsOpen[T a_1] (u a_1) ∧ a a_1 ∈ u a_1) ∧ univ.pi u ⊆ s) →\n ∃ I t, (∀ (i : ι), ∃ t_1 ⊆ t i, IsO... | [] | · exact fun ⟨u, ⟨h1, _⟩⟩ =>
⟨Finset.univ, u, ⟨fun i => ⟨u i, ⟨rfl.subset, h1 i⟩⟩, by rwa [Finset.coe_univ]⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Bases | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 9
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type u_1\nt : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nT : Set (Set β)\nhf : IsInducing f\nh : IsTopologicalBasis T\na : α\ns : Set α\n⊢ s ∈ preimage f '' T ∧ a ∈ s ↔ s ∈ (preimage f ∘ fun t ↦ t) '' {i | i ∈ T ∧ f a ∈ i}",
"ppTerm": "?m.102",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.Filter.Ultrafilter.Basic | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 7
} | {
"line": 48,
"column": 0
} | [
{
"pp": "case e'_1\nα : Type u\nβ : Type v\nf : Ultrafilter α\nis : Set β\nP : β → α → Prop\nhis : is.Finite\n⊢ {x | ∃ a ∈ is, P a x} = ⋃ i ∈ is, P i",
"ppTerm": "?e'_1",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"congrArg",
"Iff.rfl",
"setOf",
"Se... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Bases | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 7
} | {
"line": 270,
"column": 0
} | [
{
"pp": "case e'_4\nβ : Type u_1\nt₁ t₂ : TopologicalSpace β\nB₁ B₂ : Set (Set β)\nh₁ : IsTopologicalBasis B₁\nh₂ : IsTopologicalBasis B₂\na : β\nx✝ : Set β\n⊢ x✝ ∈ image2 (fun x1 x2 ↦ x1 ∩ x2) B₁ B₂ ∧ a ∈ x✝ ↔\n x✝ ∈ (fun i ↦ i.1 ∩ i.2) '' {i | (i.1 ∈ B₁ ∧ a ∈ i.1) ∧ i.2 ∈ B₂ ∧ a ∈ i.2}",
"ppTerm": "?e'... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.LocallyFinite | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 86
} | {
"line": 59,
"column": 0
} | [
{
"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": true,
"usedConstants": [
"congrArg",
"Function.comp",
"Eq.mp",
"Functi... | [] | simpa only [comp_def, surjInv_eq hg] using hfg.comp_injective (injective_surjInv hg) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.LocallyFinite | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 86
} | {
"line": 59,
"column": 0
} | [
{
"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": true,
"usedConstants": [
"congrArg",
"Function.comp",
"Eq.mp",
"Functi... | [] | simpa only [comp_def, surjInv_eq hg] using hfg.comp_injective (injective_surjInv hg) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinite | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 86
} | {
"line": 59,
"column": 0
} | [
{
"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": true,
"usedConstants": [
"congrArg",
"Function.comp",
"Eq.mp",
"Functi... | [] | simpa only [comp_def, surjInv_eq hg] using hfg.comp_injective (injective_surjInv hg) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Bases | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 21
} | {
"line": 645,
"column": 2
} | [
{
"pp": "case inr\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ni : ι\nU : (i : ι) → Set (X i)\ns : Finset ι\nhU : ∀ i ∈ s, U i ∈ {U | IsOpen[inst✝ i] U}\nh : ((↑s).pi U).Nonempty\n⊢ IsOpen[inst✝ i] (eval i '' (↑s).pi U)",
"ppTerm": "?inr",
"assigned": true,
"usedConstan... | [
"case pos\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ni : ι\nU : (i : ι) → Set (X i)\ns : Finset ι\nhU : ∀ i ∈ s, U i ∈ {U | IsOpen[inst✝ i] U}\nh : ((↑s).pi U).Nonempty\nhi : i ∈ s\n⊢ IsOpen[inst✝ i] (eval i '' (↑s).pi U)",
"case neg\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι)... | by_cases hi : i ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 208,
"column": 68
} | {
"line": 210,
"column": 75
} | {
"line": 212,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : SigmaCompactSpace X\n⊢ ⋃ n, compactCovering X n = univ",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"compactCovering",
"congrArg",
"Set.univ",
"Preorder.toLE",
"compactCovering.eq_1"... | [] | by
rw [compactCovering, iUnion_accumulate]
exact (Classical.choose_spec SigmaCompactSpace.exists_compact_covering).2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Bases | {
"line": 1018,
"column": 2
} | {
"line": 1020,
"column": 39
} | {
"line": 1021,
"column": 2
} | [
{
"pp": "case refine_1\nα : 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' : IsTopologicalB... | [
"case refine_2\nα : 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 :... | · apply s'_count.biUnion
intro u hu
exact Finite.countable (f_fin u hu) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Compactness.Compact | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 44
} | {
"line": 325,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nι : Type v\nhι : Nonempty ι\nt : ι → Set X\nhtd : Directed (fun x1 x2 ↦ x1 ⊇ x2) t\nhtc : ∀ (i : ι), IsCompact (t i)\nhtcl : ∀ (i : ι), IsClosed[inst✝] (t i)\ni₀ : ι := hι.some\nhtn : ∀ (i : ι), (t i).Nonempty\ni j : ι\nhji₀ : t i₀ ⊇ t j\nhji : t i ⊇ t j\n⊢ (t i₀... | [] | exact (htn j).mono (subset_inter hji₀ hji) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Bases | {
"line": 1052,
"column": 2
} | {
"line": 1052,
"column": 7
} | {
"line": 1054,
"column": 0
} | [
{
"pp": "case e'_4\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (E i)\ns : (i : ι) → Set (Set (E i))\nhs : ∀ (i : ι), IsTopologicalBasis (s i)\na : (i : ι) × E i\nx✝ : Set ((i : ι) × E i)\n⊢ x✝ ∈ ⋃ i, (fun u ↦ Sigma.mk i '' u) '' s i ∧ a ∈ x✝ ↔\n x✝ ∈ (fun i ↦ Sigma.mk a.fst '' i) '' {... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Compactness.Compact | {
"line": 367,
"column": 25
} | {
"line": 367,
"column": 38
} | {
"line": 367,
"column": 38
