module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 69,
"column": 2
} | {
"line": 70,
"column": 13
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case inl\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ c ∉ Set.range e, 1 ≤ g c\nh : ∀ (i : ι), f i ≤ g (... | [
"case inr\nι : Type u_1\nκ : Type u_2\nα : Type u_3\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₁ a₂ : α\ng : κ → α\ne : ι → κ\nhe : Injective e\nhs : ∀ c ∉ Set.range e, 1 ≤ g c\nh✝ : ∀ (i : ι), f i ≤ g (e i)\nhf : ... | · rw [he.extend_apply]
exact h _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Order.Bornology | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 85
} | {
"line": 125,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Bornology α\ninst✝² : Nonempty α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderBornology α\ns : Set α\n⊢ s ∈ Filter.atBot ⊔ Filter.atTop → s ∈ cobounded α",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.... | [
"α : Type u_1\ninst✝³ : Bornology α\ninst✝² : Nonempty α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderBornology α\ns : Set α\n⊢ ((∃ i, True ∧ Iic i ⊆ s) ∧ ∃ i, True ∧ Ici i ⊆ s) → BddBelow sᶜ ∧ BddAbove sᶜ"
] | rw [Filter.mem_sup, Filter.atTop_basis.mem_iff, Filter.atBot_basis.mem_iff,
← compl_compl s, ← isBounded_def, isBounded_iff_bddBelow_bddAbove, compl_compl s] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Operations | {
"line": 283,
"column": 66
} | {
"line": 283,
"column": 95
} | {
"line": 284,
"column": 0
} | [
{
"pp": "a : EReal\n⊢ 0 < -a ↔ a < 0",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Preorder.toLT",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"id",
"instZeroEReal... | [] | by rw [lt_neg_comm, neg_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.EReal.Operations | {
"line": 521,
"column": 2
} | {
"line": 521,
"column": 74
} | {
"line": 522,
"column": 2
} | [
{
"pp": "a b c : EReal\nh₁ : a ≠ ⊥ ∨ b ≠ ⊤\nh₂ : a ≠ ⊤ ∨ b ≠ ⊥\nh : ∀ a' > a, ∀ b' > b, c ≤ a' + b'\n⊢ -a - b ≤ -c",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"EReal.le_neg_of_le_neg",
... | [
"a b c : EReal\nh₁ : a ≠ ⊥ ∨ b ≠ ⊤\nh₂ : a ≠ ⊤ ∨ b ≠ ⊥\nh : ∀ a' > a, ∀ b' > b, c ≤ a' + b'\na' : EReal\nha' : a' < -a\nb' : EReal\nhb' : b' < -b\n⊢ c ≤ -(a' + b')"
] | refine add_le_of_forall_lt fun a' ha' b' hb' ↦ EReal.le_neg_of_le_neg ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Order.Compact | {
"line": 480,
"column": 4
} | {
"line": 480,
"column": 25
} | {
"line": 481,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁴ : ConditionallyCompleteLinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : γ → β → α\nhf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f\nhK : IsCompact ∅\n⊢ Conti... | [
"case inl\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝⁴ : ConditionallyCompleteLinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nf : γ → β → α\nhf : Continuous[instTopologicalSpaceProd, inst✝³] ↿f\nhK : IsCompact ∅\n⊢ Continuous[inst✝,... | simp_rw [image_empty] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 17
} | {
"line": 127,
"column": 17
} | [
{
"pp": "z : ℝ≥0\n⊢ nndist z 0 = z",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"congrArg",
"PseudoMetricSpace.toNNDist",
"nndist_comm",
"id",
"NNReal",
"instPseudoMetricSpaceNNReal",
"NNReal.instZero",
... | [
"z : ℝ≥0\n⊢ nndist 0 z = z"
] | nndist_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 601,
"column": 17
} | {
"line": 601,
"column": 59
} | {
"line": 603,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx : α\nε : ℝ\ny : α\nh : y ∈ ball x (ε / 2)\n⊢ dist y x ≤ ε - ε / 2",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHDiv",
"AddGroupWithOne.toAdd... | [] | by rw [sub_self_div_two]; exact le_of_lt h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.IntermediateValue | {
"line": 436,
"column": 12
} | {
"line": 436,
"column": 26
} | {
"line": 436,
"column": 26
} | [
{
"pp": "case inr\nα : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns t : Set α\nhs : IsClosed[inst✝³] s\nht : IsClosed[inst✝³] t\nhab : Icc a b ⊆ s ∪ t\nx : α\nhx : x ∈ Icc a b ∩ s\ny : α\nhy : y ∈ Icc a b ∩ t\nh ... | [
"case inr\nα : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns t : Set α\nhs : IsClosed[inst✝³] s\nht : IsClosed[inst✝³] t\nhab : Icc a b ⊆ s ∪ t\nx : α\nhx : x ∈ Icc a b ∩ s\ny : α\nhy : y ∈ Icc a b ∩ t\nh : y ≤ x\n⊢ (... | inter_comm s t | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 43
} | {
"line": 913,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\na : α\n⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ici ia, u x ∈ ball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
... | [] | simp only [true_and, mem_ball, mem_Ici] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 43
} | {
"line": 913,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\na : α\n⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ici ia, u x ∈ ball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
... | [] | simp only [true_and, mem_ball, mem_Ici] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 43
} | {
"line": 913,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\na : α\n⊢ (∀ (ib : ℝ), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ici ia, u x ∈ ball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
... | [] | simp only [true_and, mem_ball, mem_Ici] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Cauchy | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 47
} | {
"line": 144,
"column": 4
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : ℕ → α\nhs : CauchySeq s\nS : ℕ → Set ℝ := fun N ↦ (fun p ↦ dist (s p.1) (s p.2)) '' {p | p.1 ≥ N ∧ p.2 ≥ N}\nhS : ∀ (N : ℕ), ∃ x, ∀ y ∈ S N, y ≤ x\nub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)\nS0m : ∀ (n : ℕ), 0 ∈ S n\n⊢ ∃ b, (∀ (n : ℕ)... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns : ℕ → α\nhs : CauchySeq s\nS : ℕ → Set ℝ := fun N ↦ (fun p ↦ dist (s p.1) (s p.2)) '' {p | p.1 ≥ N ∧ p.2 ≥ N}\nhS : ∀ (N : ℕ), ∃ x, ∀ y ∈ S N, y ≤ x\nub : ∀ (m n N : ℕ), N ≤ m → N ≤ n → dist (s m) (s n) ≤ sSup (S N)\nS0m : ∀ (n : ℕ), 0 ∈ S n\nS0 : ∀ (n : ℕ), 0 ≤ sSup (S n... | have S0 := fun n => le_csSup (hS n) (S0m n) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 11
