module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Maps.Basic | {
"line": 157,
"column": 49
} | {
"line": 157,
"column": 87
} | {
"line": 159,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace X\nhf : IsInducing f\ns : Set X\n⊢ IsOpen[inst✝] s ↔ ∃ t, IsOpen[inst✝¹] t ∧ f ⁻¹' t = s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
... | [] | rw [hf.eq_induced, isOpen_induced_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Maps.Basic | {
"line": 157,
"column": 49
} | {
"line": 157,
"column": 87
} | {
"line": 159,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace X\nhf : IsInducing f\ns : Set X\n⊢ IsOpen[inst✝] s ↔ ∃ t, IsOpen[inst✝¹] t ∧ f ⁻¹' t = s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
... | [] | rw [hf.eq_induced, isOpen_induced_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Maps.Basic | {
"line": 157,
"column": 49
} | {
"line": 157,
"column": 87
} | {
"line": 159,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace X\nhf : IsInducing f\ns : Set X\n⊢ IsOpen[inst✝] s ↔ ∃ t, IsOpen[inst✝¹] t ∧ f ⁻¹' t = s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
... | [] | rw [hf.eq_induced, isOpen_induced_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 82
} | {
"line": 632,
"column": 2
} | [
{
"pp": "α : Type u_1\ns : Set (Set α)\nt : Set α\nht : GenerateOpen s t\n⊢ generateFrom (insert t s) = generateFrom s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"TopologicalSpace.generateFrom_anti",
"Insert.insert",
"TopologicalSpace.generateFrom",
"Topological... | [
"α : Type u_1\ns : Set (Set α)\nt : Set α\nht : GenerateOpen s t\n⊢ ∀ s_1 ∈ insert t s, IsOpen[generateFrom s] s_1"
] | refine le_antisymm (generateFrom_anti <| subset_insert t s) (le_generateFrom ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Order | {
"line": 831,
"column": 63
} | {
"line": 832,
"column": 52
} | {
"line": 834,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nι : Sort u_2\nt₁ : ι → TopologicalSpace α\nt₂ : TopologicalSpace β\n⊢ Continuous[iSup t₁, t₂] f ↔ ∀ (i : ι), Continuous[t₁ i, t₂] f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Continuous",
"congrArg",
"iSup",
"PartialO... | [] | by
simp only [continuous_iff_le_induced, iSup_le_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Maps.Basic | {
"line": 534,
"column": 2
} | {
"line": 536,
"column": 22
} | {
"line": 538,
"column": 0
} | [
{
"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 : IsClosedMap g\nhf : IsClosedMap f\n⊢ IsClosedMap (g ∘ f)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | intro s hs
rw [image_comp]
exact hg _ (hf _ hs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Maps.Basic | {
"line": 534,
"column": 2
} | {
"line": 536,
"column": 22
} | {
"line": 538,
"column": 0
} | [
{
"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 : IsClosedMap g\nhf : IsClosedMap f\n⊢ IsClosedMap (g ∘ f)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | intro s hs
rw [image_comp]
exact hg _ (hf _ hs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Finite | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 13
} | {
"line": 68,
"column": 14
} | [
{
"pp": "case mp.univ\nα : Type u\ns : Set (Set α)\nU : Set α\n⊢ ∃ t ⊆ s, t.Finite ∧ ⋂₀ t ⊆ univ",
"ppTerm": "?mp.univ",
"assigned": true,
"usedConstants": [
"Set.finite_empty",
"Set.subset_univ",
"Set.univ",
"Set.Finite",
"HasSubset.Subset",
"And",
"And.int... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.UniformSpace.Defs | {
"line": 613,
"column": 47
} | {
"line": 613,
"column": 63
} | {
"line": 613,
"column": 63
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\na b : α\n⊢ ((𝓤 α).lift' fun s ↦ {y | (y, a) ∈ s}) ×ˢ (𝓤 α).lift' (ball b) =\n (𝓤 α).lift fun s ↦ (𝓤 α).lift' fun t ↦ {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mp... | [
"α : Type ua\ninst✝ : UniformSpace α\na b : α\n⊢ ((𝓤 α).lift fun s ↦ (𝓤 α).lift' fun t ↦ {y | (y, a) ∈ s} ×ˢ ball b t) =\n (𝓤 α).lift fun s ↦ (𝓤 α).lift' fun t ↦ {y | (y, a) ∈ s} ×ˢ {y | (b, y) ∈ t}",
"case hg₁\nα : Type ua\ninst✝ : UniformSpace α\na b : α\n⊢ Monotone fun s ↦ {y | (y, a) ∈ s}",
"case hg₂... | prod_lift'_lift' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Bases.Finite | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 50
} | {
"line": 103,
"column": 2
} | [
{
"pp": "α : Type u_1\nI : Type u_6\nl : I → Filter α\nι : I → Sort u_7\np : (i : I) → ι i → Prop\ns : (i : I) → ι i → Set α\nS : Set I\nhd : S.PairwiseDisjoint l\nhS : S.Finite\nh : ∀ (i : I), (l i).HasBasis (p i) (s i)\n⊢ ∃ ind, (∀ (i : I), p i (ind i)) ∧ S.PairwiseDisjoint fun i ↦ s i (ind i)",
"ppTerm":... | [
"α : Type u_1\nI : Type u_6\nl : I → Filter α\nι : I → Sort u_7\np : (i : I) → ι i → Prop\ns : (i : I) → ι i → Set α\nS : Set I\nhd✝ : S.PairwiseDisjoint l\nhS : S.Finite\nh : ∀ (i : I), (l i).HasBasis (p i) (s i)\nt : I → Set α\nhtl : ∀ (i : I), t i ∈ l i\nhd : S.PairwiseDisjoint t\n⊢ ∃ ind, (∀ (i : I), p i (ind i... | rcases hd.exists_mem_filter hS with ⟨t, htl, hd⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.Filter.Finite | {
"line": 131,
"column": 2
} | {
"line": 133,
"column": 54
} | {
"line": 135,
"column": 0
} | [
{
"pp": "ι : Sort u_2\ninst✝ : Finite ι\nα : Type u_3\nf : ι → Filter α\ns : Set α\n⊢ s ∈ ⨅ i, f i ↔ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ i, t i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"iInf",
"Set.iInter",
"Filter.in... | [] | refine ⟨exists_iInter_of_mem_iInf, ?_⟩
rintro ⟨t, ht, rfl⟩
exact iInter_mem.2 fun i => mem_iInf_of_mem i (ht i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Finite | {
"line": 131,
"column": 2
} | {
"line": 133,
"column": 54
} | {
"line": 135,
"column": 0
} | [
{
"pp": "ι : Sort u_2\ninst✝ : Finite ι\nα : Type u_3\nf : ι → Filter α\ns : Set α\n⊢ s ∈ ⨅ i, f i ↔ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ i, t i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"iInf",
"Set.iInter",
"Filter.in... | [] | refine ⟨exists_iInter_of_mem_iInf, ?_⟩
