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.Algebra.InfiniteSum.Ring | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 44
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : HasSum (fun x ↦ 1 / f x) a L\nb : α\nthis : HasSum (fun i ↦ b * (1 / f i)) (b * a) L\n⊢ HasSum (fun i ↦ b / f i) (b * a) L",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 329,
"column": 4
} | {
"line": 329,
"column": 44
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsOrderedAddMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x ↦ b\nH : ∀ (s : Finset ι), #s • b ≤ ∑' (x : ι), b\nn : ℕ\nhn : ∑' (x : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 44
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\n⊢ HasSum (fun i ↦ a₂ / f i) (a₂ * a₁) L ↔ HasSum (1 / f) a₁ L",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 44
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\n⊢ Summable (fun i ↦ a / f i) L ↔ Summable (1 / f) L",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoid... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 40
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\ninst✝⁴ : CommRing α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nf : ι → α\nx : α\nhfx : HasProd f x\n⊢ HasProd (fun x ↦ |f x|) |x|",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 50
} | [
{
"pp": "α : Type u_3\ninst✝³ : Ring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalRing α\ninst✝ : T2Space α\nx : α\nh : Summable fun x_1 ↦ x ^ x_1\n⊢ Tendsto (fun n ↦ ∑ i ∈ range n, x ^ i * (1 - x)) atTop (nhds 1)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Ring.toNonAssocRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 287,
"column": 2
} | {
"line": 287,
"column": 50
} | [
{
"pp": "α : Type u_3\ninst✝³ : Ring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalRing α\ninst✝ : T2Space α\nx : α\nh : Summable fun x_1 ↦ x ^ x_1\n⊢ Tendsto (fun n ↦ ∑ i ∈ range n, (1 - x) * x ^ i) atTop (nhds 1)",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Ring.toNonAssocRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 552,
"column": 2
} | {
"line": 552,
"column": 36
} | [
{
"pp": "G : Type u_2\ninst✝³ : CommGroup G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\nf : ℕ → G\nhf : Multipliable f\n⊢ f 0 * ∏' (n : ℕ+), f ↑n = ∏' (n : ℕ), f n",
"usedConstants": [
"PNat.val",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 569,
"column": 48
} | {
"line": 569,
"column": 59
} | [
{
"pp": "G : Type u_2\ninst✝⁴ : CommGroup G\ninst✝³ : UniformSpace G\ninst✝² : IsUniformGroup G\ninst✝¹ : CompleteSpace G\ninst✝ : T2Space G\nf : ℤ → G\nhf2 : Multipliable f\nh1 : Multipliable fun n ↦ f ↑n\nh2 : Multipliable fun n ↦ f (-↑n)\nh3 : Multipliable fun n ↦ f ↑↑n\nh4 : Multipliable fun n ↦ f (-↑↑n)\nt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 42
} | [
{
"pp": "G : Type u_2\ninst✝⁴ : CommGroup G\ninst✝³ : UniformSpace G\ninst✝² : IsUniformGroup G\ninst✝¹ : CompleteSpace G\ninst✝ : T2Space G\nf : ℤ → G\nhf : Function.Even f\nhf2 : Multipliable f\n⊢ ∏' (n : ℤ), f n = f 0 * (∏' (n : ℕ+), f ↑↑n) ^ 2",
"usedConstants": [
"PNat.val",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Bornology | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 38
} | [
{
"pp": "α : Type u_1\ns✝ t : Set α\ninst✝² : Bornology α\ninst✝¹ : Preorder α\ninst✝ : IsOrderBornology α\ns : Set αᵒᵈ\n⊢ IsBounded (⇑toDual ⁻¹' s) ↔ BddAbove (⇑toDual ⁻¹' s) ∧ BddBelow (⇑toDual ⁻¹' s)",
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"Equiv.instEquivLike",
"congrA... | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 93,
"column": 6
} | {
"line": 93,
"column": 38
} | [
{
"pp": "α : Type u_1\ns✝ t : Set α\ninst✝⁵ : Bornology α\ninst✝⁴ : Preorder α\ninst✝³ : IsOrderBornology α\nβ : Type u_2\ninst✝² : Preorder β\ninst✝¹ : Bornology β\ninst✝ : IsOrderBornology β\ns : Set (α × β)\n⊢ IsBounded (fst '' s) ∧ IsBounded (snd '' s) ↔\n (BddBelow (fst '' s) ∧ BddAbove (fst '' s)) ∧ Bd... | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 109,
"column": 40
} | {
"line": 109,
"column": 72
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : Bornology α\ninst✝³ : Nonempty α\ninst✝² : LinearOrder α\ninst✝¹ : IsOrderBornology α\ninst✝ : NoMaxOrder α\ns : Set α\n⊢ IsBounded sᶜ → sᶜᶜ ∈ Filter.atTop",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Compl.compl",