} | {
"line": 57,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ AntilipschitzWith K f ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Pseud... | [] | simp only [AntilipschitzWith, edist_nndist]
norm_cast | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 11
} | {
"line": 57,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ AntilipschitzWith K f ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"Pseud... | [] | simp only [AntilipschitzWith, edist_nndist]
norm_cast | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 25
} | {
"line": 226,
"column": 2
} | [
{
"pp": "β : Type u_2\ninst✝² : PseudoMetricSpace β\nα : Type u_4\ninst✝¹ : MetricSpace α\nK : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K f\nf_cont :\n Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nhf : Function.Surj... | [
"β : Type u_2\ninst✝² : PseudoMetricSpace β\nα : Type u_4\ninst✝¹ : MetricSpace α\nK : ℝ≥0\nf : α → β\ninst✝ : ProperSpace α\nhK : AntilipschitzWith K f\nf_cont :\n Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f\nhf : Function.Surjective f\nx₀... | refine ⟨fun x₀ r => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 65
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\n⊢ Tendsto m f (𝓝 ∞) ↔ ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
"Set.Ioi",
"Preorder.toLT",
"iInf",
"ENNReal.nhds_top'",
"congrArg",
... | [] | simp only [nhds_top', tendsto_iInf, tendsto_principal, mem_Ioi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 65
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\n⊢ Tendsto m f (𝓝 ∞) ↔ ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
"Set.Ioi",
"Preorder.toLT",
"iInf",
"ENNReal.nhds_top'",
"congrArg",
... | [] | simp only [nhds_top', tendsto_iInf, tendsto_principal, mem_Ioi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 65
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\n⊢ Tendsto m f (𝓝 ∞) ↔ ∀ (x : ℝ≥0), ∀ᶠ (a : α) in f, ↑x < m a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
"Set.Ioi",
"Preorder.toLT",
"iInf",
"ENNReal.nhds_top'",
"congrArg",
... | [] | simp only [nhds_top', tendsto_iInf, tendsto_principal, mem_Ioi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 64
} | {
"line": 204,
"column": 6
} | [
{
"pp": "case inr.refine_1\nx : ℝ≥0∞\nxt : x ≠ ∞\nx0 : 0 < x\na b : ℝ≥0∞\nha : (a, b).1 < x\nhb : x < (a, b).2\n⊢ ∃ i', i' ≠ 0 ∧ Icc (x - i') (x + i') ⊆ Ioo (a, b).1 (a, b).2",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"ENNReal.i... | [
"case inr.refine_1\nx : ℝ≥0∞\nxt : x ≠ ∞\nx0 : 0 < x\na b : ℝ≥0∞\nha : (a, b).1 < x\nhb : x < (a, b).2\nε : ℝ≥0∞\nε0 : 0 < ε\nhε : ε < x - (a, b).1\n⊢ ∃ i', i' ≠ 0 ∧ Icc (x - i') (x + i') ⊆ Ioo (a, b).1 (a, b).2"
] | rcases exists_between (tsub_pos_of_lt ha) with ⟨ε, ε0, hε⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 247,
"column": 18
} | {
"line": 247,
"column": 28
} | {
"line": 248,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : OuterMeasure α\na : α\nx✝¹ x✝ : Set α\nh : x✝¹ ⊆ x✝\n⊢ x✝¹.indicator (fun x ↦ 1) a ≤ x✝.indicator (fun x ↦ 1) a",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"ENNReal.instIsOrderedRing",
"le_refl",
"Nat.ble",
"CommSemiring.... | [] | by grw [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 97
} | {
"line": 308,
"column": 4
} | [
{
"pp": "a b : ℝ≥0\nx✝ : ↑a ≠ ∞ ∨ ↑b ≠ ∞\n⊢ Tendsto (fun p ↦ p.1 - p.2) (𝓝 (↑a, ↑b)) (𝓝 (↑a - ↑b))",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"congrArg",
"Filter.map",
"HSub.hSub",
... | [
"a b : ℝ≥0\nx✝ : ↑a ≠ ∞ ∨ ↑b ≠ ∞\n⊢ Tendsto (fun a ↦ a.1 - a.2) (𝓝 (a, b)) (𝓝 (a - b))"
] | simp only [nhds_coe_coe, tendsto_map'_iff, ← ENNReal.coe_sub, Function.comp_def, tendsto_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 79
} | {
"line": 429,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\na_ne_top : a ≠ ∞\nx : ℝ≥0∞\n⊢ (a, x) ∈ {p | p ≠ (∞, ∞)}",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"false_and",
"Prod.mk",
"And",
"_private.Mathlib.Topology.Instances.ENNReal.Lemmas.0.ENNReal... | [] | simp only [a_ne_top, Ne, mem_setOf_eq, Prod.mk_inj, false_and, not_false_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 711,
"column": 56
} | {
"line": 711,
"column": 67
} | {
"line": 711,
"column": 67
} | [
{
"pp": "case inr\na b : ℝ\nh : a < b\n⊢ ENNReal.ofReal (b - sInf (Ioo a b)) = ENNReal.ofReal (b - a)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"MulZeroClass.toMul",
"ENNReal.ofReal",
"congrArg",
"Real.in... | [
"case inr\na b : ℝ\nh : a < b\n⊢ ENNReal.ofReal (b - a) = ENNReal.ofReal (b - a)"
] | csInf_Ioo h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 422,
"column": 4
} | {
"line": 422,
"column": 19
} | {
"line": 423,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : α → Type u_4\nf : (x : α) × β x → ℝ≥0\n⊢ ((∀ (x : α), Summable fun y ↦ f ⟨x, y⟩) ∧ Summable fun x ↦ ∑' (y : β x), f ⟨x, y⟩) → Summable f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal",
"And.cases... | [
"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"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 543,
"column": 2
} | {
"line": 545,
"column": 19
} | {
"line": 547,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nhf : Summable f\n⊢ ∑' (i : α), ENNReal.ofReal (f i) < ∞",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.instAddCommMonoid",
"ENNReal.... | [] | unfold ENNReal.ofReal
rw [lt_top_iff_ne_top, ENNReal.tsum_coe_ne_top_iff_summable]
exact hf.toNNReal | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 543,
"column": 2
} | {
"line": 545,
"column": 19
} | {
"line": 547,
"column": 0
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nhf : Summable f\n⊢ ∑' (i : α), ENNReal.ofReal (f i) < ∞",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.instAddCommMonoid",
"ENNReal.... | [] | unfold ENNReal.ofReal
rw [lt_top_iff_ne_top, ENNReal.tsum_coe_ne_top_iff_summable]
exact hf.toNNReal | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Inv | {
"line": 77,
"column": 30
} | {
"line": 77,
"column": 38
} | {
"line": 77,
"column": 39
} | [
{
"pp": "case top_pos\nx✝ : ℝ\nh : 0 < x✝\n⊢ ⊤.abs = ⊤.abs * (↑x✝).abs",
"ppTerm": "?top_pos",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"EReal",
