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 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 432,
"column": 34
} | {
"line": 432,
"column": 61
} | {
"line": 432,
"column": 61
} | [
{
"pp": "X : Type u_1\nY✝ : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set (OnePoint X)\nY : Type u_3\ninst✝ : TopologicalSpace Y\nf : ℕ → Y\ny : Y\nh : Tendsto f atTop (𝓝 y)\n⊢ Tendsto f cofinite (𝓝 y)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 432,
"column": 34
} | {
"line": 432,
"column": 61
} | {
"line": 432,
"column": 61
} | [
{
"pp": "X : Type u_1\nY✝ : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set (OnePoint X)\nY : Type u_3\ninst✝ : TopologicalSpace Y\nf : ℕ → Y\ny : Y\nh : Tendsto f atTop (𝓝 y)\n⊢ Tendsto f cofinite (𝓝 y)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 432,
"column": 34
} | {
"line": 432,
"column": 61
} | {
"line": 432,
"column": 61
} | [
{
"pp": "X : Type u_1\nY✝ : Type u_2\ninst✝¹ : TopologicalSpace X\ns : Set (OnePoint X)\nY : Type u_3\ninst✝ : TopologicalSpace Y\nf : ℕ → Y\ny : Y\nh : Tendsto f atTop (𝓝 y)\n⊢ Tendsto f cofinite (𝓝 y)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 442,
"column": 35
} | {
"line": 442,
"column": 62
} | {
"line": 442,
"column": 62
} | [
{
"pp": "X : Type u_1\nY✝ : Type u_2\ninst✝² : TopologicalSpace X\ns : Set (OnePoint X)\nY : Type u_3\ninst✝¹ : TopologicalSpace Y\ninst✝ : T2Space Y\nx✝ : { f // ∃ L, Tendsto (fun x ↦ f x) atTop (𝓝 L) }\nf : ℕ → Y\nhf : ∃ L, Tendsto (fun x ↦ f x) atTop (𝓝 L)\n⊢ ∃ L, Tendsto (fun x ↦ f x) cofinite (𝓝 L)",
... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 442,
"column": 35
} | {
"line": 442,
"column": 62
} | {
"line": 442,
"column": 62
} | [
{
"pp": "X : Type u_1\nY✝ : Type u_2\ninst✝² : TopologicalSpace X\ns : Set (OnePoint X)\nY : Type u_3\ninst✝¹ : TopologicalSpace Y\ninst✝ : T2Space Y\nx✝ : { f // ∃ L, Tendsto (fun x ↦ f x) atTop (𝓝 L) }\nf : ℕ → Y\nhf : ∃ L, Tendsto (fun x ↦ f x) atTop (𝓝 L)\n⊢ ∃ L, Tendsto (fun x ↦ f x) cofinite (𝓝 L)",
... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 442,
"column": 35
} | {
"line": 442,
"column": 62
} | {
"line": 442,
"column": 62
} | [
{
"pp": "X : Type u_1\nY✝ : Type u_2\ninst✝² : TopologicalSpace X\ns : Set (OnePoint X)\nY : Type u_3\ninst✝¹ : TopologicalSpace Y\ninst✝ : T2Space Y\nx✝ : { f // ∃ L, Tendsto (fun x ↦ f x) atTop (𝓝 L) }\nf : ℕ → Y\nhf : ∃ L, Tendsto (fun x ↦ f x) atTop (𝓝 L)\n⊢ ∃ L, Tendsto (fun x ↦ f x) cofinite (𝓝 L)",
... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 129,
"column": 10
} | {
"line": 129,
"column": 12
} | {
"line": 130,
"column": 2
} | [
{
"pp": "f : ℂ → ℂ\nh : meromorphicOrderAt f 0 < 0\na : ℂ\n⊢ a ∈ sphere 0 |1| →\n (fun a ↦ log ‖meromorphicTrailingCoeffAt (fun x ↦ f x - a) 0‖) a = (fun x ↦ log ‖meromorphicTrailingCoeffAt f 0‖) a",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRin... | [
"f : ℂ → ℂ\nh : meromorphicOrderAt f 0 < 0\na : ℂ\nha : a ∈ sphere 0 |1|\n⊢ (fun a ↦ log ‖meromorphicTrailingCoeffAt (fun x ↦ f x - a) 0‖) a = (fun x ↦ log ‖meromorphicTrailingCoeffAt f 0‖) a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 184,
"column": 14
} | {
"line": 184,
"column": 16
} | {
"line": 185,
"column": 6
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nhR : R ≠ 0\na : ℂ\n⊢ a ∈ sphere 0 |1| →\n (fun a ↦ circleAverage (fun x ↦ log ‖f x - a‖) 0 R + logCounting f ⊤ R) a =\n ((fun x ↦ logCounting f (↑x) R) + fun a ↦ log ‖meromorphicTrailingCoeffAt (fun x ↦ f x - a) 0‖) a",
"ppTerm": "?m.198",
"assigned"... | [
