module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.ContinuousMap.Sigma | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 15
} | [
{
"pp": "X : Type u_1\nι : Type u_2\nY : ι → Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : (i : ι) → TopologicalSpace (Y i)\ninst✝ : Nonempty X\ni : ι\ng g' : C(X, Y i)\nh : (fun g ↦ (sigmaMk g.fst).comp g.snd) ⟨i, g⟩ = (fun g ↦ (sigmaMk g.fst).comp g.snd) ⟨i, g'⟩\nhg : ⇑g ≍ ⇑g'\n⊢ ⟨i, g⟩ = ⟨i, g'⟩",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 80
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace F\nA✝ B : Set E\ninst✝¹ : SequentialSpace E\ninst✝ : CountablyCompactSpace E\nx : ℕ → E\nhx : ∀ (x_1 : E) (x_2 : ℕ → ℕ), StrictMono x_2 → ¬Tendsto (x ∘ x_2) atTop (𝓝 x_1)\nA : Set E := ⋃ i, closure[inst✝³]... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 388,
"column": 13
} | {
"line": 388,
"column": 24
} | [
{
"pp": "case empty\nι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nf : ι → Set E\nhf : ∀ i ∈ ∅, IsCountablyCompact (f i)\n⊢ IsCountablyCompact (⋃ i ∈ ∅, f i)",
"usedConstants": [
"Eq.mpr",
"False",
"Iff.of_eq",
"congrArg",
"Finset",
"Membership.mem",
"id",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Compactness.CountablyCompact | {
"line": 389,
"column": 25
} | {
"line": 389,
"column": 60
} | [
{
"pp": "case insert\nι : Type u_1\nE : Type u_2\ninst✝ : TopologicalSpace E\nf : ι → Set E\na : ι\ns : Finset ι\nha : a ∉ s\nih : (∀ i ∈ s, IsCountablyCompact (f i)) → IsCountablyCompact (⋃ i ∈ s, f i)\nhf : ∀ i ∈ insert a s, IsCountablyCompact (f i)\n⊢ IsCountablyCompact (⋃ i ∈ insert a s, f i)",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Convenient.ContinuousMapGeneratedBy | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 41
} | [
{
"pp": "ι : Type t\nX : ι → Type u\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝ : TopologicalSpace Y\n⊢ ContinuousGeneratedBy X _root_.id",
"usedConstants": [
"Eq.mpr",
"Continuous",
"Equiv.instEquivLike",
"Equiv.symm_comp_self",
"congrArg",
"Function.co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Convenient.HomSpace | {
"line": 102,
"column": 8
} | {
"line": 103,
"column": 51
} | [
{
"pp": "ι : Type t\nX : ι → Type u\ninst✝⁵ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝⁴ : TopologicalSpace Y\nZ : Type v'\ninst✝³ : TopologicalSpace Z\nT : Type v''\ninst✝² : TopologicalSpace T\ninst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)\ninst✝ : ∀ (i j : ι), IsGeneratedBy X (X i × X j)\ng : Conti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Convenient.HomSpace | {
"line": 107,
"column": 8
} | {
"line": 108,
"column": 51
} | [
{
"pp": "ι : Type t\nX : ι → Type u\ninst✝⁵ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝⁴ : TopologicalSpace Y\nZ : Type v'\ninst✝³ : TopologicalSpace Z\nT : Type v''\ninst✝² : TopologicalSpace T\ninst✝¹ : ∀ (i : ι), LocallyCompactSpace (X i)\ninst✝ : ∀ (i j : ι), IsGeneratedBy X (X i × X j)\ng : Conti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DerivedSet | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 26
} | [
{
"pp": "case h₁\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nβ : Type u_2\ninst✝ : TopologicalSpace β\nF : Filter X\nx : X\nh : AccPt x F\nf : X → β\nhf1 : ContinuousAt f x\nhf2 : Function.Injective f\n⊢ ∀ᶠ (x_1 : X) in 𝓝[≠] x, f x_1 ∈ {f x}ᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.DerivedSet | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 18
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nU : Set X\n⊢ Perfect U ↔ U = derivedSet U",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Perfect",
"Preperfect",
"id",
"derivedSet",
"IsClosed",
"perfect_def",
"And",
"Iff",
"propext",
"Eq... | perfect_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Convenient.OpenClosed | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 67
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type u_3\ninst✝² : TopologicalSpace Y\ninst✝¹ : ∀ (i : ι) (U : TopologicalSpace.Opens (X i)), IsGeneratedBy X ↥U\ninst✝ : IsGeneratedBy X Y\nU : Set Y\nhU : IsOpen[inst✝²] U\nW : (a : (i : ι) × C(X i, Y)) → TopologicalSpace.... | have hg (a) : Continuous (g a) := a.2.continuous.restrictPreimage | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Convenient.OpenClosed | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 67
} | [
{
