module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Connected.Basic | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 26
} | {
"line": 195,
"column": 27
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nι : Type u_3\ns : ι → Set α\nH : ∀ (i : ι), IsPreconnected (s i)\nK : ∀ (i j : ι), ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty) i j\ni : ι\nx✝¹ : i ∈ univ\nj : ι\nx✝ : j ∈ univ\n⊢ ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty ∧ i ∈ univ) i j",
"ppTerm": "?m.3... | [
"α : Type u\ninst✝ : TopologicalSpace α\nι : Type u_3\ns : ι → Set α\nH : ∀ (i : ι), IsPreconnected (s i)\nK : ∀ (i j : ι), ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty) i j\ni : ι\nx✝¹ : i ∈ univ\nj : ι\nx✝ : j ∈ univ\n⊢ ReflTransGen (fun i j ↦ (s i ∩ s j).Nonempty) i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CountableInter | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 52
} | {
"line": 64,
"column": 4
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nl : Filter α\ninst✝¹ : CountableInterFilter l\ninst✝ : Countable ι\np : α → ι → Prop\n⊢ (∀ᶠ (x : α) in l, ∀ (i : ι), p x i) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p x i",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",... | [
"ι : Sort u_1\nα : Type u_2\nl : Filter α\ninst✝¹ : CountableInterFilter l\ninst✝ : Countable ι\np : α → ι → Prop\n⊢ ⋂ i, {x | p x i} ∈ l ↔ ∀ (i : ι), {x | p x i} ∈ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CountableInter | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 52
} | {
"line": 70,
"column": 4
} | [
{
"pp": "α : Type u_2\nl : Filter α\ninst✝ : CountableInterFilter l\nι : Type u_4\nS : Set ι\nhS : S.Countable\np : α → (i : ι) → i ∈ S → Prop\n⊢ (∀ᶠ (x : α) in l, ∀ (i : ι) (hi : i ∈ S), p x i hi) ↔ ∀ (i : ι) (hi : i ∈ S), ∀ᶠ (x : α) in l, p x i hi",
"ppTerm": "?m.15",
"assigned": true,
"usedConsta... | [
"α : Type u_2\nl : Filter α\ninst✝ : CountableInterFilter l\nι : Type u_4\nS : Set ι\nhS : S.Countable\np : α → (i : ι) → i ∈ S → Prop\n⊢ ⋂ i, ⋂ (i_1 : i ∈ S), {x | p x i i_1} ∈ l ↔ ∀ (i : ι) (hi : i ∈ S), {x | p x i hi} ∈ l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.CountableInter | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 31
} | {
"line": 217,
"column": 32
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl✝ : Filter α\ninst✝¹ : CountableInterFilter l✝\nl : Filter β\ninst✝ : CountableInterFilter l\nf : α → β\nS : Set (Set α)\nhSc : S.Countable\nt : Set α → Set β\nhtl : ∀ s ∈ S, t s ∈ l\nht : ∀ s ∈ S, f ⁻¹' t s ⊆ s\nthis : ⋂ s ∈ S, t s ∈ l\n⊢ f ⁻¹' ⋂ s ∈ S, t s ⊆... | [
"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl✝ : Filter α\ninst✝¹ : CountableInterFilter l✝\nl : Filter β\ninst✝ : CountableInterFilter l\nf : α → β\nS : Set (Set α)\nhSc : S.Countable\nt : Set α → Set β\nhtl : ∀ s ∈ S, t s ∈ l\nht : ∀ s ∈ S, f ⁻¹' t s ⊆ s\nthis : ⋂ s ∈ S, t s ∈ l\n⊢ ∀ t' ∈ S, ⋂ i ∈ S, f ⁻¹' t i ⊆ t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 48
} | {
"line": 112,
"column": 49
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns t : Set α\n⊢ IsOpen[inst✝¹] s → IsOpen[inst✝¹] t → s ∪ t = univ → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns t : Set α\n⊢ IsOpen[inst✝¹] s → IsOpen[inst✝¹] t → s ∪ t = univ → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 117,
"column": 47
} | {
"line": 117,
"column": 82
} | {
"line": 117,
"column": 83
} | [
{
"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",
"id"... | [
"α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : Set α\nhs : IsClopen s\nh : ¬(s = ∅ ∨ s = univ)\n⊢ ¬s = ∅ ∧ ¬s = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 88
} | {
"line": 150,
"column": 89
} | [
{
"pp": "α : Type u\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : ι → Set α\nh_nonempty : ∀ (i : ι), (s i).Nonempty\nh_disj : Pairwise (Disjoint on s)\nh_clopen : ∀ (i : ι), IsClopen (s i)\ni j : ι\nh_ne : i ≠ j\n⊢ s i ∩ s j = ∅",
"ppTerm": "?m.48",
"assigned": true,
"... | [
"α : Type u\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : ι → Set α\nh_nonempty : ∀ (i : ι), (s i).Nonempty\nh_disj : Pairwise (Disjoint on s)\nh_clopen : ∀ (i : ι), IsClopen (s i)\ni j : ι\nh_ne : i ≠ j\n⊢ Disjoint (s i) (s j)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 55
} | {
"line": 166,
"column": 56
} | [
{
"pp": "case inr\nα : Type u\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : ι → Set α\nh_nonempty : ∀ (i : ι), (s i).Nonempty\nh_disj : Pairwise (Disjoint on s)\nh_open : ∀ (i : ι), IsOpen[inst✝¹] (s i)\nh_Union : ⋃ i, s i = univ\ni j : ι\nh_ne : i ≠ j\n⊢ IsOpen[inst✝¹] (s j \\ s ... | [
"case inr\nα : Type u\nι : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PreconnectedSpace α\ns : ι → Set α\nh_nonempty : ∀ (i : ι), (s i).Nonempty\nh_disj : Pairwise (Disjoint on s)\nh_open : ∀ (i : ι), IsOpen[inst✝¹] (s i)\nh_Union : ⋃ i, s i = univ\ni j : ι\nh_ne : i ≠ j\n⊢ IsOpen[inst✝¹] (s j)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 55
} | {
"line": 178,
"column": 56
} | [
{
"pp": "case inr\nα : Type u\nι : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ns : ι → Set α\nh_nonempty : ∀ (i : ι), (s i).Nonempty\nh_disj : Pairwise (Disjoint on s)\ninst✝ : Finite ι\nh_closed : ∀ (i : ι), IsClosed[inst✝²] (s i)\nh_Union : ⋃ i, s i = univ\ni j : ι\nh_ne : i ≠ j\n⊢ Is... | [
"case inr\nα : Type u\nι : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : PreconnectedSpace α\ns : ι → Set α\nh_nonempty : ∀ (i : ι), (s i).Nonempty\nh_disj : Pairwise (Disjoint on s)\ninst✝ : Finite ι\nh_closed : ∀ (i : ι), IsClosed[inst✝²] (s i)\nh_Union : ⋃ i, s i = univ\ni j : ι\nh_ne : i ≠ j\n⊢ IsClosed[inst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 361,
"column": 6
} | {
"line": 361,
"column": 41
} | {
"line": 361,
"column": 42
} | [
{
"pp": "case refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nhs : IsPreconnected s\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set α\nhu : IsOpen[inst✝¹] u\nhv : IsOpen[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹... | [
"case refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nhs : IsPreconnected s\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set α\nhu : IsOpen[inst✝¹] u\nhv : IsOpen[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹' s ∩ v).Non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 45
} | {
"line": 362,
"column": 46
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nhs : IsPreconnected s\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set α\nhu : IsOpen[inst✝¹] u\nhv : IsOpen[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹... | [
"case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nhs : IsPreconnected s\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set α\nhu : IsOpen[inst✝¹] u\nhv : IsOpen[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹' s ∩ v).Non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 363,
"column": 6
} | {
"line": 363,
"column": 45
} | {
"line": 363,
"column": 46
} | [
{