"f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nhR : R ≠ 0\na : ℂ\nha : a ∈ sphere 0 |1|\n⊢ (fun a ↦ circleAverage (fun x ↦ log ‖f x - a‖) 0 R + logCounting f ⊤ R) a =\n ((fun x ↦ logCounting f (↑x) R) + fun a ↦ log ‖meromorphicTrailingCoeffAt (fun x ↦ f x - a) 0‖) a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 90
} | {
"line": 187,
"column": 0
} | [
{
"pp": "f : ℂ → ℂ\nR : ℝ\nh : Meromorphic f\nhR : R ≠ 0\na : ℂ\nha : a ∈ sphere 0 |1|\n⊢ (fun a ↦ circleAverage (fun x ↦ log ‖f x - a‖) 0 R + logCounting f ⊤ R) a =\n ((fun x ↦ logCounting f (↑x) R) + fun a ↦ log ‖meromorphicTrailingCoeffAt (fun x ↦ f x - a) 0‖) a",
"ppTerm": "?m.200",
"assigned": t... | [] | simp [logCounting_add_log_trailingCoeff_eq_circleAverage_add_logCounting_top h hR a] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 217,
"column": 10
} | {
"line": 217,
"column": 12
} | {
"line": 217,
"column": 13
} | [
{
"pp": "f : ℂ → ℂ\nh : Meromorphic f\na : ℝ\n⊢ a ∈ Ioi 0 → ∀ ⦃b : ℝ⦄, b ∈ Ioi 0 → a ≤ b → characteristic f ⊤ a ≤ characteristic f ⊤ b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Real.instZero",
"Membership.mem",
"Zero.toOfNat0",
... | [
"f : ℂ → ℂ\nh : Meromorphic f\na : ℝ\nha : a ∈ Ioi 0\n⊢ ∀ ⦃b : ℝ⦄, b ∈ Ioi 0 → a ≤ b → characteristic f ⊤ a ≤ characteristic f ⊤ b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 166,
"column": 10
} | {
"line": 166,
"column": 12
} | {
"line": 166,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nhD : 0 ≤ D\na : ℝ\n⊢ a ∈ Ioi 0 → ∀ ⦃b : ℝ⦄, b ∈ Ioi 0 → a ≤ b → logCounting D a ≤ logCounting D b",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Real.... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\nhD : 0 ≤ D\na : ℝ\nha : a ∈ Ioi 0\n⊢ ∀ ⦃b : ℝ⦄, b ∈ Ioi 0 → a ≤ b → logCounting D a ≤ logCounting D b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.ConstantSpeed | {
"line": 98,
"column": 56
} | {
"line": 98,
"column": 93
} | {
"line": 98,
"column": 93
} | [
{
"pp": "case mpr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nh : LocallyBoundedVariationOn f s ∧ ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → variationOnFromTo f s x y = ↑l * (y - x)\nx : ℝ\nxs : x ∈ s\ny : ℝ\nys : y ∈ s\nxy : x ≤ y\n⊢ eVariationOn f (s ∩ Icc x y) = ENNReal.ofReal (eVa... | [
"case mpr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl : ℝ≥0\nh : LocallyBoundedVariationOn f s ∧ ∀ ⦃x : ℝ⦄, x ∈ s → ∀ ⦃y : ℝ⦄, y ∈ s → variationOnFromTo f s x y = ↑l * (y - x)\nx : ℝ\nxs : x ∈ s\ny : ℝ\nys : y ∈ s\nxy : x ≤ y\n⊢ eVariationOn f (s ∩ Icc x y) = eVariationOn f (s ∩ Icc x y)"
] | ENNReal.ofReal_toReal (h.1 x y xs ys) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic | {
"line": 69,
"column": 10
} | {
"line": 69,
"column": 12
} | {
"line": 70,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsuppWithin univ ℤ\nh : 0 ≤ D\nh₁ : ¬D = 0\ne : E\nhe : single e 1 ≤ D\na : ℝ\n⊢ a > 0 → ∃ᶠ (x : ℝ) in atTop, ‖1 x‖ < a * ‖logCounting D x‖",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Real"... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nD : locallyFinsuppWithin univ ℤ\nh : 0 ≤ D\nh₁ : ¬D = 0\ne : E\nhe : single e 1 ≤ D\na : ℝ\nha : a > 0\n⊢ ∃ᶠ (x : ℝ) in atTop, ‖1 x‖ < a * ‖logCounting D x‖"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 208,
"column": 12
} | {
"line": 208,
"column": 14
} | {
"line": 208,
"column": 15
} | [
{
"pp": "case hf\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\n⊢ a ∈ Ioi ‖e‖ → ∀ ⦃b : ℝ⦄, b ∈ Ioi ‖e‖ → a < b → logCounting (single e 1) a < logCounting (single e 1) b",
"ppTerm": "?hf",
"assigned":... | [