"EReal.abs_top",
"instTopEReal",
... | [
"case top_pos\nx✝ : ℝ\nh : 0 < x✝\n⊢ ∞ = ∞ * (↑x✝).abs"
] | abs_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 62
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case mp\nx y : EReal\nh : x ≤ y\n⊢ sign x < sign y ∨\n sign x = neg ∧ sign y = neg ∧ y.abs ≤ x.abs ∨\n sign x = zero ∧ sign y = zero ∨ sign x = pos ∧ sign y = pos ∧ x.abs ≤ y.abs",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"EReal.abs",
"Preorder.toLT",
... | [
"case mp\nx y : EReal\nh : x ≤ y\nhs : sign x = sign y\n⊢ sign x = neg ∧ sign y = neg ∧ y.abs ≤ x.abs ∨\n sign x = zero ∧ sign y = zero ∨ sign x = pos ∧ sign y = pos ∧ x.abs ≤ y.abs"
] | refine (sign.monotone h).lt_or_eq.imp_right (fun hs => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 151,
"column": 60
} | {
"line": 154,
"column": 72
} | {
"line": 156,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : TopologicalSpace α\nr : α → β → Prop\ns : Set α\n⊢ Semicontinuous (s.restrict r) ↔ SemicontinuousOn r s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Filter.map",
"nhdsWithin",
"Filter.Eventu... | [] | by
rw [SemicontinuousOn, Semicontinuous, SetCoe.forall]
refine forall₂_congr fun a ha ↦ forall₂_congr fun b _ ↦ ?_
simp only [nhdsWithin_eq_map_subtype_coe ha, eventually_map, restrict] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.EReal.Inv | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 38
} | {
"line": 150,
"column": 6
} | [
{
"pp": "x y z : EReal\n⊢ x * y * z = x * (y * z)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"EReal.abs",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"EReal.sign_eq_and_abs_eq_iff_eq",
"congrArg",
"PartialOrder.toPreorder",
"Sign... | [
"x y z : EReal\n⊢ (x * y * z).abs = (x * (y * z)).abs ∧ sign (x * y * z) = sign (x * (y * z))"
] | rw [← sign_eq_and_abs_eq_iff_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 107,
"column": 2
} | {
"line": 110,
"column": 43
} | {
"line": 112,
"column": 0
} | [
{
"pp": "⊢ (𝓝 ⊤).HasBasis (fun x ↦ True) fun x ↦ Ioi ↑x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Preorder.toLT",
"trivial",
"EReal.instTopologicalSpace",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
... | [] | refine (nhds_top_basis (α := EReal)).to_hasBasis (fun x hx => ?_)
fun _ _ ↦ ⟨_, coe_lt_top _, Subset.rfl⟩
rcases exists_rat_btwn_of_lt hx with ⟨y, hxy, -⟩
exact ⟨_, trivial, Ioi_subset_Ioi hxy.le⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 107,
"column": 2
} | {
"line": 110,
"column": 43
} | {
"line": 112,
"column": 0
} | [
{
"pp": "⊢ (𝓝 ⊤).HasBasis (fun x ↦ True) fun x ↦ Ioi ↑x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Preorder.toLT",
"trivial",
"EReal.instTopologicalSpace",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
... | [] | refine (nhds_top_basis (α := EReal)).to_hasBasis (fun x hx => ?_)
fun _ _ ↦ ⟨_, coe_lt_top _, Subset.rfl⟩
rcases exists_rat_btwn_of_lt hx with ⟨y, hxy, -⟩
exact ⟨_, trivial, Ioi_subset_Ioi hxy.le⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 267,
"column": 2
} | {
"line": 267,
"column": 85
} | {
"line": 268,
"column": 2
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu v : α → EReal\nh : limsup u f ≠ ⊥ ∨ limsup v f ≠ ⊤\nh' : limsup u f ≠ ⊤ ∨ limsup v f ≠ ⊥\n⊢ limsup (u + v) f ≤ limsup u f + limsup v f",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"Filter.isBounded_le_o... | [
"α : Type u_3\nf : Filter α\nu v : α → EReal\nh : limsup u f ≠ ⊥ ∨ limsup v f ≠ ⊤\nh' : limsup u f ≠ ⊤ ∨ 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 + b\n⊢ ∀ᶠ (a : α) in f, (u + v) a < c"
] | refine le_add_of_forall_gt h h' fun a a_u b b_v ↦ (limsup_le_iff).2 fun c c_ab ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 55
} | {
"line": 359,
"column": 4
} | [
{
"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... | [
"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 * b\nx : α\... | apply (mul_nonpos_iff.2 (.inr ⟨hux.le, v_0⟩)).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 1264,
"column": 2
} | {
"line": 1264,
"column": 17
} | {
"line": 1265,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf : α → γ\n⊢ LowerSemicontinuousWithinAt f s x ∧ UpperSemicontinuousWithinAt f s x → ContinuousWithinAt f s x",
"ppTerm": "?m.39",
"assigned":... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ns : Set α\nx : α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf : α → γ\nh₁ : LowerSemicontinuousWithinAt f s x\nh₂ : UpperSemicontinuousWithinAt f s x\n⊢ ContinuousWithinAt f s x"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 599,
"column": 4
} | {
"line": 600,
"column": 25
} | {
"line": 601,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι✝ : Type u_3\nε : ℝ\nhε : 0 < ε\nι : Type ?u.11\ninst✝ : Encodable ι\nf : ℕ → ℝ := fun n ↦ ε / 2 / 2 ^ n\nhf : HasSum f ε\nf0 : ∀ (n : ℕ), 0 < f n\nc : ℝ\nhg : HasSum (f ∘ Encodable.encode) c\n⊢ ∀ c ∉ Set.range Encodable.encode, 0 ≤ f c",
"ppTerm": "?refi... | [] | intro i _
exact le_of_lt (f0 _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 599,
"column": 4
} | {
"line": 600,
"column": 25
} | {
"line": 601,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nι✝ : Type u_3\nε : ℝ\nhε : 0 < ε\nι : Type ?u.11\ninst✝ : Encodable ι\nf : ℕ → ℝ := fun n ↦ ε / 2 / 2 ^ n\nhf : HasSum f ε\nf0 : ∀ (n : ℕ), 0 < f n\nc : ℝ\nhg : HasSum (f ∘ Encodable.encode) c\n⊢ ∀ c ∉ Set.range Encodable.encode, 0 ≤ f c",
"ppTerm": "?refi... | [] | intro i _
exact le_of_lt (f0 _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 17
} | {
"line": 247,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂),... | [
"case mp\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m... | intro h t _ht | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 100,
"column": 16
} | {
"line": 100,
"column": 32
} | {
"line": 100,
"column": 32
} | [
{
"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⊢ map f (c • μa) ≪ c • μb",
"ppTerm": "?m.41",
"assigned": tr... | [
"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"
] | Measure.map_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Map | {
"line": 204,
"column": 18
} | {
"line": 204,
"column": 74
} | {