"pp": "case refine_3\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nhs : IsPreconnected s\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set α\nhu : IsOpen[inst✝¹] u\nhv : IsOpen[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹... | [
"case refine_3\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\ns : Set β\nhs : IsPreconnected s\nhinj : Injective f\nhf : IsOpenMap f\nu v : Set α\nhu : IsOpen[inst✝¹] u\nhv : IsOpen[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹' s ∩ v).Non... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 373,
"column": 8
} | {
"line": 373,
"column": 43
} | {
"line": 373,
"column": 44
} | [
{
"pp": "case refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed[inst✝¹] u\nhv : IsClosed[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv :... | [
"case refine_1\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed[inst✝¹] u\nhv : IsClosed[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹' s ∩ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 31
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : IsPreconnected s\nP : α → α → Prop\nh : ∀ x ∈ s, ∀ᶠ (y : α) in 𝓝[s] x, P x y\nh' : ∀ (x y z : α), x ∈ s → y ∈ s → z ∈ s → P x y → P y z → P x z\nh'' : ∀ (x y : α), x ∈ s → y ∈ s → P x y → P y x\nx y : α\nhx : x ∈ s\nhy : y ∈ s\nz : α\nhz : z ∈ s\... | [] | exact ⟨ha, h'' z a hz h'a ha⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Basic | {
"line": 374,
"column": 8
} | {
"line": 374,
"column": 47
} | {
"line": 374,
"column": 48
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed[inst✝¹] u\nhv : IsClosed[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv :... | [
"case refine_2\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed[inst✝¹] u\nhv : IsClosed[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹' s ∩ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 375,
"column": 8
} | {
"line": 375,
"column": 47
} | {
"line": 375,
"column": 48
} | [
{
"pp": "case refine_3\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed[inst✝¹] u\nhv : IsClosed[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv :... | [
"case refine_3\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns : Set β\nhs : IsPreconnected s\nf : α → β\nhinj : Injective f\nhf : IsClosedMap f\nu v : Set α\nhu : IsClosed[inst✝¹] u\nhv : IsClosed[inst✝¹] v\nhsuv : f ⁻¹' s ⊆ u ∪ v\nhsu : (f ⁻¹' s ∩ u).Nonempty\nhsv : (f ⁻¹' s ∩ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 39
} | {
"line": 405,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ns : Set α\ninst✝² : LinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderClosedTopology β\nf : α → β\nb : β\nhs : IsPreconnected s\nhf : ContinuousOn f s\nhfb : ∀ x ∈ s, f x ≠ b\n⊢ MapsTo f s (Ioi b) ∨ MapsTo f s (Iio b)",
"ppTerm": "?m.22",
... | [
"α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ns : Set α\ninst✝² : LinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderClosedTopology β\nf : α → β\nb : β\nhs : IsPreconnected s\nhf : ContinuousOn f s\nhfb : ∀ x ∈ s, f x ≠ b\n⊢ s ⊆ f ⁻¹' Ioi b ∨ s ⊆ f ⁻¹' Iio b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 468,
"column": 4
} | {
"line": 468,
"column": 15
} | {
"line": 468,
"column": 16
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 282,
"column": 32
} | {
"line": 282,
"column": 43
} | {
"line": 282,
"column": 44
} | [
{
"pp": "α : 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\nhne : s.Nonempty\nh : ∀ (u v : Set α), IsOpen[inst✝] u → IsOpen[inst✝] v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nhU : ∀ (u v : Se... | [
"α : 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\nhne : s.Nonempty\nh : ∀ (u v : Set α), IsOpen[inst✝] u → IsOpen[inst✝] v → s ⊆ u ∪ v → s ∩ (u ∩ v) = ∅ → s ⊆ u ∨ s ⊆ v\nhU : ∀ (u v : Set α), u ∈ ∅ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Regular | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 29
} | {
"line": 209,
"column": 30
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\nB : Set (Set X)\nhB : IsTopologicalBasis B\nx : X\n⊢ (𝓝 x).HasBasis (fun s ↦ x ∈ s ∧ s ∈ B) closure[inst✝¹]",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\nB : Set (Set X)\nhB : IsTopologicalBasis B\nx : X\n⊢ (𝓝 x).HasBasis (fun s ↦ x ∈ s ∧ s ∈ B) closure[inst✝¹]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Regular | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 43
} | {
"line": 214,
"column": 44
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\nB : Set (Set X)\nhB : IsTopologicalBasis B\nx : X\ns : Set X\nh : s ∈ 𝓝 x\n⊢ ∃ t ∈ B, x ∈ t ∧ closure[inst✝¹] t ⊆ s",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\nB : Set (Set X)\nhB : IsTopologicalBasis B\nx : X\ns : Set X\nh : s ∈ 𝓝 x\n⊢ ∃ t ∈ B, x ∈ t ∧ closure[inst✝¹] t ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Regular | {
"line": 261,
"column": 2
} | {
"line": 262,
"column": 39
} | {
"line": 262,
"column": 40
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝¹] t\nhst : Disjoint s t\n⊢ SeparatedNhds s t",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Topology.Separation.Regular.0.... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : RegularSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝¹] t\nhst : Disjoint s t\n⊢ ∀ x ∈ s, x ∉ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 57
} | {
"line": 290,
"column": 58
} | [
{
"pp": "case refine_2\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh :\n ∀ (U : Finset (Set α)),\n (∀ (u v : Set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) →\n (∀ u ∈ U, IsOpen[inst✝] u) → s ⊆ ⋃₀ ↑U → ∃ u ∈ U, s ⊆ u\n⊢ s.Nonempty",
"ppTerm": "?refine_2",
"assigned": true,
"... | [
"case refine_2\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh :\n ∀ (U : Finset (Set α)),\n (∀ (u v : Set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) →\n (∀ u ∈ U, IsOpen[inst✝] u) → s ⊆ ⋃₀ ↑U → ∃ u ∈ U, s ⊆ u\n⊢ ¬s = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 293,
"column": 4
} | {
"line": 293,
"column": 39
} | {
"line": 293,
"column": 40
} | [
{
"pp": "case refine_3\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh :\n ∀ (U : Finset (Set α)),\n (∀ (u v : Set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) →\n (∀ u ∈ U, IsOpen[inst✝] u) → s ⊆ ⋃₀ ↑U → ∃ u ∈ U, s ⊆ u\nu v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhs : s ⊆ u ∪... | [
"case refine_3\nα : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh :\n ∀ (U : Finset (Set α)),\n (∀ (u v : Set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).Nonempty → u = v) →\n (∀ u ∈ U, IsOpen[inst✝] u) → s ⊆ ⋃₀ ↑U → ∃ u ∈ U, s ⊆ u\nu v : Set α\nhu : IsOpen[inst✝] u\nhv : IsOpen[inst✝] v\nhs : s ⊆ u ∪ v\nhsuv : ¬... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Basic | {
"line": 757,
"column": 4
} | {
"line": 757,
"column": 15
} | {
"line": 757,
"column": 16
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh : PreconnectedSpace ↑s\n⊢ IsPreconnected s",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nh : PreconnectedSpace ↑s\n⊢ IsPreconnected s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.Regular | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 43
