module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.EReal.Operations | {
"line": 628,
"column": 2
} | {
"line": 628,
"column": 21
} | {
"line": 629,
"column": 2
} | [
{
"pp": "x : EReal\nh : 0 < x\n⊢ ⊥ * x = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"id",
"Bot.bot",
"EReal.mul_comm",
"Eq",
"instBotEReal",
"EReal.instMul",
"instHMul"
... | [
"x : EReal\nh : 0 < x\n⊢ x * ⊥ = ⊥"
] | rw [EReal.mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Operations | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 21
} | {
"line": 633,
"column": 2
} | [
{
"pp": "x : EReal\nh : x < 0\n⊢ ⊥ * x = ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"EReal",
"instTopEReal",
"id",
"Bot.bot",
"EReal.mul_comm",
"Top.top",
"Eq",
"instBotEReal",
... | [
"x : EReal\nh : x < 0\n⊢ x * ⊥ = ⊤"
] | rw [EReal.mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 430,
"column": 4
} | {
"line": 434,
"column": 26
} | {
"line": 435,
"column": 4
} | [
{
"pp": "case inl\nι : Type u_1\nα : Type u_7\nβ : Type u_8\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : CompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nu : ι → α → β\nc : β\nh_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)\nh_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofi... | [
"case inr\nι : Type u_1\nα : Type u_7\nβ : Type u_8\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : CompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nu : ι → α → β\nc : β\nh_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)\nh_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u i r) cofinite) atTop ... | · refine ⟨b', ⟨le_rfl, hb'b⟩, ?_⟩
have h_lt := eventually_lt_of_limsup_lt ((hr r le_rfl).trans_lt hb'b)
filter_upwards [h_lt] with n hn
contrapose! h
exact ⟨u n r, h, hn⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 335,
"column": 2
} | {
"line": 337,
"column": 40
} | {
"line": 339,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\ns : Set α\nf : α → β\ng : α → γ\nKf Kg : ℝ≥0\nhf : LipschitzOnWith Kf f s\nhg : LipschitzOnWith Kg g s\n⊢ LipschitzOnWith (max Kf Kg) (fun x ↦ (f x, g x)) s",
"ppTerm": "?... | [] | intro _ hx _ hy
rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul]
exact max_le_max (hf hx hy) (hg hx hy) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 335,
"column": 2
} | {
"line": 337,
"column": 40
} | {
"line": 339,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\ns : Set α\nf : α → β\ng : α → γ\nKf Kg : ℝ≥0\nhf : LipschitzOnWith Kf f s\nhg : LipschitzOnWith Kg g s\n⊢ LipschitzOnWith (max Kf Kg) (fun x ↦ (f x, g x)) s",
"ppTerm": "?... | [] | intro _ hx _ hy
rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul]
exact max_le_max (hf hx hy) (hg hx hy) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 22
} | {
"line": 49,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nL : SummationFilter β\ninst✝ : IsEmpty β\n⊢ HasProd f 1 L",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"HEq.refl",... | [] | convert! hasProd_one | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 22
} | {
"line": 49,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nL : SummationFilter β\ninst✝ : IsEmpty β\n⊢ HasProd f 1 L",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"HEq.refl",... | [] | convert! hasProd_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 22
} | {
"line": 49,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nL : SummationFilter β\ninst✝ : IsEmpty β\n⊢ HasProd f 1 L",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"HEq.refl",... | [] | convert! hasProd_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 22
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nL : SummationFilter β\ninst✝ : Subsingleton α\n⊢ HasProd f 1 L",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"Lean.... | [] | convert! hasProd_one | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 22
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nL : SummationFilter β\ninst✝ : Subsingleton α\n⊢ HasProd f 1 L",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"Lean.... | [] | convert! hasProd_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 22
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nL : SummationFilter β\ninst✝ : Subsingleton α\n⊢ HasProd f 1 L",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"Lean.... | [] | convert! hasProd_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 462,
"column": 44
} | {
"line": 464,
"column": 84
} | {
"line": 466,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\n⊢ ∏'[L] (x : β), 1 = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"if_true",
"Monoid.toMulOneClass",
"congrArg",
... | [] | by
rw [tprod_def, dif_pos multipliable_one, mulSupport_fun_one, Set.empty_inter,
Set.mulIndicator_one, finprod_one, eq_true_intro hasProd_one, if_true, ite_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 414,
"column": 6
} | {
"line": 414,
"column": 45
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case inr.inr\nβ : Type u_2\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\na : G\nhβ : Infinite β\nha : a ≠ 1\n⊢ ¬Multipliable fun b ↦ a",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"In... | [] | simpa [multipliable_const_iff] using ha | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Instances.NNReal.Lemmas | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 77
} | {
"line": 155,
"column": 4
} | [
{
"pp": "case inl.mp\nα : Type u_2\nL : SummationFilter α\nf : α → ℝ≥0\nh✝ : L.NeBot\n⊢ Summable (fun a ↦ ↑(f a)) L → Summable f L",
"ppTerm": "?inl.mp",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Real.instLE",
"Real",
"_private.Mathlib.Topology.Inst... | [
"case inl.mpr\nα : Type u_2\nL : SummationFilter α\nf : α → ℝ≥0\nh✝ : L.NeBot\n⊢ Summable f L → Summable (fun a ↦ ↑(f a)) L"
] | · exact fun ⟨a, ha⟩ => ⟨⟨a, ha.nonneg fun x => (f x).2⟩, hasSum_coe.1 ha⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 42
} | {
"line": 128,
"column": 2
} | [
{
"pp": "case h\nX : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.... | [
"case h.inl\nX : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.length =... | rcases eq_or_ne (d x y) 0 with hd₀ | hd₀ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Sequences | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 43