rintro ⟨t, ht, rfl⟩
exact iInter_mem.2 fun i => mem_iInf_of_mem i (ht i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.CountablyGenerated | {
"line": 196,
"column": 2
} | {
"line": 201,
"column": 36
} | {
"line": 203,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : Filter α\n⊢ f.IsCountablyGenerated ↔ ∃ x, f.HasAntitoneBasis x",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"Filter.HasAntitoneBasis.toHasBasis",
"congrArg",
"Filter.instInfSet",
"Exists",
"id",
... | [] | constructor
· intro h
exact f.exists_antitone_basis
· rintro ⟨x, h⟩
rw [h.1.eq_iInf]
exact isCountablyGenerated_seq x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CountablyGenerated | {
"line": 196,
"column": 2
} | {
"line": 201,
"column": 36
} | {
"line": 203,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : Filter α\n⊢ f.IsCountablyGenerated ↔ ∃ x, f.HasAntitoneBasis x",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"Filter.HasAntitoneBasis.toHasBasis",
"congrArg",
"Filter.instInfSet",
"Exists",
"id",
... | [] | constructor
· intro h
exact f.exists_antitone_basis
· rintro ⟨x, h⟩
rw [h.1.eq_iInf]
exact isCountablyGenerated_seq x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Finite | {
"line": 248,
"column": 55
} | {
"line": 249,
"column": 63
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort u_2\ninst✝ : Finite ι\nl : Filter α\np : ι → α → Prop\n⊢ (∀ᶠ (x : α) in l, ∀ (i : ι), p i x) ↔ ∀ (i : ι), ∀ᶠ (x : α) in l, p i x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Set.iInter",... | [] | by
simpa only [Filter.Eventually, setOf_forall] using iInter_mem | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Heyting.Boundary | {
"line": 99,
"column": 50
} | {
"line": 99,
"column": 85
} | {
"line": 99,
"column": 86
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ a ⊔ (a ⊔ b)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"CoheytingAlgebra.toHNot",
"Lattice.toSemilatticeSup",
"CoheytingAlgebra.toDistribLattice",
"SemilatticeSup.toMax",
... | [] | exact le_sup_of_le_left inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Heyting.Boundary | {
"line": 99,
"column": 50
} | {
"line": 99,
"column": 85
} | {
"line": 99,
"column": 86
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ a ⊔ ￢(a ⊔ b)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"CoheytingAlgebra.toHNot",
"Lattice.toSemilatticeSup",
"CoheytingAlgebra.toDistribLattice",
"SemilatticeSup.toMax",... | [] | exact le_sup_of_le_left inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Heyting.Boundary | {
"line": 99,
"column": 50
} | {
"line": 99,
"column": 85
} | {
"line": 99,
"column": 86
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ a ⊔ b",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"CoheytingAlgebra.toHNot",
"Lattice.toSemilatticeSup",
"le_sup_of_le_left",
"SemilatticeInf.toMin",
"HNot.hnot",
... | [] | exact le_sup_of_le_left inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Heyting.Boundary | {
"line": 99,
"column": 50
} | {
"line": 99,
"column": 85
} | {
"line": 99,
"column": 86
} | [
{
"pp": "case refine_4\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ ￢(a ⊔ b) ⊔ b",
"ppTerm": "?refine_4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_4\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ ￢(a ⊔ b) ⊔ b"
] | exact le_sup_of_le_left inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Heyting.Boundary | {
"line": 99,
"column": 50
} | {
"line": 99,
"column": 85
} | {
"line": 99,
"column": 86
} | [
{
"pp": "case refine_5\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ a ⊔ (b ⊔ ￢(a ⊓ b))",
"ppTerm": "?refine_5",
"assigned": true,
"usedConstants": [
"CoheytingAlgebra.toHNot",
"Lattice.toSemilatticeSup",
"SemilatticeSup.toMax",
"le_sup_of_le_left",
"Se... | [] | exact le_sup_of_le_left inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Heyting.Boundary | {
"line": 99,
"column": 50
} | {
"line": 99,
"column": 85
} | {
"line": 99,
"column": 86
} | [
{
"pp": "case refine_6\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ ￢(a ⊔ b) ⊔ ￢(a ⊓ b)",
"ppTerm": "?refine_6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_6\nα : Type u_1\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ a ⊓ ￢a ≤ ￢(a ⊔ b) ⊔ ￢(a ⊓ b)"
] | exact le_sup_of_le_left inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.ContinuousOn | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 82
} | {
"line": 303,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nx : α\n⊢ ContinuousWithinAt f {x}ᶜ x ↔ ContinuousAt f x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ContinuousWithinAt",
"congrArg",
"ContinuousAt... | [] | rw [compl_eq_univ_sdiff, continuousWithinAt_sdiff_self, continuousWithinAt_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.ContinuousOn | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 82
} | {
"line": 303,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nx : α\n⊢ ContinuousWithinAt f {x}ᶜ x ↔ ContinuousAt f x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ContinuousWithinAt",
"congrArg",
"ContinuousAt... | [] | rw [compl_eq_univ_sdiff, continuousWithinAt_sdiff_self, continuousWithinAt_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousOn | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 82
} | {
"line": 303,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\nx : α\n⊢ ContinuousWithinAt f {x}ᶜ x ↔ ContinuousAt f x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ContinuousWithinAt",
"congrArg",
"ContinuousAt... | [] | rw [compl_eq_univ_sdiff, continuousWithinAt_sdiff_self, continuousWithinAt_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Basic | {
"line": 511,
"column": 2
} | {
"line": 512,
"column": 39
} | {
"line": 514,
"column": 0
} | [
{
"pp": "α : Type ua\ns : Set (UniformSpace α)\n⊢ (sInf s).toTopologicalSpace = ⨅ i ∈ s, i.toTopologicalSpace",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"UniformSpace",
"Eq.mpr",
"iInf",
"congrArg",
"Membership.mem",