"Partia... | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 124,
"column": 38
} | {
"line": 124,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝³ : Bornology α\ninst✝² : Nonempty α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderBornology α\ns : Set α\n⊢ ((∃ i, True ∧ Iic i ⊆ sᶜᶜ) ∧ ∃ i, True ∧ Ici i ⊆ sᶜᶜ) → IsBounded sᶜ",
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"congrArg",
"Compl.compl",
"PartialO... | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Bornology | {
"line": 140,
"column": 38
} | {
"line": 140,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Bornology α\ninst✝⁴ : Nonempty α\ninst✝³ : LinearOrder α\ninst✝² : IsOrderBornology α\ninst✝¹ : NoMaxOrder α\ninst✝ : OrderBot α\ns : Set α\n⊢ (∃ i, True ∧ Ici i ⊆ sᶜᶜ) → IsBounded sᶜ",
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"congrArg",
"Compl.compl",... | isBounded_iff_bddBelow_bddAbove, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 30,
"column": 2
} | {
"line": 30,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na : α\nh : (Ioi a).Nonempty\n⊢ closure[inst✝³] (Ioi a) = Ici a",
"usedConstants": [
"Set.Subset.antisymm",
"Set.Ioi",
"Set.Ici",
"PartialOrder.toPreorder",
... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ioo a b) = Icc a b",
"usedConstants": [
"Set.Subset.antisymm",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder"... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ioc a b) = Icc a b",
"usedConstants": [
"Set.Subset.antisymm",
"Set.Ioc",
"PartialOrder.toPreorder",
"SemilatticeIn... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\nhab : a ≠ b\n⊢ closure[inst✝³] (Ico a b) = Icc a b",
"usedConstants": [
"Set.Subset.antisymm",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder"... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 31
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\n⊢ Ico a b ⊆ closure[inst✝³] (interior (Ico a b))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 134,
"column": 18
} | {
"line": 134,
"column": 29
} | [
{
"pp": "x : ℝ\nx✝ : EReal\nh : ↑x + x✝ ≤ ↑x + ⊥\n⊢ x✝ ≤ ⊥",
"usedConstants": [
"Eq.mpr",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"EReal",
"Preorder.toLE",
"id",
"Bot.bot",
"LE.le",
"instCompleteLinearOrderEReal",
"le_bot_iff._simp_1",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 75
} | [
{
"pp": "x y z : ℝ\nh : ↑x + ↑y ≤ ↑x + ↑z\n⊢ ↑y ≤ ↑z",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"PartialOrder.toPreorder",
"EReal",
"Preorder.toLE",
"id",
"LE.le",
"_private.Mathlib.Data.EReal.Operations.0.EReal.addLECancellable_coe._simp_1_7"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\ninst✝¹ : SeparableSpace α\ninst✝ : Nontrivial α\n⊢ ∃ s, s.Countable ∧ Dense s ∧ (∀ (x : α), IsBot x → x ∉ s) ∧ ∀ (x : α), IsTop x → x ∉ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 24
} | [
{
"pp": "x y : EReal\nh : x < y\nz : ℝ\n⊢ ↑z + x < ↑z + y",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"id",
"instAddCommMonoidEReal",
"add_comm",
"instHAdd",
"HAdd.hAdd",
"congr",
"LT.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 312,
"column": 28
} | {
"line": 312,
"column": 39
} | [
{
"pp": "motive : EReal → Sort u_1\ncoe : (x : ℝ≥0∞) → motive ↑x\nneg_coe : (x : ℝ≥0∞) → 0 < x → motive (-↑x)\nx : EReal\nhx : ¬0 ≤ x\n⊢ 0 < -x",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"id",
"instZeroEReal"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Compact | {
"line": 530,
"column": 2
} | {
"line": 532,
"column": 37
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : TopologicalSpace β\ninst✝² : DenselyOrdered α\ninst✝¹ : ConditionallyCompleteLinearOrder β\ninst✝ : OrderTopology β\nf : α → β\na b : α\nh : ContinuousOn f [[a, b]]\n... | refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm
refine csInf_le_csSup (nonempty_uIcc.image _) ?_ ?_ <;> rw [h.image_uIcc_eq_Icc]
exacts [bddBelow_Icc, bddAbove_Icc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Compact | {
"line": 530,
"column": 2
} | {
"line": 532,
"column": 37