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nf : Filter X\nhf : f.NeBot\nhfs : f ≤ 𝓟 s\nU : ↑s → Set X\nhU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen[inst✝] (U x)\nhUf : ∀ (x : ↑s), (U x)ᶜ ∈ f\nt : Finset ↑s\nht : s ⊆ ⋃ i ∈ t, U i\n⊢ s ⊆ (⋂ i ∈ t, (U i)ᶜ)ᶜ",
"ppTerm": "?m.104",
"assigned": true,
... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nf : Filter X\nhf : f.NeBot\nhfs : f ≤ 𝓟 s\nU : ↑s → Set X\nhU : ∀ (x : ↑s), ↑x ∈ U x ∧ IsOpen[inst✝] (U x)\nhUf : ∀ (x : ↑s), (U x)ᶜ ∈ f\nt : Finset ↑s\nht : s ⊆ ⋃ i ∈ t, U i\n⊢ s ⊆ ⋃ i ∈ t, (U i)ᶜᶜ"
] | compl_iInter₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactness.Compact | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 13
} | {
"line": 555,
"column": 2
} | [
{
"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\n⊢ ∀ (f : Ultrafilter X), s ∈ f → ∃ x ∈ s, ↑f ≤ 𝓝 x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Ultraf... | [
"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\n⊢ ∃ x ∈ s, ↑F ≤ 𝓝 x"
] | intro F hsF | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Compactness.Compact | {
"line": 573,
"column": 42
} | {
"line": 573,
"column": 47
} | {
"line": 573,
"column": 47
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ (ι : Type u) (U : ι → ↑S), s ⊆ ⋃ i, ↑(U i) → ∃ J, J.Finite ∧ s ⊆ ⋃ i ∈ J, ↑(U i)\nP : Set (Set X)\nhP : P ⊆ S\nhs : s ⊆ ⋃₀ P\nJ : Set ↑P\nhJ : J.Finite\ncover : s ⊆ ⋃ i ∈ J, ↑⟨↑i, ⋯⟩\n⊢ s ⊆ ⋃₀ ((fun x ↦ ↑x) ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Compactness.Compact | {
"line": 573,
"column": 42
} | {
"line": 573,
"column": 47
} | {
"line": 573,
"column": 47
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ (ι : Type u) (U : ι → ↑S), s ⊆ ⋃ i, ↑(U i) → ∃ J, J.Finite ∧ s ⊆ ⋃ i ∈ J, ↑(U i)\nP : Set (Set X)\nhP : P ⊆ S\nhs : s ⊆ ⋃₀ P\nJ : Set ↑P\nhJ : J.Finite\ncover : s ⊆ ⋃ i ∈ J, ↑⟨↑i, ⋯⟩\n⊢ s ⊆ ⋃₀ ((fun x ↦ ↑x) ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 573,
"column": 42
} | {
"line": 573,
"column": 47
} | {
"line": 573,
"column": 47
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ (ι : Type u) (U : ι → ↑S), s ⊆ ⋃ i, ↑(U i) → ∃ J, J.Finite ∧ s ⊆ ⋃ i ∈ J, ↑(U i)\nP : Set (Set X)\nhP : P ⊆ S\nhs : s ⊆ ⋃₀ P\nJ : Set ↑P\nhJ : J.Finite\ncover : s ⊆ ⋃ i ∈ J, ↑⟨↑i, ⋯⟩\n⊢ s ⊆ ⋃₀ ((fun x ↦ ↑x) ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Piecewise | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 31
} | {
"line": 156,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns s' t : Set α\nht : s ∩ frontier t = s' ∩ frontier t\n⊢ t.ite s s' ∩ closure[inst✝] tᶜ = s' ∩ closure[inst✝] tᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"frontier_compl",
"frontier",
"Set.ite",
... | [] | rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]
rwa [frontier_compl, eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Piecewise | {
"line": 153,
"column": 2
} | {
"line": 154,
"column": 31
} | {
"line": 156,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns s' t : Set α\nht : s ∩ frontier t = s' ∩ frontier t\n⊢ t.ite s s' ∩ closure[inst✝] tᶜ = s' ∩ closure[inst✝] tᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"frontier_compl",
"frontier",
"Set.ite",
... | [] | rw [← ite_compl, ite_inter_closure_eq_of_inter_frontier_eq]
rwa [frontier_compl, eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 1176,
"column": 4
} | {
"line": 1176,
"column": 47
} | {
"line": 1177,
"column": 2
} | [
{
"pp": "case mp\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : Set ((i : ι) → X i)\nhs : IsClosed[topologicalSpace] s\nH : IsCompact s\n⊢ ∀ (i : ι), IsCompact (eval i '' s)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Pi.topologicalSpace",
"Fu... | [] | exact fun i ↦ H.image <| continuous_apply i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.Compact | {
"line": 1176,
"column": 4
} | {
"line": 1176,
"column": 47
} | {
"line": 1177,
"column": 2
} | [
{
"pp": "case mp\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : Set ((i : ι) → X i)\nhs : IsClosed[topologicalSpace] s\nH : IsCompact s\n⊢ ∀ (i : ι), IsCompact (eval i '' s)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Pi.topologicalSpace",
"Fu... | [] | exact fun i ↦ H.image <| continuous_apply i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 1176,
"column": 4
} | {
"line": 1176,
"column": 47
} | {
"line": 1177,
"column": 2
} | [
{
"pp": "case mp\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : Set ((i : ι) → X i)\nhs : IsClosed[topologicalSpace] s\nH : IsCompact s\n⊢ ∀ (i : ι), IsCompact (eval i '' s)",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Pi.topologicalSpace",
"Fu... | [] | exact fun i ↦ H.image <| continuous_apply i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Irreducible | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 46
} | {
"line": 289,
"column": 4
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsIrreducible s\nU : Finset (Set X)\nhu : ∀ u ∈ U, IsOpen[inst✝] u\nhU : ∀ u ∈ U, (s ∩ u).Nonempty\n⊢ (s ∩ ⋂₀ ↑U).Nonempty",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"IsPreirreducible",
"E... | [] | induction U using Finset.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Irreducible | {
"line": 358,
"column": 13
} | {
"line": 363,
"column": 54
} | {
"line": 365,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nZ : Set X\nhZ : Z ∈ irreducibleComponents X\nS : Set (Set X)\nhS : S.Finite\nhSα : S ⊆ irreducibleComponents X\nhZS : Z ⊆ ⋃₀ S\n⊢ Z ∈ S",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"isClosed_of_mem_irreducibleComponents",
"S... | [] | by
obtain ⟨W, hWS, hZW⟩ := isIrreducible_iff_sUnion_isClosed.mp hZ.1 hS.toFinset
