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.Instances.ENNReal.Lemmas | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 85
} | [
{
"pp": "⊢ 𝓟 (Icc (∞ - 1) (∞ + 1)) ≤ 𝓝 ∞",
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"ENNReal.instAdd",
"Set.Icc_self",
"ENNReal.ofNNReal",
"congrArg",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreorder",
"LinearOrderedAddCommMonoidWithTop.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 210,
"column": 6
} | {
"line": 210,
"column": 82
} | [
{
"pp": "case neg\nX : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 204,
"column": 2
} | {
"line": 210,
"column": 84
} | [
{
"pp": "X : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else... | have hd₀ : ∀ {x y}, d x y = 0 ↔ Inseparable x y := by
intro x y
refine Iff.trans ?_ hB.inseparable_iff_uniformity.symm
simp only [d, true_imp_iff]
split_ifs with h
· simp [h, pow_eq_zero_iff']
· simpa only [not_exists, Classical.not_not, eq_self_iff_true, true_iff] using h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 211,
"column": 34
} | {
"line": 211,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ✝ : Type u_2\nm✝ : OuterMeasure α\nβ : Type ?u.23043\nf : α → β\nm : OuterMeasure α\ns : ℕ → Set β\nx✝ : Pairwise (Disjoint on s)\n⊢ m (f ⁻¹' ⋃ i, s i) ≤ ∑' (i : ℕ), m (f ⁻¹' s i)",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAddCommMonoid",
"congrArg",
"Measure... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 298,
"column": 27
} | {
"line": 298,
"column": 31
} | [
{
"pp": "b : ℝ≥0\nx✝ : ∞ ≠ ∞ ∨ ↑b ≠ ∞\nx : ℝ≥0\ny : ℝ≥0∞ × ℝ≥0∞\nhy : ↑(b + 1 + x) < y.1 ∧ y.2 ≤ ↑(b + 1)\n⊢ y.2 + ↑x < y.1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"le_refl",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.instAddCommMonoid",
"covari... | hy.2 | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 279,
"column": 34
} | {
"line": 279,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ✝ : Type u_2\nm✝ : OuterMeasure α\nβ : Type ?u.36286\nf : α → β\nm : OuterMeasure β\ns : ℕ → Set α\nx✝ : Pairwise (Disjoint on s)\n⊢ m (f '' ⋃ i, s i) ≤ ∑' (i : ℕ), m (f '' s i)",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAddCommMonoid",
"congrArg",
"MeasureTh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 332,
"column": 6
} | {
"line": 333,
"column": 13
} | [
{
"pp": "case coe.top\na✝ b : ℝ≥0∞\nht : ∀ (b : ℝ≥0∞), b ≠ 0 → Tendsto (fun p ↦ p.1 * p.2) (𝓝 (∞, b)) (𝓝 ∞)\na : ℝ≥0\nhb : ∞ ≠ 0 ∨ ↑a ≠ ∞\nha : ¬↑a = 0\n⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (↑a, ∞)) (𝓝 (↑a * ∞))",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"HMul.hMul",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 334,
"column": 58
} | {
"line": 334,
"column": 69
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nma : OuterMeasure α\nmb : OuterMeasure β\nf : α → β\nh : (map f) ma ≤ mb\ns : Set β\n⊢ ((map f) ma) s ≤ ((restrict (range f)) mb) s",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"IsScalarTower.right",
"congrArg",
"CommSemiring.toSemiri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 165,
"column": 67
} | {
"line": 165,
"column": 78
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Icc 0 1)\n⊢ 0 ≤ 1 - ↑x",
"usedConstants": [
"Eq.mpr",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAddGroup",
"covariant_swap_add_of_covariant_add",
"PartialOrder.toPreorder",
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 167,
"column": 67
} | {
"line": 167,
"column": 78
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Icc 0 1)\n⊢ 1 - ↑x ≤ 1",
"usedConstants": [
"Eq.mpr",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAddGroup",
"covariant_swap_add_of_covarian... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 360,
"column": 66
} | {
"line": 360,
"column": 77
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Ioo 0 1)\n⊢ 0 < 1 - ↑x",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Eq.mpr",