"case hf\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\nha : a ∈ Ioi ‖e‖\n⊢ ∀ ⦃b : ℝ⦄, b ∈ Ioi ‖e‖ → a < b → logCounting (single e 1) a < logCounting (single e 1) b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 213,
"column": 12
} | {
"line": 213,
"column": 14
} | {
"line": 213,
"column": 15
} | [
{
"pp": "case hg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\n⊢ a ∈ Ioi ‖e‖ → ∀ ⦃b : ℝ⦄, b ∈ Ioi ‖e‖ → a ≤ b → logCounting (D - single e 1) a ≤ logCounting (D - single e 1) b",
"ppTerm": "?hg",
"as... | [
"case hg\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : DecidableEq E\ninst✝ : ProperSpace E\nD : locallyFinsupp E ℤ\ne : E\nhD : single e 1 ≤ D\na : ℝ\nha : a ∈ Ioi ‖e‖\n⊢ ∀ ⦃b : ℝ⦄, b ∈ Ioi ‖e‖ → a ≤ b → logCounting (D - single e 1) a ≤ logCounting (D - single e 1) b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 54
} | {
"line": 398,
"column": 4
} | [
{
"pp": "case neg.e_6\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfg : f =ᶠ[codiscrete ℂ] g\na : WithTop E\nh : ¬a = ⊤\n⊢ divisor (fun x ↦ f x - a.untop₀) univ = divisor (fun x ↦ g x - a.untop₀) univ",
"ppTerm": "?neg.e_6✝",
"assigned": true,
"usedConstants":... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfg : f =ᶠ[codiscrete ℂ] g\na : WithTop E\nh : ¬a = ⊤\n⊢ (fun x ↦ f x - a.untop₀) =ᶠ[codiscreteWithin univ] fun x ↦ g x - a.untop₀"
] | apply divisor_congr_codiscreteWithin _ isOpen_univ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 584,
"column": 2
} | {
"line": 590,
"column": 19
} | {
"line": 592,
"column": 0
} | [
{
"pp": "R : ℝ\nf : ℂ → ℂ\nh : Meromorphic f\nhR : R ≠ 0\n⊢ logCounting (divisor f univ) R = circleAverage (fun x ↦ log ‖f x‖) 0 R - log ‖meromorphicTrailingCoeffAt f 0‖",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"Norm.norm",
"Int.cast",
"... | [] | have h₁f : MeromorphicOn f (closedBall 0 |R|) := by tauto
simp only [MeromorphicOn.circleAverage_log_norm hR h₁f, logCounting, AddMonoidHom.coe_mk,
ZeroHom.coe_mk, zero_sub, norm_neg, add_sub_cancel_right]
congr 1
· simp_all
· rw [divisor_apply, divisor_apply]
all_goals aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 584,
"column": 2
} | {
"line": 590,
"column": 19
} | {
"line": 592,
"column": 0
} | [
{
"pp": "R : ℝ\nf : ℂ → ℂ\nh : Meromorphic f\nhR : R ≠ 0\n⊢ logCounting (divisor f univ) R = circleAverage (fun x ↦ log ‖f x‖) 0 R - log ‖meromorphicTrailingCoeffAt f 0‖",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"Norm.norm",
"Int.cast",
"... | [] | have h₁f : MeromorphicOn f (closedBall 0 |R|) := by tauto
simp only [MeromorphicOn.circleAverage_log_norm hR h₁f, logCounting, AddMonoidHom.coe_mk,
ZeroHom.coe_mk, zero_sub, norm_neg, add_sub_cancel_right]
congr 1
· simp_all
· rw [divisor_apply, divisor_apply]
all_goals aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Triplewise | {
"line": 90,
"column": 20
} | {
"line": 90,
"column": 41
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case nil\nα : Type u_1\nl₁ l₂ : List α\np : α → α → α → Prop\n⊢ Triplewise p ([] ++ l₂) ↔\n Triplewise p [] ∧\n Triplewise p l₂ ∧ (∀ (a : α), a ∈ [] → Pairwise (p a) l₂) ∧ ∀ (a : α), a ∈ l₂ → Pairwise (fun x y ↦ p x y a) []",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
... | [] | grind [pairwise_cons] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Data.List.Triplewise | {
"line": 90,
"column": 20
} | {
"line": 90,
"column": 41
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nl₁ l₂ : List α\np : α → α → α → Prop\nhead✝ : α\ntail✝ : List α\ntail_ih✝ :\n Triplewise p (tail✝ ++ l₂) ↔\n Triplewise p tail✝ ∧\n Triplewise p l₂ ∧\n (∀ (a : α), a ∈ tail✝ → Pairwise (p a) l₂) ∧ ∀ (a : α), a ∈ l₂ → Pairwise (fun x y ↦ p x y a) tail✝\n⊢ Triplew... | [] | grind [pairwise_cons] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.Convex.BetweenList | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 55
