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.Order.Compact | {
"line": 128,
"column": 47
} | {
"line": 128,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\n⊢ b ≤ a → IsCompact (Ioc a b)",
"usedConstants": [
"Set.Ioc",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.Compact | {
"line": 133,
"column": 47
} | {
"line": 133,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\n⊢ b ≤ a → IsCompact (Ioo a b)",
"usedConstants": [
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Set.Finite",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Order.Compact | {
"line": 133,
"column": 47
} | {
"line": 133,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\n⊢ b ≤ a → IsCompact (Ioo a b)",
"usedConstants": [
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Set.Finite",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Compact | {
"line": 133,
"column": 47
} | {
"line": 133,
"column": 55
} | [
{
"pp": "α : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\n⊢ b ≤ a → IsCompact (Ioo a b)",
"usedConstants": [
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Set.Finite",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Operations | {
"line": 303,
"column": 64
} | {
"line": 303,
"column": 72
} | [
{
"pp": "x y : EReal\nh1 : x ≠ ⊥ ∨ y ≠ ⊥\nh2 : x ≠ ⊤ ∨ y ≠ ⊤\n⊢ x ≠ ⊥ ∨ -y ≠ ⊤",
"usedConstants": [
"congrArg",
"EReal",
"instTopEReal",
"id",
"Ne",
"Bot.bot",
"EReal.neg_eq_top_iff._simp_1",
"SubNegZeroMonoid.toSubNegMonoid",
"True",
"SubNegMonoid... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.EReal.Operations | {
"line": 303,
"column": 64
} | {
"line": 303,
"column": 72
} | [
{
"pp": "x y : EReal\nh1 : x ≠ ⊥ ∨ y ≠ ⊥\nh2 : x ≠ ⊤ ∨ y ≠ ⊤\n⊢ x ≠ ⊤ ∨ -y ≠ ⊥",
"usedConstants": [
"congrArg",
"EReal",
"instTopEReal",
"id",
"Ne",
"Bot.bot",
"SubNegZeroMonoid.toSubNegMonoid",
"True",
"EReal.neg_eq_bot_iff._simp_1",
"SubNegMonoid... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.EReal.Operations | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 34
} | [
{
"pp": "x : ℝ\n⊢ ↑x = ↑↑x.toNNReal - ↑↑(-x).toNNReal",
"usedConstants": [
"Real",
"Real.instZero",
"le_total",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.linearOrder"
]
}
] | rcases le_total 0 x with (h | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.EReal.Operations | {
"line": 621,
"column": 4
} | {
"line": 631,
"column": 64
} | [
{
"pp": "⊢ ∀ {a b : EReal}, a * b = 0 → a = 0 ∨ b = 0",
"usedConstants": [
"Eq.mpr",
"lt_or_gt_of_ne",
"False",
"Real",
"IsDomain.to_noZeroDivisors",
"Preorder.toLT",
"HMul.hMul",
"eq_false",
"MulZeroClass.toMul",
"Real.instZero",
"congrArg",... | intro a b h
contrapose! h
cases a <;> cases b <;> try {· simp_all [← EReal.coe_mul]}
· rcases lt_or_gt_of_ne h.2 with (h | h)
<;> simp [EReal.bot_mul_of_neg, EReal.bot_mul_of_pos, h]
· rcases lt_or_gt_of_ne h.1 with (h | h)
<;> simp [EReal.mul_bot_of_pos, EReal.mul_bot_of_neg, h]
· r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Operations | {
"line": 621,
"column": 4
} | {
"line": 631,
"column": 64
} | [
{
"pp": "⊢ ∀ {a b : EReal}, a * b = 0 → a = 0 ∨ b = 0",
"usedConstants": [
"Eq.mpr",
"lt_or_gt_of_ne",
"False",
"Real",
"IsDomain.to_noZeroDivisors",
"Preorder.toLT",
"HMul.hMul",
"eq_false",
"MulZeroClass.toMul",
"Real.instZero",
"congrArg",... | intro a b h
contrapose! h
cases a <;> cases b <;> try {· simp_all [← EReal.coe_mul]}
· rcases lt_or_gt_of_ne h.2 with (h | h)
<;> simp [EReal.bot_mul_of_neg, EReal.bot_mul_of_pos, h]
· rcases lt_or_gt_of_ne h.1 with (h | h)
<;> simp [EReal.mul_bot_of_pos, EReal.mul_bot_of_neg, h]
· r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 345,
"column": 47
} | {
"line": 345,
"column": 55
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Order.IntermediateValue | {
"line": 345,
"column": 47
} | {
"line": 345,
"column": 55
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (t... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 345,
"column": 47
} | {
"line": 345,
"column": 55
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (t... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 345,
"column": 61
} | {
"line": 345,
"column": 69
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Order.IntermediateValue | {
"line": 345,
"column": 61
} | {
"line": 345,
"column": 69
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (t... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 345,
"column": 61
} | {
"line": 345,
"column": 69
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (t... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 12
} | [
{
"pp": "case inl\nα : 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 : b < a\n⊢ Icc a b ⊆ s",