"pp": "ι : Type u_1\nX : ι → Type u_2\ninst✝³ : (i : ι) → TopologicalSpace (X i)\nY : Type u_3\ninst✝² : TopologicalSpace Y\ninst✝¹ : ∀ (i : ι) (F : TopologicalSpace.Closeds (X i)), IsGeneratedBy X ↥F\ninst✝ : IsGeneratedBy X Y\nF : Set Y\nhF : IsClosed[inst✝²] F\nW : (a : (i : ι) × C(X i, Y)) → TopologicalSp... | have hg (a) : Continuous (g a) := a.2.continuous.restrictPreimage | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Filter | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 34
} | [
{
"pp": "α : Type u_2\ns : Set α\n⊢ IsOpen {l | s ∈ l}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Filter | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 53
} | [
{
"pp": "α : Type u_2\nl : Filter α\n⊢ 𝓝 l = l.lift' fun s ↦ {l' | s ∈ l'}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Filter | {
"line": 100,
"column": 54
} | {
"line": 100,
"column": 86
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\n⊢ (𝓝 l).HasBasis p fun i ↦ {l' | s i ∈ l'}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Filter | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 33
} | [
{
"pp": "α : Type u_2\nl₁ l₂ : Filter α\n⊢ 𝓝 (l₁ ⊓ l₂) = 𝓝 l₁ ⊓ 𝓝 l₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HSpaces | {
"line": 197,
"column": 13
} | {
"line": 198,
"column": 35
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nx y : X\nθ : ↑I\nγ : Path x y\n⊢ γ (qRight (0, θ)) = x",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Icc.instZero",
"congrArg",
"Real.semiring",
"Set.Elem",
"id... | by
rw [qRight_zero_left, γ.source] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.ENatENNReal | {
"line": 26,
"column": 4
} | {
"line": 26,
"column": 15
} | [
{
"pp": "case refine_1\na : ENNReal\n⊢ IsOpen (toENNReal ⁻¹' Set.Ioi a)",
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
"PartialOrder.toPreorder",
"instPreorderENat",
"id",
"ENat.toENNReal",
"ENat.preimage_toENNReal_Ioi",
"ENat.floor",
"Set.p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.ENatENNReal | {
"line": 27,
"column": 4
} | {
"line": 27,
"column": 15
} | [
{
"pp": "case refine_2\na : ENNReal\n⊢ IsOpen (toENNReal ⁻¹' Set.Iio a)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"instPreorderENat",
"id",
"ENat.ceil",
"ENat.toENNReal",
"ENat.preimage_toENNReal_Iio",
"Set.preimage",
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 37
} | [
{
"pp": "case succ.refine_1\nf : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯\ng : ℝ ≃ₜ ℝ := (Homeomorph.addLeft 2).trans f\nn : ℕ\nih : IsClosed (preCantorSet n)\n⊢ IsClosed ((fun x ↦ x / 3) '' preCantorSet n)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Divisio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 40
} | [
{
"pp": "case succ.refine_2\nf : ℝ ≃ₜ ℝ := Homeomorph.mulLeft₀ (1 / 3) ⋯\ng : ℝ ≃ₜ ℝ := (Homeomorph.addLeft 2).trans f\nn : ℕ\nih : IsClosed (preCantorSet n)\n⊢ IsClosed ((fun x ↦ (2 + x) / 3) '' preCantorSet n)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 174,
"column": 46
} | {
"line": 174,
"column": 57
} | [
{
"pp": "a b : ℕ → Fin 3\nha : ∀ (n : ℕ), a n ≠ 1\nhb : ∀ (n : ℕ), b n ≠ 1\nh✝ : ofDigits a = ofDigits b\nh : ∃ a_1, a a_1 ≠ b a_1\nn0 : ℕ := Nat.find h\nn : ℕ\nhn : n < n0\n⊢ a n = b n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 175,
"column": 30
} | {
"line": 175,
"column": 41
} | [
{
"pp": "a b : ℕ → Fin 3\nha : ∀ (n : ℕ), a n ≠ 1\nhb : ∀ (n : ℕ), b n ≠ 1\nh✝ : ofDigits a = ofDigits b\nh : ∃ a_1, a a_1 ≠ b a_1\nn0 : ℕ := Nat.find h\nh1 : ∀ n < n0, a n = b n\n⊢ a n0 ≠ b n0",
"usedConstants": [
"id",
"Ne",
"instOfNatNat",
"Nat",
"OfNat.ofNat",
"Fin"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.RatLemmas | {
"line": 57,
"column": 2
} | {
"line": 62,
"column": 28
} | [
{
"pp": "⊢ ¬(cocompact ℚ).IsCountablyGenerated",
"usedConstants": [
"Rat.instOfNat",
"False",
"Filter.tendsto_inf",
"congrArg",
"Filter.Inf.isCountablyGenerated",
"TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology",
"Rat",
"IsCompact.compl_mem_coco... | intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.RatLemmas | {
"line": 57,
"column": 2
} | {
"line": 62,
"column": 28
} | [
{
"pp": "⊢ ¬(cocompact ℚ).IsCountablyGenerated",
"usedConstants": [
"Rat.instOfNat",
"False",
"Filter.tendsto_inf",
"congrArg",
"Filter.Inf.isCountablyGenerated",
"TopologicalSpace.PseudoMetrizableSpace.firstCountableTopology",
"Rat",
"IsCompact.compl_mem_coco... | intro H
rcases exists_seq_tendsto (cocompact ℚ ⊓ 𝓝 0) with ⟨x, hx⟩
rw [tendsto_inf] at hx; rcases hx with ⟨hxc, hx0⟩
obtain ⟨n, hn⟩ : ∃ n : ℕ, x n ∉ insert (0 : ℚ) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr ⟨n, rfl⟩) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 38