} | {
"line": 384,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T25Space X\nx y : X\nh : x ≠ y\n⊢ ∃ u, x ∈ u ∧ IsOpen[inst✝¹] u ∧ ∃ v, y ∈ v ∧ IsOpen[inst✝¹] v ∧ Disjoint (closure[inst✝¹] u) (closure[inst✝¹] v)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T25Space X\nx y : X\nh : x ≠ y\n⊢ ∃ u, x ∈ u ∧ IsOpen[inst✝¹] u ∧ ∃ v, y ∈ v ∧ IsOpen[inst✝¹] v ∧ Disjoint (closure[inst✝¹] u) (closure[inst✝¹] v)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DenseEmbedding | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 48
} | {
"line": 183,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ni : α → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nb : β\nf : α → γ\ng : β → γ\ndi : IsDenseInducing i\nhf : ∀ᶠ (x : α) in comap i (𝓝 b), g (i x) = f x\nhg : ContinuousAt g b\ns : Set γ\nhs : s ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ni : α → β\ninst✝¹ : TopologicalSpace γ\ninst✝ : T2Space γ\nb : β\nf : α → γ\ng : β → γ\ndi : IsDenseInducing i\nhf : ∀ᶠ (x : α) in comap i (𝓝 b), g (i x) = f x\nhg : ContinuousAt g b\ns : Set γ\nhs : s ∈ 𝓝 (g b)\n... | suffices ∀ᶠ x : α in comap i (𝓝 b), g (i x) ∈ s from
hf.mp (this.mono fun x hgx hfx => hfx ▸ hgx) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.Separation.Regular | {
"line": 551,
"column": 23
} | {
"line": 551,
"column": 59
} | {
"line": 551,
"column": 60
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : T4Space X\ns✝ : Set X\na✝ : X\nhs : IsClosed[inst✝¹] s✝\nhxs : a✝ ∉ s✝\n⊢ Disjoint (𝓝ˢ s✝) (𝓝 a✝)",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : T4Space X\ns✝ : Set X\na✝ : X\nhs : IsClosed[inst✝¹] s✝\nhxs : a✝ ∉ s✝\n⊢ Disjoint (𝓝ˢ s✝) (𝓝 a✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 34
} | {
"line": 217,
"column": 35
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\nl : Filter X\ninst✝ : CountableInterFilter l\nhs : IsLindelof s\n⊢ Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x)",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\nl : Filter X\ninst✝ : CountableInterFilter l\nhs : IsLindelof s\n⊢ Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 29
} | {
"line": 422,
"column": 2
} | [
{
"pp": "α : 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 : Set β\nht : IsClosed[inst✝] t\nht' : IsConnected t\nhf : Surjective f\nhT : IsClosed[inst✝¹] (f ⁻¹' t)\n⊢ ∀ (u v : Set α), IsClose... | [
"α : 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 : Set β\nht : IsClosed[inst✝] t\nht' : IsConnected t\nhf : Surjective f\nhT : IsClosed[inst✝¹] (f ⁻¹' t)\nu v : Set α\nhu : IsClosed[inst✝¹] u\n... | intro u v hu hv huv uv_disj | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Compactness.Lindelof | {
"line": 315,
"column": 21
} | {
"line": 315,
"column": 59
} | {
"line": 315,
"column": 60
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nx : X\nx✝¹ : Filter X\nhf : x✝¹.NeBot\nx✝ : CountableInterFilter x✝¹\nhfa : x✝¹ ≤ 𝓟 {x}\n⊢ 𝓟 {x} ≤ 𝓝 x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
... | [
"X : Type u\ninst✝ : TopologicalSpace X\nx : X\nx✝¹ : Filter X\nhf : x✝¹.NeBot\nx✝ : CountableInterFilter x✝¹\nhfa : x✝¹ ≤ 𝓟 {x}\n⊢ pure x ≤ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.TotallyDisconnected | {
"line": 108,
"column": 72
} | {
"line": 112,
"column": 30
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\n⊢ TotallyDisconnectedSpace α ↔ ∀ (x : α), connectedComponent x = {x}",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"mem_connectedComponent",
"Set.instSingletonSet",
"id",... | [] | by
rw [totallyDisconnectedSpace_iff_connectedComponent_subsingleton]
refine forall_congr' fun x => ?_
rw [subsingleton_iff_singleton]
exact mem_connectedComponent | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.TotallyDisconnected | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 15
} | {
"line": 363,
"column": 16
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\nS : Set α\nhS : IsPreconnected S\nh : IsDiscrete S\nthis✝¹ : DiscreteTopology ↑S\nthis✝ : PreconnectedSpace ↑S\nthis : Subsingleton ↑S\n⊢ S.Subsingleton",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"α : Type u\ninst✝ : TopologicalSpace α\nS : Set α\nhS : IsPreconnected S\nh : IsDiscrete S\nthis✝¹ : DiscreteTopology ↑S\nthis✝ : PreconnectedSpace ↑S\nthis : Subsingleton ↑S\n⊢ S.Subsingleton"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.GDelta.Basic | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 31
} | {
"line": 109,
"column": 32
} | [
{
"pp": "X : Type u_1\nι' : Sort u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : Countable ι'\nT : ι' → Set (Set X)\nhTo : ∀ (i : ι'), ∀ t ∈ T i, IsOpen[inst✝¹] t\nhTc : ∀ (i : ι'), (T i).Countable\nhTs : ∀ (i : ι'), (fun i ↦ ⋂₀ T i) i = ⋂₀ T i\n⊢ ∀ t ∈ ⋃ i, T i, IsOpen[inst✝¹] t",
"ppTerm": "?m.56",
"assigne... | [
"X : Type u_1\nι' : Sort u_4\ninst✝¹ : TopologicalSpace X\ninst✝ : Countable ι'\nT : ι' → Set (Set X)\nhTo : ∀ (i : ι'), ∀ t ∈ T i, IsOpen[inst✝¹] t\nhTc : ∀ (i : ι'), (T i).Countable\nhTs : ∀ (i : ι'), (fun i ↦ ⋂₀ T i) i = ⋂₀ T i\n⊢ ∀ (t : Set X) (x : ι'), t ∈ T x → IsOpen[inst✝¹] t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.GDelta.Basic | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 38
} | {
"line": 120,
"column": 39
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nh : ∀ s ∈ S, IsGδ s\nhS : S.Countable\n⊢ IsGδ (⋂₀ S)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.iInter",
"Membership.mem",
"id",
"Set.sInter_eq_biInter",
... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nh : ∀ s ∈ S, IsGδ s\nhS : S.Countable\n⊢ IsGδ (⋂ i ∈ S, i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 13
} | {
"line": 497,
"column": 14
} | [
{
"pp": "X : Type u\ninst✝³ : TopologicalSpace X\ninst✝² : LindelofSpace X\nf : Filter X\ninst✝¹ : f.NeBot\ninst✝ : CountableInterFilter f\n⊢ ∃ x, ClusterPt x f",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u\ninst✝³ : TopologicalSpace X\ninst✝² : LindelofSpace X\nf : Filter X\ninst✝¹ : f.NeBot\ninst✝ : CountableInterFilter f\n⊢ ∃ x, ClusterPt x f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 509,
"column": 74
} | {
"line": 509,
"column": 85
} | {
"line": 509,
"column": 86
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nh :\n ∀ {ι : Type u} (t : ι → Set X),\n (∀ (i : ι), IsClosed[inst✝] (t i)) → ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ ⋂ i ∈ u, t i = ∅\nι✝ : Type u\nt : ι✝ → Set X\n⊢ (∀ (i : ι✝), IsClosed[inst✝] (t i)) → univ ∩ ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ univ ∩ ⋂ i ∈ u, t i ... | [
"X : Type u\ninst✝ : TopologicalSpace X\nh :\n ∀ {ι : Type u} (t : ι → Set X),\n (∀ (i : ι), IsClosed[inst✝] (t i)) → ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ ⋂ i ∈ u, t i = ∅\nι✝ : Type u\nt : ι✝ → Set X\n⊢ (∀ (i : ι✝), IsClosed[inst✝] (t i)) → ⋂ i, t i = ∅ → ∃ u, u.Countable ∧ ⋂ i ∈ u, t i = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 594,
"column": 2
} | {
"line": 594,
"column": 13
} | {
"line": 594,
"column": 14
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : Continuous[inst✝¹, inst✝] f\nt : Set X\nht : IsLindelof t\n⊢ f ⁻¹' (f '' t)ᶜ ⊆ tᶜ",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Compl.compl",