} | {
"line": 126,
"column": 6
} | [
{
"pp": "case cons.refine_2\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead : P\ntail : List P\nih : List.Wbtw R tail ∧ Pairwise (fun x1 x2 ↦ x1 ≠ x2) tail ↔ Triplewise (Sbtw ... | [
"case cons.refine_2\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead : P\ntail : List P\nih : List.Wbtw R tail ∧ Pairwise (fun x1 x2 ↦ x1 ≠ x2) tail ↔ Triplewise (Sbtw R) tail ∧ ∀ ... | have ht' : tail.Wbtw R := ht.imp _root_.Sbtw.wbtw | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Hall.Finite | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 8
} | {
"line": 135,
"column": 0
} | [
{
"pp": "α : Type v\ninst✝ : DecidableEq α\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\ns' : Finset ↑↑s\n⊢ #s' ≤ #(s'.biUnion fun a' ↦ t ↑a')",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Iff.of_eq",
... | [] | classical
rw [← card_image_of_injective s' Subtype.coe_injective]
convert! ht (s'.image fun z => z.1) using 1
apply congr_arg
ext y
simp | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.Hall.Finite | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 8
} | {
"line": 135,
"column": 0
} | [
{
"pp": "α : Type v\ninst✝ : DecidableEq α\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\ns' : Finset ↑↑s\n⊢ #s' ≤ #(s'.biUnion fun a' ↦ t ↑a')",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Iff.of_eq",
... | [] | classical
rw [← card_image_of_injective s' Subtype.coe_injective]
convert! ht (s'.image fun z => z.1) using 1
apply congr_arg
ext y
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Hall.Finite | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 8
} | {
"line": 135,
"column": 0
} | [
{
"pp": "α : Type v\ninst✝ : DecidableEq α\nι : Type u\nt : ι → Finset α\ns : Finset ι\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\ns' : Finset ↑↑s\n⊢ #s' ≤ #(s'.biUnion fun a' ↦ t ↑a')",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Iff.of_eq",
... | [] | classical
rw [← card_image_of_injective s' Subtype.coe_injective]
convert! ht (s'.image fun z => z.1) using 1
apply congr_arg
ext y
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.BetweenList | {
"line": 223,
"column": 22
} | {
"line": 223,
"column": 24
} | {
"line": 224,
"column": 14
} | [
{
"pp": "case neg.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\nl'' : List R\nhl''0 : ∀ a ∈ l'', 0 ≤ a\nhl''s : l''.SortedLE\ntail : List P\nht2 : ¬tail ... | [
"case neg.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\nl'' : List R\nhl''0 : ∀ a ∈ l'', 0 ≤ a\nhl''s : l''.SortedLE\ntail : List P\nht2 : ¬tail = []\nr : R\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.BetweenList | {
"line": 229,
"column": 24
} | {
"line": 229,
"column": 26
} | {
"line": 230,
"column": 16
} | [
{
"pp": "case neg.refine_2.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\nl'' : List R\nhl''0 : ∀ a ∈ l'', 0 ≤ a\nhl''s : l''.SortedLE\ntail : List P\nht2... | [
"case neg.refine_2.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\nl'' : List R\nhl''0 : ∀ a ∈ l'', 0 ≤ a\nhl''s : l''.SortedLE\ntail : List P\nht2 : ¬tail = [... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.Hall.Finite | {
"line": 254,
"column": 8
} | {
"line": 254,
"column": 18
} | {
"line": 254,
"column": 19
} | [
{