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"usedConstants": [
"Finset",
"Finset.instSDiff",
"PairReduction.logSizeBallStruct... | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 38
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"usedConstants": [
"Finset",
"Finset.instSDiff",
"PairReduction.logSizeBallStruct... | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 38
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"usedConstants": [
"Finset",
"Finset.instSDiff",
"PairReduction.logSizeBallStruct... | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 42
} | [
{
"pp": "case h₁\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\nh : (logSizeBallSeq J hJ a c i).finset.Nonempty\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"usedConstants": [
"Finset",
... | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 42
} | [
{
"pp": "case h₁\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\nh : (logSizeBallSeq J hJ a c i).finset.Nonempty\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"usedConstants": [
"Finset",
... | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 42
} | [
{
"pp": "case h₁\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\nh : (logSizeBallSeq J hJ a c i).finset.Nonempty\n⊢ (logSizeBallSeq J hJ a c (i + 1)).finset ⊆ (logSizeBallSeq J hJ a c i).finset",
"usedConstants": [
"Finset",
... | simp [finset_logSizeBallSeq_add_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 338,
"column": 4
} | {
"line": 339,
"column": 39
} | [
{
"pp": "case zero\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\nha : 1 < a\n⊢ ↑(#(pairSetSeq J a c 0)) ≤\n (if (logSizeBallSeq J hJ a c 0).finset.Nonempty then 1 else 0) * a ^ (logSizeBallSeq J hJ a c 0).radius",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 342,
"column": 6
} | {
"line": 343,
"column": 13
} | [
{
"pp": "case pos\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\nha : 1 < a\ni : ℕ\nih :\n ↑(#(pairSetSeq J a c i)) ≤\n (if (logSizeBallSeq J hJ a c i).finset.Nonempty then 1 else 0) * a ^ (logSizeBallSeq J hJ a c i).radius\nh : (logSizeBallSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 351,
"column": 4
} | {
"line": 352,
"column": 11
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni : ℕ\nha : 1 < a\n⊢ (if (logSizeBallSeq J hJ a c 0).finset.Nonempty then 1 else 0) * a ^ ((logSizeBallSeq J hJ a c 0).radius - 1) ≤\n ↑(#((logSizeBallSeq J hJ a c 0).smallBall c))",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 355,
"column": 6
} | {
"line": 356,
"column": 13
} | [
{
"pp": "case pos\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nhJ : J.Nonempty\ni✝ : ℕ\nha : 1 < a\ni : ℕ\nh : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty\n⊢ (if (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty then 1 else 0) *\n a ^ ((logSizeBallSeq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 379,
"column": 68
} | {
"line": 379,
"column": 89
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nn : ℕ\nJ : Finset T\ninst✝ : DecidableEq T\nha : 1 < a\nhJ_card : ↑(#J) ≤ a ^ n\ns t : T\nh : (s, t) ∈ pairSet J a c\n⊢ ∃ i < #J, (s, t) ∈ pairSetSeq J a c i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : x ∈ cantorSet\nn : ℕ\nh_mem :\n 0 ≤ (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * 3 ^ n ∧\n (x - ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x).get i) * 3 ^ n ≤ 1\n⊢ ∑ i ∈ Finset.range n, ofDigitsTerm (cantorToTernary x) i ≤ x",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.CantorSet | {
"line": 293,
"column": 4
} | {
"line": 293,
"column": 57
} | [
{
"pp": "x : ℝ\nhx : x ∈ cantorSet\n⊢ Summable fun i ↦ ‖ofDigitsTerm (cantorToTernary x).get i‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"cantorToTernary",
"congrArg",
"SummationFilter",
"PseudoMetricSpace.toUn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 47
} | [
{
"pp": "T : Type u_1\ninst✝² : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝¹ : DecidableEq T\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha : 1 < a\nf : T → E\ns : T\nhs : s ∈ J\nt : T\nht : t ∈ J\nhst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c\nhJ : J.Nonempty\nP : ℕ → Prop := ⋯\nl : ℕ := ⋯\n⊢ P 0",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 160,
"column": 7
} | {
"line": 160,
"column": 18
