module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Nat.Count | {
"line": 52,
"column": 81
} | {
"line": 54,
"column": 5
} | {
"line": 56,
"column": 0
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\n⊢ count p n = #({x ∈ range n | p x})",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.countP",
"congrArg",
"id",
"List.range",
"Finset.range",
"List.countP_eq_length_filter",
... | [] | by
rw [count, List.countP_eq_length_filter]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Periodic | {
"line": 37,
"column": 67
} | {
"line": 37,
"column": 75
} | {
"line": 37,
"column": 75
} | [
{
"pp": "α : Type u_1\nf : ℕ → α\na : ℕ\nhf : Periodic f a\nn : ℕ\n| f (n % a + (n / a) • a)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"instHSMul",
"instHDiv",
"Function.Periodic.nsmul",
"AddMonoid.toAddSemigroup",
"congrArg",
"AddMonoid.toNSMul",
... | [
"α : Type u_1\nf : ℕ → α\na : ℕ\nhf : Periodic f a\nn : ℕ\n| f (n % a)"
] | hf.nsmul | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Data.Nat.Totient | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 25
} | {
"line": 225,
"column": 2
} | [
{
"pp": "case a\nhp : 0 < 1\nh : 1 = 0\n⊢ False",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"instOfNatNat",
"AddZero.toZero",
"Nat.instNeZeroSucc",
"Nat",
"Nat.instAddComm... | [] | exact one_ne_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.Lift | {
"line": 335,
"column": 2
} | {
"line": 335,
"column": 22
} | {
"line": 336,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf g : Filter α\ns : Set α → Set β\nhs : ∀ (t₁ t₂ : Set α), s (t₁ ∩ t₂) = s t₁ ∩ s t₂\n⊢ ⨅ i, (bif i then f else g).lift' s = ⨅ b, bif b then f.lift' s else g.lift' s",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"cond",
"Filter.instInfSet"... | [
"α : Type u_1\nβ : Type u_2\nf g : Filter α\ns : Set α → Set β\nhs : ∀ (t₁ t₂ : Set α), s (t₁ ∩ t₂) = s t₁ ∩ s t₂\n⊢ ∀ (i : Bool), (bif i then f else g).lift' s = bif i then f.lift' s else g.lift' s"
] | refine iInf_congr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Continuous | {
"line": 136,
"column": 74
} | {
"line": 136,
"column": 100
} | {
"line": 138,
"column": 0
} | [
{
"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\nx : X\ny : Y\ng : Y → Z\nhg : ContinuousAt g y\nhf : ContinuousAt f x\nhy : f x = y\n⊢ ContinuousAt (g ∘ f) x",
"ppTerm": "?m.17",
"assigned": true,
"us... | [] | subst hy; exact hg.comp hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Continuous | {
"line": 136,
"column": 74
} | {
"line": 136,
"column": 100
} | {
"line": 138,
"column": 0
} | [
{
"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\nx : X\ny : Y\ng : Y → Z\nhg : ContinuousAt g y\nhf : ContinuousAt f x\nhy : f x = y\n⊢ ContinuousAt (g ∘ f) x",
"ppTerm": "?m.17",
"assigned": true,
"us... | [] | subst hy; exact hg.comp hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Closure | {
"line": 444,
"column": 13
} | {
"line": 444,
"column": 15
} | {
"line": 445,
"column": 4
} | [
{
"pp": "case mp\nX : Type u\ninst✝ : TopologicalSpace X\nx : X\nhd : Dense {x}ᶜ\n⊢ ¬IsOpen {x}",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Set.instSingletonSet",
"IsOpen",
"Singleton.singleton",
"Set"
],
"usedFVars": [
"X",
"inst✝",
"x"... | [
"case mp\nX : Type u\ninst✝ : TopologicalSpace X\nx : X\nhd : Dense {x}ᶜ\nho : IsOpen {x}\n⊢ False"
] | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Closure | {
"line": 570,
"column": 23
} | {
"line": 570,
"column": 40
} | {
"line": 570,
"column": 40
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) ⊆ interior s\nC : interior (frontier s) ⊆ frontier s\n⊢ interior (frontier s) ⊆ s \\ interior s",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa [A] using C | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Closure | {
"line": 570,
"column": 23
} | {
"line": 570,
"column": 40
} | {
"line": 570,
"column": 40
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) ⊆ interior s\nC : interior (frontier s) ⊆ frontier s\n⊢ interior (frontier s) ⊆ s \\ interior s",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa [A] using C | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Closure | {
"line": 570,
"column": 23
} | {
"line": 570,
"column": 40
} | {
"line": 570,
"column": 40
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nh : IsClosed s\nA : frontier s = s \\ interior s\nB : interior (frontier s) ⊆ interior s\nC : interior (frontier s) ⊆ frontier s\n⊢ interior (frontier s) ⊆ s \\ interior s",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.... | [] | simpa [A] using C | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order | {
"line": 866,
"column": 2
} | {
"line": 866,
"column": 54
} | {
"line": 868,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nι : Sort u_2\nt₁ : TopologicalSpace α\nt₂ : ι → TopologicalSpace β\n⊢ Continuous[t₁, iInf t₂] f ↔ ∀ (i : ι), Continuous[t₁, t₂ i] f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"iInf",
"Continuous",
"congrArg",
"PartialO... | [] | simp only [continuous_iff_coinduced_le, le_iInf_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order | {
"line": 866,
"column": 2
} | {
"line": 866,
"column": 54
} | {
"line": 868,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nι : Sort u_2\nt₁ : TopologicalSpace α\nt₂ : ι → TopologicalSpace β\n⊢ Continuous[t₁, iInf t₂] f ↔ ∀ (i : ι), Continuous[t₁, t₂ i] f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"iInf",
"Continuous",
"congrArg",
"PartialO... | [] | simp only [continuous_iff_coinduced_le, le_iInf_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order | {
"line": 866,
"column": 2
} | {
"line": 866,
"column": 54
} | {
"line": 868,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nι : Sort u_2\nt₁ : TopologicalSpace α\nt₂ : ι → TopologicalSpace β\n⊢ Continuous[t₁, iInf t₂] f ↔ ∀ (i : ι), Continuous[t₁, t₂ i] f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"iInf",
"Continuous",
"congrArg",
"PartialO... | [] | simp only [continuous_iff_coinduced_le, le_iInf_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Basic | {
"line": 477,
"column": 2
} | {
"line": 477,
"column": 49
} | {
"line": 478,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nhf : IsOpenMap f\nx : X\nl : Filter Y\nh : ClusterPt (f x) (𝓟 {f x}ᶜ ⊓ l)\n⊢ comap f (𝓟 {f x}ᶜ ⊓ l) ≤ 𝓟 {x}ᶜ ⊓ comap f l",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nhf : IsOpenMap f\nx : X\nl : Filter Y\nh : ClusterPt (f x) (𝓟 {f x}ᶜ ⊓ l)\n⊢ 𝓟 (f ⁻¹' {f x})ᶜ ⊓ comap f l ≤ 𝓟 {x}ᶜ ⊓ comap f l"
] | rw [comap_inf, comap_principal, preimage_compl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Maps.Basic | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 15