"CompleteLattice.toConditionally... | [] | rw [sInf_eq_iInf]
simp only [← toTopologicalSpace_iInf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Basic | {
"line": 511,
"column": 2
} | {
"line": 512,
"column": 39
} | {
"line": 514,
"column": 0
} | [
{
"pp": "α : Type ua\ns : Set (UniformSpace α)\n⊢ (sInf s).toTopologicalSpace = ⨅ i ∈ s, i.toTopologicalSpace",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"UniformSpace",
"Eq.mpr",
"iInf",
"congrArg",
"Membership.mem",
"CompleteLattice.toConditionally... | [] | rw [sInf_eq_iInf]
simp only [← toTopologicalSpace_iInf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Ultrafilter.Defs | {
"line": 389,
"column": 69
} | {
"line": 389,
"column": 82
} | {
"line": 389,
"column": 82
} | [
{
"pp": "α : Type u\nβ : Type v\nm : α → β\ns : Set α\ng : Ultrafilter β\nh : m '' s ∈ g\nf : Filter α := Filter.comap m ↑g ⊓ 𝓟 s\nthis : f.NeBot\n⊢ Filter.map m (Filter.comap m ↑g) ⊓ Filter.map m (𝓟 s) = Filter.map m (Filter.comap m ↑g) ⊓ 𝓟 (m '' s)",
"ppTerm": "?m.82",
"assigned": true,
"usedCo... | [
"α : Type u\nβ : Type v\nm : α → β\ns : Set α\ng : Ultrafilter β\nh : m '' s ∈ g\nf : Filter α := Filter.comap m ↑g ⊓ 𝓟 s\nthis : f.NeBot\n⊢ Filter.map m (Filter.comap m ↑g) ⊓ 𝓟 (m '' s) = Filter.map m (Filter.comap m ↑g) ⊓ 𝓟 (m '' s)"
] | map_principal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Torsor.Defs | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 48
} | {
"line": 166,
"column": 0
} | [
{
"pp": "G : Type u_1\nP : Type u_2\ninst✝ : Group G\nT : Torsor G P\np₁ p₂ p₃ : P\n⊢ ((p₁ /ₛ p₂) * (p₂ /ₛ p₃)) • p₃ = (p₁ /ₛ p₃) • p₃",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
"Monoid.toMulOneClass"... | [] | rw [mul_smul, sdiv_smul, sdiv_smul, sdiv_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Constructions | {
"line": 70,
"column": 20
} | {
"line": 70,
"column": 33
} | {
"line": 70,
"column": 33
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nA : Set α\ncond : FiniteInter S\nP s✝ : Set α\nh✝ : s✝ ∈ S\n⊢ s✝ ∈ S",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Constructions | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 11
} | {
"line": 71,
"column": 12
} | [
{
"pp": "case univ\nα : Type u_1\nS : Set (Set α)\nA : Set α\ncond : FiniteInter S\nP : Set α\n⊢ Set.univ ∈ S ∨ ∃ Q ∈ S, Set.univ = A ∩ Q",
"ppTerm": "?univ",
"assigned": true,
"usedConstants": [
"Set.univ",
"Membership.mem",
"Exists",
"Set.instInter",
"Inter.inter",
... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Order.Filter.Pointwise | {
"line": 578,
"column": 4
} | {
"line": 578,
"column": 21
} | {
"line": 579,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\nf : Filter α\ns : Set α\nhs : s ∈ f\n⊢ s ^ 0 ∈ 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"Filter.one_mem_one",
"MulOneClass.toMulOne"
],
"usedFVars": [
"α",
... | [] | exact one_mem_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.AtTopBot.CountablyGenerated | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 60
} | {
"line": 149,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\nx : ι → α\nf : Filter α\nl : Filter ι\ninst✝ : l.IsCountablyGenerated\nhxy : ¬Tendsto x l f\ns : Set α\nhs : s ∈ f\nhfreq : ∃ᶠ (n : ι) in l, x n ∉ s\ny : ℕ → ι\nhy_tendsto : Tendsto y atTop l\nhy_freq : ∀ (n : ℕ), x (y n) ∉ s\nms : ℕ → ℕ\nhms_tendsto : Tendsto (fun n ↦ x (y ... | [
"α : Type u_1\nι : Type u_3\nx : ι → α\nf : Filter α\nl : Filter ι\ninst✝ : l.IsCountablyGenerated\nhxy : ¬Tendsto x l f\ns : Set α\nhs : s ∈ f\nhfreq : ∃ᶠ (n : ι) in l, x n ∉ s\ny : ℕ → ι\nhy_tendsto : Tendsto y atTop l\nhy_freq : ∀ (n : ℕ), x (y n) ∉ s\nms : ℕ → ℕ\nhms_tendsto : Tendsto (fun n ↦ x (y (ms n))) atT... | rcases (hms_tendsto.eventually_mem hs).exists with ⟨n, hn⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.Filter.Pointwise | {
"line": 660,
"column": 4
} | {
"line": 660,
"column": 19
} | {
"line": 662,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝ : DivisionMonoid α\na : α\nha : IsUnit a\n⊢ IsUnit (pure a)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"DivInvMonoid.toMonoid",
"DivisionMonoid.toDivInvMonoid",
"IsUnit.filter"
],
"usedFVars": [
"α",
"inst✝",
... | [] | exact ha.filter | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.LocallyFinite | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 52
} | {
"line": 86,
"column": 4
} | [
{
"pp": "ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\na : X\nU : Set X\nhaU : U ∈ 𝓝 a\nhfin : {i | (f i ∩ U).Nonempty}.Finite\n⊢ 𝓝[⋃ i, f i ∩ U] a = 𝓝[⋃ i ∈ {j | (f j ∩ U).Nonempty}, f i ∩ U] a",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants":... | [] | simp only [mem_setOf_eq, iUnion_nonempty_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.LocallyFinite | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 52
} | {
"line": 86,
"column": 4
} | [
{
"pp": "ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\na : X\nU : Set X\nhaU : U ∈ 𝓝 a\nhfin : {i | (f i ∩ U).Nonempty}.Finite\n⊢ 𝓝[⋃ i, f i ∩ U] a = 𝓝[⋃ i ∈ {j | (f j ∩ U).Nonempty}, f i ∩ U] a",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants":... | [] | simp only [mem_setOf_eq, iUnion_nonempty_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.LocallyFinite | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 52
} | {
"line": 86,
"column": 4
} | [
{
"pp": "ι : Type u_1\nX : Type u_4\ninst✝ : TopologicalSpace X\nf : ι → Set X\nhf : LocallyFinite f\na : X\nU : Set X\nhaU : U ∈ 𝓝 a\nhfin : {i | (f i ∩ U).Nonempty}.Finite\n⊢ 𝓝[⋃ i, f i ∩ U] a = 𝓝[⋃ i ∈ {j | (f j ∩ U).Nonempty}, f i ∩ U] a",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants":... | [] | simp only [mem_setOf_eq, iUnion_nonempty_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 48
} | {
"line": 57,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nhc : S.Countable\nhcomp : ∀ s ∈ S, IsCompact s\n⊢ IsSigmaCompact (⋃₀ S)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.Elem",
"Set.countable_coe_iff",
"Countable",