} | {
"line": 186,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nx : ℕ → X\ninst✝ : SequentialSpace X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\ny : ℕ → X\nl : X\nhy : ∀ (n : ℕ), y n ∈ ⋃ i, closure[inst✝¹] {x i}\nhy' : Tendsto y atTop (𝓝 l)\nhm : ∀ (m : ℕ), ∀ᶠ (x_1 : ℕ) in atTop, y x_1 ∉ clo... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nx : ℕ → X\ninst✝ : SequentialSpace X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\ny : ℕ → X\nl : X\nhy : ∀ (n : ℕ), y n ∈ ⋃ i, closure[inst✝¹] {x i}\nhy' : Tendsto y atTop (𝓝 l)\nhm : ∀ (m : ℕ), ∀ᶠ (x_1 : ℕ) in atTop, y x_1 ∉ closure[inst✝¹]... | refine frequently_atTop.2 fun a => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 98,
"column": 7
} | {
"line": 98,
"column": 46
} | {
"line": 98,
"column": 46
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf g : α → ℝ≥0∞\nhfi : ∀ᶠ (x : α) in l, f x ≠ ∞\nhgi : ∀ᶠ (x : α) in l, g x ≠ ∞\nhfg : (fun x ↦ (f x).toReal) =ᶠ[l] fun x ↦ (g x).toReal\na✝¹ : α\nhfx : f a✝¹ ≠ ∞\nhgx : g a✝¹ ≠ ∞\na✝ : (f a✝¹).toReal = (g a✝¹).toReal\n⊢ f a✝¹ = g a✝¹",
"ppTerm": "?m.50",
"assigned": ... | [
"α : Type u_1\nl : Filter α\nf g : α → ℝ≥0∞\nhfi : ∀ᶠ (x : α) in l, f x ≠ ∞\nhgi : ∀ᶠ (x : α) in l, g x ≠ ∞\nhfg : (fun x ↦ (f x).toReal) =ᶠ[l] fun x ↦ (g x).toReal\na✝¹ : α\nhfx : f a✝¹ ≠ ∞\nhgx : g a✝¹ ≠ ∞\na✝ : (f a✝¹).toReal = (g a✝¹).toReal\n⊢ (f a✝¹).toReal = (g a✝¹).toReal"
] | ← ENNReal.toReal_eq_toReal_iff' hfx hgx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sequences | {
"line": 184,
"column": 2
} | {
"line": 197,
"column": 47
} | {
"line": 199,
"column": 0
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nx : ℕ → X\ninst✝ : SequentialSpace X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\ny : ℕ → X\nl : X\nhy : ∀ (n : ℕ), y n ∈ ⋃ i, closure[inst✝¹] {x i}\nhy' : Tendsto y atTop (𝓝 l)\nhm : ∀ (m : ℕ), ∀ᶠ (x_1 : ℕ) in atTop, y... | [] | · have (j : ℕ) : ∃ᶠ k in atTop, ∃ n ≥ j, y n ∈ closure {x k} := by
refine frequently_atTop.2 fun a => ?_
have := (Filter.eventually_all_finite (by simp : (Iic a).Finite)).2 fun i hi => hm i
simp only [mem_Iic, eventually_atTop] at this
obtain ⟨c, hc⟩ := this
obtain ⟨b, hb⟩ := mem_iUnion.1 ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 346,
"column": 45
} | {
"line": 348,
"column": 44
} | {
"line": 349,
"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)\n⊢ {x | ‖x⁻¹ * a‖ < b 0} ∈ 𝓝 a",
"ppTerm": "?m.1... | [] | by
simp_rw [← dist_eq_norm_inv_mul]
exact Metric.ball_mem_nhds _ (b_pos _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 522,
"column": 6
} | {
"line": 522,
"column": 28
} | {
"line": 522,
"column": 29
} | [
{
"pp": "ι : Type u_4\nl : Filter ι\nx : ι → ℝ\nhf : liminf (fun i ↦ ↑(Real.nnabs (x i))) l ≠ ∞\nhbdd : ¬IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun i ↦ |x i|\n⊢ ∃ a b, a < b ∧ (∃ᶠ (i : ι) in l, x i < ↑a) ∧ ∃ᶠ (i : ι) in l, ↑b < x i",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Rea... | [
"ι : Type u_4\nl : Filter ι\nx : ι → ℝ\nhf : liminf (fun i ↦ ↑(Real.nnabs (x i))) l ≠ ∞\nhbdd : ¬(IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l x ∧ IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) l x)\n⊢ ∃ a b, a < b ∧ (∃ᶠ (i : ι) in l, x i < ↑a) ∧ ∃ᶠ (i : ι) in l, ↑b < x i"
] | isBoundedUnder_le_abs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 84
} | {
"line": 224,
"column": 4
} | [
{
"pp": "X : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else... | [
"X : Type u_2\ninst✝¹ : UniformSpace X\ninst✝ : (𝓤 X).IsCountablyGenerated\nU : ℕ → SetRel X X\nhU_symm : ∀ (n : ℕ), (U n).IsSymm\nhU_comp : ∀ ⦃m n : ℕ⦄, m < n → U n ○ (U n ○ U n) ⊆ U m\nhB : (𝓤 X).HasAntitoneBasis U\nd : X → X → ℝ≥0 := fun x y ↦ if h : ∃ n, (x, y) ∉ U n then (1 / 2) ^ Nat.find h else 0\nhd₀ : ∀ ... | refine PseudoMetricSpace.le_two_mul_dist_ofPreNNDist _ _ _ fun x₁ x₂ x₃ x₄ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 373,
"column": 4
} | {
"line": 373,
"column": 20
} | {
"line": 374,
"column": 4
} | [
{
"pp": "case refine_3\nE : 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‖... | [
"case refine_3\nE : 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 : ... | rw [← norm_inv'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.UniformSpace.Completion | {
"line": 257,
"column": 6
} | {
"line": 257,
"column": 22
} | {
"line": 257,
"column": 23
} | [
{
"pp": "α : Type u\ninst✝ : UniformSpace α\nf g : CauchyFilter α\na b : α\nha : Tendsto id (↑f) (𝓝 a)\nhb : Tendsto id (↑g) (𝓝 b)\n⊢ Inseparable a b ↔ Inseparable f g",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cauchy",
"SProd.sprod",
"congrArg",
... | [
"α : Type u\ninst✝ : UniformSpace α\nf g : CauchyFilter α\na b : α\nha : Tendsto id (↑f) (𝓝 a)\nhb : Tendsto id (↑g) (𝓝 b)\n⊢ Inseparable a b ↔ ↑f ×ˢ ↑g ≤ 𝓤 α"
] | inseparable_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 77,
"column": 4
} | {
"line": 80,
"column": 31
} | {
"line": 82,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\ne : X ≃ᵈ Y\n⊢ ∃ r, r ≠ 0 ∧ ∀ (x y : Y), edist (e.symm.toFun x) (e.symm.toFun y) = ↑r * edist x y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
... | [] | refine ⟨(ratio e)⁻¹, inv_ne_zero <| ratio_ne_zero e, e.surjective.forall₂.2 fun x y ↦ ?_⟩
simp_rw [Equiv.toFun_as_coe, Equiv.symm_apply_apply, coe_toEquiv, edist_eq]
rw [← mul_assoc, ← ENNReal.coe_mul, inv_mul_cancel₀ (ratio_ne_zero e),
ENNReal.coe_one, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 77,
"column": 4
} | {
"line": 80,
"column": 31
} | {
"line": 82,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\ne : X ≃ᵈ Y\n⊢ ∃ r, r ≠ 0 ∧ ∀ (x y : Y), edist (e.symm.toFun x) (e.symm.toFun y) = ↑r * edist x y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
... | [] | refine ⟨(ratio e)⁻¹, inv_ne_zero <| ratio_ne_zero e, e.surjective.forall₂.2 fun x y ↦ ?_⟩
simp_rw [Equiv.toFun_as_coe, Equiv.symm_apply_apply, coe_toEquiv, edist_eq]
rw [← mul_assoc, ← ENNReal.coe_mul, inv_mul_cancel₀ (ratio_ne_zero e),
ENNReal.coe_one, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.MulAction | {
"line": 181,
"column": 41
} | {
"line": 181,
"column": 65
} | {