} | {
"line": 505,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nh : ∀ (x : X) (l : Filter Y), ClusterPt (f x) l → ClusterPt x (comap f l)\nx : X\ns : Set X\nhs : s ∈ 𝓝 x\n⊢ f '' s ∈ 𝓝 (f x)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Fi... | [
"X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nh : ∀ (x : X) (l : Filter Y), ClusterPt (f x) l → ClusterPt x (comap f l)\nx : X\ns : Set X\nhs : f '' s ∉ 𝓝 (f x)\n⊢ s ∉ 𝓝 x"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Topology.Maps.Basic | {
"line": 779,
"column": 69
} | {
"line": 779,
"column": 96
} | {
"line": 779,
"column": 96
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nf : X → Y\ng : Y → Z\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nhg : IsOpenEmbedding g\n⊢ (∀ (x : X), 𝓝 (f x) ≤ map f (𝓝 x)) ↔ ∀ (x : X), map g (𝓝 (f x)) ≤ map g (map f (𝓝 x))",
"ppTerm": "?m.20",
"assigne... | [] | map_le_map_iff hg.injective | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Constructions.SumProd | {
"line": 546,
"column": 25
} | {
"line": 549,
"column": 57
} | {
"line": 551,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y → Z\nx : X\ny : Y\ns : Set X\nt : Set Y\nu : Set Z\nhf₁ : ∀ (x : X), Continuous[inst✝¹, inst✝] (f x)\nhf₂ : ∀ (y : Y), Continuous[inst✝², inst✝] fun x ↦ f x y\nhx : x ∈ ... | [] | by
rw [← isClosed_closure.closure_eq]
apply map_mem_closure (hf₁ x) hy fun b hb ↦ ?_
apply map_mem_closure (hf₂ b) hx fun a ha ↦ h a ha b hb | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousOn | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 54
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns t : Set α\nx : α\nh : t ∈ 𝓝[s] x\n⊢ ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"ContinuousWithinAt",
"congr... | [] | simp [ContinuousWithinAt, nhdsWithin_restrict'' s h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.ContinuousOn | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 54
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns t : Set α\nx : α\nh : t ∈ 𝓝[s] x\n⊢ ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"ContinuousWithinAt",
"congr... | [] | simp [ContinuousWithinAt, nhdsWithin_restrict'' s h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousOn | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 54
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns t : Set α\nx : α\nh : t ∈ 𝓝[s] x\n⊢ ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"ContinuousWithinAt",
"congr... | [] | simp [ContinuousWithinAt, nhdsWithin_restrict'' s h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Constructions | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 33
} | {
"line": 337,
"column": 33
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nα : Type u_5\nβ : Type u_6\nf : α → X\ng : β → Y\nla : Filter α\nlb : Filter β\nx : X\ny : Y\nhf : MapClusterPt x la f\nhg : MapClusterPt y lb g\n⊢ MapClusterPt (x, y) (la.curry lb) (Prod.map f g)",
"ppTerm": "?m.18",
... | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nα : Type u_5\nβ : Type u_6\nf : α → X\ng : β → Y\nla : Filter α\nlb : Filter β\nx : X\ny : Y\nhf : ∀ s ∈ 𝓝 x, ∃ᶠ (a : α) in la, f a ∈ s\nhg : ∀ s ∈ 𝓝 y, ∃ᶠ (a : β) in lb, g a ∈ s\n⊢ MapClusterPt (x, y) (la.curry lb) (Prod.map f g)"
... | mapClusterPt_iff_frequently | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Bornology.Basic | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 91
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nx✝ : Bornology α\ninst✝ : Bornology β\nf : α → β\nh : comap f (cobounded β) ≤ cobounded α\ns : Set α\nhs : IsBounded s\nt : Set β\nht : t ∈ cobounded β\nhts : s ⊆ f ⁻¹' tᶜ\n⊢ IsBounded (f '' s)",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Comp... | [] | exact (IsCobounded.compl ht).subset ((image_mono hts).trans <| image_preimage_subset _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.Ultrafilter.Defs | {
"line": 251,
"column": 10
} | {
"line": 251,
"column": 19
} | {
"line": 251,
"column": 20
} | [
{
"pp": "α : Type u\nβ : Type v\nm : α → β\na : α\ninj : Injective m\nlarge : range m ∈ pure (m a)\n⊢ 𝓟 (m ⁻¹' {m a}) = ↑(pure a)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"Ultrafilter.coe_pure",
"congrArg",
"Set.instSingletonSet"... | [
"α : Type u\nβ : Type v\nm : α → β\na : α\ninj : Injective m\nlarge : range m ∈ pure (m a)\n⊢ 𝓟 (m ⁻¹' {m a}) = pure a"
] | coe_pure, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Bases | {
"line": 312,
"column": 24
} | {
"line": 312,
"column": 54
} | {
"line": 312,
"column": 54
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsEmpty α\n⊢ ⋃₀ ∅ = univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Set.sUnion",
"Set.univ_eq_empty_iff",
"IsEmpty",
"congr",
"True",
"eq_self",
... | [] | simp [Set.univ_eq_empty_iff.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Bases | {
"line": 312,
"column": 24
} | {
"line": 312,
"column": 54
} | {
"line": 312,
"column": 54
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsEmpty α\n⊢ ⋃₀ ∅ = univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Set.sUnion",
"Set.univ_eq_empty_iff",
"IsEmpty",
"congr",
"True",
"eq_self",
... | [] | simp [Set.univ_eq_empty_iff.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 312,
"column": 24
} | {
"line": 312,
"column": 54
} | {
"line": 312,
"column": 54
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsEmpty α\n⊢ ⋃₀ ∅ = univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Set.sUnion",
"Set.univ_eq_empty_iff",
"IsEmpty",
"congr",
"True",
"eq_self",
... | [] | simp [Set.univ_eq_empty_iff.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Bases | {
"line": 317,
"column": 24
} | {
"line": 317,
"column": 54
} | {
"line": 317,
"column": 54
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsEmpty α\n⊢ ⋃₀ {∅} = univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Set.sUnion",
"Set.instSingletonSet",
"Set.univ_eq_empty_iff",
"Set.sUnion_singleton",
... | [] | simp [Set.univ_eq_empty_iff.2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Bases | {
"line": 317,
"column": 24
} | {
"line": 317,
"column": 54
} | {
"line": 317,
"column": 54
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsEmpty α\n⊢ ⋃₀ {∅} = univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Set.sUnion",
"Set.instSingletonSet",
"Set.univ_eq_empty_iff",
"Set.sUnion_singleton",
... | [] | simp [Set.univ_eq_empty_iff.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bases | {
"line": 317,
"column": 24
} | {
"line": 317,
"column": 54
} | {
"line": 317,
"column": 54