"Set.Counta... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nhc : S.Countable\nhcomp : ∀ s ∈ S, IsCompact s\nthis : Countable ↑S\n⊢ IsSigmaCompact (⋃₀ S)"
] | have : Countable S := countable_coe_iff.mpr hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 48
} | {
"line": 78,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nhc : S.Countable\nhcomp : ∀ (s : ↑S), IsSigmaCompact ↑s\n⊢ IsSigmaCompact (⋃₀ S)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.Elem",
"Set.countable_coe_iff",
"Countable",
"Se... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nS : Set (Set X)\nhc : S.Countable\nhcomp : ∀ (s : ↑S), IsSigmaCompact ↑s\nthis : Countable ↑S\n⊢ IsSigmaCompact (⋃₀ S)"
] | have : Countable S := countable_coe_iff.mpr hc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 408,
"column": 40
} | {
"line": 408,
"column": 89
} | {
"line": 408,
"column": 89
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nK : CompactExhaustion X\nx : X\n⊢ ¬K.shiftr.find x ≤ K.find x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CompactExhaustion.find_shiftr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"CompactEx... | [] | simp only [find_shiftr, not_le, Nat.lt_succ_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 408,
"column": 40
} | {
"line": 408,
"column": 89
} | {
"line": 408,
"column": 89
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nK : CompactExhaustion X\nx : X\n⊢ ¬K.shiftr.find x ≤ K.find x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CompactExhaustion.find_shiftr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"CompactEx... | [] | simp only [find_shiftr, not_le, Nat.lt_succ_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 408,
"column": 40
} | {
"line": 408,
"column": 89
} | {
"line": 408,
"column": 89
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nK : CompactExhaustion X\nx : X\n⊢ ¬K.shiftr.find x ≤ K.find x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CompactExhaustion.find_shiftr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"CompactEx... | [] | simp only [find_shiftr, not_le, Nat.lt_succ_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 82,
"column": 55
} | {
"line": 82,
"column": 76
} | {
"line": 82,
"column": 76
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsCompact s\np : Set X → Prop\nhe : p ∅\nhmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s\nhunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)\nhnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t\nf : Filter X := comk p he ⋯ ⋯\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f",
"ppTer... | [] | simpa [f] using hnhds | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Compactness.Compact | {
"line": 82,
"column": 55
} | {
"line": 82,
"column": 76
} | {
"line": 82,
"column": 76
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsCompact s\np : Set X → Prop\nhe : p ∅\nhmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s\nhunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)\nhnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t\nf : Filter X := comk p he ⋯ ⋯\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f",
"ppTer... | [] | simpa [f] using hnhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 82,
"column": 55
} | {
"line": 82,
"column": 76
} | {
"line": 82,
"column": 76
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns : Set X\nhs : IsCompact s\np : Set X → Prop\nhe : p ∅\nhmono : ∀ ⦃s t : Set X⦄, s ⊆ t → p t → p s\nhunion : ∀ ⦃s t : Set X⦄, p s → p t → p (s ∪ t)\nhnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t\nf : Filter X := comk p he ⋯ ⋯\n⊢ ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f",
"ppTer... | [] | simpa [f] using hnhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 87,
"column": 2
} | {
"line": 92,
"column": 28
} | {
"line": 94,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝] t\n⊢ IsCompact (s ∩ t)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"ClusterPt.mono",
"Filter.le_principal_iff",
"Filter.... | [] | intro f hnf hstf
obtain ⟨x, hsx, hx⟩ : ∃ x ∈ s, ClusterPt x f :=
hs (le_trans hstf (le_principal_iff.2 inter_subset_left))
have : x ∈ t := ht.mem_of_nhdsWithin_neBot <|
hx.mono <| le_trans hstf (le_principal_iff.2 inter_subset_right)
exact ⟨x, ⟨hsx, this⟩, hx⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 87,
"column": 2
} | {
"line": 92,
"column": 28
} | {
"line": 94,
"column": 0
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\ns t : Set X\nhs : IsCompact s\nht : IsClosed[inst✝] t\n⊢ IsCompact (s ∩ t)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"ClusterPt.mono",
"Filter.le_principal_iff",
"Filter.... | [] | intro f hnf hstf
obtain ⟨x, hsx, hx⟩ : ∃ x ∈ s, ClusterPt x f :=
hs (le_trans hstf (le_principal_iff.2 inter_subset_left))
have : x ∈ t := ht.mem_of_nhdsWithin_neBot <|
hx.mono <| le_trans hstf (le_principal_iff.2 inter_subset_right)
exact ⟨x, ⟨hsx, this⟩, hx⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Subset | {
"line": 94,
"column": 56
} | {
"line": 94,
"column": 82
} | {
"line": 94,
"column": 83
} | [
{
"pp": "α : Type u_2\nA : Set α\nD : Set ↑A\n⊢ range Subtype.val \\ Subtype.val '' D = A \\ Subtype.val '' D",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"setOf",
"Membership.mem",
"id",
"Subtype",
"Subtype.range_coe_subty... | [
"α : Type u_2\nA : Set α\nD : Set ↑A\n⊢ {x | x ∈ A} \\ Subtype.val '' D = A \\ Subtype.val '' D"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.CountableInter | {
"line": 69,
"column": 2
} | {
"line": 70,
"column": 67
} | {
"line": 72,
"column": 0
} | [
{
"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... | [] | simpa only [Filter.Eventually, setOf_forall] using
@countable_bInter_mem _ l _ _ _ hS fun i hi => { x | p x i hi } | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Filter.CountableInter | {
"line": 69,
"column": 2
} | {
"line": 70,
"column": 67
} | {
"line": 72,
"column": 0
} | [
{
"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... | [] | simpa only [Filter.Eventually, setOf_forall] using