"usedCo... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Order.IntermediateValue | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 12
} | [
{
"pp": "case inl\nα : 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 : b < a\n⊢ Icc a b ⊆ s",
"usedCo... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 12
} | [
{
"pp": "case inl\nα : 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 : b < a\n⊢ Icc a b ⊆ s",
"usedCo... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 395,
"column": 4
} | {
"line": 395,
"column": 72
} | [
{
"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... | refine ⟨⟨by order, ht.le.trans (min_le_right _ _)⟩, fun t' ht' ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Order.IntermediateValue | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 12
} | [
{
"pp": "case pos\nα : Type u\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : DenselyOrdered α\nδ : Type u_1\ninst✝³ : LinearOrder δ\ninst✝² : TopologicalSpace δ\ninst✝¹ : OrderClosedTopology δ\ninst✝ : BoundedOrder α\nf : α → δ\nhf_c : Continuous[in... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.Order.IntermediateValue | {
"line": 696,
"column": 6
} | {
"line": 696,
"column": 14
} | [
{
"pp": "case neg\nα : Type u\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : DenselyOrdered α\nδ : Type u_1\ninst✝³ : LinearOrder δ\ninst✝² : TopologicalSpace δ\ninst✝¹ : OrderClosedTopology δ\ninst✝ : BoundedOrder α\nf : α → δ\nhf_c : Continuous[in... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 49
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nhU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅\n⊢ ⋃ i, (fun i ↦ ∅) i = ∅ ∧\n ∀ (i : ℕ),\n Bornology.IsBounded ((fun i ↦ ∅) i) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((fun i ↦ ∅) i)",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 49
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nhU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅\n⊢ ⋃ i, (fun i ↦ ∅) i = ∅ ∧\n ∀ (i : ℕ),\n Bornology.IsBounded ((fun i ↦ ∅) i) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((fun i ↦ ∅) i)",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 49
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nhU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅\n⊢ ⋃ i, (fun i ↦ ∅) i = ∅ ∧\n ∀ (i : ℕ),\n Bornology.IsBounded ((fun i ↦ ∅) i) ∧ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ((fun i ↦ ∅) i)",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 50
} | [
{
"pp": "case inr.refine_3\nα : Type u\ninst✝ : PseudoMetricSpace α\nU : Set α\nhU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nx : α\ni : ℕ\n⊢ Bornology.IsBounded ((fun i ↦ U ∩ ball x ↑i) i)",
"usedConstants": [
"Metric.isBounded_ball",
"Real",
"PseudoMetricSpace.toBor... | exact isBounded_ball.subset inter_subset_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 50
} | [
{
"pp": "case inr.refine_3\nα : Type u\ninst✝ : PseudoMetricSpace α\nU : Set α\nhU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nx : α\ni : ℕ\n⊢ Bornology.IsBounded ((fun i ↦ U ∩ ball x ↑i) i)",
"usedConstants": [
"Metric.isBounded_ball",
"Real",
"PseudoMetricSpace.toBor... | exact isBounded_ball.subset inter_subset_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 50
} | [
{
"pp": "case inr.refine_3\nα : Type u\ninst✝ : PseudoMetricSpace α\nU : Set α\nhU : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nx : α\ni : ℕ\n⊢ Bornology.IsBounded ((fun i ↦ U ∩ ball x ↑i) i)",
"usedConstants": [
"Metric.isBounded_ball",
"Real",
"PseudoMetricSpace.toBor... | exact isBounded_ball.subset inter_subset_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 189,
"column": 89
} | {
"line": 190,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\nf : α → β\ninst✝ : Subsingleton α\nK : ℝ≥0\nx y : α\n⊢ edist x y ≤ ↑K * edist (f x) (f y)",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | by
simp only [Subsingleton.elim x y, edist_self, zero_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 97,
"column": 89
} | {
"line": 98,
"column": 46
} | [
{
"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": [
"ENNReal.instAdd",
"congrArg",
"Set.instUnion",
"Eq.mp",
"LE.le",
"Set.instInter",
"Inter.int... | by
simpa using measure_union_le (s ∩ t) (s \ t) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 32
} | [
{
"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⊢ ↑x < y.1 - y.2",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"le_refl",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.instOrderedSub",
... | grw [lt_tsub_iff_left, hy.2] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 58
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\nd : ℕ → ℝ\nhf : ∀ (n : ℕ), dist (f n) (f n.succ) ≤ d n\nhd : Summable d\n⊢ CauchySeq f",