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nh : IsEmpty α\n⊢ ⋃₀ {∅} = univ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"Set.univ",
"Set.sUnion",
"Set.instSingletonSet",
"Set.univ_eq_empty_iff",
"Set.sUnion_singleton",
... | [] | simp [Set.univ_eq_empty_iff.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.LocallyFinite | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 73
} | {
"line": 120,
"column": 2
} | [
{
"pp": "ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\nx : X\ns : Set X\nhsx : s ∈ 𝓝 x\nhsf : {i | (f i ∩ s).Nonempty}.Finite\n⊢ ∃ t ∈ 𝓝 x, {i | ((fun i ↦ closure[inst✝] (f i)) i ∩ t).Nonempty}.Finite",
"ppTerm": "?m.29",
"assigned": true,
"usedConsta... | [
"ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\nx : X\ns : Set X\nhsx : s ∈ 𝓝 x\nhsf : {i | (f i ∩ s).Nonempty}.Finite\ni : ι\nhi : i ∈ {i | ((fun i ↦ closure[inst✝] (f i)) i ∩ interior s).Nonempty}\n⊢ i ∈ {i | (f i ∩ s).Nonempty}"
] | refine ⟨interior s, interior_mem_nhds.2 hsx, hsf.subset fun i hi => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.LocallyFinite | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 91
} | {
"line": 127,
"column": 0
} | [
{
"pp": "ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nh : LocallyFinite f\nx : X\n⊢ x ∈ closure[inst✝] (⋃ i, f i) ↔ x ∈ ⋃ i, closure[inst✝] (f i)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Filter.instSupSet",
"congrArg",
"iSup",
"Filt... | [] | simp only [mem_closure_iff_nhdsWithin_neBot, h.nhdsWithin_iUnion, iSup_neBot, mem_iUnion] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.LocallyFinite | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 44
} | {
"line": 158,
"column": 2
} | [
{
"pp": "X : Type u_4\ninst✝ : TopologicalSpace X\nπ : X → Sort u_6\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), {i | ((fun n ↦ {x | f (n + 1) x ≠ f n x}) i ∩ U x).Nonempty}.Finite\nN : X → ℕ\nhN : ∀ (x : X), ∀ n ≥ N x + 1, ∀ y ∈ U x, f n y = f (N x + 1) y\nx : X\n⊢ ∀ a ∈ ... | [
"X : Type u_4\ninst✝ : TopologicalSpace X\nπ : X → Sort u_6\nf : ℕ → (x : X) → π x\nU : X → Set X\nhUx : ∀ (x : X), U x ∈ 𝓝 x\nhU : ∀ (x : X), {i | ((fun n ↦ {x | f (n + 1) x ≠ f n x}) i ∩ U x).Nonempty}.Finite\nN : X → ℕ\nhN : ∀ (x : X), ∀ n ≥ N x + 1, ∀ y ∈ U x, f n y = f (N x + 1) y\nx : X\nn : ℕ\ny : X\nhn : N... | rintro ⟨n, y⟩ ⟨hn : N x < n, hy : y ∈ U x⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 209,
"column": 23
} | {
"line": 209,
"column": 40
} | {
"line": 209,
"column": 40
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : SigmaCompactSpace X\n⊢ ⋃ n, accumulate ⋯.choose n = univ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Preorder.toLE",
"id",
"instLENat",
"Set.iUnion_a... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : SigmaCompactSpace X\n⊢ ⋃ x, ⋯.choose x = univ"
] | iUnion_accumulate | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactness.Compact | {
"line": 252,
"column": 2
} | {
"line": 253,
"column": 74
} | {
"line": 254,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nl : Filter X\nhs : IsCompact s\nH : ∀ x ∈ s, Disjoint (𝓝 x) l\nU : X → Set X\nhUl : ∀ x ∈ s, (U x)ᶜ ∈ l\nhxU : ∀ x ∈ s, x ∈ U x\nhUo : ∀ x ∈ s, IsOpen[inst✝] (U x)\nt : Finset X\nhts : ∀ x ∈ t, x ∈ s\nhst : s ⊆ ⋃ x ∈ t, U x\n⊢ Disjoint (𝓝ˢ s) l",
... | [
"X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nl : Filter X\nhs : IsCompact s\nH : ∀ x ∈ s, Disjoint (𝓝 x) l\nU : X → Set X\nhUl : ∀ x ∈ s, (U x)ᶜ ∈ l\nhxU : ∀ x ∈ s, x ∈ U x\nhUo : ∀ x ∈ s, IsOpen[inst✝] (U x)\nt : Finset X\nhts : ∀ x ∈ t, x ∈ s\nhst : s ⊆ ⋃ x ∈ t, U x\n⊢ (⋃ x ∈ t, U x)ᶜ ∈ l"
] | refine (hasBasis_nhdsSet _).disjoint_iff_left.2
⟨⋃ x ∈ t, U x, ⟨isOpen_biUnion fun x hx => hUo x (hts x hx), hst⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Bases | {
"line": 1022,
"column": 6
} | {
"line": 1022,
"column": 47
} | {
"line": 1023,
"column": 6
} | [
{
"pp": "case refine_2.a\nα : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\nt' : Set (Set α) := ⋯\nthis : IsTopologicalBasis t'\ns' : Set (Set α)\ns't' : s' ⊆ t'\ns'_count : s'.Countable\nhs' : IsTopologicalBasis s'\nf : Set α → Set (Set α)\nf_f... | [
"case refine_2.a\nα : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\nt' : Set (Set α) := ⋯\nthis : IsTopologicalBasis t'\ns' : Set (Set α)\ns't' : s' ⊆ t'\ns'_count : s'.Countable\nhs' : IsTopologicalBasis s'\nf : Set α → Set (Set α)\nf_fin : ∀ u ∈ s... | apply le_generateFrom_iff_subset_isOpen.2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Bases | {
"line": 1029,
"column": 6
} | {
"line": 1029,
"column": 47
} | {
"line": 1030,
"column": 6
} | [
{
"pp": "case a\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'... | [
"case a\nα : Type u_1\nts : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nt : Set (Set α)\nht : ts = generateFrom t\nt' : Set (Set α) := ⋯\nthis : IsTopologicalBasis t'\ns' : Set (Set α)\ns't' : s' ⊆ t'\ns'_count : s'.Countable\nhs' : IsTopologicalBasis s'\nf : Set α → Set (Set α)\nf_fin : ∀ u ∈ s', (f u).... | apply le_generateFrom_iff_subset_isOpen.2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Compactness.Compact | {
"line": 879,
"column": 2
} | {
"line": 879,
"column": 15
} | {
"line": 879,
"column": 15
} | [
{
"pp": "X : Type u\nY : Type v\nι : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\nf : X → Y\ninst✝ : NoncompactSpace X\ni✝ : Set X\nhs : IsCompact i✝\n⊢ i✝ᶜ.Nonempty",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Compl.compl",
"Set.instComp... | [
"X : Type u\nY : Type v\nι : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns t : Set X\nf : X → Y\ninst✝ : NoncompactSpace X\ni✝ : Set X\nhs : ¬i✝ᶜ.Nonempty\n⊢ ¬IsCompact i✝"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Topology.Compactness.Compact | {
"line": 899,
"column": 30
} | {
"line": 899,
"column": 96
} | {
"line": 902,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nι : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns t : Set X\nf : X → Y\n⊢ (cocompact ℤ).NeBot",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"congrArg",
"Filter.NeBot",
"instDiscreteTopologyInt",
"instTopologic... | [] | by simp only [Filter.cocompact_eq_cofinite, Filter.cofinite_neBot] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.DiscreteSubset | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 68
} | {
"line": 279,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\n⊢ (∀ x ∈ U, (U \\ s)ᶜ ∈ 𝓝[≠] x) ↔ ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \\ s) = ∅",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Compl.compl",
"nhdsWithin",