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.28395\nx : X\ne : M ≃ N\np : ↑(Ω^ M X x)\ny : N → ↑I\nx✝ : y ∈ Cube.boundary N\nn : N\nhn : y n = 0 ∨ y n = 1\n⊢ ((↑p).comp { toFun := fun t m ↦ t (e m), continuous_toFun := ⋯ }) y = x",
"usedConstants": [
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 160,
"column": 39
} | {
"line": 160,
"column": 50
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.28395\nx : X\ne : M ≃ N\np : ↑(Ω^ M X x)\ny : N → ↑I\nx✝ : y ∈ Cube.boundary N\nn : N\nhn : y n = 0 ∨ y n = 1\n⊢ y (e (e.symm n)) = 0 ∨ y (e (e.symm n)) = 1",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 15
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.28395\nx : X\ne : M ≃ N\np : ↑(Ω^ N X x)\ny : M → ↑I\nx✝ : y ∈ Cube.boundary M\nm : M\nhm : y m = 0 ∨ y m = 1\n⊢ ((↑p).comp { toFun := fun t n ↦ t (e.symm n), continuous_toFun := ⋯ }) y = x",
"usedConstants": [
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 162,
"column": 31
} | {
"line": 162,
"column": 42
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nx✝¹ : X\nM : Type ?u.28395\nx : X\ne : M ≃ N\np : ↑(Ω^ N X x)\ny : M → ↑I\nx✝ : y ∈ Cube.boundary M\nm : M\nhm : y m = 0 ∨ y m = 1\n⊢ y (e.symm (e m)) = 0 ∨ y (e.symm (e m)) = 1",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 420,
"column": 6
} | {
"line": 420,
"column": 24
} | [
{
"pp": "T : Type u_1\ninst✝² : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝¹ : DecidableEq T\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha : 1 < a\nf : T → E\ns : T\nhs : s ∈ J\nt : T\nht : t ∈ J\nhst : edist ⟨s, hs⟩ ⟨t, ht⟩ ≤ c\nhJ : J.Nonempty\nP : ℕ → Prop := fun l ↦ s ∈ (logSizeBallSeq J hJ a c ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 15
} | [
{
"pp": "case inr\nN : Type u_1\nX : Type u_2\ninst✝ : TopologicalSpace X\nM : Type u_3\nx : X\np : ↑(Ω^ M (↑(Ω^ N X x)) const)\ny : M ⊕ N → ↑I\nhy : y ∈ Cube.boundary (M ⊕ N)\nhN : y ∘ Sum.inr ∈ Cube.boundary N\n⊢ (p (y ∘ Sum.inl)) (y ∘ Sum.inr) = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.List | {
"line": 38,
"column": 8
} | {
"line": 38,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nas : List α\na : α\nl : List α\nih : pure l ≤ traverse 𝓝 l\nthis : List.cons <$> pure a <*> pure l ≤ List.cons <$> 𝓝 a <*> traverse 𝓝 l\n⊢ pure (a :: l) ≤ traverse 𝓝 (a :: l)",
"usedConstants": [
"Pure.pure",
"PartialOrder.toPreorder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 348,
"column": 26
} | {
"line": 348,
"column": 35
} | [
{
"pp": "case refine_1\nN : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\ni✝ : N\np q : ↑(Ω^ N X x)\nH : (↑p).HomotopyRel (↑q) (Cube.boundary N)\nt : ↑I × ↑I\ny : { j // j ≠ i✝ } → ↑I\ni : { j // j ≠ i✝ }\niH : y i = 0 ∨ y i = 1\n⊢ H (t.1, (Cube.insertAt i✝) (t.2, y)) = x",
... | H.eq_fst, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.List | {
"line": 40,
"column": 2
} | {
"line": 66,
"column": 79
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : TopologicalSpace α\nas : List α\n⊢ ∀ (a : List α), ∀ s ∈ traverse 𝓝 a, ∀ᶠ (y : List α) in traverse 𝓝 a, s ∈ traverse 𝓝 y",
"usedConstants": [
"Filter.instMembership",
"List.Forall₂.cons",
"Eq.mpr",
"Unit.unit",
"Filter.mem_traver... | · intro l s hs
rcases (mem_traverse_iff _ _).1 hs with ⟨u, hu, hus⟩
clear as hs
have : ∃ v : List (Set α), l.Forall₂ (fun a s => IsOpen s ∧ a ∈ s) v ∧ sequence v ⊆ s := by
induction hu generalizing s with
| nil =>
exists []
simp only [List.forall₂_nil_left_iff]
exact ⟨tri... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.List | {
"line": 148,
"column": 41
} | {
"line": 148,
"column": 58
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\na : α\nl : List α\n⊢ Tendsto (fun p ↦ (p.1 :: p.2).eraseIdx 0) (𝓝 a ×ˢ 𝓝 l) (𝓝 ((a :: l).eraseIdx 0))",
"usedConstants": [
"nhds",
"List",
"Filter.tendsto_snd",
"instTopologicalSpaceList"
]
}
] | exact tendsto_snd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.BundledFun | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 13
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : AddCommMonoid R\ninst✝² : LinearOrder R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : IsOrderedAddMonoid R\nY : Type u_3\nf : Y → PseudoMetric X R\ns : Finset Y\nhs : s.Nonempty\n⊢ ⇑(s.sup f) = ⇑(s.sup' hs fun x ↦ f x)",
"usedConstants": [
"Eq.mpr",
"Lattic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 493,
"column": 8
} | {
"line": 493,
"column": 60
} | [
{