"compl_le_c... | [
"X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nhf : Continuous[inst✝¹, inst✝] f\nt : Set X\nht : IsLindelof t\n⊢ t ⊆ f ⁻¹' f '' t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 53
} | {
"line": 758,
"column": 2
} | [
{
"pp": "X : 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)\n⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exists... | [
"X : 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)\nt : Set ι\nhtc : t.Countable\nhtu : ⋃ i ∈ t, U i = ⋃ i, U i\n⊢ ∃ k, ⋃ n, U (k n) = ⋃ i, U i"
] | obtain ⟨t, htc, htu⟩ := eq_open_union_countable U h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Connected.Clopen | {
"line": 583,
"column": 73
} | {
"line": 584,
"column": 44
} | {
"line": 584,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\nι✝ : Type u_1\nX : ι✝ → Type u_2\ninst✝ : TopologicalSpace α\ns t u v : Set α\nι : Type u_3\nU : ι → Set α\nhclopen : ∀ (i : ι), IsClopen (U i)\nhdisj : Pairwise (Disjoint on U)\nhunion : ⋃ i, U i = univ\nhconn : ∀ (i : ι), IsPreconnected (U i)\nheq : ∀ {x : α} {i : ι} (hx : x ∈... | [] | by
simp [← heq x.2, ← heq y.2, hxy] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.GDelta.Basic | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 18
} | {
"line": 312,
"column": 19
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : s = ∅\n⊢ IsMeagre s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsMeagre",
"congrArg",
"id",
"Set.instEmptyCollection",
"EmptyCollection.emptyCollection",
"Eq",
... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\nhs : s = ∅\n⊢ IsMeagre ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 648,
"column": 69
} | {
"line": 648,
"column": 80
} | {
"line": 648,
"column": 81
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\ns : Set α\nU : Fin n → Set α\nh₃ : Pairwise (Disjoint on (Equiv.piCongrLeft (fun x ↦ Set α) (Equiv.swap 0 i)).symm (Fin.cons s U))\nh₁ : ∀ (i_1 : Fin (n + 1)), IsClopen (Fin.cons ... | [
"α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\ns : Set α\nU : Fin n → Set α\nh₃ : Pairwise (Disjoint on (Equiv.piCongrLeft (fun x ↦ Set α) (Equiv.swap 0 i)).symm (Fin.cons s U))\nh₁ : ∀ (i_1 : Fin (n + 1)), IsClopen (Fin.cons s U ((Equiv.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 649,
"column": 44
} | {
"line": 649,
"column": 55
} | {
"line": 649,
"column": 56
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\ns : Set α\nU : Fin n → Set α\nh₃ : Pairwise (Disjoint on (Equiv.piCongrLeft (fun x ↦ Set α) (Equiv.swap 0 i)).symm (Fin.cons s U))\nh₁ : ∀ (i_1 : Fin (n + 1)), IsClopen (Fin.cons ... | [
"α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\ns : Set α\nU : Fin n → Set α\nh₃ : Pairwise (Disjoint on (Equiv.piCongrLeft (fun x ↦ Set α) (Equiv.swap 0 i)).symm (Fin.cons s U))\nh₁ : ∀ (i_1 : Fin (n + 1)), IsClopen (Fin.cons s U ((Equiv.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 651,
"column": 6
} | {
"line": 651,
"column": 42
} | {
"line": 651,
"column": 43
} | [
{
"pp": "case inr.succ.succ.cons.refine_1\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint... | [
"case inr.succ.succ.cons.refine_1\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint on (Equiv.p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homeomorph.Lemmas | {
"line": 69,
"column": 15
} | {
"line": 70,
"column": 9
} | {
"line": 70,
"column": 10
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nh : X ≃ₜ Y\nhs : IsPreconnected (⇑h '' s)\n⊢ IsPreconnected s",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ns : Set X\nh : X ≃ₜ Y\nhs : IsPreconnected (⇑h '' s)\n⊢ IsPreconnected s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Rel.Separated | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 40
} | {
"line": 60,
"column": 41
} | [
{
"pp": "X : Type u_1\nR : SetRel X X\ns : Set X\nx : X\ninst✝ : R.IsSymm\nthis : Std.Symm fun x y ↦ ¬(x, y) ∈ R\n⊢ R.IsSeparated (insert x s) ↔ R.IsSeparated s ∧ ∀ (y : X), y ∈ s → (x, y) ∈ R → x = y",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"SetRel",
"Membership.mem",
... | [
"X : Type u_1\nR : SetRel X X\ns : Set X\nx : X\ninst✝ : R.IsSymm\nthis : Std.Symm fun x y ↦ ¬(x, y) ∈ R\n⊢ ((insert x s).Pairwise fun x y ↦ ¬(x, y) ∈ R) ↔\n (s.Pairwise fun x y ↦ ¬(x, y) ∈ R) ∧ ∀ (y : X), y ∈ s → (x, y) ∈ R → x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 652,
"column": 6
} | {
"line": 652,
"column": 42
} | {
"line": 652,
"column": 43
} | [
{
"pp": "case inr.succ.succ.cons.refine_2\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint... | [
"case inr.succ.succ.cons.refine_2\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint on (Equiv.p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homeomorph.Lemmas | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 26
} | {
"line": 124,
"column": 27
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nh : X ≃ₜ Y\n⊢ (coclosedCompact X).HasBasis (fun s ↦ IsClosed[inst✝] s ∧ IsCompact s) fun i ↦ ⇑h ⁻¹' iᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Compl.compl",
"id",
"Filte... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nh : X ≃ₜ Y\n⊢ (coclosedCompact X).HasBasis (fun s ↦ IsClosed[inst✝] s ∧ IsCompact s) fun i ↦ (⇑h ⁻¹' i)ᶜ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 654,
"column": 8
} | {
"line": 654,
"column": 27
} | {
"line": 654,
"column": 28
} | [
{
"pp": "α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint on (Equiv.piCongrLeft (fun x ↦ Se... | [
"α : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint on (Equiv.piCongrLeft (fun x ↦ Set α) (Equiv.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Baire.Lemmas | {
"line": 259,
"column": 2
} | {
"line": 259,
"column": 13
} | {
"line": 259,
"column": 14
} | [
{
"pp": "X : Type u_1\nι : Sort u_3\ninst✝³ : TopologicalSpace X\ninst✝² : BaireSpace X\ninst✝¹ : Nonempty X\ninst✝ : Countable ι\nf : ι → Set X\nhc : ∀ (i : ι), IsClosed[inst✝³] (f i)\nhU : ⋃ i, f i = univ\n⊢ ∃ i, (interior (f i)).Nonempty",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": []... | [
"X : Type u_1\nι : Sort u_3\ninst✝³ : TopologicalSpace X\ninst✝² : BaireSpace X\ninst✝¹ : Nonempty X\ninst✝ : Countable ι\nf : ι → Set X\nhc : ∀ (i : ι), IsClosed[inst✝³] (f i)\nhU : ⋃ i, f i = univ\n⊢ ∃ i, (interior (f i)).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 655,
"column": 6
} | {
"line": 656,
"column": 42
} | {
"line": 657,
"column": 8
} | [
{
"pp": "case inr.succ.succ.cons.refine_3\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint... | [
"case inr.succ.succ.cons.refine_3\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint on (Equiv.p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homeomorph.Lemmas | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 72
} | {
"line": 230,
"column": 73
} | [
{
"pp": "X : Type u_1\nY✝ : Type u_2\nW : Type u_3\nZ : Type u_4\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace W\ninst✝³ : TopologicalSpace Z\nX' : Type u_5\nY' : Type u_6\ninst✝² : TopologicalSpace X'\ninst✝¹ : TopologicalSpace Y'\nι : Type u_7\nι' : Type u_8\nY : ι' → T... | [