"usedConstants": [
"Real.instLE",
"Real",
"NNReal.canLift",
"Real.instZero",
"dist_nonneg",
"NNReal",
"LE.le",
... | lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n) | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 627,
"column": 2
} | {
"line": 634,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\n⊢ Continuous[instTopologicalSpaceProd, _] fun p ↦ edist p.1 p.2",
"usedConstants": [
"le_max_right",
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"instDecidableNot",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.ins... | apply continuous_of_le_add_edist 2 (by decide)
rintro ⟨x, y⟩ ⟨x', y'⟩
calc
edist x y ≤ edist x x' + edist x' y' + edist y' y := edist_triangle4 _ _ _ _
_ = edist x' y' + (edist x x' + edist y y') := by rw [edist_comm y y']; abel
_ ≤ edist x' y' + (edist (x, y) (x', y') + edist (x, y) (x', y')) := by
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 627,
"column": 2
} | {
"line": 634,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\n⊢ Continuous[instTopologicalSpaceProd, _] fun p ↦ edist p.1 p.2",
"usedConstants": [
"le_max_right",
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"instDecidableNot",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.ins... | apply continuous_of_le_add_edist 2 (by decide)
rintro ⟨x, y⟩ ⟨x', y'⟩
calc
edist x y ≤ edist x x' + edist x' y' + edist y' y := edist_triangle4 _ _ _ _
_ = edist x' y' + (edist x x' + edist y y') := by rw [edist_comm y y']; abel
_ ≤ edist x' y' + (edist (x, y) (x', y') + edist (x, y) (x', y')) := by
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Basic | {
"line": 49,
"column": 14
} | {
"line": 49,
"column": 22
} | [
{
"pp": "case zero\nn : ℕ\nhn : Even n\nhs : zero ≠ 0\n⊢ zero ^ n = 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"SignType.instCommGroupWithZero",
"SignType.instOne",
"congrArg",
"False.elim",
"Eq.mp",
"not_true_eq_false",
"Ne",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Basic | {
"line": 49,
"column": 14
} | {
"line": 49,
"column": 22
} | [
{
"pp": "case neg\nn : ℕ\nhn : Even n\nhs : neg ≠ 0\n⊢ neg ^ n = 1",
"usedConstants": [
"one_pow",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"SignType.instHasDistribNeg",
"SignType.instCommGroupWithZero",
"Monoid.toMulOneClass",
"SignType.instOne",
"co... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Basic | {
"line": 49,
"column": 14
} | {
"line": 49,
"column": 22
} | [
{
"pp": "case pos\nn : ℕ\nhn : Even n\nhs : pos ≠ 0\n⊢ pos ^ n = 1",
"usedConstants": [
"one_pow",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"SignType.instCommGroupWithZero",
"Monoid.toMulOneClass",
"SignType.instOne",
"congrArg",
"SignType.pos",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inl.inl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = -1\nhsb : sign b = -1\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"SignType.instHasDistribNeg",
"NeZe... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inl.inr.inl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = -1\nhsb : sign b = 0\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"False",
"eq_false",
"sign_eq_zero_iff._simp_1",
"False.elim",
"Parti... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inl.inr.inr\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = -1\nhsb : sign b = 1\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"NeZero.one",
"SignType.instComm... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inr.inl.inl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = 0\nhsb : sign b = -1\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"False",
"eq_false",
"sign_eq_zero_iff._simp_1",
"False.elim",
"Parti... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inr.inl.inr.inl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = 0\nhsb : sign b = 0\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"False",
"eq_false",
"sign_eq_zero_iff._simp_1",
"False.elim",
"Pa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inr.inl.inr.inr\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = 0\nhsb : sign b = 1\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"False",
"eq_false",
"sign_eq_zero_iff._simp_1",
"False.elim",
"Pa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inr.inr.inl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = 1\nhsb : sign b = -1\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"SignType.instHasDistribNeg",
"N... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inr.inr.inr.inl\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = 1\nhsb : sign b = 0\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"False",
"eq_false",
"sign_eq_zero_iff._simp_1",
"False.elim",