"pp": "case refine_2\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na : ℝ≥0∞\nn : ℕ\nJ : Finset T\nhJ_card : ↑(#J) ≤ a ^ n\nc : ℝ≥0∞\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha1 : a ≤ 1\nhJ : Nonempty ↥J\n⊢ 1 ≤ #J",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrde... | rwa [Finset.one_le_card, ← Finset.nonempty_coe_sort] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 493,
"column": 8
} | {
"line": 493,
"column": 60
} | [
{
"pp": "case refine_2\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na : ℝ≥0∞\nn : ℕ\nJ : Finset T\nhJ_card : ↑(#J) ≤ a ^ n\nc : ℝ≥0∞\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha1 : a ≤ 1\nhJ : Nonempty ↥J\n⊢ 1 ≤ #J",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrde... | rwa [Finset.one_le_card, ← Finset.nonempty_coe_sort] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 493,
"column": 8
} | {
"line": 493,
"column": 60
} | [
{
"pp": "case refine_2\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na : ℝ≥0∞\nn : ℕ\nJ : Finset T\nhJ_card : ↑(#J) ≤ a ^ n\nc : ℝ≥0∞\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nha1 : a ≤ 1\nhJ : Nonempty ↥J\n⊢ 1 ≤ #J",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrde... | rwa [Finset.one_le_card, ← Finset.nonempty_coe_sort] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 623,
"column": 4
} | {
"line": 625,
"column": 58
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\na b : π_ 1 X x\np q : ↑(Ω^ (Fin 1) X x)\n⊢ pi1EquivFundamentalGroup.toFun (⟦p⟧ * ⟦q⟧) = pi1EquivFundamentalGroup.toFun ⟦p⟧ * pi1EquivFundamentalGroup.toFun ⟦q⟧",
"usedConstants": [
"Eq.mpr",
"Inhabite... | simp only [HomotopyGroup.mul_spec (i := (0 : Fin 1))]
apply Quotient.sound
rw [Unique.eq_default 0, genLoopEquivOfUnique_transAt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 623,
"column": 4
} | {
"line": 625,
"column": 58
} | [
{
"pp": "N : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\na b : π_ 1 X x\np q : ↑(Ω^ (Fin 1) X x)\n⊢ pi1EquivFundamentalGroup.toFun (⟦p⟧ * ⟦q⟧) = pi1EquivFundamentalGroup.toFun ⟦p⟧ * pi1EquivFundamentalGroup.toFun ⟦q⟧",
"usedConstants": [
"Eq.mpr",
"Inhabite... | simp only [HomotopyGroup.mul_spec (i := (0 : Fin 1))]
apply Quotient.sound
rw [Unique.eq_default 0, genLoopEquivOfUnique_transAt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : ∀ i ⊆ A, IsSeparated (↑ε) i → i = ∅\nx : X\nhx : x ∈ A\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\ns✝ t✝ : Set α\nδ : ℝ≥0∞\nh✝ : ⨆ x ∈ s✝, infEDist x t✝ < δ ∧ ⨆ y ∈ t✝, infEDist y s✝ < δ\ns t : Set α\nh : ⨆ x ∈ s, infEDist x t < δ\nx : α\nhx : x ∈ s\n⊢ x ∈ SetRel.preimage {p | edist p.1 p.2 < δ} t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 315,
"column": 54
} | {
"line": 315,
"column": 65
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : packingNumber ε A ≠ ⊤\nx : X\nhxA : x ∈ A\nh_dist : ∀ y ∈ maximalSeparatedSet ε A, (x, y) ∉ {x | edist x.1 x.2 ≤ ↑ε}\nC : Set X := {x} ∪ maximalSeparatedSet ε A\n⊢ x ∉ maximalSeparatedSet ε A",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 323,
"column": 43
} | {
"line": 323,
"column": 54
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA : Set X\nε : ℝ≥0\nh : packingNumber ε A ≠ ⊤\nx : X\nhxA : x ∈ A\nh_dist : ∀ y ∈ maximalSeparatedSet ε A, (x, y) ∉ {x | edist x.1 x.2 ≤ ↑ε}\nC : Set X := {x} ∪ maximalSeparatedSet ε A\nhx_not_mem : x ∉ maximalSeparatedSet ε A\n⊢ ∀ y ∈ maximalSeparatedSet ε A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 353,
"column": 8
} | {
"line": 353,
"column": 37
} | [
{
"pp": "case h₂\nX : Type u_1\ninst✝ : PseudoEMetricSpace X\nε : ℝ≥0\nA C : Set X\nhC_cover : IsCover ε A C\nD : Set X\nhD_subset : D ⊆ A\nhD_separated : IsSeparated (2 * ↑ε) D\nf : ↑D → ↑C := ⋯\nhf' : ∀ (x : ↑D), edist ↑x ↑(f x) ≤ ↑ε\nx y : ↑D\nhxy : f x = f y\n⊢ edist ↑(f x) ↑y ≤ ↑ε",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 402,
"column": 44
} | {
"line": 402,
"column": 55
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nA : Set X\nε : ℝ≥0\nf : X → Y\nhf : Isometry f\nhf_inj : InjOn f A\nC : Set Y\nhC_subset : C ⊆ f '' A\nhC_cover : IsCover ε (f '' A) C\nx : ↑C\n⊢ ∃ y ∈ A, f y = ↑x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 329,
"column": 27
} | {
"line": 329,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : EMetricSpace α\ninst✝¹ : EMetricSpace β\ns✝ : Set α\ninst✝ : SecondCountableTopology α\ns : Set α\ncs : s.Countable\ns_dense : Dense s\nv0 : Set (Set α) := {t | t.Finite ∧ t ⊆ s}\nv : Set (NonemptyCompacts α) := {t | ↑t ∈ v0}\nt : NonemptyCompacts α\nε : ℝ≥0∞\nεpos ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Kuratowski | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 25