"pp": "case mpr\nι : Type u_1\nα : Type u_2\ninst✝¹ : Finite ι\ninst✝ : DecidableEq α\nt : ι → Finset α\nf : ι → α\nhf₁ : Function.Injective f\nhf₂ : ∀ (x : ι), f x ∈ t x\ns : Finset ι\nx✝ : α\n⊢ x✝ ∈ image f s → x✝ ∈ s.biUnion t",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case mpr\nι : Type u_1\nα : Type u_2\ninst✝¹ : Finite ι\ninst✝ : DecidableEq α\nt : ι → Finset α\nf : ι → α\nhf₁ : Function.Injective f\nhf₂ : ∀ (x : ι), f x ∈ t x\ns : Finset ι\nx✝ : α\n⊢ (∃ a ∈ s, f a = x✝) → x✝ ∈ s.biUnion t"
] | mem_image, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.PEquiv | {
"line": 377,
"column": 10
} | {
"line": 377,
"column": 24
} | {
"line": 377,
"column": 24
} | [
{
"pp": "case some\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf g : α ≃. β\nfg : ∀ (a : α) (b : β), b ∈ f a → b ∈ g a\ngf : ∀ (a : α) (b : β), b ∈ g a → b ∈ f a\na : α\nb : β\nh : g a = some b\n⊢ f a = some b",
"ppTerm": "?some",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [] | exact gf _ _ h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.PEquiv | {
"line": 377,
"column": 10
} | {
"line": 377,
"column": 24
} | {
"line": 377,
"column": 24
} | [
{
"pp": "case some\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf g : α ≃. β\nfg : ∀ (a : α) (b : β), b ∈ f a → b ∈ g a\ngf : ∀ (a : α) (b : β), b ∈ g a → b ∈ f a\na : α\nb : β\nh : g a = some b\n⊢ f a = some b",
"ppTerm": "?some",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [] | exact gf _ _ h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.PEquiv | {
"line": 377,
"column": 10
} | {
"line": 377,
"column": 24
} | {
"line": 377,
"column": 24
} | [
{
"pp": "case some\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf g : α ≃. β\nfg : ∀ (a : α) (b : β), b ∈ f a → b ∈ g a\ngf : ∀ (a : α) (b : β), b ∈ g a → b ∈ f a\na : α\nb : β\nh : g a = some b\n⊢ f a = some b",
"ppTerm": "?some",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [] | exact gf _ _ h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 111,
"column": 22
} | {
"line": 111,
"column": 24
} | {
"line": 111,
"column": 25
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : Matrix n n R\nhx : x ∈ ↑(rowStochastic R n)\ny : Matrix n n R\nhy : y ∈ ↑(rowStochastic R n)\na b : R\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ ↑(rowSt... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : Matrix n n R\nhx : x ∈ ↑(rowStochastic R n)\ny : Matrix n n R\nhy : y ∈ ↑(rowStochastic R n)\na b : R\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x + b • y ∈ ↑(rowStochastic... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Between | {
"line": 787,
"column": 4
} | {
"line": 787,
"column": 19
} | {
"line": 788,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsTorsionFree R V\nt : Affine.Triangle R P\ni₁ i₂ i₃ : Fin 3\nh₁₂ : i₁ ≠ i₂\np₁ p₂ p : P\nh₁ : Sbtw R (t.point... | [] | exact hle _ h₁' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 205,
"column": 22
} | {
"line": 205,
"column": 24
} | {
"line": 205,
"column": 25
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : Matrix n n R\nhx : x ∈ ↑(colStochastic R n)\ny : Matrix n n R\nhy : y ∈ ↑(colStochastic R n)\na b : R\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ ↑(colSt... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : Matrix n n R\nhx : x ∈ ↑(colStochastic R n)\ny : Matrix n n R\nhy : y ∈ ↑(colStochastic R n)\na b : R\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x + b • y ∈ ↑(colStochastic... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.DoublyStochasticMatrix | {
"line": 103,
"column": 22
} | {
"line": 103,
"column": 24
} | {
"line": 103,