"Pa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Sign.Defs | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 12
} | [
{
"pp": "case inr.inr.inr.inr\nα : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na b : α\nha : a ≠ 0\nhb : b ≠ 0\nhsa : sign a = 1\nhsb : sign b = 1\n⊢ sign a = sign b ∨ sign a = -sign b",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"NeZero.one",
"SignType.instC... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.EReal.Inv | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 57
} | [
{
"pp": "case top\n⊢ ⊤.abs = 0 ↔ ⊤ = 0",
"usedConstants": [
"EReal.abs",
"False",
"congrArg",
"_private.Mathlib.Data.EReal.Inv.0.EReal.abs_eq_zero_iff._simp_1_5",
"EReal",
"instTopEReal",
"instZeroEReal",
"iff_self",
"_private.Mathlib.Data.EReal.Inv.0.ER... | simp only [abs_top, ENNReal.top_ne_zero, top_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.EReal.Inv | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 57
} | [
{
"pp": "case top\n⊢ ⊤.abs = 0 ↔ ⊤ = 0",
"usedConstants": [
"EReal.abs",
"False",
"congrArg",
"_private.Mathlib.Data.EReal.Inv.0.EReal.abs_eq_zero_iff._simp_1_5",
"EReal",
"instTopEReal",
"instZeroEReal",
"iff_self",
"_private.Mathlib.Data.EReal.Inv.0.ER... | simp only [abs_top, ENNReal.top_ne_zero, top_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Inv | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 57
} | [
{
"pp": "case top\n⊢ ⊤.abs = 0 ↔ ⊤ = 0",
"usedConstants": [
"EReal.abs",
"False",
"congrArg",
"_private.Mathlib.Data.EReal.Inv.0.EReal.abs_eq_zero_iff._simp_1_5",
"EReal",
"instTopEReal",
"instZeroEReal",
"iff_self",
"_private.Mathlib.Data.EReal.Inv.0.ER... | simp only [abs_top, ENNReal.top_ne_zero, top_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Inv | {
"line": 191,
"column": 39
} | {
"line": 191,
"column": 47
} | [
{
"pp": "case inl\nx y : EReal\nh : x ≤ y\n⊢ min (-x) (-y) = -max x y",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"SemilatticeSup.toMax",
"inf_of_le_right",
"Semilatti... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.EReal.Inv | {
"line": 191,
"column": 39
} | {
"line": 191,
"column": 47
} | [
{
"pp": "case inr\nx y : EReal\nh : y ≤ x\n⊢ min (-x) (-y) = -max x y",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"inf_of_le_left",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"SemilatticeSup.toMax",
"Semilattic... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.EReal.Inv | {
"line": 194,
"column": 39
} | {
"line": 194,
"column": 47
} | [
{
"pp": "case inl\nx y : EReal\nh : x ≤ y\n⊢ max (-x) (-y) = -min x y",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"inf_of_le_left",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"SemilatticeSup.toMax",
"Semilattic... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.EReal.Inv | {
"line": 194,
"column": 39
} | {
"line": 194,
"column": 47
} | [
{
"pp": "case inr\nx y : EReal\nh : y ≤ x\n⊢ max (-x) (-y) = -min x y",
"usedConstants": [
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"SemilatticeSup.toMax",
"inf_of_le_right",
"Semilatti... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.EReal.Inv | {
"line": 442,
"column": 12
} | {
"line": 442,
"column": 59
} | [
{
"pp": "a b c : EReal\nh : b > 0\nh' : b ≠ ⊤\n⊢ a ≤ c / b ↔ a * b ≤ c",
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"Preorder.toLE",
"id",
"HDiv.hDiv",
"inst... | ← @mul_div_cancel a b (ne_bot_of_gt h) h' h.ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 452,
"column": 12
} | {
"line": 452,
"column": 59
} | [
{
"pp": "a b c : EReal\nh : 0 < b\nh' : b ≠ ⊤\n⊢ a < c / b ↔ a * b < c",
"usedConstants": [
"Eq.mpr",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"id",
"HDiv.hDiv",
"inst... | ← @mul_div_cancel a b (ne_bot_of_gt h) h' h.ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 457,
"column": 2
} | {
"line": 459,
"column": 52
} | [
{
"pp": "a b c : EReal\nh : 0 < c\nh' : c ≠ ⊤\n⊢ b / c < a ↔ b < a * c",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"EReal.strictMono_div_rig... | nth_rw 1 [← @mul_div_cancel a c (ne_bot_of_gt h) h' h.ne']
rw [EReal.mul_div c a c, mul_comm a c]
exact (strictMono_div_right_of_pos h h').lt_iff_lt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Inv | {
"line": 457,
"column": 2
} | {
"line": 459,
"column": 52
} | [
{
"pp": "a b c : EReal\nh : 0 < c\nh' : c ≠ ⊤\n⊢ b / c < a ↔ b < a * c",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"EReal.instDivInvMonoid",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"EReal.strictMono_div_rig... | nth_rw 1 [← @mul_div_cancel a c (ne_bot_of_gt h) h' h.ne']
rw [EReal.mul_div c a c, mul_comm a c]
exact (strictMono_div_right_of_pos h h').lt_iff_lt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 333,