@countable_bInter_mem _ l _ _ _ hS fun i hi => { x | p x i hi } | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CountableInter | {
"line": 69,
"column": 2
} | {
"line": 70,
"column": 67
} | {
"line": 72,
"column": 0
} | [
{
"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... | [] | simpa only [Filter.Eventually, setOf_forall] using
@countable_bInter_mem _ l _ _ _ hS fun i hi => { x | p x i hi } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 558,
"column": 45
} | {
"line": 558,
"column": 67
} | {
"line": 558,
"column": 67
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\nU : X → Set X\nhxU : ∀ x ∈ s, x ∈ U x\nhSU : ∀ x ∈ s, U x ∈ S\nhUF : ∀ x ∈ s, U x ∉ F\n⊢ U '' s ⊆ S",... | [] | simpa [Set.subset_def] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Compactness.Compact | {
"line": 558,
"column": 45
} | {
"line": 558,
"column": 67
} | {
"line": 558,
"column": 67
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\nU : X → Set X\nhxU : ∀ x ∈ s, x ∈ U x\nhSU : ∀ x ∈ s, U x ∈ S\nhUF : ∀ x ∈ s, U x ∉ F\n⊢ U '' s ⊆ S",... | [] | simpa [Set.subset_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 558,
"column": 45
} | {
"line": 558,
"column": 67
} | {
"line": 558,
"column": 67
} | [
{
"pp": "X : Type u\nT : TopologicalSpace X\nS : Set (Set X)\nhTS : T = generateFrom S\ns : Set X\nh : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q\nF : Ultrafilter X\nhsF : s ∈ F\nhF : ¬∃ x ∈ s, ↑F ≤ 𝓝 x\nU : X → Set X\nhxU : ∀ x ∈ s, x ∈ U x\nhSU : ∀ x ∈ s, U x ∈ S\nhUF : ∀ x ∈ s, U x ∉ F\n⊢ U '' s ⊆ S",... | [] | simpa [Set.subset_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.CountableInter | {
"line": 233,
"column": 6
} | {
"line": 233,
"column": 14
} | {
"line": 233,
"column": 14
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝² : CountableInterFilter l\nl₁ l₂ : Filter α\ninst✝¹ : CountableInterFilter l₁\ninst✝ : CountableInterFilter l₂\nS : Set (Set α)\nhSc : S.Countable\ns t : (s : Set α) → s ∈ S → Set α\nhst : ∀ (s_1 : Set α) (a : s_1 ∈ S), s_1 = s s_1 a ∩ t s_1... | [
"ι : Sort u_1\nα : Type u_2\nβ : Type u_3\nl : Filter α\ninst✝² : CountableInterFilter l\nl₁ l₂ : Filter α\ninst✝¹ : CountableInterFilter l₁\ninst✝ : CountableInterFilter l₂\nS : Set (Set α)\nhSc : S.Countable\ns t : (s : Set α) → s ∈ S → Set α\nhst : ∀ (s_1 : Set α) (a : s_1 ∈ S), s_1 = s s_1 a ∩ t s_1 a\nhs : ⋂ i... | hst i hi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.CountableInter | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 13
} | {
"line": 277,
"column": 14
} | [
{
"pp": "case mp.univ\nα : Type u_2\ng : Set (Set α)\ns : Set α\n⊢ ∃ S ⊆ g, S.Countable ∧ ⋂₀ S ⊆ univ",
"ppTerm": "?mp.univ",
"assigned": true,
"usedConstants": [
"subset_refl._simp_1",
"Set.sInter_empty",
"congrArg",
"and_self",
"Set.univ",
"Set.instReflSubset",
... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Order.Filter.CountableInter | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 11
} | {
"line": 297,
"column": 12
} | [
{
"pp": "case mpr.univ\nα : Type u_2\ng : Set (Set α)\nf : Filter α\ninst✝ : CountableInterFilter f\nh : g ⊆ f.sets\ns : Set α\n⊢ univ ∈ f",
"ppTerm": "?mpr.univ",
"assigned": true,
"usedConstants": [
"Filter.univ_mem"
],
"usedFVars": [
"α",
"f"
],
"usedGoals": []
... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Baire.Lemmas | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 23
} | {
"line": 54,
"column": 4
} | [
{
"pp": "case insert\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (Set X)\na✝ : Set X\ns✝ : Set (Set X)\nha : a✝ ∉ s✝\nhsf : s✝.Finite\nih : (∀ t ∈ s✝, IsOpen[inst✝] t) → (∀ t ∈ s✝, Dense t) → Dense (⋂₀ s✝)\nho : ∀ t ∈ insert a✝ s✝, IsOpen[inst✝] t\nhd : ∀ t ∈ insert a✝ s✝, Dense t\n⊢ Dense (⋂₀ insert a✝ ... | [] | | insert ha hsf ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Compactness.Compact | {
"line": 962,
"column": 2
} | {
"line": 962,
"column": 50
} | {
"line": 964,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : CompactSpace X\nf : X → Y\nhf : Continuous[inst✝², inst✝¹] f\n⊢ IsCompact (range f)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"congrArg",
... | [] | rw [← image_univ]; exact isCompact_univ.image hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 962,
"column": 2
} | {
"line": 962,
"column": 50
} | {
"line": 964,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : CompactSpace X\nf : X → Y\nhf : Continuous[inst✝², inst✝¹] f\n⊢ IsCompact (range f)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"congrArg",
... | [] | rw [← image_univ]; exact isCompact_univ.image hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 961,
"column": 98
} | {
"line": 962,
"column": 50
} | {
"line": 964,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : CompactSpace X\nf : X → Y\nhf : Continuous[inst✝², inst✝¹] f\n⊢ IsCompact (range f)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"congrArg",
... | [] | by
rw [← image_univ]; exact isCompact_univ.image hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Inseparable | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 12
} | {
"line": 474,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\n⊢ (x ~ᵢ y) ↔ ∀ (s : Set X), IsClosed[inst✝] s → (x ∈ s ↔ y ∈ s)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.Inseparable.0.inseparable_iff_forall_isClosed._simp_1_3",
"Specializes",
... | [] | simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Inseparable | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 12
} | {
"line": 474,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\n⊢ (x ~ᵢ y) ↔ ∀ (s : Set X), IsClosed[inst✝] s → (x ∈ s ↔ y ∈ s)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.Inseparable.0.inseparable_iff_forall_isClosed._simp_1_3",
"Specializes",
... | [] | simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Inseparable | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 12
} | {