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : TopologicalSpace β\ninst✝² : DenselyOrdered α\ninst✝¹ : ConditionallyCompleteLinearOrder β\ninst✝ : OrderTopology β\nf : α → β\na b : α\nh : ContinuousOn f [[a, b]]\n... | refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm
refine csInf_le_csSup (nonempty_uIcc.image _) ?_ ?_ <;> rw [h.image_uIcc_eq_Icc]
exacts [bddBelow_Icc, bddAbove_Icc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 29
} | [
{
"pp": "α : Type v\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\ns : Set α\nhs : IsPreconnected s\na b : α\nha : a ∈ s\nhb : b ∈ s\n⊢ Icc a b ⊆ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 41
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nM : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nf : α ⊕ β → M\na b : M\nh₁ : HasProd (f ∘ Sum.inl) a\nh₂ : HasProd (f ∘ Sum.inr) b\nthis : Tendsto ((fun x ↦ ∏ b ∈ x, f b) ∘ ⇑sumEquiv.symm) (Filter.map (⇑sumEquiv) atTop) (𝓝 (a * b))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 10
} | [
{
"pp": "case right\nα : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : RegularSpace α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd f a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\ns : Set α\nhs : s ∈ 𝓝 a\nhsc : Is... | intro bs | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 154,
"column": 22
} | {
"line": 154,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : T3Space α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd g a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\nhf' : Multipliable f\n⊢ HasProd f a",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : UniformSpace α\ninst✝ : IsUniformGroup α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\nhg : HasProd g a\nh : CauchySeq fun s ↦ ∏ i ∈ s, f i\nu : Set α\nhu : u ∈ 𝓝 a\ns : Finset ((... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝³ : AddCommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : StarAddMonoid α\ninst✝ : ContinuousStar α\nf : β → α\na : α\nh : HasSum f a L\n⊢ HasSum (fun b ↦ Star.star (f b)) (Star.star a) L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝³ : AddCommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : StarAddMonoid α\ninst✝ : ContinuousStar α\nf : β → α\nhf : Summable (fun b ↦ Star.star (f b)) L\n⊢ Summable f L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 36
} | [
{
"pp": "case inr.inr\nα✝ : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\nα : Type u_3\ndist : α → α → ℝ\ndist_comm : ∀ (x y : α), dist x y = dist y x\ndist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z\ns : Set α\nhs : s ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}\nt : Set α\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 730,
"column": 4
} | {
"line": 731,
"column": 61
} | [
{
"pp": "case coe_coe\nx y : ℝ\n⊢ ↑x * ↑y = ⊤ ↔ ↑x = ⊥ ∧ ↑y < 0 ∨ ↑x < 0 ∧ ↑y = ⊥ ∨ ↑x = ⊤ ∧ 0 < ↑y ∨ 0 < ↑x ∧ ↑y = ⊤",
"usedConstants": [
"Eq.mpr",
"False",
"Real",
"Preorder.toLT",
"HMul.hMul",
"iff_false",
"Real.instZero",
"congrArg",
"PartialOrder.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Basic | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 13
} | [
{
"pp": "X : Type u_2\ninst✝ : PseudoMetricSpace X\ns : Set X\nε : ℝ\nhs : IsCompact (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s)\nhε : 0 < ε\nt : Finset ↑s\nhst : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s ⊆ ⋃ i ∈ t, ball (↑i) ε\n⊢ s ⊆ ⋃ x ∈ ↑(Finset.map { toFun := Subty... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IntermediateValue | {
"line": 389,
"column": 21
} | {
"line": 389,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nhs : IsClosed[inst✝³] (s ∩ Icc a b)\nha : a ∈ s\nh : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t\nhab : a ≤ b\nA : Set α := {t | t ∈ Icc a b ∧ Icc a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 24
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\nh : x ∈ ball y ε\n⊢ ∃ ε' < ε, x ∈ ball y ε'",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instLT",
"Membership.mem",
"Exists",
"id",
"Metric.ball",
"funext",
"And",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 433,
"column": 31
} | {
"line": 433,
"column": 42
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\nh : y ∈ sphere x ε\nhε : ε ≠ 0\n⊢ x ∉ sphere x ε",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Membership.mem",