"line": 206,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\ng : β → γ\nf : α → β\nhg : Measurable g\nhf : Measurable f\ns : Set γ\nhs : MeasurableSet s\n⊢ (map g (map f μ)) s = (map (g ∘ f) μ) s",
"ppTerm": "?m.33",
"assigned"... | [] | by simp [hf, hg, hs, hg hs, hg.comp hf, ← preimage_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 80
} | {
"line": 123,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nμ : Measure β\nhfi : Injective f\nhf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ\ns t : Set α\nhst : (comap f μ) {a | ¬s a = t a} = 0\nh_eq_α : {a | ¬s a = t a} = s \\ t ∪ t \\ s\nh_eq_β : {a | ¬(... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β\nμ : Measure β\nhfi : Injective f\nhf : ∀ (s : Set α), MeasurableSet s → NullMeasurableSet (f '' s) μ\ns t : Set α\nhst : (comap f μ) {a | ¬s a = t a} = 0\nh_eq_α : {a | ¬s a = t a} = s \\ t ∪ t \\ s\nh_eq_β : {a | ¬(f '' s) a = ... | rw [← Set.image_sdiff hfi, ← Set.image_sdiff hfi, ← Set.image_union] at h_eq_β | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Comap | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 77
} | {
"line": 196,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nma : MeasurableSpace α\nmb : MeasurableSpace β\nμ : Measure α\ne : α ≃ᵐ β\ns : Set β\nhs : MeasurableSet s\n⊢ (Measure.comap (⇑e.symm) μ) s = (Measure.map (⇑e) μ) s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MeasurableEquiv.injective",
... | [
"α : Type u_1\nβ : Type u_2\nma : MeasurableSpace α\nmb : MeasurableSpace β\nμ : Measure α\ne : α ≃ᵐ β\ns : Set β\nhs : MeasurableSet s\n⊢ ∀ (s : Set β), MeasurableSet s → MeasurableSet (⇑e.symm '' s)"
] | rw [e.map_apply, Measure.comap_apply _ e.symm.injective _ _ hs, image_symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 285,
"column": 33
} | {
"line": 285,
"column": 51
} | {
"line": 286,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nhst : s ⊆ t\nhs' : μ s ≠ ∞\nε : ℝ≥0∞\nh : μ t < μ s + ε\n⊢ μ t - μ s < ε",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"ENNRea... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nhst : s ⊆ t\nhs' : μ s ≠ ∞\nε : ℝ≥0∞\nh : μ t < ε + μ s\n⊢ μ t - μ s < ε"
] | rw [add_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 312,
"column": 18
} | {
"line": 312,
"column": 54
} | {
"line": 313,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns₁ s₂ s₃ : Set α\nh12 : s₁ ⊆ s₂\nh23 : s₂ ⊆ s₃\nh_nullsdiff : μ (s₃ \\ s₁) = 0\nle12 : μ s₁ ≤ μ s₂\nle23 : μ s₂ ≤ μ s₃\n⊢ μ (s₃ \\ s₁) + μ s₁ = μ s₁",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"... | [] | by simp only [h_nullsdiff, zero_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 45
} | {
"line": 169,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nhμ : IsFiniteMeasure μ\nf : ℕ → Set α\nhf₁ : ∀ (i : ℕ), MeasurableSet (f i)\nhf₂ : Pairwise (Disjoint on f)\n⊢ ∑' (x : ℕ), μ (f x) ≠ ∞",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure"... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nhμ : IsFiniteMeasure μ\nf : ℕ → Set α\nhf₁ : ∀ (i : ℕ), MeasurableSet (f i)\nhf₂ : Pairwise (Disjoint on f)\n⊢ μ (⋃ i, f i) ≠ ∞"
] | rw [← MeasureTheory.measure_iUnion hf₂ hf₁] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 20
} | {
"line": 479,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nhs : MeasurableSet s\nh's : s ⊆ u\nh't : t ⊆ u\nh : μ u < μ s + μ t\n⊢ (s ∩ t).Nonempty",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"ENNReal.instAddCo... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t u : Set α\nhs : MeasurableSet s\nh's : s ⊆ u\nh't : t ⊆ u\nh : μ u < μ t + μ s\n⊢ (s ∩ t).Nonempty"
] | rw [add_comm] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 212,
"column": 4
} | {
"line": 213,
"column": 67
} | {
"line": 214,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nht : NullMeasurableSet t μ\nhs' : μ s ≠ ∞\nht' : μ t ≠ ∞\nhst : μ (s \\ t) ≠ ∞\nhts : μ (t \\ s) ≠ ∞\nthis : (μ s).toReal - (μ t).toReal = (μ (s \\ t)).toReal - (μ (t \\ s)).toReal\n⊢ |(μ s).toReal - (μ t).toR... | [] | rw [this, measure_symmDiff_eq hs ht, ENNReal.toReal_add hst hts]
convert! abs_sub (μ (s \ t)).toReal (μ (t \ s)).toReal <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 212,
"column": 4
} | {
"line": 213,
"column": 67
} | {
"line": 214,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nht : NullMeasurableSet t μ\nhs' : μ s ≠ ∞\nht' : μ t ≠ ∞\nhst : μ (s \\ t) ≠ ∞\nhts : μ (t \\ s) ≠ ∞\nthis : (μ s).toReal - (μ t).toReal = (μ (s \\ t)).toReal - (μ (t \\ s)).toReal\n⊢ |(μ s).toReal - (μ t).toR... | [] | rw [this, measure_symmDiff_eq hs ht, ENNReal.toReal_add hst hts]
convert! abs_sub (μ (s \ t)).toReal (μ (t \ s)).toReal <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 500,
"column": 2
} | {
"line": 500,
"column": 36
} | {
"line": 501,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nt : ℕ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) t\nT : ℕ → Set α := fun n ↦ toMeasurable μ (t n)\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"disjointed",
"BooleanAlgebra.toGener... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nt : ℕ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) t\nT : ℕ → Set α := fun n ↦ toMeasurable μ (t n)\nTd : ℕ → Set α := disjointed T\n⊢ μ (⋃ n, t n) ≤ ⨆ n, μ (t n)"
] | set Td : ℕ → Set α := disjointed T | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 393,
"column": 6
} | {
"line": 393,
"column": 29
} | {
"line": 393,
"column": 29
} | [
{
"pp": "α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ ν : Measure α\ninst✝ : Countable ι\ns : ι → Set α\nh : ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)\nt : Set α\nht : MeasurableSet t\nD : Directed (fun x1 x2 ↦ x1 ⊆ x2) fun t ↦ ⋃ i ∈ t, s i\n⊢ (μ.restrict (⋃ i, s i)) t = (ν.restrict (⋃ i, s i)) t"... | [
"α : Type u_2\nι : Type u_6\nm0 : MeasurableSpace α\nμ ν : Measure α\ninst✝ : Countable ι\ns : ι → Set α\nh : ∀ (i : ι), μ.restrict (s i) = ν.restrict (s i)\nt : Set α\nht : MeasurableSet t\nD : Directed (fun x1 x2 ↦ x1 ⊆ x2) fun t ↦ ⋃ i ∈ t, s i\n⊢ (μ.restrict (⋃ t, ⋃ i ∈ t, s i)) t = (ν.restrict (⋃ t, ⋃ i ∈ t, s ... | iUnion_eq_iUnion_finset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 109,
"column": 68
} | {
"line": 113,
"column": 66
} | {