(fun W hW ↦ isClosed_of_mem_irreducibleComponents W (hSα (hS.mem_toFinset.mp hW)))
(hS.coe_toFinset.symm ▸ hZS)
rw [hS.mem_toFinset] at hWS
rwa [Set.Subset.antisymm hZW (hZ.2 (hSα hWS).1 hZW)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.DiscreteSubset | {
"line": 342,
"column": 68
} | {
"line": 342,
"column": 73
} | {
"line": 344,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns t : Set X\nh : t.Subsingleton\nz : X\nhz : z ∈ t\n⊢ univ \\ t ∩ (t \\ s) = ∅",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"eq_false",
"iff_false",
"Set.mem_empty_if... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.DiscreteSubset | {
"line": 342,
"column": 68
} | {
"line": 342,
"column": 73
} | {
"line": 344,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns t : Set X\nh : t.Subsingleton\nz : X\nhz : z ∈ t\n⊢ univ \\ t ∩ (t \\ s) = ∅",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"eq_false",
"iff_false",
"Set.mem_empty_if... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 342,
"column": 68
} | {
"line": 342,
"column": 73
} | {
"line": 344,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns t : Set X\nh : t.Subsingleton\nz : X\nhz : z ∈ t\n⊢ univ \\ t ∩ (t \\ s) = ∅",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Set.ext",
"False",
"eq_false",
"iff_false",
"Set.mem_empty_if... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.DiscreteSubset | {
"line": 354,
"column": 75
} | {
"line": 354,
"column": 80
} | {
"line": 354,
"column": 80
} | [
{
"pp": "X : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns : Set X\nx z : X\nhz : z ∈ s\n⊢ univ \\ {x} ∩ (s \\ {x}ᶜ) = ∅",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"eq_false",
"iff_false",
"Set.mem_empty_i... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.DiscreteSubset | {
"line": 354,
"column": 75
} | {
"line": 354,
"column": 80
} | {
"line": 354,
"column": 80
} | [
{
"pp": "X : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns : Set X\nx z : X\nhz : z ∈ s\n⊢ univ \\ {x} ∩ (s \\ {x}ᶜ) = ∅",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"eq_false",
"iff_false",
"Set.mem_empty_i... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 354,
"column": 75
} | {
"line": 354,
"column": 80
} | {
"line": 354,
"column": 80
} | [
{
"pp": "X : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns : Set X\nx z : X\nhz : z ∈ s\n⊢ univ \\ {x} ∩ (s \\ {x}ᶜ) = ∅",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"eq_false",
"iff_false",
"Set.mem_empty_i... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.DiscreteSubset | {
"line": 365,
"column": 43
} | {
"line": 365,
"column": 48
} | {
"line": 366,
"column": 4
} | [
{
"pp": "X : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\ns t✝ : Set X\nh : t✝.Finite\nτ : X\nt : Set X\nhτ : τ ∉ t\nh₁t : t.Finite\nh₂t : tᶜ ∈ codiscreteWithin s\n⊢ (insert τ t)ᶜ = {τ}ᶜ ∩ tᶜ",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Compactness.Compact | {
"line": 1244,
"column": 4
} | {
"line": 1244,
"column": 35
} | {
"line": 1245,
"column": 2
} | [
{
"pp": "case refine_2\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nS : Set X\nhS : IsClosed[inst✝¹] S\nhne : S.Nonempty\nopens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}\nU : Set X\nh : Maximal (fun x ↦ x ∈ opens) U\nUc : Sᶜ ⊆ U\nUo : IsOpen[inst✝¹] U\nUcne : Uᶜ.Nonempt... | [] | · simp only [compl_compl, V'ne] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.SuccPred.Relation | {
"line": 41,
"column": 42
} | {
"line": 43,
"column": 49
} | {
"line": 45,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : IsSuccArchimedean α\nr : α → α → Prop\nn m : α\nh : ∀ i ∈ Ico m n, r (succ i) i\nhmn : m ≤ n\n⊢ ReflTransGen r n m",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Relation.reflTransGen_swap... | [] | by
rw [← reflTransGen_swap]
exact reflTransGen_of_succ_of_le (swap r) h hmn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Separation.Hausdorff | {
"line": 201,
"column": 32
} | {
"line": 207,
"column": 54
} | {
"line": 209,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T2Space X\ns t : Set X\nhs : IsCompact s\nht : IsCompact t\n⊢ 𝓝ˢ (s ∩ t) = 𝓝ˢ s ⊓ 𝓝ˢ t",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"iSup₂_le",
"Filter.instSupSet",
"Iff.of_... | [] | by
refine le_antisymm (nhdsSet_inter_le _ _) ?_
simp_rw [hs.nhdsSet_inf_eq_biSup, ht.inf_nhdsSet_eq_biSup, nhdsSet, sSup_image]
refine iSup₂_le fun x hxs ↦ iSup₂_le fun y hyt ↦ ?_
rcases eq_or_ne x y with (rfl | hne)
· exact le_iSup₂_of_le x ⟨hxs, hyt⟩ (inf_idem _).le
· exact (disjoint_nhds_nhds.mpr hne).eq... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Clopen | {
"line": 141,
"column": 47
} | {
"line": 142,
"column": 60
} | {
"line": 144,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nU : Set X\n⊢ Continuous[inst✝, _] U.boolIndicator ↔ IsClopen U",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Continuous",
"congrArg",
"Iff.rfl",
"instTopologicalSpaceBool",
"continuous_bool_rng... | [] | by
rw [continuous_bool_rng true, preimage_boolIndicator_true] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Separation.Basic | {
"line": 1100,
"column": 9
} | {
"line": 1100,
"column": 34
} | {
"line": 1100,
"column": 34
} | [
{
"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... | [
"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\nhxy : Insepar... | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Separation.Basic | {
"line": 1125,
"column": 81
} | {
"line": 1128,
"column": 99
} | {