"id",
"dist_self",
"Metric.mem_sphere._simp_1",
"Zero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Operations | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 35
} | [
{
"pp": "x : EReal\nhx_nonneg : 0 ≤ x\nhx_ne_top : x ≠ ⊤\ny z : EReal\n⊢ (y + z) * x = y * x + z * x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"id",
"instAddCommMonoidEReal",
"instHAdd",
"HAdd.hAdd",
"EReal.mul_comm",
"congr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 732,
"column": 27
} | {
"line": 732,
"column": 38
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n_ε : ℝ\nε0 : 0 < _ε\nn : ℕ\nhn : 1 / (↑n + 1) < _ε\n⊢ 1 / ↑(n + 1) ≤ _ε",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"AddMonoid.toAddSemigroup",
"congrArg",
"Real.inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 30
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : Dense s\nx : α\nε : ℝ\nhε : 0 < ε\nthis : (ball x ε).Nonempty\n⊢ ∃ y ∈ s, dist x y < ε",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 974,
"column": 8
} | {
"line": 974,
"column": 19
} | [
{
"pp": "α : Type u\nβ : Type v\nX : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nx y z : α\n⊢ 0 ≤ dist x y + dist y z",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1069,
"column": 20
} | {
"line": 1069,
"column": 78
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx : X\n⊢ dist x x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1070,
"column": 22
} | {
"line": 1070,
"column": 50
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx y : X\n⊢ dist x y = dist y x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1072,
"column": 4
} | {
"line": 1072,
"column": 66
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\nx y z : X\n⊢ dist x z ≤ dist x y + dist y z",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1077,
"column": 4
} | {
"line": 1078,
"column": 11
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nX : Type u_3\ne : PseudoEMetricSpace X\ndist : X → X → ℝ\ndist_nonneg : ∀ (x y : X), 0 ≤ dist x y\nh : ∀ (x y : X), edist x y = ENNReal.ofReal (dist x y)\n⊢ ⨅ ε, ⨅ (_ : ε > 0), 𝓟 {p | edist p.1 p.2 < ε} = ⨅ ε, ⨅ (_ : ε >... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Lemmas | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 29
} | [
{
"pp": "α : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : WeaklyLocallyCompactSpace α\nx : α\nthis : ∀ᶠ (r : ℝ) in 𝓝[>] 0, IsCompact (closedBall x r)\n⊢ ∃ r, 0 < r ∧ IsCompact (closedBall x r)",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"_private.Mat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1137,
"column": 6
} | {
"line": 1137,
"column": 17
} | [
{
"pp": "x r y : ℝ\n⊢ x - y < r ∧ y - x < r ↔ x - r < y ∧ y - x < r",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"sub_lt_comm",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"Real.instSub",
"AddMonoid.toAddZeroClass... | sub_lt_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Lemmas | {
"line": 122,
"column": 2
} | {
"line": 123,
"column": 9
} | [
{
"pp": "α : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nι : Sort u_3\nc : ι → Set α\nhs : IsCompact s\nhc₁ : ∀ (i : ι), IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (c i)\nhc₂ : s ⊆ ⋃ i, c i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ i, ball x δ ⊆ c i",
"usedConstants": [
"Eq.mpr",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Lemmas | {
"line": 127,
"column": 32
} | {
"line": 127,
"column": 43
} | [
{
"pp": "α : Type u_2\ninst✝ : PseudoMetricSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ t ∈ c, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t\nhc₂ : s ⊆ ⋃ i, ↑i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, ball x δ ⊆ t",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1192,
"column": 2
} | {
"line": 1192,
"column": 32
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y z : α\n⊢ dist (dist x y) (dist x z) ≤ dist y z",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"id",
"dist_comm",
"LE.le",
"congr",
"congrFun'",
"Real.pseudoMetricSpace",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1228,
"column": 2
} | {
"line": 1228,
"column": 34