"line": 182,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : MulActionWithZero α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\nh : ¬r = 0\n⊢ ‖r‖ * ‖x‖ = ‖r‖ * ‖r⁻¹ • r • x‖",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtra... | [] | by rw [inv_smul_smul₀ h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Update | {
"line": 115,
"column": 73
} | {
"line": 116,
"column": 14
} | {
"line": 118,
"column": 0
} | [
{
"pp": "ι : Type u_1\nπ : ι → Sort u_2\nx : (i : ι) → π i\ninst✝ : DecidableEq ι\ns t : Finset ι\ny : (i : ↥s) → π ↑i\nz : (i : ↥t) → π ↑i\nh : s = t\n⊢ updateFinset x s y = updateFinset x t fun i ↦ y ⟨↑i, ⋯⟩",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Function.updateFinset",
... | [] | by
subst h; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 65
} | {
"line": 238,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_4\nh : NormedField 𝕜\nH : ∃ x, x ≠ 0 ∧ ‖x‖ ≠ 1\n⊢ DiscreteTopology 𝕜 ∨ Nonempty { h' // h'.toNormedField = h }",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"DiscreteTopology",
"PseudoMetricSpace.to... | [
"case neg\n𝕜 : Type u_4\nh : NormedField 𝕜\nH : ¬∃ x, x ≠ 0 ∧ ‖x‖ ≠ 1\n⊢ DiscreteTopology 𝕜 ∨ Nonempty { h' // h'.toNormedField = h }"
] | · exact Or.inr ⟨(⟨NontriviallyNormedField.ofNormNeOne H, rfl⟩)⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 57
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\nι : Sort u_6\nm : MeasurableSpace α\ninst✝ : Countable ι\nf : ι → Set α\nh : ∀ (b : ι), MeasurableSet (f b)\nh✝ : Nonempty ι\ne : ℕ → ι\nhe : Surjective e\n⊢ MeasurableSet (⋃ b, f b)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Meas... | [
"case inr\nα : Type u_1\nι : Sort u_6\nm : MeasurableSpace α\ninst✝ : Countable ι\nf : ι → Set α\nh : ∀ (b : ι), MeasurableSet (f b)\nh✝ : Nonempty ι\ne : ℕ → ι\nhe : Surjective e\n⊢ MeasurableSet (⋃ x, f (e x))"
] | rw [← iUnion_congr_of_surjective _ he (fun _ => rfl)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 233,
"column": 59
} | {
"line": 233,
"column": 92
} | {
"line": 235,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : OuterMeasure α\n⊢ LawfulFunctor OuterMeasure",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"LawfulFunctor.mk",
"MeasureTheory.OuterMeasure.instFunctor",
"Function.comp",
"MeasureTheory.OuterMeasure",
"id",
"Funct... | [] | by constructor <;> intros <;> rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Real | {
"line": 42,
"column": 2
} | {
"line": 42,
"column": 27
} | {
"line": 43,
"column": 2
} | [
{
"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\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : n ≤ m\n⊢ ∑ i ∈ Ico n m, d i ≤ ∑' (m : ℕ), d (n + m)",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
... | [
"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\na : α\nha : Tendsto f atTop (𝓝 a)\nn m : ℕ\nhnm : n ≤ m\n⊢ ∑ k ∈ range (m - n), d (n + k) ≤ ∑' (m : ℕ), d (n + m)"
] | rw [sum_Ico_eq_sum_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Sign.Basic | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 29
} | {
"line": 145,
"column": 2
} | [
{
"pp": "case neg\nα : Type u\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nι : Type u_1\ns : Finset ι\nf : ι → α\nhs : s.Nonempty\nh : ∀ i ∈ s, f i < 0\n⊢ ∑ i ∈ s, f i < 0",
"ppTerm": "?neg",
"assigned": true,
"usedConstants": [
"AddCommGroup.toAddCommMonoid"... | [] | exact Finset.sum_neg h hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.EReal.Inv | {
"line": 372,
"column": 35
} | {
"line": 372,
"column": 50
} | {
"line": 372,
"column": 50
} | [
{
"pp": "a b c : EReal\n⊢ b * a / c = b * (a / c)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"EReal.instDivInvMonoid",
"instHDiv",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"congrArg",
"EReal... | [
"a b c : EReal\n⊢ b * (a / c) = b * (a / c)"
] | ← mul_div b a c | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 54
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nP : Set α → Prop\nm_top : ∀ (s : Set α), ¬P s → m s = ∞\ns : Set α\nt : ℕ → Set α\nht_subset : s ⊆ iUnion t\ni : ℕ\nhti_notMem : ¬P (t i)\nhfi_top : m (t i) = ∞\n⊢ ∑' (i : ℕ), m (t i) = ∞",
"ppTerm": "?neg✝",
"assigned": true,
"us... | [] | exact ENNReal.tsum_eq_top_of_eq_top ⟨i, hfi_top⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 300,
"column": 4
} | {
"line": 303,
"column": 74
} | {
"line": 305,
"column": 0
} | [
{
"pp": "case refine_2\nr : ℝ≥0∞\n⊢ 1 < r → Tendsto (fun n ↦ r ^ n) atTop (𝓝 ∞)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Filter.Tendsto.inv",
... | [] | intro r_gt_one
have obs := @Tendsto.inv ℝ≥0∞ ℕ _ _ _ (fun n ↦ (r⁻¹) ^ n) atTop 0
simp only [ENNReal.tendsto_pow_atTop_nhds_zero_iff, inv_zero] at obs
simpa [← ENNReal.inv_pow] using obs <| ENNReal.inv_lt_one.mpr r_gt_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 300,
"column": 4
} | {
"line": 303,
"column": 74
} | {
"line": 305,
"column": 0
} | [
{
"pp": "case refine_2\nr : ℝ≥0∞\n⊢ 1 < r → Tendsto (fun n ↦ r ^ n) atTop (𝓝 ∞)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Filter.Tendsto.inv",
... | [] | intro r_gt_one
have obs := @Tendsto.inv ℝ≥0∞ ℕ _ _ _ (fun n ↦ (r⁻¹) ^ n) atTop 0
simp only [ENNReal.tendsto_pow_atTop_nhds_zero_iff, inv_zero] at obs
simpa [← ENNReal.inv_pow] using obs <| ENNReal.inv_lt_one.mpr r_gt_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 36
} | {
"line": 411,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\nβ : Type u_3\nf : α → β\nm : ι → OuterMeasure β\n⊢ (comap f) (⨅ i, m i) = ⨅ i, (comap f) (m i)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"iInf",
"Semiring.toModule",
"IsScalarTower.right",
"CommSemiring.toSemiring",
"... | [
"α : Type u_1\nι : Sort u_2\nβ : Type u_3\nf : α → β\nm : ι → OuterMeasure β\ns : Set α\nhs : s.Nonempty\n⊢ ((comap f) (⨅ i, m i)) s = (⨅ i, (comap f) (m i)) s"
] | refine ext_nonempty fun s hs => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 443,
"column": 66
} | {
"line": 443,
"column": 91
} | {
"line": 444,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\ns : Set α\nm : ι → OuterMeasure α\n⊢ (restrict s) (⨅ i, (restrict s) (m i)) = (restrict (range Subtype.val)) (⨅ i, (restrict s) (m i))",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"Semiring.toModule",
"IsScal... | [] | by rw [Subtype.range_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 758,
"column": 88