"line": 1130,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\n⊢ coclosedCompact X = cocompact X",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"subset_closure",
"Compl.compl",
"PartialOrder.toPreorder",
"Pr... | [] | by
refine le_antisymm ?_ cocompact_le_coclosedCompact
rw [hasBasis_coclosedCompact.le_basis_iff hasBasis_cocompact]
exact fun K hK ↦ ⟨closure K, ⟨isClosed_closure, hK.closure⟩, compl_subset_compl.2 subset_closure⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.Basic | {
"line": 256,
"column": 7
} | {
"line": 256,
"column": 44
} | {
"line": 256,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder β\ninst✝¹ : SuccOrder β\ninst✝ : IsSuccArchimedean β\ns : β → Set α\nt : Set β\nht : t.OrdConnected\nH : ∀ n ∈ t, IsPreconnected (s n)\nK : ∀ n ∈ t, succ n ∈ t → (s n ∩ s (succ n)).Nonempty\nh1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ... | [] | by rw [inter_comm]; exact h3 hj hi hk | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.Basic | {
"line": 304,
"column": 35
} | {
"line": 304,
"column": 41
} | {
"line": 304,
"column": 42
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nH : IsPreconnected s\nf : α → β\nhf : ContinuousOn f s\nu v : Set β\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nx : α\nxs : x ∈ s\nxu : f x ∈ u\ny : α\nys : y ∈ s\nyv : f y ∈ v\nu' : Set α\nhu' : IsOpen[inst✝¹]... | [
"α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set α\nH : IsPreconnected s\nf : α → β\nhf : ContinuousOn f s\nu v : Set β\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nx : α\nxs : x ∈ s\nxu : f x ∈ u\ny : α\nys : y ∈ s\nyv : f y ∈ v\nu' : Set α\nhu' : IsOpen[inst✝¹] u'\nu'_eq :... | u'_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.Clopen | {
"line": 117,
"column": 47
} | {
"line": 117,
"column": 84
} | {
"line": 118,
"column": 6
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : Set α\nhs : IsClopen s\nh : ¬(s = ∅ ∨ s = univ)\n⊢ s.Nonempty ∧ sᶜ.Nonempty",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.univ",
"Eq.... | [] | simpa [nonempty_iff_ne_empty] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Connected.Clopen | {
"line": 117,
"column": 47
} | {
"line": 117,
"column": 84
} | {
"line": 118,
"column": 6
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : Set α\nhs : IsClopen s\nh : ¬(s = ∅ ∨ s = univ)\n⊢ s.Nonempty ∧ sᶜ.Nonempty",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.univ",
"Eq.... | [] | simpa [nonempty_iff_ne_empty] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Clopen | {
"line": 117,
"column": 47
} | {
"line": 117,
"column": 84
} | {
"line": 118,
"column": 6
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : Set α\nhs : IsClopen s\nh : ¬(s = ∅ ∨ s = univ)\n⊢ s.Nonempty ∧ sᶜ.Nonempty",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.univ",
"Eq.... | [] | simpa [nonempty_iff_ne_empty] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 59
} | {
"line": 86,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns t : Set X\nhs : IsLindelof s\nht : IsClosed[inst✝¹] t\nf : Filter X\nhnf : f.NeBot\ninst✝ : CountableInterFilter f\nhstf : f ≤ 𝓟 s ∧ f ≤ 𝓟 t\n⊢ ∃ x ∈ s ∩ t, ClusterPt x f",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Filter.ins... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ns t : Set X\nhs : IsLindelof s\nht : IsClosed[inst✝¹] t\nf : Filter X\nhnf : f.NeBot\ninst✝ : CountableInterFilter f\nhstf : f ≤ 𝓟 s ∧ f ≤ 𝓟 t\nx : X\nhsx : x ∈ s\nhx : ClusterPt x f\n⊢ ∃ x ∈ s ∩ t, ClusterPt x f"
] | obtain ⟨x, hsx, hx⟩ : ∃ x ∈ s, ClusterPt x f := hs hstf.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Connected.Clopen | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 46
} | {
"line": 282,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nx✝ : s.Nonempty ∧ ∀ (u v : Set α), IsOpen[inst✝] u → IsOpen[inst✝] v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nU : Finset (Set α)\nhU : ∀ (u v : Set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v\nhUo : ∀ u ∈ U, IsOpen[inst✝] u\... | [] | induction U using Finset.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Separation.Regular | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 85
} | {
"line": 276,
"column": 2
} | [
{
"pp": "case inl\nX : Type u_1\ninst✝² : TopologicalSpace X\ns t : Set X\ninst✝¹ : LindelofSpace X\ninst✝ : RegularSpace X\ns_cl : IsClosed[inst✝²] s\nt_cl : IsClosed[inst✝²] t\nst_dis : Disjoint s t\nempty_X : IsEmpty X\n⊢ _root_.HasSeparatingCover ∅ t",
"ppTerm": "?inl",
"assigned": true,
"usedCo... | [] | exact hasSeparatingCovers_iff_separatedNhds.mpr (SeparatedNhds.empty_left t) |>.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.Lindelof | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 53
} | {
"line": 157,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nι : Type v\nhs : IsLindelof s\nU : ι → Set X\nhUo : ∀ (i : ι), IsOpen[inst✝] (U i)\nhsU : s ⊆ ⋃ i, U i\nhmono : ∀ ⦃s t : Set X⦄, s ⊆ t → (∃ r, r.Countable ∧ t ⊆ ⋃ i ∈ r, U i) → ∃ r, r.Countable ∧ s ⊆ ⋃ i ∈ r, U i\nhcountable_union :\n ∀ (S : Set (Set ... | [] | exact hs.induction_on hmono hcountable_union h_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Separation.Regular | {
"line": 298,
"column": 96
} | {
"line": 303,
"column": 50
} | {
"line": 305,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\nx y : X\nh : ¬Inseparable x y\n⊢ ∃ U₁ ∈ 𝓝 x,\n ∃ V₁ ∈ 𝓝 x,\n ∃ U₂ ∈ 𝓝 y,\n ∃ V₂ ∈ 𝓝 y,\n IsClosed[inst✝¹] V₁ ∧\n IsClosed[inst✝¹] V₂ ∧ IsOpen[inst✝¹] U₁ ∧ IsOpen[inst✝¹] U₂ ∧ V₁ ⊆ U₁ ∧ V₂ ⊆ U₂ ∧ Di... | [] | by
rcases r1_separation h with ⟨U₁, U₂, U₁_op, U₂_op, x_in, y_in, H⟩