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nhs : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\na : α\n⊢ a ∈ s ↔ ∀ ε > 0, ∃ b ∈ s, dist a b < ε",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\ni : PseudoMetricSpace ((b : β) → X b) :=\n PseudoEMetricSpace.toPseudoMetricSpaceOfDist (fun f g ↦ ↑(Finset.univ.sup fun b ↦ nndist (f b) (g b))) ⋯ ⋯\ns : Set ((b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.ProperSpace | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 59
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nh : ∀ (b : β), ProperSpace (X b)\n⊢ ProperSpace ((b : β) → X b)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",... | refine .of_isCompact_closedBall_of_le 0 fun x r hr => ?_
rw [closedBall_pi _ hr]
exact isCompact_univ_pi fun _ => isCompact_closedBall _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.ProperSpace | {
"line": 132,
"column": 2
} | {
"line": 134,
"column": 59
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\nι : Type u_2\ninst✝² : PseudoMetricSpace α\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nh : ∀ (b : β), ProperSpace (X b)\n⊢ ProperSpace ((b : β) → X b)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",... | refine .of_isCompact_closedBall_of_le 0 fun x r hr => ?_
rw [closedBall_pi _ hr]
exact isCompact_univ_pi fun _ => isCompact_closedBall _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Fintype β\ninst✝ : Nonempty β\na b : α\n⊢ (dist (fun x ↦ a) fun x ↦ b) = dist a b",
"usedConstants": [
"Eq.mpr",
"Real",
"pseudoMetricSpacePi",
"congrArg",
"id",
"ENNReal.toReal",
"congr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Defs | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 32
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ dist x y ≠ 0 ↔ x ≠ y",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"id",
"Ne",
"Iff",
"Zero.toOfNat0",
"MetricSpace.toPseudoMetricSpace",
"Dist.dist",
"PseudoMetricSpace.toDist",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Defs | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 44
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ dist x y ≤ 0 ↔ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 19
} | [
{
"pp": "case refine_2\nβ : Type u_2\ninst✝³ : Fintype β\nY : Type u_4\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : Zero Y\ninst✝ : DecidableEq β\ni j : β\na b : Y\nh : i ≠ j\n⊢ nndist a 0 ≤ nndist (Pi.single i a) (Pi.single j b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Defs | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 27
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ 0 < dist x y ↔ x ≠ y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 32
} | [
{
"pp": "case refine_3\nβ : Type u_2\ninst✝³ : Fintype β\nY : Type u_4\ninst✝² : PseudoMetricSpace Y\ninst✝¹ : Zero Y\ninst✝ : DecidableEq β\ni j : β\na b : Y\nh : i ≠ j\n⊢ nndist b 0 ≤ nndist (Pi.single i a) (Pi.single j b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Cauchy | {
"line": 155,
"column": 2
} | {
"line": 161,
"column": 50
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nu : ℕ → α\nhu : CauchySeq u\nb : ℕ → ℝ\nhb : ∀ (n : ℕ), 0 < b n\n⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n",
"usedConstants": [
"Metric.cauchySeq_iff",
"Eq.mpr",
"Nat.instLattice",
"Real",
"Lattice.... | rw [cauchySeq_iff] at hu
have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
intro k
rw [eventually_atTop]
obtain ⟨N, hN⟩ := hu (b k) (hb k)
exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩
exact Filter.extraction_forall_of_eventually hu' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Cauchy | {
"line": 155,
"column": 2
} | {
"line": 161,
"column": 50
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nu : ℕ → α\nhu : CauchySeq u\nb : ℕ → ℝ\nhb : ∀ (n : ℕ), 0 < b n\n⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n",
"usedConstants": [
"Metric.cauchySeq_iff",
"Eq.mpr",
"Nat.instLattice",
"Real",
"Lattice.... | rw [cauchySeq_iff] at hu
have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
intro k
rw [eventually_atTop]
obtain ⟨N, hN⟩ := hu (b k) (hb k)
exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩
exact Filter.extraction_forall_of_eventually hu' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 68,
"column": 12
} | {
"line": 68,
"column": 32