"Membe... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nh : ∀ z ∈ U, ∃ t ∈ 𝓝[≠] z, t ∩ (U \\ s) = ∅\nz : X\nhz : z ∈ U\n⊢ (U \\ s)ᶜ ∈ 𝓝[≠] z"
] | refine ⟨fun h z hz ↦ ⟨(U \ s)ᶜ, h z hz, by simp⟩, fun h z hz ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Separation.Hausdorff | {
"line": 235,
"column": 29
} | {
"line": 244,
"column": 39
} | {
"line": 246,
"column": 0
} | [
{
"pp": "X : Type u_3\nY : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : T2Space Y\nf : X → Y\ns : Set X\ninj : InjOn f s\nsc : IsCompact s\nfc : ∀ x ∈ s, ContinuousAt f x\nloc : ∀ x ∈ s, ∃ u ∈ 𝓝 x, InjOn f u\n⊢ ∃ t ∈ 𝓝ˢ s, InjOn f t",
"ppTerm": "?m.39",
"assigned": true,... | [] | by
have : ∀ x ∈ s ×ˢ s, ∀ᶠ y in 𝓝 x, f y.1 = f y.2 → y.1 = y.2 := fun (x, y) ⟨hx, hy⟩ ↦ by
rcases eq_or_ne x y with rfl | hne
· rcases loc x hx with ⟨u, hu, hf⟩
exact Filter.mem_of_superset (prod_mem_nhds hu hu) <| forall_prod_set.2 hf
· suffices ∀ᶠ z in 𝓝 (x, y), f z.1 ≠ f z.2 from this.mono fun ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.OrderClosed | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 17
} | {
"line": 219,
"column": 17
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : ClosedIicTopology α\na : α\n⊢ IsOpen[inst✝²] (Ioi a)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
... | [
"α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : ClosedIicTopology α\na : α\n⊢ IsOpen[inst✝²] (Iic a)ᶜ"
] | ← compl_Iic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.Basic | {
"line": 132,
"column": 2
} | {
"line": 135,
"column": 21
} | {
"line": 137,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns t : Set α\nH : (s ∩ t).Nonempty\nHs : IsConnected s\nHt : IsConnected t\n⊢ IsConnected (s ∪ t)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"IsPreconnected.union",
"IsConnected",
"Membership.mem",
"Set.mem_of_mem... | [] | rcases H with ⟨x, hx⟩
refine ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, ?_⟩
exact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx)
Ht.isPreconnected | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Basic | {
"line": 132,
"column": 2
} | {
"line": 135,
"column": 21
} | {
"line": 137,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns t : Set α\nH : (s ∩ t).Nonempty\nHs : IsConnected s\nHt : IsConnected t\n⊢ IsConnected (s ∪ t)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"IsPreconnected.union",
"IsConnected",
"Membership.mem",
"Set.mem_of_mem... | [] | rcases H with ⟨x, hx⟩
refine ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, ?_⟩
exact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx)
Ht.isPreconnected | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.LocallyConnected | {
"line": 155,
"column": 82
} | {
"line": 159,
"column": 70
} | {
"line": 161,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nι : Type u_1\nX : ι → Type u_2\ninst✝¹ : TopologicalSpace α\ns t u v : Set α\ninst✝ : LocallyConnectedSpace α\n⊢ DiscreteTopology (ConnectedComponents α)",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"congrArg",
"ConnectedCompo... | [] | by
refine discreteTopology_iff_isOpen_singleton.mpr fun c ↦ ?_
obtain ⟨x, rfl⟩ := ConnectedComponents.surjective_coe c
simp [← ConnectedComponents.isQuotientMap_coe.isOpen_preimage,
connectedComponents_preimage_singleton, isOpen_connectedComponent] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.Basic | {
"line": 332,
"column": 6
} | {
"line": 332,
"column": 52
} | {
"line": 333,
"column": 6
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ (∀ (t t' : Set α),\n IsClosed[inst✝] t →\n IsClosed[inst✝] t' → s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty) →\n IsPreconnected s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [
"α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh :\n ∀ (t t' : Set α),\n IsClosed[inst✝] t → IsClosed[inst✝] t' → s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty\nu v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhuv : s ⊆ u ∪ v\nx : α\nxs : x ∈ s\nxu : x ∈ u\ny : α\nys :... | rintro h u v hu hv huv ⟨x, xs, xu⟩ ⟨y, ys, yv⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Connected.Basic | {
"line": 467,
"column": 4
} | {
"line": 467,
"column": 30
} | {
"line": 468,
"column": 4
} | [
{
"pp": "case empty\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : (i : ι) → Set (X i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → X i)\nuo : IsOpen[Pi.topologicalSpace] u\nvo : IsOpen[Pi.topologicalSpace] v\nhsuv : univ.pi s ⊆ u ∪ v\nf : (i : ι) → X i\nhfs : f ∈ ... | [
"case empty\nι : Type u_1\nX : ι → Type u_2\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : (i : ι) → Set (X i)\nhs : ∀ (i : ι), IsPreconnected (s i)\nu v : Set ((i : ι) → X i)\nuo : IsOpen[Pi.topologicalSpace] u\nvo : IsOpen[Pi.topologicalSpace] v\nhsuv : univ.pi s ⊆ u ∪ v\nf : (i : ι) → X i\nhfs : f ∈ univ.pi s\nh... | refine ⟨g, hgs, ⟨?_, hgv⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Connected.Clopen | {
"line": 268,
"column": 4
} | {
"line": 269,
"column": 34
} | {
"line": 271,
"column": 0
} | [
{
"pp": "case mpr.inr\nα : Type u\ninst✝ : TopologicalSpace α\ns u v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhs : s ⊆ u ∪ v\nhsu : (s ∩ u).Nonempty\nhsv : (s ∩ v).Nonempty\nH : ¬(s ∩ (u ∩ v)).Nonempty\nh : s ⊆ v\n⊢ (s ∩ (u ∩ v)).Nonempty",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedCo... | [] | · rcases hsu with ⟨x, hxs, hxu⟩
exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Connected.Clopen | {
"line": 324,
"column": 4
} | {
"line": 325,
"column": 34
} | {
"line": 327,
"column": 0
} | [
{
"pp": "case mpr.inr\nα : Type u\ninst✝ : TopologicalSpace α\ns u v : Set α\nhu : IsClosed[inst✝] u\nhv : IsClosed[inst✝] v\nhs : s ⊆ u ∪ v\nhsu : (s ∩ u).Nonempty\nhsv : (s ∩ v).Nonempty\nH : ¬(s ∩ (u ∩ v)).Nonempty\nh : s ⊆ v\n⊢ (s ∩ (u ∩ v)).Nonempty",
"ppTerm": "?mpr.inr",
"assigned": true,
"us... | [] | · rcases hsu with ⟨x, hxs, hxu⟩
exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Connected.TotallyDisconnected | {
"line": 309,
"column": 2
} | {
"line": 310,
"column": 66
} | {
"line": 312,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\nβ : Type u_3\ninst✝ : TopologicalSpace β\nf : α → β\nhf : IsCoinducing f\n⊢ IsCoinducing ⋯.connectedComponentsMap",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"ConnectedComponents.mk",
"Topol... | [] | rw [← ConnectedComponents.isQuotientMap_coe.isCoinducing.of_comp_iff]
exact ConnectedComponents.isQuotientMap_coe.isCoinducing.comp hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.TotallyDisconnected | {
"line": 309,
"column": 2
} | {
"line": 310,
"column": 66
} | {