rcases exists_mem_nhds_isClosed_subset (U₁_op.mem_nhds x_in) with ⟨V₁, V₁_in, V₁_closed, h₁⟩
rcases exists_mem_nhds_isClosed_subset (U₂_op.mem_nhds y_in) with ⟨V₂, V₂_in, V₂_closed, h₂⟩
exact ⟨U₁, mem_of_superset V₁_in h₁, V₁, V₁_in, U₂, mem_o... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.DenseEmbedding | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 31
} | {
"line": 86,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ninst✝ : T2Space β\ndi : IsDenseInducing i\nhd : Dense (range i)ᶜ\ns : Set α\nhs : IsCompact s\nx : α\nhx : x ∈ interior s\n⊢ False",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ni : α → β\ninst✝ : T2Space β\ndi : IsDenseInducing i\nhd : Dense (range i)ᶜ\ns : Set α\nhs : IsCompact s\nx : α\nhx : s ∈ 𝓝 x\n⊢ False"
] | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Separation.Regular | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 64
} | {
"line": 422,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : T3Space X\nx y : X\nhne : x ≠ y\nthis : x ∉ closure[inst✝¹] {y} ∨ y ∉ closure[inst✝¹] {x}\n⊢ Disjoint (𝓝 x) (𝓝 y)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"congrArg",
"Filter.instCompleteLatticeF... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : T3Space X\nx y : X\nhne : x ≠ y\nthis : Disjoint (𝓝 x) (𝓝 y) ∨ Disjoint (𝓝 y) (𝓝 x)\n⊢ Disjoint (𝓝 x) (𝓝 y)"
] | simp only [← disjoint_nhds_nhdsSet, nhdsSet_singleton] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Compactness.Lindelof | {
"line": 398,
"column": 6
} | {
"line": 402,
"column": 84
} | {
"line": 403,
"column": 4
} | [
{
"pp": "case mp.refine_1\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsLindelof (b i)\nY : Type u\nf' : Y → ι\nh₁ : IsLindelof (⋃ i, (b ∘ f') i)\nh₂ : IsOpen[inst✝] (⋃ i, (b ∘ f') i)\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\nt : Set Y... | [] | refine Set.Subset.trans ht.2 ?_
simp only [Set.iUnion_subset_iff]
intro i hi
rw [← Set.iUnion_subtype (fun x : ι => x ∈ t.image f') fun i => b i.1]
exact Set.subset_iUnion (fun i : t.image f' => b i) ⟨_, mem_image_of_mem _ hi⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Lindelof | {
"line": 398,
"column": 6
} | {
"line": 402,
"column": 84
} | {
"line": 403,
"column": 4
} | [
{
"pp": "case mp.refine_1\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsLindelof (b i)\nY : Type u\nf' : Y → ι\nh₁ : IsLindelof (⋃ i, (b ∘ f') i)\nh₂ : IsOpen[inst✝] (⋃ i, (b ∘ f') i)\nhf' : ∀ (i : Y), b (f' i) = (b ∘ f') i\nt : Set Y... | [] | refine Set.Subset.trans ht.2 ?_
simp only [Set.iUnion_subset_iff]
intro i hi
rw [← Set.iUnion_subtype (fun x : ι => x ∈ t.image f') fun i => b i.1]
exact Set.subset_iUnion (fun i : t.image f' => b i) ⟨_, mem_image_of_mem _ hi⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 501,
"column": 2
} | {
"line": 503,
"column": 20
} | {
"line": 505,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : LindelofSpace X\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\n⊢ ∃ t, t.Countable ∧ ⋃ x ∈ t, U x = univ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.univ",
"Membership.mem",
"Exists",
"CompleteLattice.to... | [] | obtain ⟨t, tc, -, s⟩ := IsLindelof.elim_nhds_subcover isLindelof_univ U fun x _ => hU x
use t, tc
apply top_unique s | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Lindelof | {
"line": 501,
"column": 2
} | {
"line": 503,
"column": 20
} | {
"line": 505,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : LindelofSpace X\nU : X → Set X\nhU : ∀ (x : X), U x ∈ 𝓝 x\n⊢ ∃ t, t.Countable ∧ ⋃ x ∈ t, U x = univ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.univ",
"Membership.mem",
"Exists",
"CompleteLattice.to... | [] | obtain ⟨t, tc, -, s⟩ := IsLindelof.elim_nhds_subcover isLindelof_univ U fun x _ => hU x
use t, tc
apply top_unique s | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.Clopen | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 40
} | {
"line": 486,
"column": 2
} | [
{
"pp": "case ht'\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : IsCoinducing f\nt : β\n⊢ IsConnected (connectedComponent t)",
"ppTerm": "?ht'",
"assigned": true,
"usedConstants": [
"isCo... | [
"case connected_fibers\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nconnected_fibers : ∀ (t : β), IsConnected (f ⁻¹' {t})\nhcl : IsCoinducing f\nt : β\n⊢ ∀ (t : β), IsConnected (f ⁻¹' {t})"
] | · exact isConnected_connectedComponent | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Connected.Clopen | {
"line": 643,
"column": 6
} | {
"line": 643,
"column": 60
} | {
"line": 644,
"column": 6
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\nU : Fin (n + 1) → Set α\nh₁ : ∀ (i : Fin (n + 1)), IsClopen (U i)\nh₂ : ∀ (i : Fin (n + 1)), (U i).Nonempty\nh₃ : Pairwise (Disjoint on U)\nh₄ : ⋃ i, U i = univ\n⊢ ∃ i, ¬IsConnected (U i)",
"p... | [
"α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\nU : Fin (n + 1) → Set α\nh₁ : ∀ (i : Fin (n + 1)), IsClopen (U i)\nh₂ : ∀ (i : Fin (n + 1)), (U i).Nonempty\nh₃ : Pairwise (Disjoint on U)\nh₄ : ⋃ i, U i = univ\n⊢ ¬∀ (x : Fin (n + 1)), ConnectedSpace ↑(U x)"... | simp_rw [isConnected_iff_connectedSpace, ← not_forall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 240,
"column": 6
} | {
"line": 240,
"column": 51
} | {
"line": 240,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\np : Filter ι\nh : TendstoUniformly F f p\ng : γ → α\n⊢ TendstoUniformly (fun n ↦ F n ∘ g) (f ∘ g) p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"tendstoUniformly_iff... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\np : Filter ι\nh : TendstoUniformlyOnFilter F f p ⊤\ng : γ → α\n⊢ TendstoUniformlyOnFilter (fun n ↦ F n ∘ g) (f ∘ g) p ⊤"