} | [
{
"pp": "α : Type u\ninst✝ : MetricSpace α\nx : ℕ → α\nH : DenseRange x\na b : α\ne : ℝ\nepos : 0 < e\nn : ℕ\nhn : dist a (x n) < e / 2\nC : dist b (x n) - dist a (x n) = ↑(embeddingOfSubset x b) n - ↑(embeddingOfSubset x a) n\nthis : dist a b ≤ dist (embeddingOfSubset x b) (embeddingOfSubset x a) + e\n⊢ dist a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 44
} | [
{
"pp": "case refine_2.inr\nα : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\nh✝ : s.Nonempty\nu : Set (Set α)\nhu₁ : u ⊆ {U | IsOpen[inst✝] U}\nhu₂ : u.Finite\nhu : powerset '' u ⊆ powerset '' {U | IsOpen[inst✝] U}\nv : Set (Set α)\nhv₁ : v ⊆ {U | IsOpen[inst✝] U}\nhv₂ : v.Finite\nhv : (fun V ↦ {s | (s ∩ V)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nu : Set (Set α)\nhu₁ : u.Finite\nhu₂ : ∀ U ∈ u, IsOpen[inst✝] U\ns : Set α\nhs₁ : s ⊆ ⋃₀ u\nhs₂ : ∀ U ∈ u, (s ∩ U).Nonempty\nf : Set α → Set α\nhfB : ∀ U ∈ u, f U ∈ B\nhfU : ∀ U ∈ u, f U ⊆ U\nhfs : ∀ U ... | exact ht₂ hU | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 205,
"column": 44
} | {
"line": 209,
"column": 75
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nι : Type u_3\ninst✝ : Finite ι\n⊢ Continuous[Pi.topologicalSpace, TopologicalSpace.vietoris α] range",
"usedConstants": [
"Eq.mpr",
"Continuous",
"Pi.topologicalSpace",
"congrArg",
"Set.iInter",
"TopologicalSpace.vietori... | by
simp_rw [continuous_iff, powerset, preimage_setOf_eq, range_subset_iff, setOf_forall]
exact ⟨
fun U hU => isOpen_iInter_of_finite fun i => hU.preimage <| continuous_apply i,
fun F hF => isClosed_iInter fun i => hF.preimage <| continuous_apply i⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 15
} | [
{
"pp": "case h\nα : Type u_1\ninst✝ : UniformSpace α\nF : Set α\nleft✝ : (∅, F).1 ⊆ SetRel.preimage Set.univ (∅, F).2\nhF : (∅, F).2 ⊆ SetRel.image Set.univ (∅, F).1\n⊢ F ∈ {∅}",
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Set.instSingletonSet",
"id",
"Set.instEmptyCollect... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nK : Set α\nhK : IsCompact K\n⊢ IsCompact (𝒫 K)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 470,
"column": 23
} | {
"line": 471,
"column": 9
} | [
{
"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\ninst✝ : CompactSpace α\n⊢ IsCompact Set.univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 24
} | [
{
"pp": "case a\nX : Type u_2\ninst✝² : EMetricSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\ns : Set X\ni : ℝ≥0\nhi : μH[↑i] s = 0\nj : ℝ≥0\nhj : μH[↑j] s = ∞\nhij : i < j\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 539,
"column": 6
} | {
"line": 539,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\ninst✝ : CompactSpace α\n⊢ IsCompact univ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 49
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nr : ℝ≥0\nf : X → Y\nhr : 0 < r\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, HolderOnWith C r f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, HolderOnWith C r f t",
"usedConstants": [
"Fil... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 13
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : EMetricSpace X\ninst✝ : EMetricSpace Y\nK : ℝ≥0\nf : X → Y\ns : Set X\nh : LipschitzOnWith K f s\n⊢ dimH (f '' s) ≤ dimH s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 39
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\n⊢ ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t",
"usedConstants": [
"Filter.instMembership",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 45
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\ns : Set X\nhf : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, LipschitzOnWith C f t\nthis : ∀ x ∈ s, ∃ C, ∃ t ∈ 𝓝[s] x, HolderOnWith C 1 f t\n⊢ dimH (f '' s) ≤ dimH s",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 49
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝² : EMetricSpace X\ninst✝¹ : EMetricSpace Y\ninst✝ : SecondCountableTopology X\nf : X → Y\nhf : ∀ (x : X), ∃ C, ∃ s ∈ 𝓝 x, LipschitzOnWith C f s\nx : X\nx✝ : x ∈ univ\n⊢ ∃ C, ∃ t ∈ 𝓝[univ] x, LipschitzOnWith C f t",
"usedConstants": [
"Filter.instMembership"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 41
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ne : E ≃L[𝕜] F\ns : Set E\n⊢ dimH s ≤ dimH (⇑e '' s)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 470,
"column": 4
} | {
"line": 470,
"column": 49
} | [
{