"sub_pos._simp_1",
"Preorder.toLT",
"AddLeftCancelSemigroup.toIsLeftCance... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 362,
"column": 69
} | {
"line": 362,
"column": 80
} | [
{
"pp": "β : Type u_2\ninst✝² : Ring β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedRing β\nx : ↑(Ioo 0 1)\n⊢ 1 - ↑x < 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Preorder.toLT",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAddGroup",
"instIsLeftCancelAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 29
} | [
{
"pp": "α : Type u_1\nf : Filter α\nm : α → ℝ≥0∞\na b : ℝ≥0∞\nhm : Tendsto m f (𝓝 a)\nha : a ≠ 0 ∨ b ≠ ∞\n⊢ Tendsto (fun x ↦ m x * b) f (𝓝 (a * b))",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.toSemiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\n⊢ (map f) ⊤ = ⊤",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
"congrArg",
"CommSemiring.toSemiring",
"Set.u... | rw [map_top, hf.range_eq, restrict_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\n⊢ (map f) ⊤ = ⊤",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
"congrArg",
"CommSemiring.toSemiring",
"Set.u... | rw [map_top, hf.range_eq, restrict_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Surjective f\n⊢ (map f) ⊤ = ⊤",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
"congrArg",
"CommSemiring.toSemiring",
"Set.u... | rw [map_top, hf.range_eq, restrict_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 467,
"column": 21
} | {
"line": 467,
"column": 32
} | [
{
"pp": "n : ℕ\n⊢ Continuous fun x ↦ x ^ Int.negSucc n",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Continuous",
"congrArg",
"zpow_negSucc",
"DivInvMonoid.toZPow",
"id",
"DivInvMonoid.toMonoid",
"instOfNatNat",
"Int",
"Monoid.toPow",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n\nhd : Summable d\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ tsum d",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\na : α\nh : Summable fun n ↦ dist (f n) (f n.succ)\nha : Tendsto f atTop (𝓝 a)\n⊢ dist (f 0) a ≤ ∑' (n : ℕ), dist (f n) (f n.succ)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 13
} | [
{
"pp": "f : ℕ → ℝ≥0\n⊢ (¬Summable fun i ↦ ↑(f i)) ↔ Tendsto (fun n ↦ ∑ i ∈ range n, (fun i ↦ ↑(f i)) i) atTop atTop",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Finset",
"PseudoMetricSpace.toUniformSpace",
"Membership.mem",
"id",
"Finset.range",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 13
} | [
{
"pp": "α : Type u_4\nβ : α → Type u_3\nf : (x : α) × β x → ℝ≥0\n⊢ (Summable fun i ↦ ↑(f i)) ↔\n (∀ (x : α), Summable fun y ↦ (fun i ↦ ↑(f i)) ⟨x, y⟩) ∧ Summable fun x ↦ ∑' (y : β x), (fun i ↦ ↑(f i)) ⟨x, y⟩",
"usedConstants": [
"Real",
"PseudoMetricSpace.toUniformSpace",
"id",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 57
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nf : β → ℝ\nhf : 0 ≤ f\ns : α → Set β\nhs : ∀ (i : β), ∃! j, i ∈ s j\n⊢ Summable f ↔ (∀ (j : α), Summable fun i ↦ f ↑i) ∧ Summable fun j ↦ ∑' (i : ↑(s j)), f ↑i",
"usedConstants": [
"Eq.mpr",
"Real",
"Equiv.instEquivLike",
"congrArg",
"Pseudo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 850,
"column": 53
} | {
"line": 850,
"column": 86
} | [
{
"pp": "ι : Type u_4\nf : Filter ι\nu : ι → ℝ≥0∞\ninst✝ : f.NeBot\nb : ℝ≥0∞\nb_ne_top : b ≠ ∞\nle_b : ∀ᶠ (i : ι) in f, u i ≤ b\nliminf_le : liminf u f ≤ b\naux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)\naux' : (liminf u f).toReal = b.truncateToReal (liminf u f)\n⊢ ∀ᶠ (x : ℝ≥0∞) in map u f, (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 866,
"column": 17
} | {
"line": 866,
"column": 50
} | [
{
"pp": "ι : Type u_4\nf : Filter ι\nu : ι → ℝ≥0∞\ninst✝ : f.NeBot\nb : ℝ≥0∞\nb_ne_top : b ≠ ∞\nle_b : ∀ᶠ (i : ι) in f, u i ≤ b\naux : ∀ᶠ (i : ι) in f, (u i).toReal = b.truncateToReal (u i)\naux' : (limsup u f).toReal = b.truncateToReal (limsup u f)\n⊢ ∀ᶠ (x : ℝ≥0∞) in map u f, (fun x1 x2 ↦ x1 ≤ x2) x b",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sign.Defs | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh : (if a < 0 then -1 else 0) = 1\nhn : ¬0 < a\n⊢ False",