"line": 115,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\ns : Set α\nhm : m ≤ m0\nμ : Measure α\nhs : MeasurableSet s\n⊢ (μ.trim hm).restrict s = (μ.restrict s).trim hm",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"MeasurableSet",
"congr... | [] | by
refine @Measure.ext _ m _ _ (fun t ht => ?_)
rw [@Measure.restrict_apply α m _ _ _ ht, trim_measurableSet_eq hm ht,
Measure.restrict_apply (hm t ht),
trim_measurableSet_eq hm (@MeasurableSet.inter α m t s ht hs)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.MemPartition | {
"line": 84,
"column": 41
} | {
"line": 84,
"column": 50
} | {
"line": 85,
"column": 4
} | [
{
"pp": "α : Type u_1\nf : ℕ → Set α\nn : ℕ\nih : ⋃₀ memPartition f n = univ\nx : α\n⊢ x ∈ ⋃₀ memPartition f n",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Set.sUnion",
"Membership.mem",
"memPartition",... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.MemPartition | {
"line": 134,
"column": 18
} | {
"line": 134,
"column": 27
} | {
"line": 136,
"column": 0
} | [
{
"pp": "case pos\nα : Type u_1\nf : ℕ → Set α\na : α\nn : ℕ\nih : memPartitionSet f n a ∈ memPartition f n\nh✝ : a ∈ f n\n⊢ memPartitionSet f n a ∈ memPartition f n ∧\n (memPartitionSet f n a ∩ f n = memPartitionSet f n a ∩ f n ∨\n memPartitionSet f n a ∩ f n = memPartitionSet f n a \\ f n)",
"ppTe... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.MemPartition | {
"line": 134,
"column": 18
} | {
"line": 134,
"column": 27
} | {
"line": 136,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nf : ℕ → Set α\na : α\nn : ℕ\nih : memPartitionSet f n a ∈ memPartition f n\nh✝ : a ∉ f n\n⊢ memPartitionSet f n a ∈ memPartition f n ∧\n (memPartitionSet f n a \\ f n = memPartitionSet f n a ∩ f n ∨\n memPartitionSet f n a \\ f n = memPartitionSet f n a \\ f n)",
"pp... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 491,
"column": 10
} | {
"line": 491,
"column": 74
} | {
"line": 492,
"column": 8
} | [
{
"pp": "case refine_1.compl.refine_2.refine_1\nα : Type u_1\nt : ℕ → Set α\nn : ℕ\ns : Set α\nS : Finset (Set α)\nhS_subset : ↑S ⊆ memPartition t n\nht✝ : MeasurableSet (⋃₀ ↑S)\nu : Set α\nhuS : u ∈ ↑S\nv : Set α\nhuV : v ∈ memPartition t n \\ ↑S\n⊢ v ≠ u",
"ppTerm": "?refine_1.compl.refine_2.refine_1",
... | [] | exact ne_of_mem_of_not_mem huS (notMem_of_mem_sdiff huV) |>.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 109,
"column": 4
} | {
"line": 112,
"column": 40
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case refine_3\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : α → β\ninst✝ : Countable ι\nμ : ι → Measure α\nh : ∀ (i : ι), AEMeasurable f (μ i)\na✝ : Nontrivial β\ninhabited_h : Inhabited β\ns : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ... | [] | refine measure_mono_null (fun x (hx : f x ≠ g x) => ?_) (hsμ i)
contrapose hx
refine (piecewise_eq_of_notMem _ _ _ ?_).symm
exact fun h => hx (mem_iInter.1 h i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 109,
"column": 4
} | {
"line": 112,
"column": 40
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case refine_3\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : α → β\ninst✝ : Countable ι\nμ : ι → Measure α\nh : ∀ (i : ι), AEMeasurable f (μ i)\na✝ : Nontrivial β\ninhabited_h : Inhabited β\ns : ι → Set α := fun i ↦ toMeasurable (μ i) {x | f x ≠ mk f ... | [] | refine measure_mono_null (fun x (hx : f x ≠ g x) => ?_) (hsμ i)
contrapose hx
refine (piecewise_eq_of_notMem _ _ _ ?_).symm
exact fun h => hx (mem_iInter.1 h i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 41
} | {
"line": 313,
"column": 4
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : atTop.IsCountablyGenerated\nx : α\ng : α → β\ng_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\n⊢ Ioi x = ⋃... | [
"α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\nmβ : MeasurableSpace β\ninst✝¹ : LinearOrder α\ninst✝ : atTop.IsCountablyGenerated\nx : α\ng : α → β\ng_meas : ∀ t > x, AEMeasurable g (μ.restrict (Ioc x t))\nthis : Nonempty α\nu : ℕ → α\nhu_tendsto : Tendsto u atTop atTop\n⊢ ∀ (x_1 : α), x < x_1 ... | rw [iUnion_Ioc_eq_Ioi_self_iff.mpr _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 427,
"column": 4
} | {
"line": 427,
"column": 47
} | {
"line": 427,
"column": 47
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝³ : MeasurableSpace G\ninst✝² : DivInvMonoid G\ninst✝¹ : MeasurableMul₂ G\ninst✝ : MeasurableInv G\n⊢ Measurable fun p ↦ p.1 * p.2⁻¹",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"DivInvMonoid.toInv",
"Measurable.mul",
"Monoid.t... | [] | exact measurable_fst.mul measurable_snd.inv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 15
} | {
"line": 69,
"column": 16
} | [
{
"pp": "case univ\nα : Type u_1\ns : Set (Set α)\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nhs : t = TopologicalSpace.generateFrom s\nu : Set α\n⊢ MeasurableSet univ",
"ppTerm": "?univ",
"assigned": true,
"usedConstants": [
"MeasurableSpace.generateFrom",
"MeasurableSet... | [] | | univ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 75
} | {
"line": 73,
"column": 8
} | [
{
"pp": "case sUnion\nα : Type u_1\ns : Set (Set α)\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nhs : t = TopologicalSpace.generateFrom s\nu : Set α\nf : Set (Set α)\nhf : ∀ s_1 ∈ f, GenerateOpen s s_1\nih : ∀ s_1 ∈ f, MeasurableSet s_1\n⊢ MeasurableSet (⋃₀ f)",
"ppTerm": "?sUnion",
"assi... | [
"case sUnion\nα : Type u_1\ns : Set (Set α)\nt : TopologicalSpace α\ninst✝ : SecondCountableTopology α\nhs : t = TopologicalSpace.generateFrom s\nu : Set α\nf : Set (Set α)\nhf : ∀ s_1 ∈ f, GenerateOpen s s_1\nih : ∀ s_1 ∈ f, MeasurableSet s_1\nv : Set (Set α)\nhv : v.Countable\nvf : v ⊆ f\nvu : ⋃₀ v = ⋃₀ f\n⊢ Meas... | rcases isOpen_sUnion_countable f (by rwa [hs]) with ⟨v, hv, vf, vu⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.MetricSpace.Lipschitz | {
"line": 168,
"column": 2
} | {
"line": 169,
"column": 68
} | {
"line": 171,
"column": 0
} | [
{
"pp": "x y : ℝ\n⊢ dist x.toNNReal y.toNNReal ≤ ↑1 * dist x y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"LipschitzWith",
"HMul.hMul",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
"Rea... | [] | simpa only [NNReal.coe_one, dist_prod_same_right, one_mul, Real.dist_eq] using!