"line": 474,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\n⊢ (x ~ᵢ y) ↔ ∀ (s : Set X), IsClosed[inst✝] s → (x ∈ s ↔ y ∈ s)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.Inseparable.0.inseparable_iff_forall_isClosed._simp_1_3",
"Specializes",
... | [] | simp only [inseparable_iff_specializes_and, specializes_iff_forall_closed, ← forall_and, ←
iff_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 1237,
"column": 8
} | {
"line": 1237,
"column": 17
} | {
"line": 1238,
"column": 2
} | [
{
"pp": "case h\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nS : Set X\nhS : IsClosed[inst✝¹] S\nhne : S.Nonempty\nopens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen[inst✝¹] U ∧ Uᶜ.Nonempty}\nc : Set (Set X)\nhc : c ⊆ opens\nhz : IsChain (fun x1 x2 ↦ x1 ⊆ x2) c\nhcne : ¬c.Nonempty\n⊢ S.Nonempty",
... | [] | exact hne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.DiscreteSubset | {
"line": 98,
"column": 77
} | {
"line": 98,
"column": 90
} | {
"line": 98,
"column": 90
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhs : IsDiscrete s\nhf : IsOpenMap f\nhf' : Injective f\nx : X\nhx : x ∈ s\n⊢ 𝓝 (f x) ⊓ 𝓟 (f '' s) ≤ map f (𝓝 x) ⊓ map f (𝓟 s)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants"... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhs : IsDiscrete s\nhf : IsOpenMap f\nhf' : Injective f\nx : X\nhx : x ∈ s\n⊢ 𝓝 (f x) ⊓ 𝓟 (f '' s) ≤ map f (𝓝 x) ⊓ 𝓟 (f '' s)"
] | map_principal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.DiscreteSubset | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 90
} | {
"line": 159,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous[inst✝³, inst✝²] f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\n⊢ IsOpen[inst✝³] {x}",
"ppTerm": "?m.19",
"assigned": true,... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nf : X → Y\ninst✝¹ : T1Space X\ninst✝ : WeaklyLocallyCompactSpace Y\nhf' : Continuous[inst✝³, inst✝²] f\nhf : Tendsto f cofinite (cocompact Y)\nx : X\nK : Set Y\nhK : IsCompact K\nhK' : K ∈ 𝓝 (f x)\n⊢ IsOpen[inst✝³] {x}"
] | obtain ⟨K : Set Y, hK : IsCompact K, hK' : K ∈ 𝓝 (f x)⟩ := exists_compact_mem_nhds (f x) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.DiscreteSubset | {
"line": 266,
"column": 6
} | {
"line": 267,
"column": 15
} | {
"line": 268,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : ¬a = x\n⊢ a ∈ (U \\ s)ᶜ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConst... | [] | specialize ha h₂a
tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 266,
"column": 6
} | {
"line": 267,
"column": 15
} | {
"line": 268,
"column": 2
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed[inst✝] U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : ¬a = x\n⊢ a ∈ (U \\ s)ᶜ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConst... | [] | specialize ha h₂a
tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.LeftRight | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 57
} | {
"line": 98,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : LinearOrder α\na : α\n⊢ 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"nhdsWithin_univ",
"congrArg",
"Set.univ",
"nhdsWithin",
"PartialOrder... | [] | rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order.LeftRight | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 57
} | {
"line": 98,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : LinearOrder α\na : α\n⊢ 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"nhdsWithin_univ",
"congrArg",
"Set.univ",
"nhdsWithin",
"PartialOrder... | [] | rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.LeftRight | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 57
} | {
"line": 98,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : LinearOrder α\na : α\n⊢ 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"nhdsWithin_univ",
"congrArg",
"Set.univ",
"nhdsWithin",
"PartialOrder... | [] | rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Basic | {
"line": 983,
"column": 94
} | {
"line": 986,
"column": 76
} | {
"line": 988,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nK : Set X\nhK : IsCompact K\n⊢ IsCompact (closure[inst✝¹] K)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"subset_closure",
"Finset",
"Membership.mem",
"Exists",
"HasSubset.Subset.trans",... | [] | by
refine isCompact_of_finite_subcover fun U hUo hKU ↦ ?_
rcases hK.elim_finite_subcover U hUo (subset_closure.trans hKU) with ⟨t, ht⟩
exact ⟨t, hK.closure_subset_of_isOpen (isOpen_biUnion fun _ _ ↦ hUo _) ht⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.OrderClosed | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 60
} | {
"line": 338,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [... | [] | simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsLT h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Order.OrderClosed | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 60
} | {
"line": 338,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [... | [] | simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsLT h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.OrderClosed | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 60
} | {
"line": 338,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\ninst✝ : TopologicalSpace β\na b : α\nf : α → β\nh : a < b\n⊢ ContinuousWithinAt f (Ico a b) b ↔ ContinuousWithinAt f (Iio b) b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [... | [] | simp only [ContinuousWithinAt, nhdsWithin_Ico_eq_nhdsLT h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Basic | {
"line": 1096,
"column": 60
} | {
"line": 1096,
"column": 86
} | {
"line": 1097,
"column": 6
} | [
{
"pp": "X : Type u_3\nY : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : R1Space Y\nf : X → Y\nx : X\nK : Set X\ns : Set Y\nhf : Continuous[inst✝², inst✝¹] f\nhs : s ∈ 𝓝 (f x)\nhKc : IsCompact K\nhKx : K ∈ 𝓝 x\nhc : IsCompact (f '' K \\ interior s)\n⊢ Disjoint (𝓝 (f x)) (𝓝ˢ (f ... | [