"column": 2
} | {
"line": 334,
"column": 36
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\n⊢ ∑' (a : α), ↑(f a) ≠ ∞ ↔ Summable fun a ↦ ↑(f a)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.summable_coe",
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"congrArg",
"PseudoMetricSpace.... | rw [NNReal.summable_coe]
exact tsum_coe_ne_top_iff_summable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 333,
"column": 2
} | {
"line": 334,
"column": 36
} | [
{
"pp": "α : Type u_1\nf : α → ℝ≥0\n⊢ ∑' (a : α), ↑(f a) ≠ ∞ ↔ Summable fun a ↦ ↑(f a)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.summable_coe",
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"congrArg",
"PseudoMetricSpace.... | rw [NNReal.summable_coe]
exact tsum_coe_ne_top_iff_summable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 138,
"column": 47
} | {
"line": 138,
"column": 80
} | [
{
"pp": "α : Type u_2\nm : α → EReal\nf : Filter α\n⊢ (∀ (i : ℝ), True → ∀ᶠ (x : α) in f, m x ∈ Iio ↑i) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, m a < ↑x",
"usedConstants": [
"Real",
"Preorder.toLT",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"EReal",
"Members... | simp only [true_implies, mem_Iio] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 138,
"column": 47
} | {
"line": 138,
"column": 80
} | [
{
"pp": "α : Type u_2\nm : α → EReal\nf : Filter α\n⊢ (∀ (i : ℝ), True → ∀ᶠ (x : α) in f, m x ∈ Iio ↑i) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, m a < ↑x",
"usedConstants": [
"Real",
"Preorder.toLT",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"EReal",
"Members... | simp only [true_implies, mem_Iio] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 138,
"column": 47
} | {
"line": 138,
"column": 80
} | [
{
"pp": "α : Type u_2\nm : α → EReal\nf : Filter α\n⊢ (∀ (i : ℝ), True → ∀ᶠ (x : α) in f, m x ∈ Iio ↑i) ↔ ∀ (x : ℝ), ∀ᶠ (a : α) in f, m a < ↑x",
"usedConstants": [
"Real",
"Preorder.toLT",
"congrArg",
"Filter.Eventually",
"PartialOrder.toPreorder",
"EReal",
"Members... | simp only [true_implies, mem_Iio] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 334,
"column": 8
} | {
"line": 334,
"column": 19
} | [
{
"pp": "case inl\nα : Type u_3\nu v : α → EReal\nhu : ∃ᶠ (x : α) in ⊥, 0 ≤ u x\nhv : 0 ≤ᶠ[⊥] v\n⊢ limsup u ⊥ * liminf v ⊥ ≤ limsup (u * v) ⊥",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"Filter.liminf",
"CompleteLattice.toLattice",
"congrArg",
... | limsup_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 334,
"column": 20
} | {
"line": 334,
"column": 31
} | [
{
"pp": "case inl\nα : Type u_3\nu v : α → EReal\nhu : ∃ᶠ (x : α) in ⊥, 0 ≤ u x\nhv : 0 ≤ᶠ[⊥] v\n⊢ ⊥ * liminf v ⊥ ≤ limsup (u * v) ⊥",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"Filter.liminf",
"CompleteLattice.toLattice",
"congrArg",
"Or... | limsup_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 501,
"column": 2
} | {
"line": 506,
"column": 37
} | [
{
"pp": "case h.refine_1\nx : ℝ\n⊢ ∀ y ∈ Ioi (max (↑x) 0) ×ˢ Ioi 1, ↑x < y.1 * y.2",
"usedConstants": [
"Set.instSProd",
"MulOne.toOne",
"Set.Ioi",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"EReal.instPosMulMono",
... | · intro p p_in_prod
simp only [Set.mem_prod, Set.mem_Ioi, max_lt_iff] at p_in_prod
rcases p_in_prod with ⟨⟨p1_gt_x, p1_pos⟩, p2_gt_1⟩
have := mul_le_mul_of_nonneg_left (le_of_lt p2_gt_1) (le_of_lt p1_pos)
rw [mul_one p.1] at this
exact lt_of_lt_of_le p1_gt_x this | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 518,
"column": 2
} | {
"line": 534,
"column": 70
} | [
{
"pp": "case h.refine_1\na : ℝ\nh : 0 < a\nx : ℝ\n⊢ ∀ y ∈ Ioi ↑(2 * max (x + 1) 0 / a) ×ˢ Ioi ↑(a / 2), ↑x < y.1 * y.2",
"usedConstants": [
"Set.instSProd",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | · intro p p_in_prod
simp only [Set.mem_prod, Set.mem_Ioi] at p_in_prod
rcases p_in_prod with ⟨p1_gt, p2_gt⟩
have p1_pos : 0 < p.1 := by
apply lt_of_le_of_lt _ p1_gt
rw [EReal.coe_nonneg]
apply mul_nonneg _ (le_of_lt (inv_pos_of_pos h))
simp only [Nat.ofNat_pos, mul_nonneg_iff_of_pos_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 65,
"column": 41
} | {
"line": 90,
"column": 30
} | [
{
"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)\n⊢ ∀ (ε : ℝ≥0), 0 < ε → ∑' (i : ℕ), μ (s i) < ∞ → μ (⋃ i, s i) ≤ ∑' (i : ℕ), μ (s i) + ↑ε",
"usedConstants": [
"Iff.mpr",... | by
intro ε hε (hb : (∑' i, μ (s i)) < ∞)
rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 hε).ne' ℕ with ⟨ε', hε', hl⟩
grw [← hl]
rw [← ENNReal.tsum_add]
choose f hf using
show ∀ i, ∃ f : ℕ → Set α, (s i ⊆ ⋃ i, f i) ∧ (∑' i, m (f i)) < μ (s i) + ε' i by
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 10
} | [
{