} | {
"line": 759,
"column": 48
} | {
"line": 761,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\n⊢ Tendsto (fun x ↦ x / n) atTop atTop",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"instHDiv",
"congrArg",
"Filter.map",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"HDiv.hDiv",
... | [] | by
rw [Tendsto, map_div_atTop_eq_nat n hn.bot_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.OuterMeasure.AE | {
"line": 151,
"column": 25
} | {
"line": 151,
"column": 46
} | {
"line": 153,
"column": 0
} | [
{
"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⊢ (∀ᵐ (x : α) ∂μ, x ∈ s → x ∈ t) ↔ μ (s \\ t) = 0",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"congrArg",
"Filt... | [] | by simp [ae_iff]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.PiSystem | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 48
} | {
"line": 481,
"column": 2
} | [
{
"pp": "α : Type u_3\nι : Type u_4\nm : ι → MeasurableSpace α\nS : Set ι\npt : Finset ι\nhpt : ↑pt ⊆ S\nft : ι → Set α\nht_m : ∀ x ∈ pt, ft x ∈ (fun n ↦ {s | MeasurableSet s}) x\n⊢ MeasurableSet (⋂ x ∈ pt, ft x)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"iSup",
"Finset",... | [
"α : Type u_3\nι : Type u_4\nm : ι → MeasurableSpace α\nS : Set ι\npt : Finset ι\nhpt : ↑pt ⊆ S\nft : ι → Set α\nht_m : ∀ x ∈ pt, ft x ∈ (fun n ↦ {s | MeasurableSet s}) x\ni : ι\nhi : i ∈ pt\n⊢ MeasurableSet (ft i)"
] | refine pt.measurableSet_biInter fun i hi => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.PiSystem | {
"line": 491,
"column": 4
} | {
"line": 491,
"column": 46
} | {
"line": 492,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_3\nι : Type u_4\nm : ι → MeasurableSpace α\nS : Set ι\ni : ι\nhi : i ∈ S\n⊢ m i ≤ generateFrom (piiUnionInter (fun n ↦ {s | MeasurableSet s}) S)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"piiUnionInter",
"MeasurableSe... | [
"case refine_2\nα : Type u_3\nι : Type u_4\nm : ι → MeasurableSpace α\nS : Set ι\ni : ι\nhi : i ∈ S\n⊢ generateFrom {s | MeasurableSet s} ≤ generateFrom (piiUnionInter (fun n ↦ {s | MeasurableSet s}) S)"
] | rw [← @generateFrom_measurableSet α (m i)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 398,
"column": 4
} | {
"line": 398,
"column": 25
} | {
"line": 399,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t\nhs : ¬ms = ∞\nthis✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀... | [
"case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nm : OuterMeasure α\ns : Set α\nms : ℝ≥0∞ := ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), m t\nhs : ¬ms = ∞\nthis✝ : ∀ r > ms, ∃ t, s ⊆ t ∧ MeasurableSet t ∧ m t < r\nt : ℕ → Set α\nhsub : ∀ (n : ℕ), s ⊆ t n\nhm : ∀ (n : ℕ), MeasurableSet (t n)\nhm' : ∀ (n : ℕ), m ... | rw [add_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 69
} | {
"line": 150,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nht : NullMeasurableSet t μ\n⊢ μ (s ∆ t) = μ (s \\ t) + μ (t \\ s)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Measure",
"Lattice.toS... | [] | simpa only [symmDiff_def, sup_eq_union]
using measure_union₀ (ht.diff hs) disjoint_sdiff_sdiff.aedisjoint | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 69
} | {
"line": 150,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nht : NullMeasurableSet t μ\n⊢ μ (s ∆ t) = μ (s \\ t) + μ (t \\ s)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Measure",
"Lattice.toS... | [] | simpa only [symmDiff_def, sup_eq_union]
using measure_union₀ (ht.diff hs) disjoint_sdiff_sdiff.aedisjoint | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 147,
"column": 2
} | {
"line": 148,
"column": 69
} | {
"line": 150,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns t : Set α\nhs : NullMeasurableSet s μ\nht : NullMeasurableSet t μ\n⊢ μ (s ∆ t) = μ (s \\ t) + μ (t \\ s)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Measure",
"Lattice.toS... | [] | simpa only [symmDiff_def, sup_eq_union]
using measure_union₀ (ht.diff hs) disjoint_sdiff_sdiff.aedisjoint | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 162,
"column": 44
} | {
"line": 165,
"column": 93
} | {
"line": 167,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns : Set β\nf : β → Set α\nhs : s.Countable\nhd : s.Pairwise (AEDisjoint μ on f)\nh : ∀ b ∈ s, NullMeasurableSet (f b) μ\n⊢ μ (⋃ b ∈ s, f b) = ∑' (p : ↑s), μ (f ↑p)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [] | by
haveI := hs.toEncodable
rw [biUnion_eq_iUnion]
exact measure_iUnion₀ (hd.on_injective Subtype.coe_injective fun x => x.2) fun x => h x x.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 48
} | {
"line": 185,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ns : Finset ι\nf : ι → Set α\nhd : (↑s).Pairwise (AEDisjoint μ on f)\nhm : ∀ b ∈ s, NullMeasurableSet (f b) μ\n⊢ μ (⋃ b ∈ s, f b) = ∑' (b : ↥s), μ (f ↑b)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Finset"... | [] | exact measure_biUnion₀ s.countable_toSet hd hm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 30
} | {
"line": 285,
"column": 30
} | [
{
"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 \\ s) < ε",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toL... | [
"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"
] | measure_sdiff hst hs hs' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 36
} | {
"line": 195,
"column": 4
} | [
{
"pp": "X : Type u_5\ninst✝¹ : MeasurableSpace X\nμ : Measure X\ninst✝ : IsFiniteMeasure μ\nEs : ℕ → Set X\nEs_mble : ∀ (i : ℕ), NullMeasurableSet (Es i) μ\nEs_disj : Pairwise fun n m ↦ Disjoint (Es n) (Es m)\ndecr : Antitone fun n ↦ ⋃ i, ⋃ (_ : i ≥ n), Es i\nx : X\nhx : ∀ (i : ℕ), ∃ i_1 ≥ i, x ∈ Es i_1\n⊢ x ∈... | [
"X : Type u_5\ninst✝¹ : MeasurableSpace X\nμ : Measure X\ninst✝ : IsFiniteMeasure μ\nEs : ℕ → Set X\nEs_mble : ∀ (i : ℕ), NullMeasurableSet (Es i) μ\nEs_disj : Pairwise fun n m ↦ Disjoint (Es n) (Es m)\ndecr : Antitone fun n ↦ ⋃ i, ⋃ (_ : i ≥ n), Es i\nx : X\nhx : ∀ (i : ℕ), ∃ i_1 ≥ i, x ∈ Es i_1\nj : ℕ\nleft✝ : j ... | obtain ⟨j, _, x_in_Es_j⟩ := hx 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 76
} | {
"line": 210,
"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 ≠ ∞\n⊢ |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure... | [