"X : Type u_1\nY✝ : Type u_2\nW : Type u_3\nZ : Type u_4\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y✝\ninst✝⁴ : TopologicalSpace W\ninst✝³ : TopologicalSpace Z\nX' : Type u_5\nY' : Type u_6\ninst✝² : TopologicalSpace X'\ninst✝¹ : TopologicalSpace Y'\nι : Type u_7\nι' : Type u_8\nY : ι' → Type u_9\nins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 658,
"column": 6
} | {
"line": 658,
"column": 67
} | {
"line": 658,
"column": 68
} | [
{
"pp": "case inr.succ.succ.cons.refine_4\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint... | [
"case inr.succ.succ.cons.refine_4\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nn : ℕ\ni : Fin (n + 1)\nU : Fin n → Set α\na b : Set α\nha : IsClopen a\nhb : IsClopen b\nha' : a.Nonempty\nhb' : b.Nonempty\nhab : Disjoint a b\nh₃ : Pairwise (Disjoint on (Equiv.p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Rel.Cover | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 21
} | {
"line": 78,
"column": 22
} | [
{
"pp": "X : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N ↦ N ⊆ s ∧ U.IsSeparated N) N\nx : X\nhx : x ∈ s\nh : ∀ (y : X), y ∈ N → ¬(x, y) ∈ U\n⊢ False",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"X : Type u_1\nU : SetRel X X\ns N : Set X\ninst✝¹ : U.IsRefl\ninst✝ : U.IsSymm\nhN : Maximal (fun N ↦ N ⊆ s ∧ U.IsSeparated N) N\nx : X\nhx : x ∈ s\nh : ∀ (y : X), y ∈ N → ¬(x, y) ∈ U\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Connected.Clopen | {
"line": 694,
"column": 2
} | {
"line": 694,
"column": 91
} | {
"line": 695,
"column": 4
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : ∀ (f : α → Bool), ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y\nu v : Set α\nu_op : IsOpen[inst✝] u\nv_op : IsOpen[inst✝] v\nhsuv : s ⊆ u ∪ v\nx : α\nx_in_s : x ∈ s\nx_in_u : x ∈ u\nH : s ∩ (u ∩ v) = ∅\ny : α\ny_in_s : y ∈ s\ny_in_v : y ∈ v\nhy ... | [
"α : Type u\ninst✝ : TopologicalSpace α\ns : Set α\nhs : ∀ (f : α → Bool), ContinuousOn f s → ∀ x ∈ s, ∀ y ∈ s, f x = f y\nu v : Set α\nu_op : IsOpen[inst✝] u\nv_op : IsOpen[inst✝] v\nhsuv : s ⊆ u ∪ v\nx : α\nx_in_s : x ∈ s\nx_in_u : x ∈ u\nH : s ∩ (u ∩ v) = ∅\ny : α\ny_in_s : y ∈ s\ny_in_v : y ∈ v\nhy : y ∉ u\nthi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 13
} | {
"line": 150,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\nx : α\np : Filter ι\np' : Filter α\nh✝ : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\ni : ι\nh : ∀ᶠ (y : α) in p', (f y, F i y) ∈ u\n⊢ i ∈ (fun x_1 ↦ (f x, F x_1 x)) ⁻¹' u... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\nx : α\np : Filter ι\np' : Filter α\nh✝ : TendstoUniformlyOnFilter F f p p'\nhx : 𝓟 {x} ≤ p'\nu : Set (β × β)\nhu : u ∈ 𝓤 β\ni : ι\nh : ∀ᶠ (y : α) in p', (f y, F i y) ∈ u\n⊢ (f x, F i x) ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 51
} | {
"line": 235,
"column": 52
} | [
{
"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\ns : Set α\np : Filter ι\nh : TendstoUniformlyOnFilter F f p (𝓟 s)\ng : γ → α\n⊢ TendstoUniformlyOnFilter (fun n ↦ F n ∘ g) (f ∘ g) p (𝓟 (g ⁻¹' s))",
"ppTerm": "?m.44",
"assigned": false,
... | [
"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\ns : Set α\np : Filter ι\nh : TendstoUniformlyOnFilter F f p (𝓟 s)\ng : γ → α\n⊢ TendstoUniformlyOnFilter (fun n ↦ F n ∘ g) (f ∘ g) p (𝓟 (g ⁻¹' s))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 47
} | {
"line": 241,
"column": 48
} | [
{
"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 : TendstoUniformlyOnFilter F f p ⊤\ng : γ → α\n⊢ TendstoUniformlyOnFilter (fun n ↦ F n ∘ g) (f ∘ g) p ⊤",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
... | [
"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 ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 37
} | {
"line": 121,
"column": 38
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : UniformSpace β\nf : Filter α\ng : Filter β\nhf : Cauchy f\nhg : Cauchy g\nthis✝ : f.NeBot\nthis : g.NeBot\n⊢ Cauchy (f ×ˢ g)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Filter.map_snd_prod",
... | [
"α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : UniformSpace β\nf : Filter α\ng : Filter β\nhf : Cauchy f\nhg : Cauchy g\nthis✝ : f.NeBot\nthis : g.NeBot\n⊢ Cauchy f ∧ Cauchy g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 179,
"column": 4
} | {
"line": 180,
"column": 55
} | {
"line": 180,
"column": 56
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : UniformSpace β\nf : Filter α\nm : α → β\nhf : Cauchy f\ns : Set α\nhm : UniformContinuousOn m s\nhfs : f ≤ 𝓟 s\nthis : Cauchy (Filter.comap Subtype.val f)\n⊢ Cauchy (map m f)",
"ppTerm": "?m.23",
"assigned": false,
"usedConstan... | [
"α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : UniformSpace β\nf : Filter α\nm : α → β\nhf : Cauchy f\ns : Set α\nhm : UniformContinuousOn m s\nhfs : f ≤ 𝓟 s\nthis : Cauchy (Filter.comap Subtype.val f)\n⊢ Cauchy (map m f)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 67
} | {
"line": 191,
"column": 68
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : Preorder β\nu : β → α\nh : CauchySeq u\n⊢ Tendsto (Prod.map u u) atTop (𝓤 α)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"uniformity",
"id",
"Prod.map",
"Filter.atTop",
"Filter.Tendst... | [
"α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝ : Preorder β\nu : β → α\nh : CauchySeq u\n⊢ map (Prod.map u u) atTop ≤ 𝓤 α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 22
} | {
"line": 200,
"column": 23
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nβ : Type u_1\ninst✝ : SemilatticeSup β\nu : β → α\nh : CauchySeq u\nV : SetRel α α\nhV : V ∈ 𝓤 α\nthis✝ : Nonempty β\nthis : Tendsto (Prod.map u u) (atTop ×ˢ atTop) (𝓤 α)\n⊢ ∃ k₀, ∀ (i j : β), k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V",
"ppTerm": "?m.52",
"as... | [
"α : Type u\nuniformSpace : UniformSpace α\nβ : Type u_1\ninst✝ : SemilatticeSup β\nu : β → α\nh : CauchySeq u\nV : SetRel α α\nhV : V ∈ 𝓤 α\nthis✝ : Nonempty β\nthis : Tendsto (Prod.map u u) (atTop ×ˢ atTop) (𝓤 α)\n⊢ ∃ k₀, ∀ (i j : β), k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 58
} | {
"line": 225,
"column": 59
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nu : ℕ → α\nf : ℕ ≃ ℕ\nH : CauchySeq (u ∘ ⇑f)\n⊢ CauchySeq u",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nuniformSpace : UniformSpace α\nu : ℕ → α\nf : ℕ ≃ ℕ\nH : CauchySeq (u ∘ ⇑f)\n⊢ CauchySeq u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 45
} | {
"line": 275,
"column": 46
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : 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'\nf' : α' → β'\np' : Filter ι'\ns' : Set α'\nh : TendstoUniformlyOnFilter F f p (𝓟 s)\nh' : Tends... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : UniformSpace β\nF : ι → α → β\nf : 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'\nf' : α' → β'\np' : Filter ι'\ns' : Set α'\nh : TendstoUniformlyOnFilter F f p (𝓟 s)\nh' : TendstoUniformlyO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 69
} | {