"pp": "case h\nα : Type u_4\nG : Type u_5\ninst✝ : GroupWithZero G\nf g : α → G\nl : Filter α\nhg : Tendsto g l (𝓟 {0}ᶜ)\nx : α\nhx : x ∈ g ⁻¹' {0}ᶜ\n⊢ f x / g x * g x = f x",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"instHDiv",
"GroupWithZero.toDivisionMonoid... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.PiSystem | {
"line": 334,
"column": 8
} | {
"line": 334,
"column": 36
} | [
{
"pp": "case refine_1.h.mp\nα : Type u_3\nβ : Type u_4\ng : β → Set (Set α)\ns : Set β\nh_pi : ∀ b ∈ s, IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s → Set α\nh_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b\nh_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)\nthis : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, ... | have h2 := h1 b h_b h_b_in_T | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.PiSystem | {
"line": 334,
"column": 8
} | {
"line": 334,
"column": 36
} | [
{
"pp": "case refine_1.h.mpr\nα : Type u_3\nβ : Type u_4\ng : β → Set (Set α)\ns : Set β\nh_pi : ∀ b ∈ s, IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s → Set α\nh_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b\nh_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)\nthis : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b,... | have h2 := h1 b h_b h_b_in_T | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 10
} | [
{
"pp": "α : Type u\nm : OuterMeasure α\ns s₁ s₂ : Set α\n⊢ ∀ (s₁ s₂ : Set α), m.caratheodoryDynkin.Has s₁ → m.caratheodoryDynkin.Has s₂ → m.caratheodoryDynkin.Has (s₁ ∩ s₂)",
"usedConstants": [
"Set"
]
}
] | intro s₁ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.MeasureTheory.PiSystem | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 32
} | [
{
"pp": "case refine_1\nα : Type u_3\nι : Type u_4\ns : ι → Set α\nS : Set ι\n⊢ ∀ t ∈ piiUnionInter (fun k ↦ {s k}) S, MeasurableSet t",
"usedConstants": [
"Set"
]
}
] | rintro _ ⟨I, hI, f, hf, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 198,
"column": 2
} | {
"line": 202,
"column": 54
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : NullMeasurableSet s μ\n⊢ ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"MeasurableSet",
"Set.union_empty",
"congrArg",
"Set.diff_subset... | rcases h with ⟨t, htm, hst⟩
refine ⟨t ∪ toMeasurable μ (s \ t), ?_, htm.union (measurableSet_toMeasurable _ _), ?_⟩
· exact diff_subset_iff.1 (subset_toMeasurable _ _)
· have : toMeasurable μ (s \ t) =ᵐ[μ] (∅ : Set α) := by simp [ae_le_set.1 hst.le]
simpa only [union_empty] using hst.symm.union this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 198,
"column": 2
} | {
"line": 202,
"column": 54
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : NullMeasurableSet s μ\n⊢ ∃ t ⊇ s, MeasurableSet t ∧ t =ᶠ[ae μ] s",
"usedConstants": [
"MeasureTheory.ae",
"MeasureTheory.Measure",
"MeasurableSet",
"Set.union_empty",
"congrArg",
"Set.diff_subset... | rcases h with ⟨t, htm, hst⟩
refine ⟨t ∪ toMeasurable μ (s \ t), ?_, htm.union (measurableSet_toMeasurable _ _), ?_⟩
· exact diff_subset_iff.1 (subset_toMeasurable _ _)
· have : toMeasurable μ (s \ t) =ᵐ[μ] (∅ : Set α) := by simp [ae_le_set.1 hst.le]
simpa only [union_empty] using hst.symm.union this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Map | {
"line": 133,
"column": 4
} | {
"line": 134,
"column": 81
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nμ : Measure α\nf : α → β\nc : ℝ≥0∞\nhc : c ≠ 0\nhf : AEMeasurable f μ\n⊢ map f (c • μ) = c • map f μ",
"usedConstants": [
"MeasureTheory... | have hfc : AEMeasurable f (c • μ) :=
⟨hf.mk f, hf.measurable_mk, (ae_ennreal_smul_measure_iff hc).2 hf.ae_eq_mk⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 100,
"column": 35
} | {
"line": 100,
"column": 52
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf : α → β\nR : Type u_5\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nhf : QuasiMeasurePreserving f μa μb\nc : R\n⊢ c • map f μa ≪ c • μb",
"usedConstants": [
"MeasureTheory... | exact hf.2.smul c | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 60
} | [
{
"pp": "G : Type u_5\nα : Type u_6\ninst✝¹ : Group G\ninst✝ : MulAction G α\nx✝ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ng : G\nh_qmp : QuasiMeasurePreserving (fun x ↦ g⁻¹ • x) μ μ\nh_ae_eq : s =ᵐ[μ] t\n⊢ g • s =ᵐ[μ] g • t",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHS... | simpa only [← preimage_smul_inv] using h_qmp.ae_eq h_ae_eq | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 60
} | [
{
"pp": "G : Type u_5\nα : Type u_6\ninst✝¹ : Group G\ninst✝ : MulAction G α\nx✝ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ng : G\nh_qmp : QuasiMeasurePreserving (fun x ↦ g⁻¹ • x) μ μ\nh_ae_eq : s =ᵐ[μ] t\n⊢ g • s =ᵐ[μ] g • t",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHS... | simpa only [← preimage_smul_inv] using h_qmp.ae_eq h_ae_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 60