"usedConstants": [
"SignType.ctorIdx",
"False",
"Preorder.toLT",
"SignType.instOne",
"congrArg",
"False.elim",
"noCon... | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Data.Sign.Defs | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 16
} | [
{
"pp": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = -1\n⊢ a < 0",
"usedConstants": [
"SignType.ctorIdx",
"False",
"Preorder.toLT",
"SignType.instOne",
"congrArg",
"False.elim",
... | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 934,
"column": 2
} | {
"line": 934,
"column": 78
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : EMetricSpace β\ninst✝ : CompleteSpace β\ns : Set α\nhs : Dense s\nf : ↑s → β\nK : ℝ≥0\nhf : LipschitzWith K f\nthis✝ : IsClosed[instTopologicalSpaceProd] {p | edist (hs.extend f p.1) (hs.extend f p.2) ≤ ↑K * edist p.1 p.2}\nthis : Dens... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Sign.Basic | {
"line": 40,
"column": 2
} | {
"line": 42,
"column": 14
} | [
{
"pp": "case inl\nz : ℤ\nhz : Odd z\n⊢ 0 ^ z = 0",
"usedConstants": [
"zero_zpow",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"GroupWithZero.toDivInvMonoid",
"SignType.instCommGroupWithZero",
"congrArg",
"False.elim",
"Odd",
"DivInvMonoi... | · rw [zero_zpow]
rintro rfl
simp at hz | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.EReal.Inv | {
"line": 80,
"column": 47
} | {
"line": 80,
"column": 60
} | [
{
"pp": "case neg_left\nx✝ y✝ : EReal\nh : (x✝ * y✝).abs = x✝.abs * y✝.abs\n⊢ (x✝ * y✝).abs = (-x✝).abs * y✝.abs",
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"EReal.abs_neg",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"EReal.instNeg",
"EReal",
"i... | EReal.abs_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 140,
"column": 12
} | {
"line": 140,
"column": 23
} | [
{
"pp": "case mp.neg.h\nx y : EReal\nh : ↑neg * ↑x.abs ≤ ↑neg * ↑y.abs\nhs : sign x = sign y\nhy : sign y = neg\n⊢ neg = neg ∧ neg = neg ∧ y.abs ≤ x.abs",
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"SignType.instOne",
"congrArg",
"id",
"SignType.instNeg",
"SignType.n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Inv | {
"line": 141,
"column": 20
} | {
"line": 141,
"column": 31
} | [
{
"pp": "case mp.pos.h.h\nx y : EReal\nh : ↑pos * ↑x.abs ≤ ↑pos * ↑y.abs\nhs : sign x = sign y\nhy : sign y = pos\n⊢ pos = pos ∧ pos = pos ∧ x.abs ≤ y.abs",
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"SignType.instOne",
"congrArg",
"SignType.pos",
"id",
"LE.le",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Inv | {
"line": 395,
"column": 42
} | {
"line": 395,
"column": 77
} | [
{
"pp": "case refine_2\na b c : EReal\nhbot : b ≠ ⊥\nhtop : b ≠ ⊤\nhzero : b ≠ 0\nh : c = a * b\n⊢ b * (a / b) = a",
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"congrArg",
"EReal",
"id",
"HDiv.hDiv",
"EReal.mul_div_ca... | @mul_div_cancel a b hbot htop hzero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 447,
"column": 2
} | {
"line": 449,
"column": 63
} | [
{
"pp": "a b c : EReal\nh : 0 < b\nh' : b ≠ ⊤\n⊢ a / b ≤ c ↔ a ≤ b * c",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"EReal.strictMono_div_right_of_pos",
"cong... | nth_rw 1 [← @mul_div_cancel c b (ne_bot_of_gt h) h' h.ne']
rw [mul_div b c b, mul_comm b]
exact StrictMono.le_iff_le (strictMono_div_right_of_pos h h') | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Inv | {
"line": 447,
"column": 2
} | {
"line": 449,
"column": 63
} | [
{
"pp": "a b c : EReal\nh : 0 < b\nh' : b ≠ ⊤\n⊢ a / b ≤ c ↔ a ≤ b * c",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"EReal.strictMono_div_right_of_pos",
"cong... | nth_rw 1 [← @mul_div_cancel c b (ne_bot_of_gt h) h' h.ne']
rw [mul_div b c b, mul_comm b]