"line": 244,
"column": 70
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\nγ : Type u_1\nδ : Type u_2\ninst✝² : UniformSpace β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nu : γ → α\nv : δ → β\nhu : CauchySeq u\nhv : CauchySeq v\n⊢ CauchySeq (Prod.map u v)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [... | [
"α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\nγ : Type u_1\nδ : Type u_2\ninst✝² : UniformSpace β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nu : γ → α\nv : δ → β\nhu : CauchySeq u\nhv : CauchySeq v\n⊢ Cauchy (map (Prod.map u v) atTop)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 366,
"column": 2
} | {
"line": 367,
"column": 9
} | {
"line": 367,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : UniformSpace β\nx : α\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace γ\nU : Set α\nhU : U ∈ 𝓝 x\nF : α → β → γ\nhF : UniformContinuousOn (↿F) (U ×ˢ univ)\n⊢ TendstoUniformly F (F x) (𝓝 x)",
"ppTerm": "?m.25",
"assigned": false,
"usedConst... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : UniformSpace β\nx : α\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace γ\nU : Set α\nhU : U ∈ 𝓝 x\nF : α → β → γ\nhF : UniformContinuousOn (↿F) (U ×ˢ univ)\n⊢ TendstoUniformly F (F x) (𝓝 x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 38
} | {
"line": 61,
"column": 38
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ng : β → γ\nhg : IsUniformInducing g\nf : α → β\nhf : IsUniformInducing f\n⊢ comap (fun x ↦ ((g ∘ f) x.1, (g ∘ f) x.2)) (𝓤 γ) = 𝓤 α",
"ppTerm": "?m.21",
"assigned": true,
"usedCons... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ng : β → γ\nhg : IsUniformInducing g\nf : α → β\nhf : IsUniformInducing f\n⊢ comap (fun x ↦ ((g ∘ f) x.1, (g ∘ f) x.2)) (𝓤 γ) = comap ((fun x ↦ (g x.1, g x.2)) ∘ fun x ↦ (f x.1, f x.2)) (𝓤 γ)"
] | rw [← hf.1, ← hg.1, comap_comap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Basic | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 69
} | {
"line": 285,
"column": 2
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nι : Type u_2\nxs : ι → α\nxs_dense : DenseRange xs\nU : SetRel α α\nhU : U ∈ 𝓤 α\n⊢ ⋃ i, ball (xs i) U = univ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Set.biUnion_range",
... | [
"α : Type ua\ninst✝ : UniformSpace α\nι : Type u_2\nxs : ι → α\nxs_dense : DenseRange xs\nU : SetRel α α\nhU : U ∈ 𝓤 α\n⊢ ⋃ x ∈ range xs, ball x U = univ"
] | rw [← biUnion_range (f := xs) (g := fun x ↦ UniformSpace.ball x U)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 391,
"column": 11
} | {
"line": 391,
"column": 56
} | {
"line": 391,
"column": 57
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompleteSpace (α × β)\nh : Nonempty β\nf✝ : Filter α\nhf : Cauchy f✝\ny : β\na : α\nb : β\nhab : f✝ ×ˢ pure y ≤ 𝓝 (a, b)\n⊢ f✝ ≤ 𝓝 a",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"us... | [
"α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompleteSpace (α × β)\nh : Nonempty β\nf✝ : Filter α\nhf : Cauchy f✝\ny : β\na : α\nb : β\nhab : f✝ ×ˢ pure y ≤ 𝓝 (a, b)\n⊢ f✝ ≤ 𝓝 a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 398,
"column": 11
} | {
"line": 398,
"column": 56
} | {
"line": 398,
"column": 57
} | [
{
"pp": "α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompleteSpace (α × β)\nh : Nonempty α\nf✝ : Filter β\nhf : Cauchy f✝\nx a : α\nb : β\nhab : pure x ×ˢ f✝ ≤ 𝓝 (a, b)\n⊢ f✝ ≤ 𝓝 b",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"α : Type u\nβ : Type v\nuniformSpace : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : CompleteSpace (α × β)\nh : Nonempty α\nf✝ : Filter β\nhf : Cauchy f✝\nx a : α\nb : β\nhab : pure x ×ˢ f✝ ≤ 𝓝 (a, b)\n⊢ f✝ ≤ 𝓝 b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 460,
"column": 82
} | {
"line": 466,
"column": 58
} | {
"line": 468,
"column": 0
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\ninst✝ : DiscreteUniformity α\nf : Filter α\nhf : Cauchy f\n⊢ ∃ x, f = pure x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.UniformSpace.Cauchy.0.DiscreteUniformity.eq_pure_of_cauchy._simp_1_2",
"Pur... | [] | by
rcases hf with ⟨f_ne_bot, f_le⟩
simp only [DiscreteUniformity.eq_principal_setRelId, le_principal_iff, mem_prod_iff] at f_le
obtain ⟨S, hS, T, hT, H⟩ := f_le
obtain ⟨x, rfl, _, _, _⟩ := SetRel.exists_eq_singleton_of_prod_subset_id
(f_ne_bot.nonempty_of_mem hS) (f_ne_bot.nonempty_of_mem hT) H
exact ⟨x, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 176,
"column": 20
} | {
"line": 176,
"column": 49
} | {
"line": 176,
"column": 50
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nx : α × α\nh : x ∈ Prod.map Sum.inl Sum.inl ⁻¹' (Prod.map Sum.inl Sum.inl '' s ∪ range (Prod.map Sum.inr Sum.inr))\n⊢ x ∈ s",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nx : α × α\nh : x ∈ Prod.map Sum.inl Sum.inl ⁻¹' (Prod.map Sum.inl Sum.inl '' s ∪ range (Prod.map Sum.inr Sum.inr))\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 182,
"column": 20
} | {
"line": 182,
"column": 49
} | {
"line": 182,
"column": 50
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nx : β × β\nh : x ∈ Prod.map Sum.inr Sum.inr ⁻¹' (range (Prod.map Sum.inl Sum.inl) ∪ Prod.map Sum.inr Sum.inr '' s)\n⊢ x ∈ s",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
... | [
"α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nx : β × β\nh : x ∈ Prod.map Sum.inr Sum.inr ⁻¹' (range (Prod.map Sum.inl Sum.inl) ∪ Prod.map Sum.inr Sum.inr '' s)\n⊢ x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 205,
"column": 17
} | {
"line": 205,
"column": 42
} | {
"line": 205,
"column": 43
} | [
{
"pp": "β : Type v\ninst✝ : UniformSpace β\nα : Type u_1\nf : α → β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nhf : Pairwise fun x y ↦ (f x, f y) ∉ s\nx y : α\n⊢ (x, y) ∈ Prod.map f f ⁻¹' s → (x, y) ∈ SetRel.id",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetRel.id",
... | [
"β : Type v\ninst✝ : UniformSpace β\nα : Type u_1\nf : α → β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nhf : Pairwise fun x y ↦ (f x, f y) ∉ s\nx y : α\n⊢ (f x, f y) ∈ s → x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Basic | {
"line": 434,
"column": 84
} | {
"line": 436,
"column": 46
} | {
"line": 438,
"column": 0
} | [
{
"pp": "α : Type u_2\n⊢ UniformSpace.comap id = id",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"UniformSpace",
"Eq.mpr",
"congrArg",
"uniformity",
"id",
"Prod.map",
"UniformSpace.ext",
"funext",
"Filter.comap_id",
"Eq.refl",
... | [] | by
ext : 2
rw [uniformity_comap, Prod.map_id, comap_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 248,
"column": 6
} | {
"line": 249,
"column": 33
} | {
"line": 250,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\np : α → Prop\ne : α → β\nue : IsUniformEmbedding e\nde : IsDenseEmbedding e\n⊢ comap (fun x ↦ (IsDenseEmbedding.subtypeEmb p e x.1, IsDenseEmbedding.subtypeEmb p e x.2))\n (𝓤 { x // x ∈ closure[inst✝.toTopologicalSpace] (e '... | [] | simp [comap_comap, Function.comp_def, IsDenseEmbedding.subtypeEmb, uniformity_subtype,
ue.comap_uniformity.symm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 248,