"line": 312,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\nβ : Type u_3\ninst✝ : TopologicalSpace β\nf : α → β\nhf : IsCoinducing f\n⊢ IsCoinducing ⋯.connectedComponentsMap",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"ConnectedComponents.mk",
"Topol... | [] | rw [← ConnectedComponents.isQuotientMap_coe.isCoinducing.of_comp_iff]
exact ConnectedComponents.isQuotientMap_coe.isCoinducing.comp hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 549,
"column": 2
} | {
"line": 549,
"column": 15
} | {
"line": 550,
"column": 2
} | [
{
"pp": "X : Type u\nY : Type v\nι : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns✝ t : Set X\ninst✝ : NonLindelofSpace X\ns : Set X\nhs : IsLindelof s\n⊢ sᶜ.Nonempty",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Compl.compl",
"Set.instCompl",
"... | [
"X : Type u\nY : Type v\nι : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ns✝ t : Set X\ninst✝ : NonLindelofSpace X\ns : Set X\nhs : ¬sᶜ.Nonempty\n⊢ ¬IsLindelof s"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Topology.GDelta.Basic | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 29
} | {
"line": 285,
"column": 2
} | [
{
"pp": "X : Type u_1\nι' : Sort u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : Countable ι'\nf : ι' → Set X\nhs : ∀ (i : ι'), IsMeagre (f i)\n⊢ IsMeagre (⋃ i, f i)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"IsMeagre",
"congrArg",
... | [
"X : Type u_1\nι' : Sort u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : Countable ι'\nf : ι' → Set X\nhs : ∀ (i : ι'), IsMeagre (f i)\n⊢ ⋂ i, (f i)ᶜ ∈ residual X"
] | rw [IsMeagre, compl_iUnion] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Compactness.Lindelof | {
"line": 763,
"column": 4
} | {
"line": 763,
"column": 9
} | {
"line": 764,
"column": 4
} | [
{
"pp": "case inr\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : HereditarilyLindelofSpace X\nι : Type u_2\ninst✝ : Nonempty ι\nU : ι → Set X\nh : ∀ (i : ι), IsOpen[inst✝²] (U i)\nk : ℕ → ι\nhtc : (range k).Countable\nhtu : ⋃ i ∈ range k, U i = ⋃ i, U i\nt_ne : (range k).Nonempty\n⊢ ∃ k, ⋃ n, U (k n) = ⋃ i,... | [
"case h\nX : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : HereditarilyLindelofSpace X\nι : Type u_2\ninst✝ : Nonempty ι\nU : ι → Set X\nh : ∀ (i : ι), IsOpen[inst✝²] (U i)\nk : ℕ → ι\nhtc : (range k).Countable\nhtu : ⋃ i ∈ range k, U i = ⋃ i, U i\nt_ne : (range k).Nonempty\n⊢ ⋃ n, U (k n) = ⋃ i, U i"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.UniformSpace.Defs | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 30
} | {
"line": 469,
"column": 2
} | [
{
"pp": "β : Type ub\nV W : SetRel β β\nx y z : β\ninst✝ : V.IsSymm\nhx : x ∈ ball z V\nhy : y ∈ ball z W\n⊢ (x, y) ∈ V ○ W",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"UniformSpace.ball",
"UniformSpace.mem_ball_symme... | [
"β : Type ub\nV W : SetRel β β\nx y z : β\ninst✝ : V.IsSymm\nhx : z ∈ ball x V\nhy : y ∈ ball z W\n⊢ (x, y) ∈ V ○ W"
] | rw [mem_ball_symmetry] at hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Defs | {
"line": 614,
"column": 6
} | {
"line": 614,
"column": 26
} | {
"line": 614,
"column": 27
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\na b : α\n⊢ 𝓝 a ×ˢ 𝓝 b = (𝓤 α).lift fun s ↦ (𝓤 α).lift' fun t ↦ {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SetRel",
"SProd.sprod",
"congrArg"... | [
"α : Type ua\ninst✝ : UniformSpace α\na b : α\n⊢ ((𝓤 α).lift' fun s ↦ {y | (y, a) ∈ s}) ×ˢ 𝓝 b =\n (𝓤 α).lift fun s ↦ (𝓤 α).lift' fun t ↦ {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}"
] | nhds_eq_uniformity', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 15
} | {
"line": 96,
"column": 2
} | [
{
"pp": "β : Type v\nu v : UniformSpace β\nF : Filter β\n⊢ Cauchy F ↔ Cauchy F ∧ Cauchy F",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"UniformSpace",
"Cauchy",
"id",
"instMinUniformSpace",
"And",
"Iff",
"Min.min"
],
"usedFVars": [
... | [
"β : Type v\nu v : UniformSpace β\nF : Filter β\n⊢ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β ↔ (F.NeBot ∧ F ×ˢ F ≤ 𝓤 β) ∧ F.NeBot ∧ F ×ˢ F ≤ 𝓤 β"
] | unfold Cauchy | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 15
} | {
"line": 102,
"column": 2
} | [
{
"pp": "β : Type v\nι : Sort u_1\ninst✝ : Nonempty ι\nu : ι → UniformSpace β\nl : Filter β\n⊢ Cauchy l ↔ ∀ (i : ι), Cauchy l",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"UniformSpace",
"iInf",
"Cauchy",
"id",
"Iff",
"instInfSetUniformSpace"
],
... | [
"β : Type v\nι : Sort u_1\ninst✝ : Nonempty ι\nu : ι → UniformSpace β\nl : Filter β\n⊢ l.NeBot ∧ l ×ˢ l ≤ 𝓤 β ↔ ∀ (i : ι), l.NeBot ∧ l ×ˢ l ≤ 𝓤 β"
] | unfold Cauchy | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 20
} | {
"line": 283,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : α → β\np : Filter ι\nι' : Type u_5\nα' : Type u_6\nβ' : Type u_7\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\nf' : α' → β'\np' : Filter ι'\nh : TendstoUniformlyOn F f p univ\nh' : TendstoUniformlyOn F' f' p' univ\n⊢ T... | [] | exact h.prodMap h' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Basic | {
"line": 156,
"column": 37
} | {
"line": 156,
"column": 51
} | {
"line": 156,
"column": 51
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nd s : SetRel α α\nhd : d ∈ 𝓤 α\ncl_d : Set (α × α) := {p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d}\nx✝ : α × α\nx y : α\nhp : (x, y) ∈ s\n⊢ cl_d ∈ (𝓤 α).lift' fun s ↦ {y | (y, x) ∈ s} ×ˢ {y_1 | (y, y_1) ∈ s}",
"ppTerm": "?m.119",
"assigned": true,... | [
"α : Type ua\ninst✝ : UniformSpace α\nd s : SetRel α α\nhd : d ∈ 𝓤 α\ncl_d : Set (α × α) := {p | ∃ x y, (p.1, x) ∈ d ∧ (x, y) ∈ s ∧ (y, p.2) ∈ d}\nx✝ : α × α\nx y : α\nhp : (x, y) ∈ s\n⊢ ∃ t ∈ 𝓤 α, {y | (y, x) ∈ t} ×ˢ {y_1 | (y, y_1) ∈ t} ⊆ cl_d",
"case hh\nα : Type ua\ninst✝ : UniformSpace α\nd s : SetRel α α\... | mem_lift'_sets | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Basic | {
"line": 260,
"column": 62
} | {
"line": 268,
"column": 22
} | {
"line": 270,
"column": 0
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\ns : Set α\n⊢ IsOpen[inst✝.toTopologicalSpace] s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, IsOpen[instTopologicalSpaceProd] V ∧ ball x V ⊆ s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"interior_subset",
... | [] | by
rw [isOpen_iff_ball_subset]
constructor <;> intro h x hx
· obtain ⟨V, hV, hV'⟩ := h x hx
exact
⟨interior V, interior_mem_uniformity hV, isOpen_interior,
(ball_mono interior_subset x).trans hV'⟩
· obtain ⟨V, hV, -, hV'⟩ := h x hx
exact ⟨V, hV, hV'⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 264,
"column": 62
} | {
"line": 264,
"column": 89
} | {
"line": 264,
"column": 89