"pp": "case refine_2\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nx : E\ns : Set E\nh : s ∈ 𝓝 x\ne : E ≃L[ℝ] Fin (finrank ℝ E) → ℝ\nthis : ⇑e '' s ∈ 𝓝 (e x)\nr : ℝ\nhr0 : 0 < r\nhr : Metric.ball (e x) r ⊆ ⇑e '' s\n⊢ ↑(finrank ℝ E) ≤ dimH (⇑e '' s)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 331,
"column": 23
} | {
"line": 331,
"column": 50
} | [
{
"pp": "X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 710,
"column": 11
} | {
"line": 710,
"column": 49
} | [
{
"pp": "case continuous_sup\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → β\n⊢ Continuous fun p ↦ p.1 ⊔ p.2",
"usedConstants": [
"Eq.mpr",
"Continuous",
"TopologicalSpace.NonemptyCompacts.toCompacts",
"TopologicalSpace.NonemptyCompacts... | isEmbedding_toCompacts.continuous_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 716,
"column": 11
} | {
"line": 716,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\n⊢ Continuous fun p ↦ p.1 ×ˢ p.2",
"usedConstants": [
"Eq.mpr",
"Continuous",
"TopologicalSpace.NonemptyCompacts.toCompacts",
"TopologicalSpace.NonemptyCompacts",
"SProd.sprod",
"... | isEmbedding_toCompacts.continuous_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.MetricSpace.HausdorffDimension | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 33
} | [
{
"pp": "E : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhcvx : Convex ℝ s\nhne : s.Nonempty\nthis : Nonempty ↑s\nφ : ↥(affineSpan ℝ s) ≃ᵃⁱ[ℝ] ↥(affineSpan ℝ s).direction := AffineIsometryEquiv.constVSub ℝ ⟨hne.some, ⋯⟩\nhs_eq : s = Subtype.val ''... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 46
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\n⊢ eHolderNorm r f ≠ ∞ ↔ MemHolder r f",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"MemHolder",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | rw [← eHolderNorm_lt_top, lt_top_iff_ne_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 46
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\n⊢ eHolderNorm r f ≠ ∞ ↔ MemHolder r f",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"MemHolder",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | rw [← eHolderNorm_lt_top, lt_top_iff_ne_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 46
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\n⊢ eHolderNorm r f ≠ ∞ ↔ MemHolder r f",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"MemHolder",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
... | rw [← eHolderNorm_lt_top, lt_top_iff_ne_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 249,
"column": 28
} | {
"line": 251,
"column": 25
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nC r : ℝ≥0\nf : X → Y\nhf : HolderWith C r f\n⊢ nnHolderNorm r f ≤ C",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"MemHolder.coe_nnHolderNorm_eq_eHolderNorm",
"congrArg",
"PartialOrder.toP... | by
rw [← ENNReal.coe_le_coe, hf.memHolder.coe_nnHolderNorm_eq_eHolderNorm]
exact hf.eHolderNorm_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 346,
"column": 23
} | {
"line": 346,
"column": 50
} | [
{
"pp": "X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.NatEmbedding | {
"line": 38,
"column": 6
} | {
"line": 38,
"column": 30
} | [
{
"pp": "case refine_1\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : Infinite X\nU : ℕ → Set X\nhne : ∀ (n : ℕ), (U n).Nonempty\nho : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhd : Pairwise (Disjoint on U)\nn i j : ℕ\nhij : U (Nat.pair n i) = U (Nat.pair n j)\n⊢ ¬(Disjoint on U) (Nat.pair n i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.NatEmbedding | {
"line": 44,
"column": 4
} | {
"line": 44,
"column": 15
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : Infinite X\nh✝ : DiscreteTopology X\nx✝¹ x✝ : ℕ\nh : x✝¹ ≠ x✝\n⊢ (Disjoint on fun n ↦ {(Infinite.natEmbedding X) n}) x✝¹ x✝",
"usedConstants": [
"Function.instEmbeddingLikeEmbedding",
"Eq.mpr",
"Funct... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.UpperLowerSetTopology | {
"line": 243,
"column": 24
} | {
"line": 243,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsUpperSet α\ns : Set α\nS : Set (Set α)\n⊢ (∀ s ∈ S, IsOpen[inst✝¹] s) → IsOpen[inst✝¹] (⋂₀ S)",
"usedConstants": [
"Eq.mpr",
"IsUpperSet",
"Preorder.toLE",
"Members... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LowerUpperTopology | {
"line": 313,
"column": 24
} | {
"line": 313,
"column": 88
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLower α\nx y : α\nh : Inseparable x y\n⊢ Ici x = Ici y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.ScottTopology | {