"column": 6
} | {
"line": 249,
"column": 33
} | {
"line": 250,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\np : α → Prop\ne : α → β\nue : IsUniformEmbedding e\nde : IsDenseEmbedding e\n⊢ comap (fun x ↦ (IsDenseEmbedding.subtypeEmb p e x.1, IsDenseEmbedding.subtypeEmb p e x.2))\n (𝓤 { x // x ∈ closure[inst✝.toTopologicalSpace] (e '... | [] | simp [comap_comap, Function.comp_def, IsDenseEmbedding.subtypeEmb, uniformity_subtype,
ue.comap_uniformity.symm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 248,
"column": 6
} | {
"line": 249,
"column": 33
} | {
"line": 250,
"column": 4
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\np : α → Prop\ne : α → β\nue : IsUniformEmbedding e\nde : IsDenseEmbedding e\n⊢ comap (fun x ↦ (IsDenseEmbedding.subtypeEmb p e x.1, IsDenseEmbedding.subtypeEmb p e x.2))\n (𝓤 { x // x ∈ closure[inst✝.toTopologicalSpace] (e '... | [] | simp [comap_comap, Function.comp_def, IsDenseEmbedding.subtypeEmb, uniformity_subtype,
ue.comap_uniformity.symm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 44
} | {
"line": 298,
"column": 45
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nhf : IsUniformInducing f\nhsurj : Surjective f\n⊢ CompleteSpace α ↔ CompleteSpace β",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsComplete",
"CompleteSpace",
"... | [
"α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nhf : IsUniformInducing f\nhsurj : Surjective f\n⊢ IsComplete (range f) ↔ CompleteSpace β"
] | completeSpace_iff_isComplete_range hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 43
} | {
"line": 468,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : IsUniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\na : α\n⊢ Tendsto f (comap e (𝓝 a)) (𝓝 (⋯.extend f a))",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ne : β → α\nh_e : IsUniformInducing e\nh_dense : DenseRange e\nf : β → γ\nh_f : UniformContinuous f\ninst✝ : CompleteSpace γ\na : α\n⊢ Tendsto f (comap e (𝓝 a)) (𝓝 ((comap e (𝓝 a)).limUnder f))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 56
} | {
"line": 529,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\ns : Set α\np : Filter ι\nγ : Type u_5\nhf : UniformCauchySeqOnFilter F p (𝓟 s)\ng : γ → α\n⊢ UniformCauchySeqOnFilter (fun n ↦ F n ∘ g) p (𝓟 (g ⁻¹' s))",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants":... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\ns : Set α\np : Filter ι\nγ : Type u_5\nhf : UniformCauchySeqOnFilter F p (𝓟 s)\ng : γ → α\n⊢ UniformCauchySeqOnFilter (fun n ↦ F n ∘ g) p (𝓟 (g ⁻¹' s))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 703,
"column": 6
} | {
"line": 703,
"column": 35
} | {
"line": 703,
"column": 36
} | [
{
"pp": "α : 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\nc : Filter α\nhcf : c ≤ f\nhc : Cauc... | [
"α : 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\nc : Filter α\nhcf : c ≤ f\nhc : Cauchy c\nm : Se... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 744,
"column": 16
} | {
"line": 744,
"column": 27
} | {
"line": 744,
"column": 28
} | [
{
"pp": "α✝ : Type u\nβ : Type v\nuniformSpace : UniformSpace α✝\nα : Type u\ninst✝¹ : UniformSpace α\ninst✝ : CompactSpace α\nf✝ : Filter α\nhf : Cauchy f✝\n⊢ ∃ x, f✝ ≤ 𝓝 x",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α✝ : Type u\nβ : Type v\nuniformSpace : UniformSpace α✝\nα : Type u\ninst✝¹ : UniformSpace α\ninst✝ : CompactSpace α\nf✝ : Filter α\nhf : Cauchy f✝\n⊢ ∃ x, f✝ ≤ 𝓝 x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 582,
"column": 6
} | {
"line": 583,
"column": 13
} | {
"line": 583,
"column": 14
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\nγ : Type u_3\ninst✝² : UniformSpace γ\ninst✝¹ : CompleteSpace β\ninst✝ : CompleteSpace γ\ni : α → β\nf : α → γ\nhid : IsDenseInducing i\nhi : IsUniformInducing i\nh : IsUniformInducing f\nsf : α → SeparationQuotient γ... | [
"case h\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\nγ : Type u_3\ninst✝² : UniformSpace γ\ninst✝¹ : CompleteSpace β\ninst✝ : CompleteSpace γ\ni : α → β\nf : α → γ\nhid : IsDenseInducing i\nhi : IsUniformInducing i\nh : IsUniformInducing f\nsf : α → SeparationQuotient γ := Separati... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 15
} | {
"line": 596,
"column": 16
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\nγ : Type u_3\ninst✝² : UniformSpace γ\ninst✝¹ : CompleteSpace β\ninst✝ : CompleteSpace γ\ni : α → β\nf : α → γ\nhid : IsDenseInducing i\nhi : IsUniformInducing i\nh : IsUniformInducing f\nsf : α → SeparationQuotient γ := Sepa... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β\nγ : Type u_3\ninst✝² : UniformSpace γ\ninst✝¹ : CompleteSpace β\ninst✝ : CompleteSpace γ\ni : α → β\nf : α → γ\nhid : IsDenseInducing i\nhi : IsUniformInducing i\nh : IsUniformInducing f\nsf : α → SeparationQuotient γ := SeparationQuotie... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 879,
"column": 6
} | {
"line": 879,
"column": 43
} | {
"line": 879,
"column": 43
} | [
{
"pp": "case refine_2\nα : Type u\nuniformSpace : UniformSpace α\nf : Filter α\nhf : Cauchy f\nU : ℕ → SetRel α α\nU_mem : ∀ (n : ℕ), U n ∈ 𝓤 α\nU_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s\na : α\nha : Tendsto (seq hf U_mem) atTop (𝓝 a)\ns : Set (α × α)\nhs : s ∈ 𝓤 α\nm : ℕ\nhm : U m ⊆ s\nn : ℕ\nhn : ∀ (b : ℕ), n ≤ b →... | [] | · exact hm (hn _ <| le_max_right m n) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.Compact | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | {
"line": 116,
"column": 14
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\nK : Set α\nS : Set (Set α)\nhK : IsCompact K\nhopen : ∀ s ∈ S, IsOpen[inst✝.toTopologicalSpace] s\nhcover : K ⊆ ⋃ i, ↑i\n⊢ ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, ball x V ⊆ s",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"u... | [
"α : Type ua\ninst✝ : UniformSpace α\nK : Set α\nS : Set (Set α)\nhK : IsCompact K\nhopen : ∀ s ∈ S, IsOpen[inst✝.toTopologicalSpace] s\nhcover : K ⊆ ⋃ i, ↑i\n⊢ ∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, ball x V ⊆ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Compact | {
"line": 129,
"column": 8
} | {
"line": 129,
"column": 72
} | {
"line": 129,
"column": 73
} | [
{
"pp": "α : Type ua\nι : Sort u_1\ninst✝ : UniformSpace α\nK : Set α\np : ι → Prop\nV : ι → Set (α × α)\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nU : Set α\nH : U ∈ 𝓝ˢ K\n⊢ K ⊆ ⋃ x, interior U",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq... | [
"α : Type ua\nι : Sort u_1\ninst✝ : UniformSpace α\nK : Set α\np : ι → Prop\nV : ι → Set (α × α)\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nU : Set α\nH : U ∈ 𝓝ˢ K\n⊢ U ∈ 𝓝ˢ K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Compact | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 19
} | {
"line": 131,
"column": 20
} | [
{
"pp": "α : Type ua\nι : Sort u_1\ninst✝ : UniformSpace α\nK : Set α\np : ι → Prop\nV : ι → Set (α × α)\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nU : Set α\nH : U ∈ 𝓝ˢ K\nHKU : K ⊆ ⋃ x, interior U\n⊢ ∃ i, p i ∧ ⋃ x ∈ K, ball x (V i) ⊆ interior U",