} | [
{
"pp": "G : Type u_5\nα : Type u_6\ninst✝¹ : Group G\ninst✝ : MulAction G α\nx✝ : MeasurableSpace α\ns t : Set α\nμ : Measure α\ng : G\nh_qmp : QuasiMeasurePreserving (fun x ↦ g⁻¹ • x) μ μ\nh_ae_eq : s =ᵐ[μ] t\n⊢ g • s =ᵐ[μ] g • t",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHS... | simpa only [← preimage_smul_inv] using h_qmp.ae_eq h_ae_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 22
} | [
{
"pp": "case h.hfi\nβ : Type u_2\nmβ : MeasurableSpace β\nμ : Measure β\ns : Set β\nhs : MeasurableSet s\n⊢ Injective fun x ↦ x",
"usedConstants": [
"Function.injective_id"
]
}
] | exact injective_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 22
} | [
{
"pp": "case h.hfi\nβ : Type u_2\nmβ : MeasurableSpace β\nμ : Measure β\ns : Set β\nhs : MeasurableSet s\n⊢ Injective fun x ↦ x",
"usedConstants": [
"Function.injective_id"
]
}
] | exact injective_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 22
} | [
{
"pp": "case h.hfi\nβ : Type u_2\nmβ : MeasurableSpace β\nμ : Measure β\ns : Set β\nhs : MeasurableSet s\n⊢ Injective fun x ↦ x",
"usedConstants": [
"Function.injective_id"
]
}
] | exact injective_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 40,
"column": 77
} | {
"line": 41,
"column": 47
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ IsFiniteMeasure (μ.restrict s) ↔ μ s ≠ ∞",
"usedConstants": [
"_private.Mathlib.MeasureTheory.Measure.Typeclasses.Finite.0.MeasureTheory.isFiniteMeasure_restrict._simp_1_2",
"MeasureTheory.Measure",
"Preorder.toLT",... | by
simp [isFiniteMeasure_iff, lt_top_iff_ne_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 53,
"column": 8
} | {
"line": 53,
"column": 19
} | [
{
"pp": "case pos.mpr.refine_1\nX : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsne : s.Nonempty\nthis : Nonempty ↑s\nhsub : s ⊆ Iio b\nh : ∀ a < b, (s ∩ Ioo a b).Nonempty\n⊢ ∀ i < b, ∃ i', True ∧ Ici i' ⊆ Subtype.val ⁻¹' Ioo i b",
"usedConstant... | intro a hab | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 36
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Finset α\ninst✝ : MeasurableSingletonClass α\n⊢ ∑ x ∈ s, μ (id ⁻¹' {x}) = μ ↑s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"ENNReal.instAddCommMonoid",
"congrArg",
"Finset",
"Set.instSingletonSet... | sum_measure_preimage_singleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 330,
"column": 85
} | {
"line": 330,
"column": 88
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nh₁ : s ≤ᶠ[ae μ] t\nhsm : NullMeasurableSet s μ\nh₂ : μ t = μ s\nht : μ s ≠ ∞\n⊢ μ t - μ s = 0",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"HSub.hSub",
"id",
"instHSub",
... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 64,
"column": 41
} | {
"line": 64,
"column": 54
} | [
{
"pp": "case inr\nα : Type u_1\nι : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : Countable ι\nm : ι → Measure α\ninst✝ : ∀ (n : ι), IsFiniteMeasure (m n)\nf : ι → ℕ\nhf : Injective f\nn : ℕ\nhn : n ∉ range f\n⊢ IsFiniteMeasure (0 n)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"c... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 164,
"column": 41
} | {
"line": 164,
"column": 55
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\n⊢ μ (univ ∩ s) = μ s",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Set.univ",
"id",
"Set.instInter",
"Set.univ_inter",
"Inter.inter",
"ENNReal",
"Eq",
... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 344,
"column": 48
} | {
"line": 344,
"column": 84
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nH : ∀ t ⊆ s, MeasurableSet t → μ t = ν t\nt : Set α\nht : MeasurableSet t\n⊢ μ (t ∩ s) = ν (t ∩ s)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
"Set.ins... | H _ inter_subset_right (ht.inter hs) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 804,
"column": 2
} | {
"line": 804,
"column": 16
} | [
{
"pp": "case hC\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nt : Set ↑s\nhs : NullMeasurableSet s μ\nt' : Set ↑s\nht' : t' ∈ {t | ∃ s_1, MeasurableSet s_1 ∧ Subtype.val ⁻¹' s_1 = t}\nht✝ : MeasurableSet t'\n⊢ NullMeasurableSet (Subtype.val '' t') μ",
"usedConstants": [
"Eq.mpr",
... | | hC t' ht' => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 34
} | [
{
"pp": "β : Type u_2\nm₁ m₂ : MeasurableSpace β\nh₁ : CountablyGenerated β\nh₂ : CountablyGenerated β\n⊢ CountablyGenerated β",
"usedConstants": []
}
] | rcases h₁ with ⟨⟨b₁, hb₁c, rfl⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 264,
"column": 2
} | {
"line": 268,
"column": 94
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nh : CountablySeparated α\n⊢ ∃ m', CountablyGenerated α ∧ SeparatesPoints α ∧ m' ≤ m",