exact StrictMono.le_iff_le (strictMono_div_right_of_pos h h') | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 20
} | [
{
"pp": "α : Type u_4\ninst✝ : Infinite α\nc : ℝ≥0∞\nhc : c ≠ 0\nA : Tendsto (fun n ↦ ↑n * c) atTop (𝓝 (∞ * c))\nn : ℕ\ns : Finset α\nhs : #s = n\n⊢ ↑n * c ≤ ∑' (x : α), c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 18
} | [
{
"pp": "α : Type u_4\ninst✝ : Infinite α\nc : ℝ≥0∞\nhc : c ≠ 0\nA : Tendsto (fun n ↦ ↑n * c) atTop (𝓝 (∞ * c))\nB : ∀ (n : ℕ), ↑n * c ≤ ∑' (x : α), c\n⊢ ∑' (x : α), c = ∞",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Algebra | {
"line": 668,
"column": 46
} | {
"line": 668,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u\ninst✝³ : TopologicalSpace A\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsSemitopologicalSemiring A\nx : A\ns : Subalgebra R A\nhs : IsClosed[inst✝³] ↑s\nhx : x ∈ s\n⊢ {x} ⊆ ↑s",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Algebra | {
"line": 691,
"column": 23
} | {
"line": 691,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u\ninst✝³ : TopologicalSpace A\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : IsSemitopologicalSemiring A\nx : A\n⊢ IsClosed[inst✝³] (range Subtype.val)",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 33
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\na : R\n⊢ ∑' (i : α), a • f i = a • ∑' (i : α), f i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 85
} | [
{
"pp": "α : Type u_4\nf : α → ℝ≥0∞\nhf : ∑' (i : α), f i ≠ ∞\n⊢ Summable (ENNReal.toNNReal ∘ f)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"congrArg",
"_private.Mathlib.Topology.Algebra.InfiniteSum.ENNRe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 287,
"column": 14
} | {
"line": 287,
"column": 25
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0∞\ns t : Set α\n⊢ ∑' (x : ↑(⋃ b, bif b then s else t)), f ↑x ≤ ∑' (x : ↑s), f ↑x + ∑' (x : ↑t), f ↑x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 23
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\nh : Summable fun a ↦ ↑(f a)\n⊢ (support f).Countable",
"usedConstants": [
"id",
"NNReal",
"NNReal.instZero",
"Function.support",
"Set.Countable"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 424,
"column": 4
} | {
"line": 425,
"column": 36
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : α → Type u_4\nf : (x : α) × β x → ℝ≥0\nh₁ : ∀ (x : α), Summable fun y ↦ f ⟨x, y⟩\nh₂ : Summable fun x ↦ ∑' (y : β x), f ⟨x, y⟩\n⊢ Summable f",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 472,
"column": 2
} | {
"line": 472,
"column": 13
} | [
{
"pp": "α : Type u_1\ng : α → ℝ≥0\nhg : Summable g\ni : α\nhi : 0 < g i\n⊢ 0 < ∑' (b : α), g b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Archimedean | {
"line": 60,
"column": 33
} | {
"line": 60,
"column": 44
} | [
{
"pp": "G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\na : G\na_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}\na_in : a ∈ H\na_pos : 1 < a\ng : G\ng_in : g ∈ H\nk : ℤ\nright✝ : ∀ (y : ℤ), (fun k ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_4\ni : β → α\nhi : Injective i\nf : α → ℝ≥0\nhf : Summable f\n⊢ tsum ((fun i ↦ ↑(f i)) ∘ i) ≤ ∑' (i : α), ↑(f i)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"PseudoMetricSpace.toUniform... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Archimedean | {
"line": 61,
"column": 49
} | {
"line": 61,
"column": 95
} | [
{
"pp": "G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\na : G\na_min : a ∈ lowerBounds {g | g ∈ H ∧ 1 < g}\na_in : a ∈ H\na_pos : 1 < a\ng : G\ng_in : g ∈ H\nk : ℤ\nright✝ : ∀ (y : ℤ), (fun k ↦ a ^ k ≤ g ∧ g < a ^ (k + 1)) y → y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 23
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\nh : ∑' (i : α), (fun i ↦ ↑(f i)) i ≠ ∞\n⊢ (support fun i ↦ ↑(f i)).Countable",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"congrArg",
"setOf",