"ppTerm": "?m.67",
"assigned": true,
"usedC... | [
"α : Type ua\nι : Sort u_1\ninst✝ : UniformSpace α\nK : Set α\np : ι → Prop\nV : ι → Set (α × α)\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nU : Set α\nH : U ∈ 𝓝ˢ K\nHKU : K ⊆ ⋃ x, interior U\n⊢ ∃ i, p i ∧ ∀ i_1 ∈ K, ball i_1 (V i) ⊆ interior U"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Support | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 70
} | {
"line": 442,
"column": 71
} | [
{
"pp": "α : Type u_9\nβ : Type u_10\ninst✝¹ : TopologicalSpace α\ninst✝ : DivisionMonoid β\nf : α → β\nhf : HasCompactMulSupport f\n⊢ HasCompactMulSupport f⁻¹",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Function.mulSupport_inv",
"Eq.mpr",
"InvOneClass.toOne",
... | [
"α : Type u_9\nβ : Type u_10\ninst✝¹ : TopologicalSpace α\ninst✝ : DivisionMonoid β\nf : α → β\nhf : HasCompactMulSupport f\n⊢ IsCompact (closure[inst✝¹] (mulSupport f))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 973,
"column": 4
} | {
"line": 973,
"column": 38
} | {
"line": 973,
"column": 39
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\nhs : ∀ U ∈ 𝓤 α, ∃ t, t.Countable ∧ ⋃ x ∈ t, ball x U = univ\n⊢ ∀ U ∈ 𝓤 α, ∃ t, t.Countable ∧ univ ⊆ ⋃ x ∈ t, ball x U",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [
"α : Type u\nuniformSpace : UniformSpace α\ninst✝ : (𝓤 α).IsCountablyGenerated\nhs : ∀ U ∈ 𝓤 α, ∃ t, t.Countable ∧ ⋃ x ∈ t, ball x U = univ\n⊢ ∀ U ∈ 𝓤 α, ∃ t, t.Countable ∧ ⋃ x ∈ t, ball x U = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformApproximation | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 74
} | {
"line": 65,
"column": 75
} | [
{
"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⊢ ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝[univ] x, ∃ F, ContinuousWithinAt F univ x ∧ ∀ y ∈ t, (f y, F y) ∈ u",
"ppTerm": "?m.56",
"assign... | [
"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⊢ ∀ u ∈ 𝓤 β, ∃ t ∈ 𝓝 x, ∃ F, ContinuousAt F x ∧ ∀ y ∈ t, (f y, F y) ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformApproximation | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 37
} | {
"line": 94,
"column": 38
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nf : α → β\nL : ∀ u ∈ 𝓤 β, ∃ F, Continuous[inst✝¹, inst✝.toTopologicalSpace] F ∧ ∀ (y : α), (f y, F y) ∈ u\n⊢ ∀ u ∈ 𝓤 β, ∃ F, ContinuousOn F univ ∧ ∀ y ∈ univ, (f y, F y) ∈ u",
"ppTerm": "?m.41",
"assigned": true,... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nf : α → β\nL : ∀ u ∈ 𝓤 β, ∃ F, Continuous[inst✝¹, inst✝.toTopologicalSpace] F ∧ ∀ (y : α), (f y, F y) ∈ u\n⊢ ∀ u ∈ 𝓤 β, ∃ F, Continuous[inst✝¹, inst✝.toTopologicalSpace] F ∧ ∀ (y : α), (f y, F y) ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Equiv | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 72
} | {
"line": 298,
"column": 6
} | [
{
"pp": "α : Type u\nβ✝ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β✝\ninst✝² : UniformSpace γ\ninst✝¹ : UniformSpace δ\nι : Type u_4\nι' : Type u_5\nβ : ι' → Type u_6\ninst✝ : (j : ι') → UniformSpace (β j)\ne : ι ≃ ι'\ni : ι\n⊢ UniformContinuous fun x ↦ (Equiv.piCong... | [
"α : Type u\nβ✝ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝⁴ : UniformSpace α\ninst✝³ : UniformSpace β✝\ninst✝² : UniformSpace γ\ninst✝¹ : UniformSpace δ\nι : Type u_4\nι' : Type u_5\nβ : ι' → Type u_6\ninst✝ : (j : ι') → UniformSpace (β j)\ne : ι ≃ ι'\ni : ι\n⊢ UniformContinuous fun x ↦ x i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Equiv | {
"line": 391,
"column": 4
} | {
"line": 391,
"column": 47
} | {
"line": 391,
"column": 48
} | [
{
"pp": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : UniformSpace δ\np : α → Prop\nq : β → Prop\ne : α ≃ᵤ β\nh : ∀ (a : α), p a ↔ q (e a)\n⊢ UniformContinuous (e.subtypeEquiv h).toFun",
"ppTerm": "?m.21",
"assig... | [
"α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : UniformSpace δ\np : α → Prop\nq : β → Prop\ne : α ≃ᵤ β\nh : ∀ (a : α), p a ↔ q (e a)\n⊢ UniformContinuous (Subtype.map ⇑e ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Equiv | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 47
} | {
"line": 393,
"column": 48
} | [
{
"pp": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : UniformSpace δ\np : α → Prop\nq : β → Prop\ne : α ≃ᵤ β\nh : ∀ (a : α), p a ↔ q (e a)\n⊢ UniformContinuous (e.subtypeEquiv h).invFun",
"ppTerm": "?m.40",
"assi... | [
"α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : UniformSpace β\ninst✝¹ : UniformSpace γ\ninst✝ : UniformSpace δ\np : α → Prop\nq : β → Prop\ne : α ≃ᵤ β\nh : ∀ (a : α), p a ↔ q (e a)\n⊢ UniformContinuous (Subtype.map ⇑e.symm ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Equiv | {
"line": 403,
"column": 65
} | {
"line": 403,
"column": 76
} | {
"line": 403,
"column": 77
} | [
{
"pp": "α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α ≃ β\nhf : IsUniformInducing ⇑f\n⊢ UniformContinuous (⇑f ∘ f.invFun)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"Eq.mpr",
"Equiv.... | [
"α : Type u\nβ : Type u_1\nγ : Type u_2\nδ : Type u_3\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α ≃ β\nhf : IsUniformInducing ⇑f\n⊢ UniformContinuous id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 88,
"column": 30
} | {
"line": 88,
"column": 41
} | {
"line": 88,
"column": 42
} | [
{
"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\nh : TendstoUniformlyOn F f p s\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx✝¹ : α\nx✝ : x✝¹ ∈ s\n⊢ ∀ᶠ (n : ι) in p, ∀ y ∈ s, (f y, F n y) ∈ u",
"ppTerm": "?m.22",
... | [
"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\nh : TendstoUniformlyOn F f p s\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx✝¹ : α\nx✝ : x✝¹ ∈ s\n⊢ ∀ᶠ (n : ι) in p, ∀ y ∈ s, (f y, F n y) ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 91,
"column": 71
} | {
"line": 91,
"column": 82
} | {
"line": 91,
"column": 83
} | [
{
"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\np : Filter ι\nh : TendstoUniformly F f p\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx✝ : α\n⊢ ∀ᶠ (n : ι) in p, ∀ y ∈ univ, (f y, F n y) ∈ u",
"ppTerm": "?m.20",
"assigned": true,
"... | [
"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\np : Filter ι\nh : TendstoUniformly F f p\nu : Set (β × β)\nhu : u ∈ 𝓤 β\nx✝ : α\n⊢ ∀ᶠ (n : ι) in p, ∀ (y : α), (f y, F n y) ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.UniformApproximation | {
"line": 212,
"column": 10
} | {
"line": 212,
"column": 48
} | {
"line": 212,
"column": 49
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nh : ∀ u ∈ 𝓤 β, ∃ F, UniformContinuous F ∧ ∀ (y : α), (f y, F y) ∈ u\n⊢ ∀ u ∈ 𝓤 β, ∃ F, UniformContinuousOn F univ ∧ ∀ y ∈ univ, (f y, F y) ∈ u",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_4\nβ : Type u_5\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\nh : ∀ u ∈ 𝓤 β, ∃ F, UniformContinuous F ∧ ∀ (y : α), (f y, F y) ∈ u\n⊢ ∀ u ∈ 𝓤 β, ∃ F, UniformContinuous F ∧ ∀ (y : α), (f y, F y) ∈ u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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