] | tendstoUniformly_iff_tendstoUniformlyOnFilter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 51
} | {
"line": 319,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\np : Filter ι\nc : β\n⊢ Tendsto (↿F) (p ×ˢ ⊤) (𝓝 c) ↔ TendstoUniformly F (fun x ↦ c) p",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"tendstoUniformly_iff_tendstoUniformlyOnFilter",
"Eq.... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\np : Filter ι\nc : β\n⊢ Tendsto (↿F) (p ×ˢ ⊤) (𝓝 c) ↔ TendstoUniformlyOnFilter F (fun x ↦ c) p ⊤"
] | tendstoUniformly_iff_tendstoUniformlyOnFilter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Separation | {
"line": 345,
"column": 2
} | {
"line": 349,
"column": 6
} | {
"line": 351,
"column": 0
} | [
{
"pp": "α : Type u_1\nu : UniformSpace α\ninst✝ : IndiscreteTopology α\n⊢ u = ⊤",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Set.ext",
"UniformSpace",
"Eq.mpr",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"uniformity",
"Membership.mem... | [] | refine UniformSpace.ext ?_
rw [top_uniformity, ← Filter.ker_eq_univ]
ext x
rw [← inseparable_iff_ker_uniformity]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Separation | {
"line": 345,
"column": 2
} | {
"line": 349,
"column": 6
} | {
"line": 351,
"column": 0
} | [
{
"pp": "α : Type u_1\nu : UniformSpace α\ninst✝ : IndiscreteTopology α\n⊢ u = ⊤",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Set.ext",
"UniformSpace",
"Eq.mpr",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"uniformity",
"Membership.mem... | [] | refine UniformSpace.ext ?_
rw [top_uniformity, ← Filter.ker_eq_univ]
ext x
rw [← inseparable_iff_ker_uniformity]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 349,
"column": 29
} | {
"line": 365,
"column": 39
} | {
"line": 367,
"column": 0
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nι : Sort u_1\ns : ι → Set α\nhs : ∀ (i : ι), IsComplete (s i)\nU : SetRel α α\nhU : U ∈ 𝓤 α\nhd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j\n⊢ IsComplete (⋃ i, s i)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Filte... | [] | by
set S := ⋃ i, s i
intro l hl hls
rw [le_principal_iff] at hls
obtain ⟨hl_ne, hl'⟩ := cauchy_iff.1 hl
obtain ⟨t, htS, htl, htU⟩ : ∃ t, t ⊆ S ∧ t ∈ l ∧ t ×ˢ t ⊆ U := by
rcases hl' U hU with ⟨t, htl, htU⟩
refine ⟨t ∩ S, inter_subset_right, inter_mem htl hls, Subset.trans ?_ htU⟩
gcongr <;> apply i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 96
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nhf : IsUniformInducing f\nF : Filter α\n⊢ Cauchy (map f F) ↔ Cauchy F",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"IsUniformInducing.comap_uniformity",
"Cauchy",
"SProd.sprod"... | [] | simp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 96
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nhf : IsUniformInducing f\nF : Filter α\n⊢ Cauchy (map f F) ↔ Cauchy F",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"IsUniformInducing.comap_uniformity",
"Cauchy",
"SProd.sprod"... | [] | simp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 96
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nhf : IsUniformInducing f\nF : Filter α\n⊢ Cauchy (map f F) ↔ Cauchy F",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"IsUniformInducing.comap_uniformity",
"Cauchy",
"SProd.sprod"... | [] | simp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap, ← hf.comap_uniformity] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Basic | {
"line": 547,
"column": 2
} | {
"line": 551,
"column": 40
} | {
"line": 553,
"column": 0
} | [
{
"pp": "α : Type ua\nβ : Type ub\nγ : Type uc\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set α\ng : β → γ\nt : Set β\nhg : UniformContinuousOn g t\nhf : UniformContinuousOn f s\nhst : MapsTo f s t\n⊢ UniformContinuousOn (g ∘ f) s",
"ppTerm": "?m.18",
"assi... | [] | change Tendsto ((fun x ↦ (g x.1, g x.2)) ∘ (fun x ↦ (f x.1, f x.2))) (𝓤 α ⊓ 𝓟 (s ×ˢ s)) (𝓤 γ)
apply Tendsto.comp hg
refine tendsto_inf.2 ⟨hf, tendsto_inf_right ?_⟩
simp only [tendsto_principal, mem_prod, eventually_principal, and_imp, Prod.forall]
exact fun a b ha hb ↦ ⟨hst ha, hst hb⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Basic | {
"line": 547,
"column": 2
} | {
"line": 551,
"column": 40
} | {
"line": 553,
"column": 0
} | [
{
"pp": "α : Type ua\nβ : Type ub\nγ : Type uc\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ns : Set α\ng : β → γ\nt : Set β\nhg : UniformContinuousOn g t\nhf : UniformContinuousOn f s\nhst : MapsTo f s t\n⊢ UniformContinuousOn (g ∘ f) s",
"ppTerm": "?m.18",
"assi... | [] | change Tendsto ((fun x ↦ (g x.1, g x.2)) ∘ (fun x ↦ (f x.1, f x.2))) (𝓤 α ⊓ 𝓟 (s ×ˢ s)) (𝓤 γ)
apply Tendsto.comp hg
refine tendsto_inf.2 ⟨hf, tendsto_inf_right ?_⟩
simp only [tendsto_principal, mem_prod, eventually_principal, and_imp, Prod.forall]
exact fun a b ha hb ↦ ⟨hst ha, hst hb⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 69
} | {
"line": 402,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\ninst✝² : UniformSpace γ\nf : α → β\ninst✝¹ : CompleteSpace α\ninst✝ : CompleteSpace β\n⊢ CompleteSpace (α ⊕ β)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsComplete",
... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\ninst✝² : UniformSpace γ\nf : α → β\ninst✝¹ : CompleteSpace α\ninst✝ : CompleteSpace β\n⊢ IsComplete (range Sum.inl ∪ range Sum.inr)"
] | rw [completeSpace_iff_isComplete_univ, ← range_inl_union_range_inr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Support | {