"usedConstants": [
"Eq.mpr",
"HasCountableSeparatingOn.casesOn",
"MeasurableSpace.instLE",
"trivial",
"MeasurableSet",
"congrArg",
"Me... | rcases h with ⟨b, bct, hbm, hb⟩
refine ⟨generateFrom b, ?_, ?_, generateFrom_le hbm⟩
· use b
rw [@separatesPoints_iff]
exact fun x y hxy ↦ hb _ trivial _ trivial fun _ hs ↦ hxy _ <| measurableSet_generateFrom hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 264,
"column": 2
} | {
"line": 268,
"column": 94
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nh : CountablySeparated α\n⊢ ∃ m', CountablyGenerated α ∧ SeparatesPoints α ∧ m' ≤ m",
"usedConstants": [
"Eq.mpr",
"HasCountableSeparatingOn.casesOn",
"MeasurableSpace.instLE",
"trivial",
"MeasurableSet",
"congrArg",
"Me... | rcases h with ⟨b, bct, hbm, hb⟩
refine ⟨generateFrom b, ?_, ?_, generateFrom_le hbm⟩
· use b
rw [@separatesPoints_iff]
exact fun x y hxy ↦ hb _ trivial _ trivial fun _ hs ↦ hxy _ <| measurableSet_generateFrom hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 707,
"column": 2
} | {
"line": 710,
"column": 20
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ ν : Measure α\ns t : Set α\na : α\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ SigmaFinite μ",
"usedConstants": [
"Filter.instMembership",
... | choose s hsx hsμ using μ.finiteAt_nhds
rcases TopologicalSpace.countable_cover_nhds hsx with ⟨t, htc, htU⟩
refine Measure.sigmaFinite_of_countable (htc.image s) (forall_mem_image.2 fun x _ => hsμ x) ?_
rwa [sUnion_image] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.SFinite | {
"line": 707,
"column": 2
} | {
"line": 710,
"column": 20
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ ν : Measure α\ns t : Set α\na : α\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ SigmaFinite μ",
"usedConstants": [
"Filter.instMembership",
... | choose s hsx hsμ using μ.finiteAt_nhds
rcases TopologicalSpace.countable_cover_nhds hsx with ⟨t, htc, htU⟩
refine Measure.sigmaFinite_of_countable (htc.image s) (forall_mem_image.2 fun x _ => hsμ x) ?_
rwa [sUnion_image] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 420,
"column": 2
} | {
"line": 421,
"column": 45
} | [
{
"pp": "α : Type u_1\nt : ℕ → Set α\nn : ℕ\n⊢ generateFrom (memPartition t n) ≤ generateFrom (range t)",
"usedConstants": [
"Eq.mpr",
"MeasurableSpace.instLE",
"congrArg",
"Eq.rec",
"memPartition",
"id",
"LE.le",
"MeasurableSpace.generateFrom",
"Measura... | conv_rhs => rw [← generateFrom_iUnion_memPartition t]
exact generateFrom_mono (subset_iUnion _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 420,
"column": 2
} | {
"line": 421,
"column": 45
} | [
{
"pp": "α : Type u_1\nt : ℕ → Set α\nn : ℕ\n⊢ generateFrom (memPartition t n) ≤ generateFrom (range t)",
"usedConstants": [
"Eq.mpr",
"MeasurableSpace.instLE",
"congrArg",
"Eq.rec",
"memPartition",
"id",
"LE.le",
"MeasurableSpace.generateFrom",
"Measura... | conv_rhs => rw [← generateFrom_iUnion_memPartition t]
exact generateFrom_mono (subset_iUnion _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 573,
"column": 2
} | {
"line": 573,
"column": 20
} | [
{
"pp": "case inr\nβ : Type u_2\nα : Type u_3\nγ : Type u_4\ninst✝ : MeasurableSpace γ\nh : CountablyGenerated γ\n⊢ CountableOrCountablyGenerated (β × α) γ",
"usedConstants": [
"MeasurableSpace.CountableOrCountablyGenerated.mk",
"Countable",
"MeasurableSpace.CountablyGenerated",
"Pro... | · exact ⟨Or.inr h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 51
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\na✝ : Nontrivial α\ninhabited_h : Inhabited α\n⊢ AEMeasurable f 0",
"usedConstants": [
"MeasureTheory.ae",
"Inhabited.default",
"MeasureTheory.Measure",
"Compl.compl",
"Measurable"... | exact ⟨fun _ => f default, measurable_const, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 317,
"column": 25
} | {
"line": 317,
"column": 41
} | [
{
"pp": "case measurableSet_diagonal\nα✝ : Type u_1\nG : Type u_2\nα : Type u_3\nβ✝ : Type u_4\ninst✝⁸ : MeasurableSpace G\ninst✝⁷ : Div G\nm : MeasurableSpace α\nmβ : MeasurableSpace β✝\nf g : α → G\nμ : Measure α\nβ : Type u_5\ninst✝⁶ : AddCommMonoid β\ninst✝⁵ : PartialOrder β\ninst✝⁴ : CanonicallyOrderedAdd ... | le_antisymm_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 50
} | [
{
"pp": "G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoEMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ≥0∞\n⊢ (fun x ↦ x * a) ⁻¹' eball b r = eball (b * a⁻¹) r",
"usedConstants": [
"MulOpposite",
"Metric.preimage_smul_eball",
"DivInvMonoid.toMonoid",
"MulOpposite.instGro... | exact preimage_smul_eball (MulOpposite.op a) b r | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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