"X : Type u_3\nY : Type u_4\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : R1Space Y\nf : X → Y\nx : X\nK : Set X\ns : Set Y\nhf : Continuous[inst✝², inst✝¹] f\nhs : s ∈ 𝓝 (f x)\nhKc : IsCompact K\nhKx : K ∈ 𝓝 x\nhc : IsCompact (f '' K \\ interior s)\n⊢ ∀ x_1 ∈ f '' K \\ interior s, Disjoint (... | hc.disjoint_nhdsSet_right, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Connected.LocallyConnected | {
"line": 158,
"column": 2
} | {
"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\nx : α\n⊢ IsOpen {ConnectedComponents.mk x}",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"congrArg",
"ConnectedComponents.mk",
... | [] | simp [← ConnectedComponents.isQuotientMap_coe.isOpen_preimage,
connectedComponents_preimage_singleton, isOpen_connectedComponent] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Compactness.Lindelof | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 70
} | {
"line": 188,
"column": 2
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\nι : Type v\ninst✝ : Nonempty ι\nhs : IsLindelof s\nU : ι → Set X\nhUo : ∀ (i : ι), IsOpen[inst✝¹] (U i)\nhsU : s ⊆ ⋃ i, U i\n⊢ ∃ f, s ⊆ ⋃ n, U (f n)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Membership.mem",
"E... | [
"X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\nι : Type v\ninst✝ : Nonempty ι\nhs : IsLindelof s\nU : ι → Set X\nhUo : ∀ (i : ι), IsOpen[inst✝¹] (U i)\nhsU : s ⊆ ⋃ i, U i\nc : Set ι\nc_count : c.Countable\nc_cov : s ⊆ ⋃ i ∈ c, U i\n⊢ ∃ f, s ⊆ ⋃ n, U (f n)"
] | obtain ⟨c, ⟨c_count, c_cov⟩⟩ := hs.elim_countable_subcover U hUo hsU | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Connected.Basic | {
"line": 699,
"column": 4
} | {
"line": 701,
"column": 95
} | {
"line": 702,
"column": 2
} | [
{
"pp": "case mp\nα : Type u\ninst✝ : TopologicalSpace α\n⊢ ConnectedSpace α → ∃ x, connectedComponent x = univ",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Set.univ",
"ConnectedSpace.casesOn",
"PreconnectedSpace",
"Exists",
"Set.eq_univ_of_univ_subset",
... | [] | rintro ⟨⟨x⟩⟩
exact
⟨x, eq_univ_of_univ_subset <| isPreconnected_univ.subset_connectedComponent (mem_univ x)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Basic | {
"line": 699,
"column": 4
} | {
"line": 701,
"column": 95
} | {
"line": 702,
"column": 2
} | [
{
"pp": "case mp\nα : Type u\ninst✝ : TopologicalSpace α\n⊢ ConnectedSpace α → ∃ x, connectedComponent x = univ",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Set.univ",
"ConnectedSpace.casesOn",
"PreconnectedSpace",
"Exists",
"Set.eq_univ_of_univ_subset",
... | [] | rintro ⟨⟨x⟩⟩
exact
⟨x, eq_univ_of_univ_subset <| isPreconnected_univ.subset_connectedComponent (mem_univ x)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Regular | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 11
} | {
"line": 147,
"column": 12
} | [
{
"pp": "case univ\nX : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (Set X)\nh : inst✝ = generateFrom s\nh' : ∀ t ∈ s, ∀ a ∈ t, Disjoint (𝓝ˢ tᶜ) (𝓝 a)\na : X\nt : Set X\nha : a ∈ univ\n⊢ Disjoint (𝓝ˢ univᶜ) (𝓝 a)",
"ppTerm": "?univ",
"assigned": true,
"usedConstants": [
"nhdsSet_empty",
... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 390,
"column": 4
} | {
"line": 390,
"column": 19
} | {
"line": 391,
"column": 4
} | [
{
"pp": "case mp\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsLindelof (b i)\nU : Set X\n⊢ IsLindelof U ∧ IsOpen[inst✝] U → ∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
... | [
"case mp\nX : Type u\nι : Type u_1\ninst✝ : TopologicalSpace X\nb : ι → Set X\nhb : IsTopologicalBasis (range b)\nhb' : ∀ (i : ι), IsLindelof (b i)\nU : Set X\nh₁ : IsLindelof U\nh₂ : IsOpen[inst✝] U\n⊢ ∃ s, s.Countable ∧ U = ⋃ i ∈ s, b i"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Connected.Clopen | {
"line": 269,
"column": 6
} | {
"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\nhsv : (s ∩ v).Nonempty\nH : ¬(s ∩ (u ∩ v)).Nonempty\nh : s ⊆ v\nx : α\nhxs : x ∈ s\nhxu : x ∈ u\n⊢ (s ∩ (u ∩ v)).Nonempty",
"ppTerm": "?mpr.inr",
"assigned": true,
... | [] | exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Clopen | {
"line": 325,
"column": 6
} | {
"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\nhsv : (s ∩ v).Nonempty\nH : ¬(s ∩ (u ∩ v)).Nonempty\nh : s ⊆ v\nx : α\nhxs : x ∈ s\nhxu : x ∈ u\n⊢ (s ∩ (u ∩ v)).Nonempty",
"ppTerm": "?mpr.inr",
"assigned": true... | [] | exact ⟨x, hxs, ⟨hxu, h hxs⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Clopen | {
"line": 418,
"column": 22
} | {
"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)\nu v : Set α\nhu : IsClosed... | [
"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... | uv_disj | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Connected.Clopen | {
"line": 636,
"column": 4
} | {
"line": 636,
"column": 14
} | {
"line": 637,
"column": 2
} | [
{
"pp": "case inr.zero\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nhn : 0 < 0\n⊢ ∃ U, (∀ (i : Fin 0), IsClopen (U i)) ∧ (∀ (i : Fin 0), (U i).Nonempty) ∧ Pairwise (Disjoint on U) ∧ ⋃ i, U i = univ",
"ppTerm": "?inr.zero",
"assigned": true,
"us... | [] | simp at hn | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Connected.Clopen | {
"line": 636,
"column": 4
} | {
"line": 636,
"column": 14
} | {
"line": 637,
"column": 2
} | [
{
"pp": "case inr.zero\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nhn : 0 < 0\n⊢ ∃ U, (∀ (i : Fin 0), IsClopen (U i)) ∧ (∀ (i : Fin 0), (U i).Nonempty) ∧ Pairwise (Disjoint on U) ∧ ⋃ i, U i = univ",
"ppTerm": "?inr.zero",
"assigned": true,
"us... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Connected.Clopen | {
"line": 636,
"column": 4
} | {
"line": 636,
"column": 14
} | {
"line": 637,
"column": 2
} | [
{
"pp": "case inr.zero\nα : Type u\ninst✝¹ : TopologicalSpace α\ninst✝ : Infinite (ConnectedComponents α)\nh✝ : Nonempty α\nhn : 0 < 0\n⊢ ∃ U, (∀ (i : Fin 0), IsClopen (U i)) ∧ (∀ (i : Fin 0), (U i).Nonempty) ∧ Pairwise (Disjoint on U) ∧ ⋃ i, U i = univ",
"ppTerm": "?inr.zero",
"assigned": true,
"us... | [] | simp at hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 65
} | {