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nr : R\nn : ℕ\nhn : -r ≤ ↑n\n⊢ ↑(-↑n) ≤ r",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Basic | {
"line": 58,
"column": 56
} | {
"line": 58,
"column": 67
} | [
{
"pp": "γ : Type w\ninst✝ : MetricSpace γ\ns : Set γ\nε : ℝ\nhε : 0 < ε\nhs : s.Pairwise fun x y ↦ ε ≤ dist x y\n⊢ s.Pairwise fun x y ↦ (x, y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"setOf",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 140,
"column": 7
} | {
"line": 140,
"column": 18
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\ns : Set α\nhs : x ∈ s ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\n⊢ Bornology.IsBounded (s ∩ ball x 1) ∧ s ∩ ball x 1 ⊆ s",
"usedConstants": [
"Eq.mpr",
"Real",
"PseudoMetricSpace.toBornology",
"and_true",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 76
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 148,
"column": 67
} | {
"line": 148,
"column": 78
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\nx✝² x✝¹ : ℝ\nhr : x✝² ≤ x✝¹\nx✝ : α\n⊢ x✝ ∈ (closedBall c x✝¹)ᶜ → x✝ ∈ (closedBall c x✝²)ᶜ",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"congrArg",
"Compl.compl",
"PartialOrder.toPreo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 156,
"column": 61
} | {
"line": 156,
"column": 72
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\nx✝² x✝¹ : ℝ\nhr : x✝² ≤ x✝¹\nx✝ : α\n⊢ x✝ ∈ (ball c x✝¹)ᶜ → x✝ ∈ (ball c x✝²)ᶜ",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"Real.instLT",
"Preorder.toLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 50
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 34
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ comap (dist c) atTop = cobounded α",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 77
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"PartialOrder... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x * r) l atBot",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 51
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nf : α → R\nr : R\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nhr : r < 0\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x * r) l atTop",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 254,
"column": 45
} | {
"line": 254,
"column": 56
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℕ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 264,
"column": 45
} | {
"line": 264,
"column": 56
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 272,
"column": 45
} | {
"line": 272,
"column": 56
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atBot\n⊢ Tendsto (fun x ↦ f x • r) l atTop",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 31
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoMetricSpace α\ninst✝ : TopologicalSpace β\nk : Set β\nf : β → α\nhk : IsCompact k\nhf : ∀ x ∈ k, ContinuousWithinAt f univ x\n⊢ ∃ t, k ⊆ t ∧ IsOpen[inst✝] t ∧ Bornology.IsBounded (f '' t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 13
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nt : Set α\nx : α\nxs : x ∈ s\nxt : x ∈ t\n⊢ diam (s ∪ t) ≤ diam s + diam t",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 42
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s n)\nh's : ∀ (n : ℕ), Bornology.IsBounded (s n)\nh : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty\nh' : Tendsto (fun n ↦ diam (s n)) atTop (𝓝 ... | apply cauchySeq_of_le_tendsto_0 _ _ h' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Algebra.Ring.Real | {
"line": 49,
"column": 28
} | {
"line": 49,
"column": 86
} | [
{
"pp": "ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℝ\nh : dist x✝¹ x✝ < ε\n⊢ dist (-x✝¹) (-x✝) < ε",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.lattice",
"abs",
"congrArg",
"Real.instSub",
"HSub.hSub",
"Real.instLT",
"id",
"Real.instAddGroup",
"Subtraction... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.NNReal.Lemmas | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 28
} | [
{
"pp": "case inl\nα : Type u_2\nL : SummationFilter α\nh✝ : L.NeBot\ny : ℝ≥0\nf : α → ℝ≥0\nhy : HasSum (fun i ↦ ↑(f i)) (↑y) L\n⊢ HasSum (fun x ↦ ((fun i ↦ ↑(f i)) x).toNNReal) (↑y).toNNReal L",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"congrArg",