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"funext",
"Function.su... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 628,
"column": 36
} | {
"line": 628,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ≥0∞\nhf : ∀ (n : ℕ), edist (f n) (f n.succ) ≤ d n\na : α\nha : Tendsto f atTop (𝓝 a)\n⊢ edist (f 0) a ≤ ∑' (m : ℕ), d m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 647,
"column": 20
} | {
"line": 647,
"column": 45
} | [
{
"pp": "α : Type u_4\n⊢ ∑' (x : ↑univ), 1 = ↑(ENat.card α)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 651,
"column": 20
} | {
"line": 651,
"column": 45
} | [
{
"pp": "α : Type u_4\nc : ℝ≥0∞\n⊢ ∑' (x : ↑univ), c = ↑(ENat.card α) * c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 153,
"column": 70
} | {
"line": 165,
"column": 43
} | [
{
"pp": "⊢ 𝓝[≠] ⊥ = map Real.toEReal atBot",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Set.Ioc",
"False",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Real.instArchimedean",
"eq_false",
"LinearOrde... | by
apply (nhdsWithin_hasBasis nhds_bot_basis_Iic _).ext (atBot_basis.map Real.toEReal)
· simp only [EReal.image_coe_Iic,
true_and]
intro x hx
by_cases hx_top : x = ⊤
· simp [hx_top]
lift x to ℝ using ⟨hx_top, hx.ne_bot⟩
refine ⟨x, fun x ⟨h1, h2⟩ ↦ ?_⟩
simp [h2, h1.ne_bot]
· simp only... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 208,
"column": 4
} | {
"line": 208,
"column": 83
} | [
{
"pp": "case pos\nx✝ : EReal\nh_top : ¬x✝ = ⊤\nx : EReal\nhx : x ∈ {⊤}ᶜ\nh_bot : x = ⊥\n⊢ ∃ i, True ∧ ∀ ⦃x : EReal⦄, x ∈ Iio ↑i → ENNReal.ofReal x.toReal = (fun x ↦ ENNReal.ofReal x.toReal) ⊥",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 58,
"column": 63
} | {
"line": 58,
"column": 74
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"id",
"LE.le",
"Nat.cast",
"Nat.dist_cast_rat",
"Real.instOne",
"funext",
"Nat",
"Pairwise",
"One.toOfNat1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 61,
"column": 58
} | {
"line": 61,
"column": 69
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"id",
"LE.le",
"Nat.cast",
"Nat.dist_cast_rat",
"Real.instOne",
"funext",
"Nat",
"Pairwise",
"One.toOfNat1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 68,
"column": 63
} | {
"line": 68,
"column": 74
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"Int.dist_cast_rat",
"Rat.instIntCast",
"id",
"Int",
"LE.le",
"Real.instOne",
"funext",
"Pairw... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 71,
"column": 58
} | {
"line": 71,
"column": 69
} | [
{
"pp": "⊢ Pairwise fun x y ↦ 1 ≤ dist ↑x ↑y",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Rat",
"Int.dist_cast_rat",
"Rat.instIntCast",
"id",
"Int",
"LE.le",
"Real.instOne",
"funext",
"Pairw... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 69
} | [
{
"pp": "ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℚ\nh : dist x✝¹ x✝ < ε\n⊢ dist (-x✝¹) (-x✝) < ε",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"DivisionRing.toRatCast",
"abs",
"congrArg",
"Real.instSub",
"Real.instRatCast",
"Rat"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 11
} | [
{
"pp": "case inr.a\nα : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ ∀ y > limsup u f * c, ∀ᶠ (a : α) in f, u a * c < y",
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"False",
"Preorder.toLT",
"instHDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 98,
"column": 25
} | {
"line": 98,
"column": 50
} | [
{
"pp": "ε : ℝ\nε0 : ε > 0\nx✝¹ x✝ : ℚ\nh : dist x✝¹ x✝ < ε\n⊢ dist |x✝¹| |x✝| ≤ dist x✝¹ x✝",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.lattice",
"DivisionRing.toRatCast",
"AddGroupWithOne.toAddGroup",
"abs",
"congrArg",
"Real.instSub",
"Real.instRatCa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.Rat | {
"line": 103,
"column": 2
} | {
"line": 104,
"column": 9
} | [
{
"pp": "a b : ℚ\n⊢ TotallyBounded (Icc a b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 11