"line": 195,
"column": 2
} | [
{
"pp": "case left\nι : Type u_5\nX : ι → Type u_6\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : (i : ι) → TopologicalSpace (Y i)\nf : (i : ι) → X i → Y i\nh :\n ∀ (i : ι),\n Continuous[inst✝¹ i, inst✝ i] (f i) ∧\n ∀ ⦃𝒰 : Ultrafilter (X i)⦄ ⦃y : Y i⦄, Tendsto (f i) (↑𝒰) (𝓝 y)... | [
"case right\nι : Type u_5\nX : ι → Type u_6\nY : ι → Type u_7\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : (i : ι) → TopologicalSpace (Y i)\nf : (i : ι) → X i → Y i\nh :\n ∀ (i : ι),\n Continuous[inst✝¹ i, inst✝ i] (f i) ∧\n ∀ ⦃𝒰 : Ultrafilter (X i)⦄ ⦃y : Y i⦄, Tendsto (f i) (↑𝒰) (𝓝 y) → ∃ x, f i... | · exact continuous_pi fun i ↦ (h i).1.comp (continuous_apply i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.DiscreteUniformity | {
"line": 69,
"column": 2
} | {
"line": 71,
"column": 20
} | {
"line": 72,
"column": 2
} | [
{
"pp": "X : Type u_1\nu : UniformSpace X\ninst✝³ : DiscreteUniformity X\nY : Type u_2\ninst✝² : Finite Y\ninst✝¹ : UniformSpace Y\ninst✝ : DiscreteTopology Y\n⊢ DiscreteUniformity Y",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Set.ext",
"SetRel.id",
"SetRel",
"c... | [
"X : Type u_1\nu : UniformSpace X\ninst✝³ : DiscreteUniformity X\nY : Type u_2\ninst✝² : Finite Y\ninst✝¹ : UniformSpace Y\ninst✝ : DiscreteTopology Y\nh : SetRel.id = ⋂ y, {p | p.2 = y → p.1 ∈ {y}}\n⊢ DiscreteUniformity Y"
] | have h : SetRel.id = ⋂ y : Y, {p | p.2 = y → p.1 ∈ ({y} : Set Y)} := by
ext x
simp [SetRel.id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 504,
"column": 2
} | {
"line": 505,
"column": 17
} | {
"line": 507,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\nι : Sort u_3\np : ι → Prop\nU : ι → Set α\nh : (𝓝 1).HasBasis p U\n⊢ (𝓤 α).HasBasis p fun i ↦ {x | x.1⁻¹ * x.2 ∈ U i}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneC... | [] | rw [uniformity_eq_comap_inv_mul_nhds_one]
exact h.comap _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.IsUniformGroup.Defs | {
"line": 504,
"column": 2
} | {
"line": 505,
"column": 17
} | {
"line": 507,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : Group α\ninst✝ : IsUniformGroup α\nι : Sort u_3\np : ι → Prop\nU : ι → Set α\nh : (𝓝 1).HasBasis p U\n⊢ (𝓤 α).HasBasis p fun i ↦ {x | x.1⁻¹ * x.2 ∈ U i}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneC... | [] | rw [uniformity_eq_comap_inv_mul_nhds_one]
exact h.comap _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 238,
"column": 70
} | {
"line": 239,
"column": 94
} | {
"line": 241,
"column": 0
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nu : ℕ → α\n⊢ CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"CauchySeq",
... | [] | by
simp only [cauchySeq_iff', Filter.eventually_atTop_prod_self', mem_preimage, Prod.map_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 80,
"column": 2
} | {
"line": 82,
"column": 30
} | {
"line": 84,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ng : β → γ\nhf : UniformContinuous f\nhg : UniformContinuous g\nhgf : IsUniformInducing (g ∘ f)\n⊢ IsUniformInducing f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstant... | [] | refine ⟨le_antisymm ?_ hf.le_comap⟩
rw [← hgf.1, ← Prod.map_def, ← Prod.map_def, ← Prod.map_comp_map f f g g, ← comap_comap]
exact comap_mono hg.le_comap | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 80,
"column": 2
} | {
"line": 82,
"column": 30
} | {
"line": 84,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : UniformSpace α\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\nf : α → β\ng : β → γ\nhf : UniformContinuous f\nhg : UniformContinuous g\nhgf : IsUniformInducing (g ∘ f)\n⊢ IsUniformInducing f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstant... | [] | refine ⟨le_antisymm ?_ hf.le_comap⟩
rw [← hgf.1, ← Prod.map_def, ← Prod.map_def, ← Prod.map_comp_map f f g g, ← comap_comap]
exact comap_mono hg.le_comap | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 63
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\n⊢ IsUniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"Eq.mpr",
"isU... | [] | rw [isUniformEmbedding_iff, and_comm, isUniformInducing_iff'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 63
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\n⊢ IsUniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"Eq.mpr",
"isU... | [] | rw [isUniformEmbedding_iff, and_comm, isUniformInducing_iff'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 63
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\n⊢ IsUniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"UniformContinuous",
"Eq.mpr",
"isU... | [] | rw [isUniformEmbedding_iff, and_comm, isUniformInducing_iff'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformEmbedding | {
"line": 373,
"column": 17
} | {
"line": 373,
"column": 30
} | {
"line": 374,
"column": 8
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nm : β → α\nhm : IsUniformInducing m\ndense : DenseRange m\nh : ∀ (f : Filter β), Cauchy f → ∃ x, map m f ≤ 𝓝 x\nf : Filter α\nhf : Cauchy f\np : Set (α × α) → Set α → Set α := fun s t ↦ {y | ∃ x ∈ t, (x, y) ∈ s}\ng : Filter α := ... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 42
} | {
"line": 503,
"column": 2
} | [
{
"pp": "α : Type u\nuniformSpace : UniformSpace α\nf : Filter α\nhf : f.TotallyBounded\ns : Set α\nhs : s ∈ f\nU : SetRel α α\nhU : U ∈ 𝓤 α\nr : Set (α × α)\nhr : r ∈ 𝓤 α\nrs : ∀ {a b : α}, (a, b) ∈ r → (b, a) ∈ r\nrU : r ○ r ⊆ U\nk : Set α\nfk : k.Finite\nks : SetRel.preimage r k ∈ f\n⊢ ∃ t ⊆ s, t.Finite ∧ ... | [
"α : Type u\nuniformSpace : UniformSpace α\nf : Filter α\nhf : f.TotallyBounded\ns : Set α\nhs : s ∈ f\nU : SetRel α α\nhU : U ∈ 𝓤 α\nr : Set (α × α)\nhr : r ∈ 𝓤 α\nrs : ∀ {a b : α}, (a, b) ∈ r → (b, a) ∈ r\nrU : r ○ r ⊆ U\nk : Set α\nfk : k.Finite\nks : SetRel.preimage r k ∈ f\nu : Set α := k ∩ {y | ∃ x ∈ s, (x,... | let u := k ∩ { y | ∃ x ∈ s, (x, y) ∈ r } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
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