"column": 25
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : Matrix n n R\nhx : x ∈ ↑(doublyStochastic R n)\ny : Matrix n n R\nhy : y ∈ ↑(doublyStochastic R n)\na b : R\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ ↑... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : Matrix n n R\nhx : x ∈ ↑(doublyStochastic R n)\ny : Matrix n n R\nhy : y ∈ ↑(doublyStochastic R n)\na b : R\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x + b • y ∈ ↑(doublyS... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 58
} | {
"line": 56,
"column": 2
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs : 0 < s\nhM : ∃ M' ∈ doublyStochastic R n, M = s • M'\n⊢ ∃ σ, ∀ (i j : n), M i j = 0 → Equiv.Perm.permMatrix R σ i j = 0",
... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs : 0 < s\nhM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s\n⊢ ∃ σ, ∀ (i j : n), M i j = 0 → Equiv.Pe... | rw [exists_mem_doublyStochastic_eq_smul_iff hs.le] at hM | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Body | {
"line": 150,
"column": 32
} | {
"line": 150,
"column": 42
} | {
"line": 150,
"column": 43
} | [
{
"pp": "V : Type u_1\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup V\ninst✝¹ : Module ℝ V\ninst✝ : ContinuousSMul ℝ V\nK : ConvexBody V\nh_zero : 0 ∈ K\na b : ℝ≥0\nh : a ≤ b\n⊢ ↑(a • K) ⊆ ↑(b • K)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"i... | [
"V : Type u_1\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup V\ninst✝¹ : Module ℝ V\ninst✝ : ContinuousSMul ℝ V\nK : ConvexBody V\nh_zero : 0 ∈ K\na b : ℝ≥0\nh : a ≤ b\n⊢ a • ↑K ⊆ ↑(b • K)"
] | coe_smul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Body | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 25
} | {
"line": 183,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : NormedSpace ℝ V\nK L : ConvexBody V\n⊢ Metric.hausdorffEDist ↑K ↑L ≠ ⊤",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"DistribMulAction.toDistribSMul",
"AddCommGroup.toAddCommMonoid",
"Normed... | [] | apply_rules [Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded, ConvexBody.nonempty,
ConvexBody.isBounded] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.Analysis.Convex.Body | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 25
} | {
"line": 183,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : NormedSpace ℝ V\nK L : ConvexBody V\n⊢ Metric.hausdorffEDist ↑K ↑L ≠ ⊤",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"DistribMulAction.toDistribSMul",
"AddCommGroup.toAddCommMonoid",
"Normed... | [] | apply_rules [Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded, ConvexBody.nonempty,
ConvexBody.isBounded] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Body | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 25
} | {
"line": 183,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : NormedSpace ℝ V\nK L : ConvexBody V\n⊢ Metric.hausdorffEDist ↑K ↑L ≠ ⊤",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"DistribMulAction.toDistribSMul",
"AddCommGroup.toAddCommMonoid",
"Normed... | [] | apply_rules [Metric.hausdorffEDist_ne_top_of_nonempty_of_bounded, ConvexBody.nonempty,
ConvexBody.isBounded] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 47
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns t : Set E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht : t ∈ 𝓝ˢ s\nU : Set E\nhU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nhU₂ : s ⊆ U\nhU₃ : U ⊆ t\nV : Set E\nhV₁... | [] | grw [hu₁, hs₁.convexHull_eq, hb₂, hV₂, hU₃] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 47