} | [
{
"pp": "case inr.a\nα : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ ∀ y < limsup u f * c, ∃ᶠ (a : α) in f, y < u a * c",
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"False",
"Preorder.toLT",
"instHDiv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 26
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ limsup (fun x ↦ c * u x) f = c * liminf u f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 49
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ liminf (fun x ↦ c * u x) f = c * liminf u f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 49
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ liminf (fun x ↦ c * u x) f = c * limsup u f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 58
} | [
{
"pp": "case inl\nα : Type u_3\nf : Filter α\nu v : α → EReal\nhu : ∃ᶠ (x : α) in f, 0 ≤ u x\nhv : 0 ≤ᶠ[f] v\nh✝ : f.NeBot\nu_0 : 0 ≤ limsup u f\nh₁ : 0 < limsup u f ∨ limsup v f ≠ ⊤\nh₂ : limsup u f ≠ ⊤ ∨ 0 < limsup v f\na : EReal\na_u : a > limsup u f\nb : EReal\nb_v : b > limsup v f\nc : EReal\nc_ab : c > a... | exact mul_nonneg (u_0.trans a_u.le) (v_0.trans x_b.le) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 399,
"column": 2
} | {
"line": 399,
"column": 53
} | [
{
"pp": "a r : ℝ\nx✝ : EReal × EReal\nh : ↑(r - (a - 1)) < x✝.1 ∧ ↑(a - 1) < x✝.2\n⊢ ↑r < x✝.1 + x✝.2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 403,
"column": 2
} | {
"line": 404,
"column": 9
} | [
{
"pp": "a : ℝ\n⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a, ⊤)",
"usedConstants": [
"ContinuousAt",
"EReal.instTopologicalSpace",
"instTopologicalSpaceProd",
"EReal",
"instTopEReal",
"id",
"Prod.mk",
"instAddCommMonoidEReal",
"Prod.fst",
"instHAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 39
} | [
{
"pp": "r : ℝ\nx✝ : EReal × EReal\nh : ↑0 < x✝.1 ∧ ↑r < x✝.2\n⊢ ↑r < x✝.1 + x✝.2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 417,
"column": 2
} | {
"line": 417,
"column": 53
} | [
{
"pp": "a r : ℝ\nx✝ : EReal × EReal\nh : x✝.1 < ↑(r - (a + 1)) ∧ x✝.2 < ↑(a + 1)\n⊢ x✝.1 + x✝.2 < ↑r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 421,
"column": 2
} | {
"line": 422,
"column": 9
} | [
{
"pp": "a : ℝ\n⊢ ContinuousAt (fun p ↦ p.1 + p.2) (↑a, ⊥)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ContinuousAt",
"EReal.instTopologicalSpace",
"instTopologicalSpaceProd",
"EReal",
"nhds",
"id",
"Prod.mk",
"Bot.bot",
"instAddCommMonoidERe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 39
} | [
{
"pp": "r : ℝ\nx✝ : EReal × EReal\nh : x✝.1 < ↑0 ∧ x✝.2 < ↑r\n⊢ x✝.1 + x✝.2 < ↑r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ns : Set α\nγ : Type u_4\ninst✝ : LinearOrder γ\nι : Type u_5\nf : ι → α → γ\nks : IsCompact s\nI : Set ι\nc : γ\nhfi : ∀ i ∈ I, LowerSemicontinuousOn (f i) s\nH : s ∩ ⋂ i ∈ I, f i ⁻¹' Iic c = ∅\nthis : ∀ i ∈ I, IsClosed[instTopologicalSpaceSubtype] (s ↓∩ (fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 451,
"column": 10
} | {
"line": 451,
"column": 25
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 75,
"column": 20
} | {
"line": 75,
"column": 31
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nμ : Set α → ℝ≥0∞ := fun s ↦ ⨅ f, ⨅ (_ : s ⊆ ⋃ i, f i), ∑' (i : ℕ), m (f i)\ns : ℕ → Set α\nx✝ : Pairwise (Disjoint on s)\nε : ℝ≥0\nhε : 0 < ε\nhb : ∑' (i : ℕ), μ (s i) < ∞\nε' : ℕ → ℝ≥0\nhε' : ∀ (i : ℕ), 0 < ε' i\nhl : ∑' (i : ℕ), ↑(ε' i) < ↑ε\ni : ℕ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 455,
"column": 10
} | {
"line": 455,
"column": 25
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 45
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : DivisionSemiring 𝕜\ninst✝³ : CharZero 𝕜\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : ContinuousSMul ℚ≥0 𝕜\ninst✝ : ContinuousMul 𝕜\nC : 𝕜\n⊢ Tendsto (fun n ↦ C / ↑n) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Div... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 588,