"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) ≠ ∞\n⊢ |(μ s).toReal - (μ t).toReal| ≤ (μ (s ∆ t)).toReal"
] | have hst : μ (s \ t) ≠ ∞ := (measure_lt_top_of_subset sdiff_subset hs').ne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 585,
"column": 6
} | {
"line": 585,
"column": 64
} | {
"line": 586,
"column": 4
} | [
{
"pp": "α : Type u_1\nι : Type u_5\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : Preorder ι\ninst✝¹ : IsCodirectedOrder ι\ninst✝ : atBot.IsCountablyGenerated\ns : ι → Set α\nhs : Monotone s\nhsm : ∀ (i : ι), NullMeasurableSet (s i) μ\nthis : Nonempty ι\nx : ℕ → ι\nhxm : Antitone x\nhx : Tendsto x atTop atBot... | [] | exact ⟨n, ne_top_of_le_ne_top hk <| measure_mono <| hs hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 46
} | {
"line": 489,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_2\nm0 : MeasurableSpace α\nμ ν : Measure α\nS T : Set (Set α)\nh_gen : m0 = generateFrom S\nh_inter : IsPiSystem S\nh_sub : T ⊆ S\nhc : T.Countable\nhU : ⋃₀ T = univ\nhtop : ∀ s ∈ T, μ s ≠ ∞\nh_eq : ∀ s ∈ S, μ s = ν s\nt : Set α\nht : t ∈ T\ns : Set α\nhs : s ∈ S\nH : (s ∩ t).Nonem... | [] | · exact h_eq _ (h_inter _ hs _ (h_sub ht) H) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 695,
"column": 37
} | {
"line": 695,
"column": 47
} | {
"line": 695,
"column": 47
} | [
{
"pp": "case refine_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : Group β\nf g : α → β\nh : f / g =ᵐ[μ] 1\nx : α\nhx : f x / g x = 1\n⊢ f x = g x",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivInvO... | [
"case refine_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : Group β\nf g : α → β\nh : f / g =ᵐ[μ] 1\nx : α\nhx : f x = g x\n⊢ f x = g x"
] | div_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 696,
"column": 37
} | {
"line": 696,
"column": 47
} | {
"line": 696,
"column": 47
} | [
{
"pp": "case refine_2\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : Group β\nf g : α → β\nh : f =ᵐ[μ] g\nx : α\nhx : f x = g x\n⊢ f x / g x = 1",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"InvOneClass.toOne",
... | [
"case refine_2\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : Group β\nf g : α → β\nh : f =ᵐ[μ] g\nx : α\nhx : f x = g x\n⊢ f x = g x"
] | div_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 702,
"column": 2
} | {
"line": 702,
"column": 55
} | {
"line": 703,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝² : Group β\ninst✝¹ : LE β\ninst✝ : MulRightMono β\nf g : α → β\nh : 1 ≤ᵐ[μ] g / f\na : α\nha : 1 a ≤ (g / f) a\n⊢ f a ≤ g a",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"instHDiv",... | [
"case refine_2\nα : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝² : Group β\ninst✝¹ : LE β\ninst✝ : MulRightMono β\nf g : α → β\nh : f ≤ᵐ[μ] g\na : α\nha : f a ≤ g a\n⊢ 1 a ≤ (g / f) a"
] | · rwa [Pi.one_apply, Pi.div_apply, one_le_div'] at ha | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1174,
"column": 10
} | {
"line": 1178,
"column": 52
} | {
"line": 1179,
"column": 8
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Type u_5\nR : Type u_6\nR' : Type u_7\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nm : Set (Measure α)\nx✝ : MeasurableSpace α\nf : ℕ → Set α\na✝¹ : ∀ (i : ℕ), M... | [] | if hne : (⋃ i, f i).Nonempty then
rw [OuterMeasure.top_apply hne]
exact le_top
else
simp_all [Set.not_nonempty_iff_eq_empty] | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacDepIfThenElse_1 | Lean.Parser.Tactic.tacDepIfThenElse |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 1202,
"column": 6
} | {
"line": 1203,
"column": 22
} | {
"line": 1204,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nt : Set α\nx✝ : μ (t ∩ s) + ν (tᶜ ∩ s) ∈ {m | ∃ t, m = μ (t ∩ s) + ν (tᶜ ∩ s)}\nt' : ℕ → Set α := fun n ↦ if n = 0 then t ∩ s else if n = 1 then tᶜ ∩ s else ∅\nht' : t' = fun n ↦ if n = 0 then t ∩ s else i... | [] | · refine mem_iUnion.2 ⟨1, ?_⟩
simp [hx, hxt] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 323,
"column": 77
} | {
"line": 326,
"column": 59
} | {
"line": 328,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns t u : Set α\ninst✝¹² : TopologicalSpace α\ninst✝¹¹ : MeasurableSpace α\ninst✝¹⁰ : OpensMeasurableSpace α\ninst✝⁹ : TopologicalSpace β\ninst✝⁸ : MeasurableSpace β\ninst✝⁷ : OpensMeasurableSpace β\ninst✝⁶ : TopologicalSp... | [] | by
rw [nhds, iInf_subtype']
refine @Filter.iInf_isMeasurablyGenerated α _ _ _ fun i => ?_
exact i.2.2.measurableSet.principal_isMeasurablyGenerated | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 86,
"column": 8
} | {
"line": 86,
"column": 19
} | {
"line": 86,
"column": 19
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\nu : α\n⊢ MeasurableSet (Ioi u)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"MeasurableSet",... | [
"case refine_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\nu : α\n⊢ MeasurableSet (Iic u)ᶜ"
] | ← compl_Iic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 318,
"column": 4
} | {
"line": 319,
"column": 50
} | {
"line": 320,
"column": 4
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns t : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x s + hausdorffEDist s t < ∞\nε0 : ↑ε / 2 ≠ 0\nthis✝ : infEDist x s < infEDist x s + ↑ε / 2\ny : α\nys : y ∈ s\ndxy : edist x y < infEDist x s + ↑ε / 2\nthis : hausdorffEDist s t < hausdorffEDist s t + ↑ε ... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\ns t : Set α\nε : ℝ≥0\nεpos : 0 < ε\nh : infEDist x s + hausdorffEDist s t < ∞\nε0 : ↑ε / 2 ≠ 0\nthis✝ : infEDist x s < infEDist x s + ↑ε / 2\ny : α\nys : y ∈ s\ndxy : edist x y < infEDist x s + ↑ε / 2\nthis : hausdorffEDist s t < hausdorffEDist s t + ↑ε / 2\nz : α\n... | obtain ⟨z : α, zt : z ∈ t, dyz : edist y z < hausdorffEDist s t + ↑ε / 2⟩ :=
exists_edist_lt_of_hausdorffEDist_lt ys this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 491,
"column": 4
} | {
"line": 492,
"column": 15
} | {
"line": 494,
"column": 0
} | [
{
"pp": "case refine_4\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMaxOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞\nh : ∀ ⦃a b : α⦄, a < b → μ (Ico... | [] | rintro _ ⟨l, u, hlt, rfl⟩
exact h hlt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 579,
"column": 4
} | {
"line": 579,
"column": 25
} | {