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns t : Set E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht : t ∈ 𝓝ˢ s\nU : Set E\nhU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nhU₂ : s ⊆ U\nhU₃ : U ⊆ t\nV : Set E\nhV₁... | [] | grw [hu₁, hs₁.convexHull_eq, hb₂, hV₂, hU₃] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 47
} | {
"line": 116,
"column": 0
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns t : Set E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht : t ∈ 𝓝ˢ s\nU : Set E\nhU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U\nhU₂ : s ⊆ U\nhU₃ : U ⊆ t\nV : Set E\nhV₁... | [] | grw [hu₁, hs₁.convexHull_eq, hb₂, hV₂, hU₃] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Continuous | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 44
} | {
"line": 176,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhf : ConvexOn ℝ C f\n⊢ LocallyLipschitzOn C f ↔ ContinuousOn f C",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"R... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ∅\nhf : ConvexOn ℝ ∅ f\n⊢ LocallyLipschitzOn ∅ f ↔ ContinuousOn f ∅",
"case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Se... | obtain rfl | hC' := C.eq_empty_or_nonempty | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 466,
"column": 53
} | {
"line": 466,
"column": 84
} | {
"line": 466,
"column": 84
} | [
{
"pp": "V : Type u_2\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nhs : Convex ℝ ∅\nh : intrinsicInterior ℝ ∅ ≠ ∅\n⊢ False",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Real",
"NormedSpace.toModule",
"PseudoMetricSpace.toUnif... | [] | exact h intrinsicInterior_empty | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Convex.ConvexSpace.Prod | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 15
} | {
"line": 99,
"column": 4
} | [
{
"pp": "I : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\nι : Type u_3\nX : Type u_4\ninst✝¹ : Zero X\ninst✝ : ConvexSpace R X\ni : ι\nw : StdSimplex R (ι →₀ X)\n⊢ ∀ (a : ι), (iConvexComb w fun x ↦ x a) ≠ 0 → a ∈ w.weights.support.biUnion support",
"p... | [
"I : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\nι : Type u_3\nX : Type u_4\ninst✝¹ : Zero X\ninst✝ : ConvexSpace R X\ni✝ : ι\nw : StdSimplex R (ι →₀ X)\ni : ι\nhi : (iConvexComb w fun x ↦ x i) ≠ 0\n⊢ i ∈ w.weights.support.biUnion support"
] | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Geometry.Convex.Set | {
"line": 119,
"column": 32
} | {
"line": 119,
"column": 66
} | {
"line": 119,
"column": 66
} | [
{
"pp": "R : Type u_3\nX : Type u_5\nY : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\ns : Set X\nhf : IsAffineMap R f\nhs : IsConvexSet R s\nw : StdSimplex R Y\nhw : ↑w.weights.support ⊆ f '' s\nu : Finset X... | [] | grw [support_onFinset_subset, hus] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Geometry.Convex.Set | {
"line": 119,
"column": 32
} | {
"line": 119,
"column": 66
} | {
"line": 119,
"column": 66
} | [
{
"pp": "R : Type u_3\nX : Type u_5\nY : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\ns : Set X\nhf : IsAffineMap R f\nhs : IsConvexSet R s\nw : StdSimplex R Y\nhw : ↑w.weights.support ⊆ f '' s\nu : Finset X... | [] | grw [support_onFinset_subset, hus] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Set | {
"line": 119,
"column": 32
} | {
"line": 119,
"column": 66
} | {
"line": 119,
"column": 66
} | [
{