"column": 2
} | {
"line": 588,
"column": 29
} | [
{
"pp": "α : Type u_2\nf : Filter α\nm : α → EReal\na b : EReal\nhm : Tendsto m f (𝓝 a)\nh₁ : a ≠ 0 ∨ b ≠ ⊥\nh₂ : a ≠ 0 ∨ b ≠ ⊤\n⊢ Tendsto (fun x ↦ m x * b) f (𝓝 (a * b))",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 465,
"column": 10
} | {
"line": 465,
"column": 25
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Dissipate | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\ns : α → Set β\nx : α\n⊢ ⋂ y, ⋂ (_ : y ≤ x), dissipate s y = dissipate s x",
"usedConstants": [
"Set.Subset.antisymm",
"Set.dissipate",
"Set.iInter",
"Preorder.toLE",
"LE.le"
]
}
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Set.Dissipate | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : α → Set β\ninst✝ : Preorder α\n⊢ ⋂ x, dissipate s x = ⋂ x, s x",
"usedConstants": [
"Set.Subset.antisymm",
"Set.dissipate",
"Set.iInter",
"Preorder.toLE"
]
}
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.PiSystem | {
"line": 117,
"column": 12
} | {
"line": 117,
"column": 23
} | [
{
"pp": "case zero\nα : Type u_1\ns : ℕ → Set α\nC : Set (Set α)\nhC : IsPiSystem C\nh : ∀ (n : ℕ), s n ∈ C\nh' : (dissipate s 0).Nonempty\n⊢ dissipate s 0 ∈ C",
"usedConstants": [
"Set.dissipate",
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"instOfNatNat",
"instL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 478,
"column": 10
} | {
"line": 478,
"column": 25
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 479,
"column": 10
} | {
"line": 479,
"column": 28
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁵ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousWithinAt f s x\nhg : LowerSemicontinuousWithin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 149,
"column": 19
} | {
"line": 149,
"column": 30
} | [
{
"pp": "case singleton\nα : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\nt : Finset (Set α)\na : Set α\nht : ∀ s ∈ {a}, s ∈ S\nh' : (⋂ s ∈ {a}, s).Nonempty\n⊢ ⋂ s ∈ {a}, s ∈ S",
"usedConstants": [
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Set.iInter",
"Finset",
"Membership.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 35
} | [
{
"pp": "case h\n𝕜 : Type u_4\ninst✝⁵ : Semifield 𝕜\ninst✝⁴ : CharZero 𝕜\ninst✝³ : TopologicalSpace 𝕜\ninst✝² : ContinuousSMul ℚ≥0 𝕜\ninst✝¹ : IsTopologicalSemiring 𝕜\ninst✝ : ContinuousInv₀ 𝕜\na b c d : 𝕜\nhd : d ≠ 0\n⊢ Tendsto (fun k ↦ (a * (↑k)⁻¹ + c) / (b * (↑k)⁻¹ + d)) atTop (𝓝 (c / d))",
"use... | apply Filter.Tendsto.div _ _ hd | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 554,
"column": 2
} | {
"line": 561,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, LowerSemic... | classical
induction a using Finset.induction_on with
| empty => exact lowerSemicontinuousWithinAt_const
| insert _ _ ia IH =>
simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
(IH fun j ja => ha j (Finset.m... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 554,
"column": 2
} | {
"line": 561,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, LowerSemic... | classical
induction a using Finset.induction_on with
| empty => exact lowerSemicontinuousWithinAt_const
| insert _ _ ia IH =>
simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
(IH fun j ja => ha j (Finset.m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 554,
"column": 2
} | {
"line": 561,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ns : Set α\nx : α\nι : Type u_4\nγ : Type u_5\ninst✝⁵ : AddCommMonoid γ\ninst✝⁴ : LinearOrder γ\ninst✝³ : IsOrderedAddMonoid γ\ninst✝² : TopologicalSpace γ\ninst✝¹ : OrderTopology γ\ninst✝ : ContinuousAdd γ\nf : ι → α → γ\na : Finset ι\nha : ∀ i ∈ a, LowerSemic... | classical
induction a using Finset.induction_on with
| empty => exact lowerSemicontinuousWithinAt_const
| insert _ _ ia IH =>
simp only [ia, Finset.sum_insert, not_false_iff]
exact
LowerSemicontinuousWithinAt.add (ha _ (Finset.mem_insert_self ..))