"line": 580,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝ : PseudoMetricSpace α\nδ : ℝ\nE : Set α\nhE : IsCompact E\nhδ : 0 ≤ δ\nhne : E.Nonempty\nx : α\nhx : x ∈ cthickening δ E\ny : α\nyE : y ∈ E\nhy : infEDist x E = edist x y\nD1 : edist x y ≤ ENNReal.ofReal δ\n⊢ dist x y ≤ δ",
"ppTerm": "?m.119",
"assigned": true,
"usedCons... | [
"α : Type u_2\ninst✝ : PseudoMetricSpace α\nδ : ℝ\nE : Set α\nhE : IsCompact E\nhδ : 0 ≤ δ\nhne : E.Nonempty\nx : α\nhx : x ∈ cthickening δ E\ny : α\nyE : y ∈ E\nhy : infEDist x E = edist x y\nD1 : ENNReal.ofReal (dist x y) ≤ ENNReal.ofReal δ\n⊢ dist x y ≤ δ"
] | rw [edist_dist] at D1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 491,
"column": 4
} | {
"line": 492,
"column": 15
} | {
"line": 494,
"column": 0
} | [
{
"pp": "case refine_4\nα : Type u_5\ninst✝⁵ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : BorelSpace α\ninst✝ : NoMaxOrder α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a < b → μ (Ico a b) ≠ ∞\nh : ∀ ⦃a b : α⦄, a < b → μ (Ico... | [] | rintro _ ⟨l, u, hlt, rfl⟩
exact h hlt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 38
} | {
"line": 137,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : Set α\nhs : ∃ R > 0, μ (cthickening R s) ≠ ∞\n⊢ Tendsto (fun r ↦ μ (cthickening r s)) (𝓝[>] 0)\n (𝓝 (μ (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s)))",
... | [
"α : Type u_1\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : Set α\nhs : ∃ R > 0, μ (cthickening R s) ≠ ∞\n⊢ Tendsto (fun r ↦ μ (cthickening r s)) (𝓝[>] 0) (𝓝 (μ (⋂ δ, ⋂ (_ : 0 < δ), cthickening δ s)))"
] | rw [closure_eq_iInter_cthickening] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.Thickening | {
"line": 708,
"column": 6
} | {
"line": 708,
"column": 36
} | {
"line": 708,
"column": 37
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nδ : ℝ\nhδ : 0 < δ\nhs : thickening δ s = s\n⊢ IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"closure_subset_iff_isClosed"... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nδ : ℝ\nhδ : 0 < δ\nhs : thickening δ s = s\n⊢ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] s ⊆ s"
] | ← closure_subset_iff_isClosed, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 564,
"column": 4
} | {
"line": 565,
"column": 15
} | {
"line": 567,
"column": 0
} | [
{
"pp": "case refine_4\nα : Type u_5\ninst✝⁴ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : BorelSpace α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞\nh : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)\ns ... | [] | rintro _ ⟨l, u, hlt, rfl⟩
exact h hlt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 564,
"column": 4
} | {
"line": 565,
"column": 15
} | {
"line": 567,
"column": 0
} | [
{
"pp": "case refine_4\nα : Type u_5\ninst✝⁴ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : BorelSpace α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞\nh : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)\ns ... | [] | rintro _ ⟨l, u, hlt, rfl⟩
exact h hlt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Prod | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 41
} | {
"line": 54,
"column": 4
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set α\nhs : s ∈ C\nt : Set β\nht : t ∈ D\n⊢ MeasurableSet s",
"ppTerm": "?m.253",
"assigned": true,
"usedConstants": [
"MeasurableSpace.measurableSet_generateFrom"... | [
"α : Type u_3\nβ : Type u_4\nC : Set (Set α)\nD : Set (Set β)\nhC : IsCountablySpanning C\nhD : IsCountablySpanning D\ns : Set α\nhs : s ∈ C\nt : Set β\nht : t ∈ D\n⊢ MeasurableSet t"
] | · exact measurableSet_generateFrom hs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 707,
"column": 55
} | {
"line": 707,
"column": 72
} | {
"line": 707,
"column": 73
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (... | [
"α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhg : ∀... | exists_range_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 750,
"column": 2
} | {
"line": 751,
"column": 94
} | {
"line": 752,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\nμ : Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf ... | [
"case inr\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\nμ : Measure δ\ninst✝ : Countable ι\nf : ι → δ → α\ng : δ → α\nhf : ∀ (i : ι),... | · simp only [IsEmpty.exists_iff, setOf_false, isLUB_empty_iff] at hg
exact aemeasurable_const' (hg.mono fun a ha => hg.mono fun b hb => (ha _).antisymm (hb _)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 912,
"column": 2
} | {
"line": 913,
"column": 35
} | {
"line": 914,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (... | [
"case inr\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Meas... | have A : MeasurableSet {b | BddAbove (range (fun i ↦ f i b))} :=
measurableSet_bddAbove_range hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.BorelSpace.Real | {
"line": 77,
"column": 7
} | {
"line": 77,
"column": 18
} | {
"line": 77,
"column": 18
} | [
{
"pp": "case refine_1\nq : ℚ\n⊢ MeasurableSet (Ioi ↑q)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"MeasurableSet",
"congrArg",
"Compl.compl",
"Real.instRatCast",
"Rat",
"PartialOrder.toPreorder",
... | [
"case refine_1\nq : ℚ\n⊢ MeasurableSet (Iic ↑q)ᶜ"
] | ← compl_Iic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 142,
"column": 2
} | {
"line": 145,
"column": 14
} | {
"line": 146,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : PseudoEMetricSpace α\ninst✝⁴ : OpensMeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : Zero α\ninst✝¹ : Preorder α\nf : β → α\nhf : 0 ≤ f\nhfm : Measurable f\ninst✝ : SeparableSpace ↑(range f ∪ {0})\nn : ℕ\n⊢ 0 ≤ approxOn f hfm (rang... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : PseudoEMetricSpace α\ninst✝⁴ : OpensMeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : Zero α\ninst✝¹ : Preorder α\nf : β → α\nhf : 0 ≤ f\nhfm : Measurable f\ninst✝ : SeparableSpace ↑(range f ∪ {0})\nn : ℕ\nthis : range f ∪ {0} ⊆ Ici 0\n⊢ 0 ≤ ap... | have : range f ∪ {0} ⊆ Set.Ici 0 := by
simp only [Set.union_singleton, Set.insert_subset_iff, Set.mem_Ici, le_refl, true_and]
rintro - ⟨x, rfl⟩
exact hf x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 143,
"column": 33
} | {
"line": 160,
"column": 30