lipschitzWith_iff_dist_le_mul.mp lipschitzWith_max (x, 0) (y, 0) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 710,
"column": 13
} | {
"line": 710,
"column": 46
} | {
"line": 710,
"column": 46
} | [
{
"pp": "case inr\nα : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\nh : x ∉ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nhs : s.Nonempty\n⊢ ¬infDist x s = 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
... | [
"case inr\nα : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nx : α\nh : x ∉ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nhs : s.Nonempty\n⊢ x ∉ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s"
] | ← mem_closure_iff_infDist_zero hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 48
} | {
"line": 44,
"column": 6
} | [
{
"pp": "case a.refine_2\nα : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhD : IsCountablySpanning D\ns : Set β\nhs : s ∈ D\nt : ℕ → Set α\nh1t : ∀ (n : ℕ), t n ∈ C\nh2t : ⋃ n, t n = univ\n⊢ MeasurableSet (Prod.snd ⁻¹' s)",
"ppTerm": "?a.refine_2✝",
"assigned": true,
"usedConstants": [... | [
"case a.refine_2\nα : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhD : IsCountablySpanning D\ns : Set β\nhs : s ∈ D\nt : ℕ → Set α\nh1t : ∀ (n : ℕ), t n ∈ C\nh2t : ⋃ n, t n = univ\n⊢ MeasurableSet (⋃ i, t i ×ˢ s)"
] | rw [← univ_prod, ← h2t, iUnion_prod_const] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 70,
"column": 10
} | {
"line": 70,
"column": 53
} | {
"line": 71,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\nthis : MeasurableSpace α := MeasurableSpace.generateFrom (range Iio)\nH : ∀ (a : α), MeasurableSet (Iio a)\na : α\nhcovBy : ¬∃ b, a ⋖ b\nt : Set α\nhat : t ⊆ Ioi a\nhtc : t.Co... | [] | simpa [CovBy, htU, subset_def] using hcovBy | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 794,
"column": 13
} | {
"line": 794,
"column": 41
} | {
"line": 794,
"column": 41
} | [
{
"pp": "case empty\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\nμ✝ : Measure α\ninst✝¹ : TopologicalSpace α\nι : Type u_3\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), (μ i).InnerRegular\n⊢ (∑ i ∈ ∅, μ i).InnerRegular",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"MeasureThe... | [
"case empty\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\nμ✝ : Measure α\ninst✝¹ : TopologicalSpace α\nι : Type u_3\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), (μ i).InnerRegular\n⊢ InnerRegular 0"
] | simp only [Finset.sum_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 251,
"column": 53
} | {
"line": 257,
"column": 81
} | {
"line": 259,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort y\ns t u : Set α\nmα : MeasurableSpace α\n⊢ MeasurableMul₂ ℝ≥0∞",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"HMul.hMul",
"... | [] | by
refine ⟨measurable_of_measurable_nnreal_nnreal ?_ ?_ ?_⟩
· simp only [← ENNReal.coe_mul, measurable_mul.coe_nnreal_ennreal]
· simp only [ENNReal.top_mul', ENNReal.coe_eq_zero]
exact measurable_const.piecewise (measurableSet_singleton _) measurable_const
· simp only [ENNReal.mul_top', ENNReal.coe_eq_zero]... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 895,
"column": 2
} | {
"line": 895,
"column": 32
} | {
"line": 896,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace α\nh : μ.InnerRegularCompactLTTop\ninst✝ : IsFiniteMeasure μ\n⊢ μ.InnerRegularWRT IsCompact MeasurableSet",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Measure... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace α\nh : μ.InnerRegularCompactLTTop\ninst✝ : IsFiniteMeasure μ\ns : Set α\n⊢ MeasurableSet s ↔ MeasurableSet s ∧ μ s ≠ ∞"
] | convert! h.innerRegular with s | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 25
} | {
"line": 355,
"column": 0
} | [
{
"pp": "α : Type u_5\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\ns t : Set α\nthis✝¹ : MeasurableSpace α := borel α\nthis✝ : BorelSpace α\na b : α\n⊢ MeasurableSet (Ico a b)",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"measurableSet_Ico",... | [] | exact measurableSet_Ico | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 41
} | {
"line": 500,
"column": 2
} | [
{
"pp": "β : Type u_6\nmβ : MeasurableSpace β\nf : EReal × EReal → β\nh_real : Measurable fun p ↦ f (↑p.1, ↑p.2)\nh_bot_left : Measurable fun r ↦ f (⊥, ↑r)\nh_top_left : Measurable fun r ↦ f (⊤, ↑r)\nh_bot_right : Measurable fun r ↦ f (↑r, ⊥)\nh_top_right : Measurable fun r ↦ f (↑r, ⊤)\n⊢ Measurable f",
"pp... | [
"case refine_1\nβ : Type u_6\nmβ : MeasurableSpace β\nf : EReal × EReal → β\nh_real : Measurable fun p ↦ f (↑p.1, ↑p.2)\nh_bot_left : Measurable fun r ↦ f (⊥, ↑r)\nh_top_left : Measurable fun r ↦ f (⊤, ↑r)\nh_bot_right : Measurable fun r ↦ f (↑r, ⊥)\nh_top_right : Measurable fun r ↦ f (↑r, ⊤)\n⊢ Measurable fun p ↦ ... | refine measurable_of_real_prod ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 968,
"column": 48
} | {
"line": 968,
"column": 67
} | {
"line": 968,
"column": 67
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : μ.InnerRegularCompactLTTop\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : R1Space α\ninst✝ : BorelSpace α\ns : Set α\nhμs : μ s ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nt : Set α\nhtm : MeasurableSet t\nhst : s =ᵐ[μ] t\n⊢ ?m... | [
"α : Type u_1\ninst✝⁵ : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : μ.InnerRegularCompactLTTop\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : R1Space α\ninst✝ : BorelSpace α\ns : Set α\nhμs : μ s ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nt : Set α\nhtm : MeasurableSet t\nhst : s =ᵐ[μ] t\n⊢ μ s ≠ ∞"
] | ← measure_congr hst | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 35
} | {
"line": 166,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ⇑φ ≤ fun a ↦ f a\nh : ∀ᵐ (a : α) ∂μ, φ a ≠ ∞\n⊢ φ.lintegral μ ≤ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants... | [
"case pos\nα : Type u_1\nm : MeasurableSpace α\nf : α → ℝ≥0∞\nμ : Measure α\nφ : α →ₛ ℝ≥0∞\nhφ : ⇑φ ≤ fun a ↦ f a\nh : ∀ᵐ (a : α) ∂μ, φ a ≠ ∞\nψ : α →ₛ ℝ≥0 := SimpleFunc.map ENNReal.toNNReal φ\n⊢ φ.lintegral μ ≤ ⨆ φ, ⨆ (_ : ∀ (x : α), ↑(φ x) ≤ f x), (SimpleFunc.map ofNNReal φ).lintegral μ"
] | let ψ := φ.map ENNReal.toNNReal | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 55