(IH fun j ja => ha j (Finset.m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.PiSystem | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 37
} | [
{
"pp": "α : Type u_1\nι : Sort u_3\nι' : Sort u_4\ninst✝ : LinearOrder α\nIxx : α → α → Set α\np : α → α → Prop\nHne : ∀ {a b : α}, (Ixx a b).Nonempty → p a b\nHi : ∀ {a₁ b₁ a₂ b₂ : α}, Ixx a₁ b₁ ∩ Ixx a₂ b₂ = Ixx (max a₁ a₂) (min b₁ b₂)\nf : ι → α\ng : ι' → α\n⊢ IsPiSystem {S | ∃ i j, p (f i) (g j) ∧ Ixx (f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 15
} | [
{
"pp": "case base\nα : Type u_3\nβ : Type u_4\ng : β → Set (Set α)\nh_pi : ∀ (b : β), IsPiSystem (g b)\nt s : Set α\nb : β\nh_s_in_t' : s ∈ (fun b ↦ g b) b\n⊢ s = ⋂ b_1 ∈ {b}, (fun x ↦ s) b_1 ∧ ∀ b_1 ∈ {b}, (fun x ↦ s) b_1 ∈ g b_1",
"usedConstants": [
"Eq.mpr",
"Iff.of_eq",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 646,
"column": 4
} | {
"line": 646,
"column": 37
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ns : Set α\nx : α\nι : Sort u_4\nδ' : Type u_6\ninst✝ : ConditionallyCompleteLinearOrder δ'\nf : ι → α → δ'\nbdd : ∀ᶠ (y : α) in 𝓝[s] x, BddAbove (range fun i ↦ f i y)\nh : ∀ (i : ι), LowerSemicontinuousWithinAt (f i) s x\nh✝ : IsEmpty ι\n⊢ LowerSemi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 232,
"column": 2
} | {
"line": 236,
"column": 44
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ c • OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c • m) ⋯",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"iInf",
"ENNReal.tsum_mul_left",
"instSMulOfMul",
"HMul.hMul",
"... | ext1 s
haveI : Nonempty { t : ℕ → Set α // s ⊆ ⋃ i, t i } := ⟨⟨fun _ => s, subset_iUnion (fun _ => s) 0⟩⟩
simp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul,
iInf_subtype']
rw [ENNReal.mul_iInf fun h => (hc h).elim] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 232,
"column": 2
} | {
"line": 236,
"column": 44
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nc : ℝ≥0∞\nhc : c ≠ ∞\n⊢ c • OuterMeasure.ofFunction m m_empty = OuterMeasure.ofFunction (c • m) ⋯",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"iInf",
"ENNReal.tsum_mul_left",
"instSMulOfMul",
"HMul.hMul",
"... | ext1 s
haveI : Nonempty { t : ℕ → Set α // s ⊆ ⋃ i, t i } := ⟨⟨fun _ => s, subset_iUnion (fun _ => s) 0⟩⟩
simp only [smul_apply, ofFunction_apply, ENNReal.tsum_mul_left, Pi.smul_apply, smul_eq_mul,
iInf_subtype']
rw [ENNReal.mul_iInf fun h => (hc h).elim] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 32
} | [
{
"pp": "case refine_2\n𝕜 : Type u_4\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nr : 𝕜\nh : |r| < 1\n⊢ Tendsto (abs ∘ fun n ↦ r ^ n) atTop (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 269,
"column": 15
} | {
"line": 269,
"column": 78
} | [
{
"pp": "r : ℝ≥0\nh : Tendsto (fun n ↦ r ^ n) atTop (𝓝 0)\n⊢ r < 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 431,
"column": 4
} | {
"line": 432,
"column": 29
} | [
{
"pp": "case refine_2\nα : Type u_1\nι : Sort u_2\nβ : Type u_3\ninst✝ : Nonempty ι\nf : α → β\nm : ι → OuterMeasure β\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\nn : ℕ\ni : ι\n⊢ f '' f ⁻¹' (fun n ↦ f '' t n ∪ (range f)ᶜ) n ⊆ f '' t n",
"usedConstants": [
"Eq.mpr",
"Set.union_empty",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 35
} | [
{
"pp": "case refine_2\nr : ℝ≥0∞\nr_gt_one : 1 < r\nobs : r⁻¹ < 1 → Tendsto (fun x ↦ (r⁻¹ ^ x)⁻¹) atTop (𝓝 ∞)\n⊢ Tendsto (fun n ↦ r ^ n) atTop (𝓝 ∞)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 360,
"column": 4
} | {
"line": 361,
"column": 11
} | [
{
"pp": "n : ℕ\n⊢ Summable fun i ↦ if n ≤ i then 2⁻¹ ^ i else 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 70,
"column": 15
} | {
"line": 70,
"column": 26
} | [
{
"pp": "α : Type u\nm : OuterMeasure α\ns : Set α\nh : m.IsCaratheodory sᶜ\n⊢ m.IsCaratheodory s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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