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nm : α → β\ns : Set α\nhm : IsUniformInducing m\nfact1 : SurjOn (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\nfact2 : MapsTo (map m) (Iic (𝓟 s)) (Iic (𝓟 (m '' s)))\n⊢ (∀ x ∈ Iic (𝓟 s), Cauchy x → ∃ x_1 ∈ s, x ≤ comap m (𝓝 (m x_1)))... | [] | hm.isInducing.nhds_eq_comap | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 644,
"column": 54
} | {
"line": 645,
"column": 87
} | {
"line": 647,
"column": 0
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\na : α\ns : Set α\n⊢ TotallyBounded (insert a s) ↔ TotallyBounded s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TotallyBounded",
"congrArg",
"Set.instUnion",
"Set.instSingletonSet",
"id",
... | [] | by
simp_rw [← singleton_union, totallyBounded_union, totallyBounded_singleton, true_and] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 690,
"column": 4
} | {
"line": 690,
"column": 16
} | {
"line": 691,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u\nuniformSpace : UniformSpace α\ng : Filter α\n⊢ (∀ (f : Filter α), f.NeBot → f ≤ g → ∃ c ≤ f, Cauchy c) → g.TotallyBounded",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Cauchy",
"SetRel",
"Filter.NeBot",
... | [
"case mpr\nα : Type u\nuniformSpace : UniformSpace α\ng : Filter α\nH : ∀ (f : Filter α), f.NeBot → f ≤ g → ∃ c ≤ f, Cauchy c\nd : SetRel α α\nhd : d ∈ 𝓤 α\n⊢ ∃ t, t.Finite ∧ d.preimage t ∈ g"
] | intro H d hd | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 698,
"column": 4
} | {
"line": 698,
"column": 45
} | {
"line": 699,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u\nuniformSpace : UniformSpace α\ng : Filter α\nd : SetRel α α\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), t.Finite → d.preimage t ∉ g\nf : Filter α := ⨅ t, g ⊓ 𝓟 (d.preimage ↑t)ᶜ\nhb : Antitone fun t ↦ g ⊓ 𝓟 (d.preimage ↑t)ᶜ\nthis : f.NeBot\n⊢ ∃ f, f.NeBot ∧ f ≤ g ∧ ∀ c ≤ f, ¬Cauchy... | [
"case mpr\nα : Type u\nuniformSpace : UniformSpace α\ng : Filter α\nd : SetRel α α\nhd : d ∈ 𝓤 α\nhd_cover : ∀ (t : Set α), t.Finite → d.preimage t ∉ g\nf : Filter α := ⨅ t, g ⊓ 𝓟 (d.preimage ↑t)ᶜ\nhb : Antitone fun t ↦ g ⊓ 𝓟 (d.preimage ↑t)ᶜ\nthis✝ : f.NeBot\nthis : f ≤ g\n⊢ ∃ f, f.NeBot ∧ f ≤ g ∧ ∀ c ≤ f, ¬Cau... | have : f ≤ g := iInf_le_of_le ∅ (by simp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 544,
"column": 43
} | {
"line": 544,
"column": 57
} | {
"line": 544,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : UniformSpace β\nF : ι → α → β\ns : Set α\np : Filter ι\nι' : Type u_5\nα' : Type u_6\nβ' : Type u_7\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\np' : Filter ι'\ns' : Set α'\nh : UniformCauchySeqOn F p s\nh' : UniformCauchySeqOn F' p' s'\nu : Set ((β × ... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : UniformSpace β\nF : ι → α → β\ns : Set α\np : Filter ι\nι' : Type u_5\nα' : Type u_6\nβ' : Type u_7\ninst✝ : UniformSpace β'\nF' : ι' → α' → β'\np' : Filter ι'\ns' : Set α'\nh : UniformCauchySeqOn F p s\nh' : UniformCauchySeqOn F' p' s'\nu : Set ((β × β') × β × β'... | Prod.map_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 901,
"column": 4
} | {
"line": 901,
"column": 32
} | {
"line": 902,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\nU : ℕ → SetRel α α\nU_mem : ∀ (n : ℕ), U n ∈ 𝓤 α\nHU : ∀ (u : ℕ → α), (∀ (N m n : ℕ), N ≤ m → N ≤ n → (u m, u n) ∈ U N) → ∃ a, Tendsto u atTop (𝓝 a)\nU' : ℕ → Set (α × α)\nhU' : ∀ {s : Set (α × α)}, s ∈ 𝓤 ... | [
"case refine_2\nα : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\nU : ℕ → SetRel α α\nU_mem : ∀ (n : ℕ), U n ∈ 𝓤 α\nHU : ∀ (u : ℕ → α), (∀ (N m n : ℕ), N ≤ m → N ≤ n → (u m, u n) ∈ U N) → ∃ a, Tendsto u atTop (𝓝 a)\nU' : ℕ → Set (α × α)\nhU' : ∀ {s : Set (α × α)}, s ∈ 𝓤 α ↔ ∃ i, U' ... | rcases hU'.1 hs with ⟨N, hN⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.UniformSpace.UniformApproximation | {
"line": 62,
"column": 24
} | {
"line": 65,
"column": 76
} | {
"line": 67,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : α\nL : ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u\n⊢ ContinuousAt f x",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"... | [] | by
rw [← continuousWithinAt_univ]
apply continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt (mem_univ _) _
simpa only [exists_prop, nhdsWithin_univ, continuousWithinAt_univ] using L | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.HeineCantor | {
"line": 84,
"column": 28
} | {
"line": 84,
"column": 44
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous[inst✝¹.toTopologicalSpace, inst✝.toTopologicalSpace] f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SetRel.IsSym... | [] | exact (h.2 h₂).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.HeineCantor | {
"line": 84,
"column": 28
} | {
"line": 84,
"column": 44
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous[inst✝¹.toTopologicalSpace, inst✝.toTopologicalSpace] f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SetRel.IsSym... | [] | exact (h.2 h₂).2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.HeineCantor | {
"line": 84,
"column": 28
} | {
"line": 84,
"column": 44
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nx : β\nh_cont : Continuous[inst✝¹.toTopologicalSpace, inst✝.toTopologicalSpace] f\nhx : Tendsto f (cocompact α) (𝓝 x)\nr : Set (β × β)\nhr : r ∈ 𝓤 β\nt : Set (β × β)\nht : t ∈ 𝓤 β\nhtsymm : SetRel.IsSym... | [] | exact (h.2 h₂).2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Equicontinuity | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 82
} | {
"line": 535,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\nF : ι → β → α\nS : Set β\n⊢ UniformEquicontinuousOn F S ↔ UniformContinuousOn (⇑ofFun ∘ swap F) S",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Set.instSProd... | [
"ι : Type u_1\nα : Type u_6\nβ : Type u_8\nuα : UniformSpace α\nuβ : UniformSpace β\nF : ι → β → α\nS : Set β\n⊢ UniformEquicontinuousOn F S ↔\n ∀ i ∈ 𝓤 α, ∀ᶠ (x : β × β) in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ((⇑ofFun ∘ swap F) x.1, (⇑ofFun ∘ swap F) x.2) ∈ UniformFun.gen ι α i"
] | rw [UniformContinuousOn, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 825,
"column": 2
} | {
"line": 825,
"column": 36
} | {
"line": 826,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 𝔗 : Set (Set α)\nh : ∀ s ∈ 𝔖, ∃ T ⊆ 𝔗, T.Finite ∧ s ⊆ ⋃₀ T\nV : Set (β × β)\nhV : V ∈ 𝓤 β\ns : Set α\nhs : s ∈ 𝔖\n⊢ ∃ I,\n I.Finite ∧\n (∀ i ∈ I, i ∈ 𝔗) ∧\n ⋂ i ∈ I, UniformOnFun.gen 𝔗 i V ⊆\n {x | ((⇑(ofFun 𝔗) ∘ ⇑(t... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : UniformSpace β\n𝔖 𝔗 : Set (Set α)\nh : ∀ s ∈ 𝔖, ∃ T ⊆ 𝔗, T.Finite ∧ s ⊆ ⋃₀ T\nV : Set (β × β)\nhV : V ∈ 𝓤 β\ns : Set α\nhs : s ∈ 𝔖\nT : Set (Set α)\nhT𝔗 : T ⊆ 𝔗\nhT : T.Finite\nhsT : s ⊆ ⋃₀ T\n⊢ ∃ I,\n I.Finite ∧\n (∀ i ∈ I, i ∈ 𝔗) ∧\n ⋂ i ∈ I, UniformOnF... | obtain ⟨T, hT𝔗, hT, hsT⟩ := h s hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.UniformConvergenceTopology | {