"line": 220,
"column": 2
} | {
"line": 221,
"column": 18
} | [
{
"pp": "case h.e'_3\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ns : Set α\ninst✝ : IsScott α univ\n⊢ ↑(lowerClosure s) = closure[upperSet α] s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Topology.IsUpperSet",
"Preorder.toLE",
"inferInstance",
"id",
... | · rw [@IsUpperSet.closure_eq_lowerClosure α _ (upperSet α) ?_ s]
infer_instance | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Order.ScottTopology | {
"line": 241,
"column": 2
} | {
"line": 242,
"column": 9
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nD : Set (Set α)\ninst✝⁵ : Preorder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsScott β univ\nf : α → β\ninst✝ : IsScott α D\nhf : Continuous[inst✝⁴, inst✝²] f\nx✝ b : α\nhab : x✝ ≤ b\nh : ¬f x✝ ≤ f b\n⊢ False",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.ScottTopology | {
"line": 247,
"column": 4
} | {
"line": 251,
"column": 83
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : Preorder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsScott β univ\nf : α → β\nD : Set (Set α)\ninst✝ : IsScott α D\nhD : ∀ (a b : α), a ≤ b → {a, b} ∈ D\nh : ScottContinuousOn D f\nu : Set β\nhu : IsOpe... | rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)]
exact ⟨(isUpperSet_of_isOpen (D := univ) hu).preimage (h.monotone D hD),
fun t h₀ hd₁ hd₂ a hd₃ ha ↦ image_inter_nonempty_iff.mp <|
(isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ).mp hu).2 trivial (Nonempty.image f hd₁)
(directedOn_ima... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.ScottTopology | {
"line": 247,
"column": 4
} | {
"line": 251,
"column": 83
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : Preorder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsScott β univ\nf : α → β\nD : Set (Set α)\ninst✝ : IsScott α D\nhD : ∀ (a b : α), a ≤ b → {a, b} ∈ D\nh : ScottContinuousOn D f\nu : Set β\nhu : IsOpe... | rw [isOpen_iff_isUpperSet_and_dirSupInaccOn (D := D)]
exact ⟨(isUpperSet_of_isOpen (D := univ) hu).preimage (h.monotone D hD),
fun t h₀ hd₁ hd₂ a hd₃ ha ↦ image_inter_nonempty_iff.mp <|
(isOpen_iff_isUpperSet_and_dirSupInaccOn (D := univ).mp hu).2 trivial (Nonempty.image f hd₁)
(directedOn_ima... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.ScottTopology | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsScott α univ\nx y : α\nh : Inseparable x y\n⊢ Iic x = Iic y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 621,
"column": 4
} | {
"line": 621,
"column": 44
} | [
{
"pp": "δ : ℝ\nδpos : δ > 0\nε : ℝ := 2 / 5 * δ\nεpos : 0 < ε\np : GHSpace\n⊢ ∃ s, s.Finite ∧ univ ⊆ ⋃ x ∈ s, ball x ε",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Hom.Esakia | {
"line": 136,
"column": 30
} | {
"line": 136,
"column": 69
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : Preorder γ\ninst✝ : Preorder δ\nf : PseudoEpimorphism α β\nf' : α → β\nh : f' = ⇑f\n⊢ ∀ ⦃a : α⦄ ⦃b : β⦄, (f.copy f' h).toFun a ≤ b → ∃ c, a ≤ c ∧ (f.copy f' h).toFun c = b",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Hom.Esakia | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 43
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\nδ : Type u_5\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : Preorder α\ninst✝⁵ : TopologicalSpace β\ninst✝⁴ : Preorder β\ninst✝³ : TopologicalSpace γ\ninst✝² : Preorder γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : Preorder δ\nf : EsakiaHom α β\nf' : α → β\nh :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.HullKernel | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 21
} | [
{
"pp": "case cons\nα : Type u_1\ninst✝¹ : SemilatticeInf α\nT : Set α\ninst✝ : OrderTop α\nhT : ∀ p ∈ T, InfPrime p\na : α\nF' : Finset α\nh✝ : a ∉ F'\nI4 : hull T (F'.inf id) = T ↓∩ ⋃ a ∈ ↑F', Set.Ici a\n⊢ hull T ((cons a F' h✝).inf id) = T ↓∩ ⋃ a_1 ∈ ↑(cons a F' h✝), Set.Ici a_1",
"usedConstants": [
... | | cons a F' _ I4 => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Order.Category.FrameAdjunction | {
"line": 117,
"column": 30
} | {
"line": 117,
"column": 41
} | [
{
"pp": "X✝ : Type u_1\ninst✝ : TopologicalSpace X✝\nL : Locale\nX : TopCat\nu : ↑(Opposite.unop (topToLocale.obj X))\n⊢ IsOpen (localePointOfSpacePoint ↑X ⁻¹' {x | x u})",
"usedConstants": [
"Locale",
"TopCat.instCategory",
"Prop.instCompleteLattice",
"setOf",
"Locale.localePo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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