"line": 188,
"column": 35
} | {
"line": 188,
"column": 59
} | {
"line": 188,
"column": 60
} | [
{
"pp": "α : Type u_2\nβ : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : One β\nf : α → β\nx : α\n⊢ (∃ x_1 ∈ 𝓝 x, Disjoint x_1 (mulSupport f)) ↔ f =ᶠ[𝓝 x] 1",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"PartialOrd... | [
"α : Type u_2\nβ : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : One β\nf : α → β\nx : α\n⊢ (∃ x_1 ∈ 𝓝 x, EqOn f 1 x_1) ↔ f =ᶠ[𝓝 x] 1"
] | disjoint_mulSupport_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Support | {
"line": 313,
"column": 2
} | {
"line": 313,
"column": 7
} | {
"line": 315,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : One β\nf : α → β\ninst✝ : TopologicalSpace β\nK : Set β\nh'f : HasCompactMulSupport f\nhf : Continuous[inst✝², inst✝] f\nhk : IsClosed[inst✝] K\nh'k : 1 ∉ K\nx : α\nhx : x ∈ f ⁻¹' K\n⊢ x ∈ mulSupport f",
"ppTerm": "?m.41",
"assig... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 922,
"column": 2
} | {
"line": 942,
"column": 64
} | {
"line": 944,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝¹ : (𝓤 α).IsCountablyGenerated\ninst✝ : SeparableSpace α\n⊢ SecondCountableTopology α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"comp_symm_of_uniformity",
"... | [] | rcases exists_countable_dense α with ⟨s, hsc, hsd⟩
obtain
⟨t : ℕ → SetRel α α, hto : ∀ i : ℕ, t i ∈ (𝓤 α).sets ∧ IsOpen (t i) ∧ (t i).IsSymm,
h_basis : (𝓤 α).HasAntitoneBasis t⟩ :=
(@uniformity_hasBasis_open_symmetric α _).exists_antitone_subbasis
choose ht_mem hto hts using hto
refine ⟨⟨⋃ x ∈ s, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 922,
"column": 2
} | {
"line": 942,
"column": 64
} | {
"line": 944,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝¹ : (𝓤 α).IsCountablyGenerated\ninst✝ : SeparableSpace α\n⊢ SecondCountableTopology α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"comp_symm_of_uniformity",
"... | [] | rcases exists_countable_dense α with ⟨s, hsc, hsd⟩
obtain
⟨t : ℕ → SetRel α α, hto : ∀ i : ℕ, t i ∈ (𝓤 α).sets ∧ IsOpen (t i) ∧ (t i).IsSymm,
h_basis : (𝓤 α).HasAntitoneBasis t⟩ :=
(@uniformity_hasBasis_open_symmetric α _).exists_antitone_subbasis
choose ht_mem hto hts using hto
refine ⟨⟨⋃ x ∈ s, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.HeineCantor | {
"line": 60,
"column": 58
} | {
"line": 70,
"column": 59
} | {
"line": 72,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nr : Set (β × β)\ns : Set α\nhs : IsCompact s\nf : α → β\nhf : ∀ a ∈ s, ContinuousAt f a\nhr : r ∈ 𝓤 β\n⊢ {x | x.1 ∈ s → (f x.1, f x.2) ∈ r} ∈ 𝓤 α",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"F... | [] | by
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
choose U hU T hT hb using fun a ha =>
exists_mem_nhds_ball_subset_of_mem_nhds ((hf a ha).preimage_mem_nhds <| mem_nhds_left _ ht)
obtain ⟨fs, hsU⟩ := hs.elim_nhds_subcover' U hU
apply mem_of_superset ((biInter_finset_mem fs).2 fun a _ => hT ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 74
} | {
"line": 295,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nhf : TendstoLocallyUniformlyOn F f p s\na : α\nha : a ∈ s\n⊢ Tendsto (fun i ↦ F i a) p (𝓝 (f a))",
"ppTerm": "?m.14",
"assigned": true,
"usedCon... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nhf : TendstoLocallyUniformlyOn F f p s\na : α\nha : a ∈ s\n⊢ 𝓟 {a} ≤ 𝓝[s] a"
] | refine ((tendstoLocallyUniformlyOn_iff_filter.mp hf) a ha).tendsto_at ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Order.Group.MinMax | {
"line": 63,
"column": 78
} | {
"line": 64,
"column": 61
} | {
"line": 66,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ max (a / c) (b / c) = max a b / c",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Lattice.toSemilatticeSup",
"instHDiv",
... | [] | by
simpa only [div_eq_mul_inv] using max_mul_mul_right a b c⁻¹ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 732,
"column": 2
} | {
"line": 734,
"column": 20
} | {
"line": 736,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nh : s ∈ 𝔖\n⊢ UniformContinuous (⇑UniformFun.ofFun ∘ s.restrict ∘ ⇑(toFun 𝔖))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"UniformSpace",
"Eq.mpr",
"i... | [] | change _ ≤ _
simp only [map_le_iff_le_comap, iInf_uniformity]
exact iInf₂_le s h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 732,
"column": 2
} | {
"line": 734,
"column": 20
} | {
"line": 736,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Set α\ninst✝ : UniformSpace β\n𝔖 : Set (Set α)\nh : s ∈ 𝔖\n⊢ UniformContinuous (⇑UniformFun.ofFun ∘ s.restrict ∘ ⇑(toFun 𝔖))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"UniformSpace",
"Eq.mpr",
"i... | [] | change _ ≤ _
simp only [map_le_iff_le_comap, iInf_uniformity]
exact iInf₂_le s h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1042,
"column": 4
} | {
"line": 1055,
"column": 71
} | {
"line": 1057,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns : Set α\np : Filter ι\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : CompleteSpace β\nh✝ : Nonempty β\n⊢ CompleteSpace (α →ᵤ[𝔖] β)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Filter.instMembership"... | [] | refine ⟨fun {F} hF ↦ ?_⟩
have := hF.1
have : ∀ x ∈ ⋃₀ 𝔖, ∃ y : β, Tendsto (toFun 𝔖 · x) F (𝓝 y) := fun x hx ↦
CompleteSpace.complete (hF.map (uniformContinuous_eval_of_mem_sUnion _ _ hx))
choose! g hg using this
use ofFun 𝔖 g
simp_rw [UniformOnFun.nhds_eq_of_basis _ _ uniformity_hasBasis_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1042,
"column": 4
} | {
"line": 1055,