} | {
"line": 215,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nι : Sort u_4\nι' : ι → Sort u_5\nf : (i : ι) → ι' i → α → ℝ≥0∞\n⊢ ∫⁻ (a : α), ⨅ i, ⨅ h, f i h a ∂μ ≤ ⨅ i, ⨅ h, ∫⁻ (a : α), f i h a ∂μ",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
... | [
"case e'_3\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nι : Sort u_4\nι' : ι → Sort u_5\nf : (i : ι) → ι' i → α → ℝ≥0∞\na : α\n⊢ ⨅ i, ⨅ h, f i h a = (⨅ i, ⨅ j, f i j) a"
] | convert! (monotone_lintegral μ).map_iInf₂_le f with a | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 89,
"column": 6
} | {
"line": 93,
"column": 33
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh_mono : Monotone f\nc : ℝ≥0 → ℝ≥0∞ := ofNNReal\nF : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a\ns : α →ₛ ℝ≥0\nhsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x\nr : ℝ≥0\nright✝ ha✝ : ↑r < 1\nha : r < 1\nrs : α →ₛ ℝ≥0 := Sim... | [] | simp only [← SimpleFunc.lintegral_eq_lintegral]
gcongr with n a
simp only [map_apply] at h_meas
simp only [coe_map, restrict_apply _ (h_meas _), (· ∘ ·)]
exact indicator_apply_le id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 89,
"column": 6
} | {
"line": 93,
"column": 33
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh_mono : Monotone f\nc : ℝ≥0 → ℝ≥0∞ := ofNNReal\nF : α → ℝ≥0∞ := fun a ↦ ⨆ n, f n a\ns : α →ₛ ℝ≥0\nhsf : ∀ (x : α), ↑(s x) ≤ ⨆ n, f n x\nr : ℝ≥0\nright✝ ha✝ : ↑r < 1\nha : r < 1\nrs : α →ₛ ℝ≥0 := Sim... | [] | simp only [← SimpleFunc.lintegral_eq_lintegral]
gcongr with n a
simp only [map_apply] at h_meas
simp only [coe_map, restrict_apply _ (h_meas _), (· ∘ ·)]
exact indicator_apply_le id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 875,
"column": 4
} | {
"line": 875,
"column": 74
} | {
"line": 876,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nn : ℕ\na : α\nb : ℕ\na✝ : b ∈ Finset.range n\nh✝ : MeasurableSet {a | ennrealRatEmbed b ≤ f a}\n⊢ (piecewise {a | ennrealRatEmbed b ≤ f a} h✝ (const α (ennrealRatEmbed b)) 0) a < ⊤",
"ppTerm": "?pos✝",
"assigned": true,
"usedC... | [
"case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nn : ℕ\na : α\nb : ℕ\na✝ : b ∈ Finset.range n\nh✝ : MeasurableSet {a | ennrealRatEmbed b ≤ f a}\n⊢ {a | ennrealRatEmbed b ≤ f a}.indicator (Function.const α (ennrealRatEmbed b)) a < ⊤"
] | simp only [coe_zero, coe_piecewise, piecewise_eq_indicator, coe_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 973,
"column": 6
} | {
"line": 973,
"column": 29
} | {
"line": 973,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ng : β → ℝ≥0∞\nf : α →ₛ β\na : α\nhb : f a ∈ f.range\n⊢ g (f a) * μ (⇑(map g f) ⁻¹' {g (f a)}) = ∑ j ∈ f.range with g j = g (f a), g j * μ (⇑f ⁻¹' {j})",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ng : β → ℝ≥0∞\nf : α →ₛ β\na : α\nhb : f a ∈ f.range\n⊢ g (f a) * μ (⇑f ⁻¹' ↑({b ∈ f.range | g b = g (f a)})) = ∑ j ∈ f.range with g j = g (f a), g j * μ (⇑f ⁻¹' {j})"
] | map_preimage_singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 50
} | {
"line": 527,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\nc : ℝ≥0∞\n⊢ ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ = c * μ s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrA... | [] | rw [lintegral_indicator₀ hs, setLIntegral_const] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 50
} | {
"line": 527,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\nc : ℝ≥0∞\n⊢ ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ = c * μ s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrA... | [] | rw [lintegral_indicator₀ hs, setLIntegral_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 50
} | {
"line": 527,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : NullMeasurableSet s μ\nc : ℝ≥0∞\n⊢ ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ = c * μ s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrA... | [] | rw [lintegral_indicator₀ hs, setLIntegral_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 847,
"column": 4
} | {
"line": 847,
"column": 26
} | {
"line": 848,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nh : ∀ x ∈ s, s ∈ 𝓝[>] x\nH : ¬∃ x ∈ s, IsTop x\n⊢ ∀ x ∈ s, ∃ y, x < y",
"ppTerm": "?neg✝",
"assign... | [
"case neg\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nh : ∀ x ∈ s, s ∈ 𝓝[>] x\nH : ¬∃ x ∈ s, ∀ (b : α), b ≤ x\n⊢ ∀ x ∈ s, ∃ y, x < y"
] | simp only [IsTop] at H | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Lebesgue.Map | {
"line": 105,
"column": 64
} | {
"line": 107,
"column": 74
} | {
"line": 109,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nt : Set ↑s\nf : α → ℝ≥0∞\n⊢ ∫⁻ (x : ↑s) in t, f ↑x ∂Measure.comap Subtype.val μ = ∫⁻ (x : α) in Subtype.val '' t, f x ∂μ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Iff.mpr",
... | [] | by
rw [(MeasurableEmbedding.subtype_coe hs).restrict_comap, lintegral_subtype_comap hs,
restrict_restrict hs, inter_eq_right.2 (Subtype.coe_image_subset _ _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1386,
"column": 2
} | {
"line": 1395,
"column": 73
} | {
"line": 1396,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nmotive : (α → ℝ≥0∞) → Prop\nindicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)\nadd :\n ∀ ⦃f g : α → ℝ≥0∞⦄,\n Disjoint (support f) (support g) → Measurable f → Measurable g → mo... | [] | refine Measurable.ennreal_induction (fun c s hs ↦ ?_) add iSup hf
convert!
iSup (f := fun n ↦ (s ∩ spanningSets μ n).indicator fun _ ↦ c)
(fun n ↦ measurable_const.indicator (hs.inter (measurableSet_spanningSets ..)))
(fun m n hmn a ↦ by dsimp; grw [hmn])
(fun n ↦
indicator _ (hs.inter (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1386,
"column": 2
} | {
"line": 1395,
"column": 73
} | {
"line": 1396,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nmotive : (α → ℝ≥0∞) → Prop\nindicator : ∀ (c : ℝ≥0∞) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)\nadd :\n ∀ ⦃f g : α → ℝ≥0∞⦄,\n Disjoint (support f) (support g) → Measurable f → Measurable g → mo... | [] | refine Measurable.ennreal_induction (fun c s hs ↦ ?_) add iSup hf
convert!
iSup (f := fun n ↦ (s ∩ spanningSets μ n).indicator fun _ ↦ c)
(fun n ↦ measurable_const.indicator (hs.inter (measurableSet_spanningSets ..)))