"line": 1048,
"column": 13
} | {
"line": 1048,
"column": 74
} | {
"line": 1048,
"column": 75
} | [
{
"pp": "case h\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\nF : Filter (α →ᵤ[𝔖] β)\nhF : Cauchy F\nthis : F.NeBot\ng : α → β\nhg : ∀ x ∈ ⋃₀ 𝔖, Tendsto (fun x_1 ↦ (toFun 𝔖) x_1 x) F (𝓝 (... | [
"case h\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\nF : Filter (α →ᵤ[𝔖] β)\nhF : Cauchy F\nthis : F.NeBot\ng : α → β\nhg : ∀ x ∈ ⋃₀ 𝔖, Tendsto (fun x_1 ↦ (toFun 𝔖) x_1 x) F (𝓝 (g x))\n⊢ F ≤... | UniformOnFun.nhds_eq_of_basis _ _ uniformity_hasBasis_closed, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Filter.NAry | {
"line": 144,
"column": 69
} | {
"line": 144,
"column": 79
} | {
"line": 144,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nf : Filter α\ng : Filter β\ninst✝ : f.NeBot\n⊢ map₂ (fun x y ↦ y) f g = g",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Filter.map₂_swap",
"Filter.map₂",
"Eq",
"Filter"
],
"... | [
"α : Type u_1\nβ : Type u_3\nf : Filter α\ng : Filter β\ninst✝ : f.NeBot\n⊢ map₂ (fun a b ↦ a) g f = g"
] | map₂_swap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.NAry | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 16
} | {
"line": 156,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\nδ : Type u_7\nf : Filter α\ng : Filter β\nm : α → γ → δ\nn : β → γ\n⊢ map₂ m f (map n g) = map₂ (fun a b ↦ m a (n b)) f g",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.map",
"id",
... | [
"α : Type u_1\nβ : Type u_3\nγ : Type u_5\nδ : Type u_7\nf : Filter α\ng : Filter β\nm : α → γ → δ\nn : β → γ\n⊢ map₂ (fun a b ↦ m b a) (map n g) f = map₂ (fun a b ↦ m a (n b)) f g"
] | map₂_swap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.Basic | {
"line": 480,
"column": 40
} | {
"line": 480,
"column": 76
} | {
"line": 480,
"column": 76
} | [
{
"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 : C(X, Y)\nhf : IsQuotientMap ⇑f\ng✝ : C(X, Z)\nh✝ : FactorsThrough ⇑g✝ ⇑f\ng : C(Y, Z)\nx✝¹ x✝ : X\nh : f x✝¹ = f x✝\n⊢ (g.comp f) x✝¹ = (g.comp f) x✝",
"ppTerm": "?m.5... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\nhf : IsQuotientMap ⇑f\ng✝ : C(X, Z)\nh✝ : FactorsThrough ⇑g✝ ⇑f\ng : C(Y, Z)\nx✝¹ x✝ : X\nh : f x✝¹ = f x✝\n⊢ g (f x✝¹) = g (f x✝)"
] | simp only [ContinuousMap.comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 24
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a * x) ⁻¹' Icc b c = Icc (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Set.Ici",
"Monoid.toMulOneClass"... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 24
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a * x) ⁻¹' Icc b c = Icc (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Set.Ici",
"Monoid.toMulOneClass"... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 24
} | {
"line": 200,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a * x) ⁻¹' Icc b c = Icc (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Set.Ici",
"Monoid.toMulOneClass"... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 24
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ x * a) ⁻¹' Icc b c = Icc (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Set.Ici",
"Monoid.toMulOneClass"... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 24
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ x * a) ⁻¹' Icc b c = Icc (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Set.Ici",
"Monoid.toMulOneClass"... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 24
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ x * a) ⁻¹' Icc b c = Icc (b / a) (c / a)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Set.Ici",
"Monoid.toMulOneClass"... | [] | simp [← Ici_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 851,
"column": 11
} | {
"line": 851,
"column": 64
} | {
"line": 853,
"column": 0
} | [
{
"pp": "case h.inr\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : α\nhb✝ : a ≠ 0\nU V : Set α\nhV : V = (fun x ↦ a * x) ⁻¹' U\naU : α\nhaU : U = Iio aU\nhb : 0 < a\n⊢ (fun x ↦ a * x) ⁻¹' Iio aU = Iio (a⁻¹ * aU)",
"ppTerm": "?h.inr",
"assigned": true,
"us... | [] | rw [Set.preimage_const_mul_Iio₀ _ hb, div_eq_inv_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 893,
"column": 19
} | {
"line": 893,
"column": 41
} | {
"line": 893,
"column": 41
} | [
{
"pp": "α : Type u_2\ninst✝³ : Monoid α\ninst✝² : Preorder α\ninst✝¹ : CanonicallyOrderedMul α\ninst✝ : MulRightMono α\na b c : α\nc_in : c ∈ Ici (a * b)\nd : α\nhd : c = a * b * d\n⊢ b * d ∈ Ici b ∧ a * (b * d) = c",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
... | [] | simp [← mul_assoc, hd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 893,
"column": 19
} | {
"line": 893,
"column": 41
} | {
"line": 893,
"column": 41
} | [
{
"pp": "α : Type u_2\ninst✝³ : Monoid α\ninst✝² : Preorder α\ninst✝¹ : CanonicallyOrderedMul α\ninst✝ : MulRightMono α\na b c : α\nc_in : c ∈ Ici (a * b)\nd : α\nhd : c = a * b * d\n⊢ b * d ∈ Ici b ∧ a * (b * d) = c",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
... | [] | simp [← mul_assoc, hd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 893,
"column": 19
} | {
"line": 893,
"column": 41
} | {
"line": 893,
"column": 41
} | [
{
"pp": "α : Type u_2\ninst✝³ : Monoid α\ninst✝² : Preorder α\ninst✝¹ : CanonicallyOrderedMul α\ninst✝ : MulRightMono α\na b c : α\nc_in : c ∈ Ici (a * b)\nd : α\nhd : c = a * b * d\n⊢ b * d ∈ Ici b ∧ a * (b * d) = c",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
... | [] | simp [← mul_assoc, hd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 248,
"column": 8
} | {
"line": 248,
"column": 37
} | {
"line": 248,
"column": 37
} | [
{
"pp": "case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous[inst✝¹, inst✝] f ∧ IsClosedMap f ∧ ∀ (y : Y), IsCompact (f ⁻¹' {y})\nℱ : Filter X\ny : Y\nhy : (ℱ.lift' closure[inst✝¹] ⊓ 𝓟 (f ⁻¹' {y})).NeBot\nx : X\nhxy : x ∈ f ⁻¹' {y}\nhx : Clus... | [
"case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous[inst✝¹, inst✝] f ∧ IsClosedMap f ∧ ∀ (y : Y), IsCompact (f ⁻¹' {y})\nℱ : Filter X\ny : Y\nhy : (ℱ.lift' closure[inst✝¹] ⊓ 𝓟 (f ⁻¹' {y})).NeBot\nx : X\nhxy : x ∈ f ⁻¹' {y}\nhx : ClusterPt x (ℱ.l... | ← clusterPt_lift'_closure_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.AtTopBot.Group | {
"line": 150,
"column": 2
} | {
"line": 152,