"SummationFilter... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Real | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 32
} | [
{
"pp": "x y z : ℝ\nh : y ∈ uIcc x z\n⊢ dist x y ≤ dist x z",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"id",
"dist_comm",
"LE.le",
"congr",
"Real.pseudoMetricSpace",
"Dist.dist",
"PseudoMetricSpace.toDist",
"Eq"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Real | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 34
} | [
{
"pp": "x y z : ℝ\nh : y ∈ uIcc x z\n⊢ dist y z ≤ dist x z",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"id",
"dist_comm",
"LE.le",
"congr",
"Real.pseudoMetricSpace",
"Dist.dist",
"PseudoMetricSpace.toDist",
"Eq"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Real | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 91
} | [
{
"pp": "x y x' y' : ℝ\nhx : x ∈ Icc x' y'\nhy : y ∈ Icc x' y'\n⊢ dist x y ≤ y' - x'",
"usedConstants": [
"Real.instLE",
"Real",
"Real.instSub",
"HSub.hSub",
"id",
"LE.le",
"instHSub",
"Real.pseudoMetricSpace",
"Dist.dist",
"PseudoMetricSpace.toDis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Pseudo.Real | {
"line": 40,
"column": 23
} | {
"line": 40,
"column": 50
} | [
{
"pp": "x y : ℝ\nhx : x ∈ Icc 0 1\nhy : y ∈ Icc 0 1\n⊢ dist x y ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\n⊢ K⁻¹ * nndist x y ≤ nndist (f x) (f y)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 59
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝¹ : EMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : AntilipschitzWith K f\nx y : α\nh : f x = f y\n⊢ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 130,
"column": 2
} | {
"line": 131,
"column": 9
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\ns : Set α\nhf : AntilipschitzWith K (s.restrict f)\ng : β → α\nt : Set β\ng_maps : MapsTo g t s\ng_inv : RightInvOn g f t\nx y : ↑t\n⊢ edist (t.restrict g x) (t.restrict g y) ≤ ↑K * edist x y",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 254,
"column": 43
} | {
"line": 254,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\nhf : LipschitzWith K f\ng : β → α\nhg : Function.RightInverse g f\nx y : β\n⊢ edist x y ≤ ↑K * edist (g x) (g y)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 84
} | [
{
"pp": "X : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nx y : X\nl : List X\na b c : X\nhab : d a b = 0\nhbc : d b c = 0\n⊢ d a c ≤ 0",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nF : Type u_3\ninst✝² : FunLike F (Set α) ℝ≥0∞\ninst✝¹ : OuterMeasureClass F α\ninst✝ : Fintype ι\nμ : F\ns : ι → Set α\n⊢ μ (⋃ i, s i) ≤ ∑ i, μ (s i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 31
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns t : Set α\n⊢ μ (s ∪ t) ≤ μ s + μ t",
"usedConstants": [
"cond",
"Eq.mpr",
"ENNReal.instAdd",
"congrArg",
"Set.instUnion",
"id",
"LE.le",
"instHAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 13
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns t : Set α\n⊢ μ s ≤ μ (s ∩ t) + μ (s \\ t)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 17
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\nI : Set ι\nhI : I.Countable\ns : ι → Set α\nh : ∀ i ∈ I, μ (s i) = 0\nx✝ : Countable ↑I\n⊢ μ (⋃ i ∈ I, s i) = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 139,
"column": 19
} | {
"line": 139,
"column": 57
} | [
{
"pp": "α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a\nn : ℕ\n⊢ ∀ᶠ (a : α) in f, ↑n < m a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 144,
"column": 59
} | {
"line": 144,
"column": 70
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝² : FunLike F (Set α) ℝ≥0∞\ninst✝¹ : OuterMeasureClass F α\nι : Type u_4\nμ : F\ns : ι → Set α\nl : Filter ι\ninst✝ : l.NeBot\nS : Set α := ⋃ n, s n\nh0 : Tendsto (fun k ↦ μ (S \\ s k)) l (𝓝 0)\nM : ℝ≥0∞ := ⨆ n, μ (s n)\nA : ∀ (k : ι), μ S ≤ M + μ (S \\ s k)\n⊢ Tendsto... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 36
} | [
{
"pp": "x : ℝ≥0∞\nxt : x ≠ ∞\n⊢ (𝓝 x).HasBasis (fun x ↦ 0 < x) fun ε ↦ Icc (x - ε) (x + ε)",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"ENNReal.instAdd",
"Preorder.toLT",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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