(fun m n hmn a ↦ by dsimp; grw [hmn])
(fun n ↦
indicator _ (hs.inter (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 414,
"column": 67
} | {
"line": 414,
"column": 72
} | {
"line": 417,
"column": 0
} | [
{
"pp": "𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝ : Group E\nf : GroupSeminorm E\nx y : E\n⊢ x⁻¹ * y = (y⁻¹ * x)⁻¹",
"ppTerm": "?m.161",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.Normed.Group.Defs.0.GroupSemi... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 413,
"column": 83
} | {
"line": 413,
"column": 88
} | {
"line": 414,
"column": 2
} | [
{
"pp": "case e'_3\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝ : Group E\nf : GroupSeminorm E\nx y z : E\n⊢ x⁻¹ * z = x⁻¹ * y * (y⁻¹ * z)",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toM... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 579,
"column": 26
} | {
"line": 579,
"column": 84
} | {
"line": 579,
"column": 84
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q✝ p q : NonarchAddGroupSeminorm E\nx : E\n⊢ (⇑p ⊔ ⇑q) (-x) = (⇑p ⊔ ⇑q) x",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Neg... | [] | simp_rw [Pi.sup_apply, map_neg_eq_map p, map_neg_eq_map q] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 579,
"column": 26
} | {
"line": 579,
"column": 84
} | {
"line": 579,
"column": 84
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q✝ p q : NonarchAddGroupSeminorm E\nx : E\n⊢ (⇑p ⊔ ⇑q) (-x) = (⇑p ⊔ ⇑q) x",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Neg... | [] | simp_rw [Pi.sup_apply, map_neg_eq_map p, map_neg_eq_map q] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 579,
"column": 26
} | {
"line": 579,
"column": 84
} | {
"line": 579,
"column": 84
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q✝ p q : NonarchAddGroupSeminorm E\nx : E\n⊢ (⇑p ⊔ ⇑q) (-x) = (⇑p ⊔ ⇑q) x",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Neg... | [] | simp_rw [Pi.sup_apply, map_neg_eq_map p, map_neg_eq_map q] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 617,
"column": 8
} | {
"line": 617,
"column": 27
} | {
"line": 618,
"column": 8
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝⁴ : Group E\ninst✝³ : SMul R ℝ\ninst✝² : SMul R ℝ≥0\ninst✝¹ : IsScalarTower R ℝ≥0 ℝ\ninst✝ : DecidableEq E\nx y : E\n⊢ (if x * y = 1 then 0 else 1) ≤ (if x = 1 then 0 else 1) + if y = 1 then 0 else 1",
"ppTerm": "?m.33",
... | [
"case pos\nR : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝⁴ : Group E\ninst✝³ : SMul R ℝ\ninst✝² : SMul R ℝ≥0\ninst✝¹ : IsScalarTower R ℝ≥0 ℝ\ninst✝ : DecidableEq E\nx y : E\nhx : x = 1\n⊢ (if x * y = 1 then 0 else 1) ≤ (if x = 1 then 0 else 1) + if y = 1 then 0 else 1",
"case neg\nR ... | by_cases hx : x = 1 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 692,
"column": 26
} | {
"line": 692,
"column": 52
} | {
"line": 692,
"column": 53
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np : NonarchAddGroupSeminorm E\nx : E\n⊢ r • p (-x) = r • p x",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
... | [] | simp_rw [map_neg_eq_map p] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 692,
"column": 26
} | {
"line": 692,
"column": 52
} | {
"line": 692,
"column": 53
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np : NonarchAddGroupSeminorm E\nx : E\n⊢ r • p (-x) = r • p x",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
... | [] | simp_rw [map_neg_eq_map p] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 692,
"column": 26
} | {
"line": 692,
"column": 52
} | {
"line": 692,
"column": 53
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np : NonarchAddGroupSeminorm E\nx : E\n⊢ r • p (-x) = r • p x",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
... | [] | simp_rw [map_neg_eq_map p] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sequences | {
"line": 320,
"column": 2
} | {
"line": 324,
"column": 59
} | {
"line": 326,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nf_cont : SeqContinuous f\nK : Set X\nK_cpt : IsSeqCompact K\n⊢ IsSeqCompact (f '' K)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"StrictMono",
"Function.comp",
"Me... | [] | intro ys ys_in_fK
choose xs xs_in_K fxs_eq_ys using ys_in_fK
obtain ⟨a, a_in_K, phi, phi_mono, xs_phi_lim⟩ := K_cpt xs_in_K
refine ⟨f a, mem_image_of_mem f a_in_K, phi, phi_mono, ?_⟩
exact (f_cont xs_phi_lim).congr fun x ↦ fxs_eq_ys (phi x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sequences | {
"line": 320,
"column": 2
} | {
"line": 324,
"column": 59
} | {
"line": 326,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nf_cont : SeqContinuous f\nK : Set X\nK_cpt : IsSeqCompact K\n⊢ IsSeqCompact (f '' K)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"StrictMono",
"Function.comp",
"Me... | [] | intro ys ys_in_fK
choose xs xs_in_K fxs_eq_ys using ys_in_fK
obtain ⟨a, a_in_K, phi, phi_mono, xs_phi_lim⟩ := K_cpt xs_in_K
refine ⟨f a, mem_image_of_mem f a_in_K, phi, phi_mono, ?_⟩
exact (f_cont xs_phi_lim).congr fun x ↦ fxs_eq_ys (phi x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 149,
"column": 41
} | {
"line": 150,
"column": 48
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_3\nβ : Type u_4\ninst✝³ : DivisionRing R\ninst✝² : UniformSpace R\ninst✝¹ : UniformContinuousConstSMul Rᵐᵒᵖ R\ninst✝ : UniformSpace β\nf : β → R\nhf : UniformContinuous f\na : R\n⊢ UniformContinuous fun x ↦ f x / a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"... | [] | by
simpa [div_eq_mul_inv] using hf.mul_const' a⁻¹ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 349,
"column": 6
} | {
"line": 349,
"column": 11
} | {
"line": 350,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\nn₀ : ℕ\nhn₀ : ∀ n ≥ n₀, ‖(u n)⁻¹ * a‖ < b 0\nz : ℕ →... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Analysis.Normed.Group.Submodule | {
"line": 53,
"column": 58
} | {
"line": 55,
"column": 10
} | {
"line": 56,
"column": 0
} | [
{
"pp": "R : Type u_3\nR' : Type u_4\nM : Type u_5\nM' : Type u_6\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R'\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : Module R M\ninst✝² : Module R' M'\nσ₁₂ : R →+* R'\nf : M →ₛₗ[σ₁₂] M'\ninst✝¹ : TopologicalSpace M\ninst✝ : TopologicalSpace M'\nhf : Continu... | [] | by
rw [coe_domRestrict]
fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 7
} | {
"line": 193,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nu v : E\n⊢ ‖u⁻¹ * v * v⁻¹‖ = ‖u⁻¹‖",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Norm.norm",
"MulOne.toOne",
"Real",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrAr... | [] | group | Mathlib.Tactic.Group._aux_Mathlib_Tactic_Group___macroRules_Mathlib_Tactic_Group_group_1 | Mathlib.Tactic.Group.group |
Mathlib.Analysis.Normed.Ring.Lemmas | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 46
} | {
"line": 214,
"column": 2
} | [
{
"pp": "⊢ LipschitzWith 2 fun p ↦ p.1 - p.2",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Prod.pseudoEMetricSpaceMax",
"Real",
"LipschitzWith",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
... | [
"⊢ LipschitzWith 2 (toReal ∘ fun p ↦ p.1 - p.2)"
] | rw [← NNReal.isometry_coe.lipschitzWith_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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