"column": 82
} | {
"line": 153,
"column": 2
} | [
{
"pp": "G : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : LinearOrder G\ninst✝ : IsOrderedMonoid G\n⊢ comap mabs atTop = atBot ⊔ atTop",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Filter.Tendsto.le_comap",
"Filter.atTop_basis",
"Lattice.toSemilatticeSup",... | [
"G : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : LinearOrder G\ninst✝ : IsOrderedMonoid G\n⊢ ∀ (i' : G × G), True ∧ True → ∃ i, True ∧ mabs ⁻¹' Ici i ⊆ Iic i'.1 ∪ Ici i'.2"
] | refine
le_antisymm (((atTop_basis.comap _).le_basis_iff (atBot_basis.sup atTop_basis)).2 ?_)
(sup_le tendsto_mabs_atBot_atTop.le_comap tendsto_mabs_atTop_atTop.le_comap) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 727,
"column": 4
} | {
"line": 727,
"column": 67
} | {
"line": 728,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : TopologicalSpace G\ninst✝¹ : MulOneClass G\ninst✝ : ContinuousMul G\ng h : G\nhg : g ∈ connectedComponent 1\nhh : h ∈ connectedComponent 1\n⊢ g ∈ (fun x ↦ g * x) '' connectedComponent 1",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
... | [] | exact ⟨(1 : G), mem_connectedComponent, by simp only [mul_one]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 94,
"column": 2
} | {
"line": 98,
"column": 75
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\n⊢ 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Set.Ioi",
"Preorder.toLT",
"Lattice.toSemilatti... | [] | by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
simp only [(nhdsGT_basis_of_exists_gt ha).eq_bot_iff, covBy_iff_Ioo_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.LeftRightNhds | {
"line": 94,
"column": 2
} | {
"line": 98,
"column": 75
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\n⊢ 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Set.Ioi",
"Preorder.toLT",
"Lattice.toSemilatti... | [] | by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
simp only [(nhdsGT_basis_of_exists_gt ha).eq_bot_iff, covBy_iff_Ioo_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 93
} | {
"line": 391,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\ninst✝¹ : (i : α) → Decidable (i ∈ s)\ninst✝ : (i : β) → Decidable (i ∈ t)\nh : e.IsImage s t\nh' : e'.IsImage s t\nx : α\nhe : x ∈ e.source\nhs : x ∈ s\n⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewi... | [] | rw [piecewise_eq_of_mem _ _ _ hs, piecewise_eq_of_mem _ _ _ ((h he).2 hs), e.left_inv he] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 93
} | {
"line": 391,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\ninst✝¹ : (i : α) → Decidable (i ∈ s)\ninst✝ : (i : β) → Decidable (i ∈ t)\nh : e.IsImage s t\nh' : e'.IsImage s t\nx : α\nhe : x ∈ e.source\nhs : x ∈ s\n⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewi... | [] | rw [piecewise_eq_of_mem _ _ _ hs, piecewise_eq_of_mem _ _ _ ((h he).2 hs), e.left_inv he] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 93
} | {
"line": 391,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\ninst✝¹ : (i : α) → Decidable (i ∈ s)\ninst✝ : (i : β) → Decidable (i ∈ t)\nh : e.IsImage s t\nh' : e'.IsImage s t\nx : α\nhe : x ∈ e.source\nhs : x ∈ s\n⊢ t.piecewise (↑e.symm) (↑e'.symm) (s.piecewi... | [] | rw [piecewise_eq_of_mem _ _ _ hs, piecewise_eq_of_mem _ _ _ ((h he).2 hs), e.left_inv he] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 396,
"column": 39
} | {
"line": 396,
"column": 91
} | {
"line": 398,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\nh : e.IsImage s t\nh' : e'.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\n⊢ e.target ∩ t = e'.target ∩ t",
"ppTerm": "?m.31",
"assigned": true,
"usedConstant... | [] | rw [← h.image_eq, ← h'.image_eq, ← hs, heq.image_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 396,
"column": 39
} | {
"line": 396,
"column": 91
} | {
"line": 398,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\nh : e.IsImage s t\nh' : e'.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\n⊢ e.target ∩ t = e'.target ∩ t",
"ppTerm": "?m.31",
"assigned": true,
"usedConstant... | [] | rw [← h.image_eq, ← h'.image_eq, ← hs, heq.image_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 396,
"column": 39
} | {
"line": 396,
"column": 91
} | {
"line": 398,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : PartialEquiv α β\ns : Set α\nt : Set β\ne' : PartialEquiv α β\nh : e.IsImage s t\nh' : e'.IsImage s t\nhs : e.source ∩ s = e'.source ∩ s\nheq : EqOn (↑e) (↑e') (e.source ∩ s)\n⊢ e.target ∩ t = e'.target ∩ t",
"ppTerm": "?m.31",
"assigned": true,
"usedConstant... | [] | rw [← h.image_eq, ← h'.image_eq, ← hs, heq.image_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.Basic | {
"line": 132,
"column": 69
} | {
"line": 132,
"column": 77
} | {
"line": 132,
"column": 78
} | [
{
"pp": "α : Type u\nts : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ ⨅ s, ⨅ (_ : (∃ x, s = Ioi x) ∧ a ∈ s ∨ (∃ x, s = Iio x) ∧ a ∈ s), 𝓟 s =\n (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"α : Type u\nts : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ ⨅ s, (⨅ (_ : (∃ x, s = Ioi x) ∧ a ∈ s), 𝓟 s) ⊓ ⨅ (_ : (∃ x, s = Iio x) ∧ a ∈ s), 𝓟 s =\n (⨅ b ∈ Iio a, 𝓟 (Ioi b)) ⊓ ⨅ b ∈ Ioi a, 𝓟 (Iio b)"
] | iInf_or, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Order.Basic | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 45
} | {
"line": 172,
"column": 4
} | [
{
"pp": "case inr.a\nα : Type u\nts : TopologicalSpace α\ninst✝² : Preorder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nhα : Nonempty α\nt : Set (Set α)\nt_subs : t ⊆ {s | ∃ a, s = Ioi a ∨ s = Iio a}\nt_count : t.Countable\nht : ts = generateFrom t\na : Set α → α\nha : ∀ s ∈ t, s = Ioi (a s)... | [
"case inr.a\nα : Type u\nts : TopologicalSpace α\ninst✝² : Preorder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nhα : Nonempty α\nt : Set (Set α)\nt_subs : t ⊆ {s | ∃ a, s = Ioi a ∨ s = Iio a}\nt_count : t.Countable\nht : ts = generateFrom t\na : Set α → α\nha : ∀ s ∈ t, s = Ioi (a s) ∨ s = Iio (... | apply le_generateFrom_iff_subset_isOpen.2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.Basic | {
"line": 559,
"column": 2
} | {
"line": 561,
"column": 39
} | {
"line": 563,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : SeparableSpace α\n⊢ SecondCountableTopology α",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"SecondCountableTopology",
"Parti... | [] | rcases exists_countable_dense α with ⟨s, hc, hd⟩
refine ⟨⟨_, ?_, hd.topology_eq_generateFrom⟩⟩
exact (hc.image _).union (hc.image _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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