"column": 71
} | {
"line": 1057,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\ns : Set α\np : Filter ι\ninst✝¹ : UniformSpace β\n𝔖 : Set (Set α)\ninst✝ : CompleteSpace β\nh✝ : Nonempty β\n⊢ CompleteSpace (α →ᵤ[𝔖] β)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Filter.instMembership"... | [] | refine ⟨fun {F} hF ↦ ?_⟩
have := hF.1
have : ∀ x ∈ ⋃₀ 𝔖, ∃ y : β, Tendsto (toFun 𝔖 · x) F (𝓝 y) := fun x hx ↦
CompleteSpace.complete (hF.map (uniformContinuous_eval_of_mem_sUnion _ _ hx))
choose! g hg using this
use ofFun 𝔖 g
simp_rw [UniformOnFun.nhds_eq_of_basis _ _ uniformity_hasBasis_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 169,
"column": 19
} | {
"line": 169,
"column": 24
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : TopologicalSpace δ\nf g : C(α, β)\nX₁ : Type u_5\nX₂ : Type u_6\nY₁ : Type u_7\nY₂ : Type u_8\ninst✝³ : TopologicalSpace X₁\ninst✝² : TopologicalSpace ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 169,
"column": 19
} | {
"line": 169,
"column": 24
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : TopologicalSpace δ\nf g : C(α, β)\nX₁ : Type u_5\nX₂ : Type u_6\nY₁ : Type u_7\nY₂ : Type u_8\ninst✝³ : TopologicalSpace X₁\ninst✝² : TopologicalSpace ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 169,
"column": 19
} | {
"line": 169,
"column": 24
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : TopologicalSpace δ\nf g : C(α, β)\nX₁ : Type u_5\nX₂ : Type u_6\nY₁ : Type u_7\nY₂ : Type u_8\ninst✝³ : TopologicalSpace X₁\ninst✝² : TopologicalSpace ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 170,
"column": 20
} | {
"line": 170,
"column": 25
} | {
"line": 172,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : TopologicalSpace δ\nf g : C(α, β)\nX₁ : Type u_5\nX₂ : Type u_6\nY₁ : Type u_7\nY₂ : Type u_8\ninst✝³ : TopologicalSpace X₁\ninst✝² : TopologicalSpace ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 170,
"column": 20
} | {
"line": 170,
"column": 25
} | {
"line": 172,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : TopologicalSpace δ\nf g : C(α, β)\nX₁ : Type u_5\nX₂ : Type u_6\nY₁ : Type u_7\nY₂ : Type u_8\ninst✝³ : TopologicalSpace X₁\ninst✝² : TopologicalSpace ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 170,
"column": 20
} | {
"line": 170,
"column": 25
} | {
"line": 172,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : TopologicalSpace δ\nf g : C(α, β)\nX₁ : Type u_5\nX₂ : Type u_6\nY₁ : Type u_7\nY₂ : Type u_8\ninst✝³ : TopologicalSpace X₁\ninst✝² : TopologicalSpace ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 70
} | {
"line": 227,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nG : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : ContinuousConstSMul G α\ninst✝ : TopologicalSpace β\nf : β → α\nc : G\n⊢ (Continuous[inst✝, inst✝⁴] fun x ↦ c • f x) ↔ Continuous[inst✝, inst✝⁴] f",
"ppTerm": "?m.17",
"assi... | [] | simp only [continuous_iff_continuousAt, continuousAt_const_smul_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 70
} | {
"line": 227,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nG : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : ContinuousConstSMul G α\ninst✝ : TopologicalSpace β\nf : β → α\nc : G\n⊢ (Continuous[inst✝, inst✝⁴] fun x ↦ c • f x) ↔ Continuous[inst✝, inst✝⁴] f",
"ppTerm": "?m.17",
"assi... | [] | simp only [continuous_iff_continuousAt, continuousAt_const_smul_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 70
} | {
"line": 227,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nG : Type u_4\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Group G\ninst✝² : MulAction G α\ninst✝¹ : ContinuousConstSMul G α\ninst✝ : TopologicalSpace β\nf : β → α\nc : G\n⊢ (Continuous[inst✝, inst✝⁴] fun x ↦ c • f x) ↔ Continuous[inst✝, inst✝⁴] f",
"ppTerm": "?m.17",
"assi... | [] | simp only [continuous_iff_continuousAt, continuousAt_const_smul_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 58
} | {
"line": 140,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nhf : Continuous[inst✝², inst✝¹] f\nhg : Continuous[inst✝¹, inst✝] g\nhgf : IsProperMap (g ∘ f)\nf_surj : Surjective f\nℱ : Filter Y\nz : Z\nh : MapCluste... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nhf : Continuous[inst✝², inst✝¹] f\nhg : Continuous[inst✝¹, inst✝] g\nhgf : IsProperMap (g ∘ f)\nf_surj : Surjective f\nℱ : Filter Y\nx : X\nhx : ClusterPt x (comap f... | rcases hgf.clusterPt_of_mapClusterPt h with ⟨x, rfl, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Group.Quotient | {
"line": 83,
"column": 39
} | {
"line": 83,
"column": 86
} | {
"line": 84,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nN : Subgroup G\n⊢ T1Space (G ⧸ N) ↔ IsClosed[inst✝²] (mk ⁻¹' {↑1})",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toIn... | [
"G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\nN : Subgroup G\n⊢ IsClosed[instTopologicalSpace N] {↑1} ↔ IsClosed[inst✝²] (mk ⁻¹' {↑1})"
] | MulAction.IsPretransitive.t1Space_iff G (mk 1), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.AtTopBot.Group | {
"line": 129,
"column": 61
} | {
"line": 130,
"column": 63
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : PartialOrder G\ninst✝ : IsOrderedMonoid G\nl : Filter α\nf : G → α\n⊢ Tendsto (fun x ↦ f x⁻¹) atBot l ↔ Tendsto f atTop l",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"DivisionCommMonoid.toDivisionMonoid",
"... | [] | by
simp [← Function.comp_def, Tendsto, ← map_map, map_inv_atBot] | [anonymous] | Lean.Parser.Term.byTactic |
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