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetrizableSpace α\ninst✝ : OpensMeasurableSpace α\ns : Set α\nh1 : IsSeparable s\nh2 : μ sᶜ = 0\n⊢ AEStronglyMeasurable id μ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Meas... | [] | by
nontriviality α
obtain ⟨a, -⟩ := exists_pair_ne α
classical
refine ⟨(closure s).piecewise id (fun _ ↦ a), ?_,
Filter.mem_of_superset h2 (fun x hx ↦ by simp [subset_closure hx])⟩
have h : StronglyMeasurable ((↑) : closure s → α) := by
have := h1.closure.secondCountableTopology
exact continuous_s... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 427,
"column": 2
} | {
"line": 432,
"column": 29
} | {
"line": 434,
"column": 0
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl : List (α → M)\nhl : ∀ f ∈ l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable l.prod μ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [] | induction l with
| nil => exact aestronglyMeasurable_one
| cons f l ihl =>
rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 427,
"column": 2
} | {
"line": 432,
"column": 29
} | {
"line": 434,
"column": 0
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl : List (α → M)\nhl : ∀ f ∈ l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable l.prod μ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [] | induction l with
| nil => exact aestronglyMeasurable_one
| cons f l ihl =>
rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 427,
"column": 2
} | {
"line": 432,
"column": 29
} | {
"line": 434,
"column": 0
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\nM : Type u_5\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nl : List (α → M)\nhl : ∀ f ∈ l, AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable l.prod μ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [] | induction l with
| nil => exact aestronglyMeasurable_one
| cons f l ihl =>
rw [List.forall_mem_cons] at hl
rw [List.prod_cons]
exact hl.1.mul (ihl hl.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 372,
"column": 2
} | {
"line": 391,
"column": 43
} | {
"line": 393,
"column": 0
} | [
{
"pp": "case right\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nh : ∫⁻ (x : α), f x ∂μ ≠ ∞\nε : ℝ≥0∞\nhε : ε ≠ 0\nε₂ : ℝ≥0∞\nhε₂0 : 0 < ε₂\nhε₂ε : ε₂ < ε\nε₁ : ℝ≥0∞\nhε₁0 : 0 < ε₁\nhε₁₂ : ε₁ < ε₂\nφ : α →ₛ ℝ≥0\nleft✝ : ∀ (x : α), ↑(φ x) ≤ f x\nhφ : ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(ψ x) ≤ f... | [] | calc
(map (↑) ψ).lintegral (μ.restrict s) ≤
(map (↑) φ).lintegral (μ.restrict s) + (map (↑) (ψ - φ)).lintegral (μ.restrict s) := by
rw [← SimpleFunc.add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add]
refine SimpleFunc.lintegral_mono (fun x => ?_) le_rfl
simp only [add_tsub_eq_max, le_ma... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 97
} | {
"line": 269,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\nS : ℕ → Set α := spanningSets (μ... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝¹ : TopologicalSpace β\ninst✝ : Zero β\nm : MeasurableSpace α\nμ : Measure α\nhf_meas : StronglyMeasurable f\nt : Set α\nht : MeasurableSet t\nhft_zero : ∀ x ∈ tᶜ, f x = 0\nhtμ this : SigmaFinite (μ.restrict t)\nS : ℕ → Set α := spanningSets (μ.restrict t)... | refine fun n => measure_biUnion_lt_top {y ∈ (fs n).range | y ≠ 0}.finite_toSet fun y hy => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 85
} | {
"line": 640,
"column": 0
} | [
{
"pp": "case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf g : α → ℝ≥0∞\ninst✝ : (j : α) → Decidable (j ∈ s)\n⊢ ∫⁻ (x : α) in sᶜ, s.piecewise f g x ∂μ = ∫⁻ (a : α) in sᶜ, g a ∂μ",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Set.piecew... | [] | exact setLIntegral_congr_fun hs.compl <| fun _ ↦ Set.piecewise_eq_of_notMem _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 85
} | {
"line": 640,
"column": 0
} | [
{
"pp": "case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf g : α → ℝ≥0∞\ninst✝ : (j : α) → Decidable (j ∈ s)\n⊢ ∫⁻ (x : α) in sᶜ, s.piecewise f g x ∂μ = ∫⁻ (a : α) in sᶜ, g a ∂μ",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Set.piecew... | [] | exact setLIntegral_congr_fun hs.compl <| fun _ ↦ Set.piecewise_eq_of_notMem _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 85
} | {
"line": 640,
"column": 0
} | [
{
"pp": "case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nf g : α → ℝ≥0∞\ninst✝ : (j : α) → Decidable (j ∈ s)\n⊢ ∫⁻ (x : α) in sᶜ, s.piecewise f g x ∂μ = ∫⁻ (a : α) in sᶜ, g a ∂μ",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Set.piecew... | [] | exact setLIntegral_congr_fun hs.compl <| fun _ ↦ Set.piecewise_eq_of_notMem _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 15
} | {
"line": 119,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure ν\nn : ℕ\nhC_lt : 1 / ↑n < ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s\nh_lt_top : ⨆ s, ⨆ (_ : MeasurableSet s), ⨆ (_ : SigmaFinite (μ.restrict s)), ν s < ∞\nt : Set α\nht_meas : Measurabl... | [] | exact ht.le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 955,
"column": 4
} | {
"line": 955,
"column": 64
} | {
"line": 957,
"column": 0
} | [
{
"pp": "case pcw\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : Nonempty β\ninst✝ : TopologicalSpace β\nP : (f : α → β) → StronglyMeasurable f → Prop\nconst : ∀ (c : β), P (fun x ↦ c) ⋯\npcw :\n ∀ ⦃f g : α → β⦄ {s : Set α} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) (hs : Measur... | [] | exact pcw f.stronglyMeasurable g.stronglyMeasurable hs Pf Pg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 461,
"column": 4
} | {
"line": 465,
"column": 37
} | {
"line": 466,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set α⦄,\n MeasurableSet s → ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ.trim hm = ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ",
"ppTerm": "?refine_1",
"assigned":... | [] | intro c s hs
rw [lintegral_indicator hs, lintegral_indicator (hm s hs), setLIntegral_const,
setLIntegral_const]
suffices h_trim_s : μ.trim hm s = μ s by rw [h_trim_s]
exact trim_measurableSet_eq hm hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 461,
"column": 4
} | {
"line": 465,
"column": 37
} | {
"line": 466,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set α⦄,\n MeasurableSet s → ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ.trim hm = ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ",
"ppTerm": "?refine_1",
"assigned":... | [] | intro c s hs
rw [lintegral_indicator hs, lintegral_indicator (hm s hs), setLIntegral_const,
setLIntegral_const]
suffices h_trim_s : μ.trim hm s = μ s by rw [h_trim_s]
exact trim_measurableSet_eq hm hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1212,
"column": 2
} | {
"line": 1212,
"column": 87
} | {
"line": 1214,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ μ (support ⇑(hf.approx n) ∪ support ⇑(hg.approx n)) ... | [] | exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1221,
"column": 2
} | {
"line": 1221,
"column": 87
} | {
"line": 1223,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf g : α → β\ninst✝³ : TopologicalSpace β\ninst✝² : Zero β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nhf : FinStronglyMeasurable f μ\nhg : FinStronglyMeasurable g μ\nn : ℕ\n⊢ μ (support ⇑(hf.approx n) ∪ support ⇑(hg.approx n)) ... | [] | exact measure_union_lt_top_iff.mpr ⟨hf.fin_support_approx n, hg.fin_support_approx n⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 34
} | {
"line": 86,
"column": 0
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nh : x ≠ y\n⊢ x (Nat.find ⋯) ≠ y (Nat.find ⋯)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"Classical.propDecidable",
"Exists",
"Ne",
"Nat",
"Function.ne_iff",
"Iff.mp",
"E... | [] | exact Nat.find_spec (ne_iff.1 h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 34
} | {
"line": 86,
"column": 0
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nh : x ≠ y\n⊢ x (Nat.find ⋯) ≠ y (Nat.find ⋯)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"Classical.propDecidable",
"Exists",
"Ne",
"Nat",
"Function.ne_iff",
"Iff.mp",
"E... | [] | exact Nat.find_spec (ne_iff.1 h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 34
} | {
"line": 86,
"column": 0
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nh : x ≠ y\n⊢ x (Nat.find ⋯) ≠ y (Nat.find ⋯)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"Classical.propDecidable",
"Exists",
"Ne",
"Nat",
"Function.ne_iff",
"Iff.mp",
"E... | [] | exact Nat.find_spec (ne_iff.1 h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Polish | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 73
} | {
"line": 192,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝ : MetricSpace α\ns : Opens α\nx✝ : s.CompleteCopy\n⊢ dist x✝ x✝ = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Real.la... | [
"case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝ : MetricSpace α\ns : Opens α\nx✝¹ x✝ : s.CompleteCopy\n⊢ dist x✝¹ x✝ = dist x✝ x✝¹",
"case refine_3\nα : Type u_1\nβ : Type u_2\ninst✝ : MetricSpace α\ns : Opens α\nx y z : s.CompleteCopy\n⊢ dist x z ≤ dist x y + dist y z",
"case refine_4\nα : Type u_1\nβ : Type... | · simp only [dist_eq, dist_self, one_div, sub_self, abs_zero, add_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UnitInterval | {
"line": 374,
"column": 6
} | {
"line": 374,
"column": 29
} | {
"line": 375,
"column": 6
} | [
{
"pp": "α : Type ?u.3\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\nh : a ≤ b\nδ : α\ns t : ↑unitInterval\n⊢ ↑s * (1 - ↑t) / (1 - ↑s * ↑t) ≤ 1",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"div_le_one_of_le₀",
"Real.partialOrder... | [
"case h\nα : Type ?u.3\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\nh : a ≤ b\nδ : α\ns t : ↑unitInterval\n⊢ ↑s * (1 - ↑t) ≤ 1 - ↑s * ↑t",
"case hb\nα : Type ?u.3\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\na b c d : α\nh : a ≤ b\... | apply div_le_one_of_le₀ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.UnitInterval | {
"line": 443,
"column": 4
} | {
"line": 443,
"column": 27
} | {
"line": 443,
"column": 28
} | [
{
"pp": "case neg\na b : ℝ\nx y z : ↑(Icc a b)\nh : ¬↑z - ↑x = 0\nhxy : ↑x ≤ ↑y\nhyz : ↑y ≤ ↑z\n⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"div_le_one_of_le₀",
"Real.partialOrder",
"Real",
"MulZeroClass.toMul",
"IsStrictOrdered... | [
"case neg.h\na b : ℝ\nx y z : ↑(Icc a b)\nh : ¬↑z - ↑x = 0\nhxy : ↑x ≤ ↑y\nhyz : ↑y ≤ ↑z\n⊢ ↑y - ↑x ≤ ↑z - ↑x",
"case hb\na b : ℝ\nx y z : ↑(Icc a b)\nh : ¬↑z - ↑x = 0\nhxy : ↑x ≤ ↑y\nhyz : ↑y ≤ ↑z\n⊢ 0 ≤ ↑z - ↑x"
] | apply div_le_one_of_le₀ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 47
} | {
"line": 642,
"column": 0
} | [
{
"pp": "case mk\nX : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\na✝¹ : InductiveLimit I\nn : ℕ\nx : X n\na✝ : Quot.mk ⇑(inseparableSetoid ((n : ℕ) × X n)) ⟨n, x⟩ ∈ univ\n⊢ Quot.mk ⇑(inseparableSetoid ((n : ℕ) × X n)) ⟨n, x⟩ ∈ ⋃ i, range (toIndu... | [] | refine mem_iUnion.2 ⟨n, mem_range.2 ⟨x, rfl⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 517,
"column": 6
} | {
"line": 517,
"column": 20
} | {
"line": 517,
"column": 21
} | [
{
"pp": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nx y : (n : ℕ) → E n\nhx : x ∉ s\nhy : y ∈ s\n⊢ firstDiff x y ≤ longestPrefix x s",
"ppTerm": "?m.19",
"assigned": true,
"use... | [
"E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nx y : (n : ℕ) → E n\nhx : x ∉ s\nhy : y ∈ s\n⊢ firstDiff x y ≤ shortestPrefixDiff x s - 1"
] | longestPrefix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 117,
"column": 37
} | {
"line": 117,
"column": 54
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case false.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : ... | [] | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 117,
"column": 37
} | {
"line": 117,
"column": 54
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case false.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {... | [
"case false.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C : Set α} →... | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 528,
"column": 6
} | {
"line": 528,
"column": 20
} | {
"line": 528,
"column": 21
} | [
{
"pp": "case neg\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∉ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : longestPrefix x s < shortestPrefixDif... | [
"case neg\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nx : (n : ℕ) → E n\nhx : x ∉ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : shortestPrefixDiff x s - 1 < shortestPrefixDiff x... | longestPrefix, | Lean.Elab.Tactic.evalRewriteSeq | null |
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