"pp": "R : Type u_3\nX : Type u_5\nY : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\ns : Set X\nhf : IsAffineMap R f\nhs : IsConvexSet R s\nw : StdSimplex R Y\nhw : ↑w.weights.support ⊆ f '' s\nu : Finset X... | [] | grw [support_onFinset_subset, hus] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Set | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 37
} | {
"line": 125,
"column": 2
} | [
{
"pp": "R : Type u_3\nX : Type u_5\nY : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\ns : Set X\nhf : IsAffineMap R f\nhs : IsConvexSet R s\nw : StdSimplex R Y\nhw : ↑w.weights.support ⊆ f '' s\nu : Finset X... | [
"case pos\nR : Type u_3\nX : Type u_5\nY : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\ns : Set X\nhf : IsAffineMap R f\nhs : IsConvexSet R s\nw : StdSimplex R Y\nhw : ↑w.weights.support ⊆ f '' s\nu : Finset X\n... | by_cases hy : y ∈ w.weights.support | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Convex.Quasiconvex | {
"line": 101,
"column": 12
} | {
"line": 101,
"column": 14
} | {
"line": 101,
"column": 15
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : SMul 𝕜 E\nβ : Type u_6\nγ : Type u_7\ninst✝¹ : LinearOrder β\ninst✝ : Preorder γ\ns : Set E\nf : E → β\ng : β → γ\nhg : Monotone g\nhf : QuasiconvexOn 𝕜 s f\nc : γ\nx : E\nhx : x ∈ s ∧ g (f... | [
"𝕜 : Type u_4\nE : Type u_5\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : SMul 𝕜 E\nβ : Type u_6\nγ : Type u_7\ninst✝¹ : LinearOrder β\ninst✝ : Preorder γ\ns : Set E\nf : E → β\ng : β → γ\nhg : Monotone g\nhf : QuasiconvexOn 𝕜 s f\nc : γ\nx : E\nhx : x ∈ s ∧ g (f x) ≤ c\ny :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Quasiconvex | {
"line": 224,
"column": 22
} | {
"line": 224,
"column": 24
} | {
"line": 224,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : LinearOrder β\ninst✝ : SMul 𝕜 E\ns : Set E\nf : E → β\nhf : QuasiconvexOn 𝕜 s f\nr : β\nx : E\nhx : x ∈ {x | x ∈ s ∧ f x < r}\ny : E\nhy : y ∈ {x | x ∈ s ∧ f x < r}\na b : 𝕜\... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : LinearOrder β\ninst✝ : SMul 𝕜 E\ns : Set E\nf : E → β\nhf : QuasiconvexOn 𝕜 s f\nr : β\nx : E\nhx : x ∈ {x | x ∈ s ∧ f x < r}\ny : E\nhy : y ∈ {x | x ∈ s ∧ f x < r}\na b : 𝕜\nha : 0 ≤ a\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 794,
"column": 10
} | {
"line": 794,
"column": 12
} | {
"line": 794,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\na b : V\nh : ¬G.Adj a b\nhn : a ≠ b\nu : Sym2 V\n⊢ u ∈ G.incidenceSet a → u ∉ G.incidenceSet b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"SimpleGraph.incidenceSet",
"Membership.mem",
"Set.instMembership",
"Sym2",
"... | [
"V : Type u\nG : SimpleGraph V\na b : V\nh : ¬G.Adj a b\nhn : a ≠ b\nu : Sym2 V\nha : u ∈ G.incidenceSet a\n⊢ u ∉ G.incidenceSet b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Side | {
"line": 269,
"column": 48
} | {
"line": 269,
"column": 61
} | {
"line": 269,
"column": 61
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v ∈ s.direction\n⊢ s.WOppSide y x ↔ s.WOppSide x y",
"ppTerm": "?... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v ∈ s.direction\n⊢ s.WOppSide x y ↔ s.WOppSide x y"
] | wOppSide_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 1116,
"column": 4
} | {
"line": 1116,
"column": 28
} | {
"line": 1117,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nv : V\nh : Gᶜ.IsUniversal v\nx : V\nhx : G.Adj v x\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"False",
"SimpleGraph.Adj.ne",
"congrArg",
"Compl.compl",
"SimpleGraph.Adj",
"False.eli... | [] | simpa [hx] using h hx.ne | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 1116,
"column": 4
} | {
"line": 1116,
"column": 28
} | {
"line": 1117,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nv : V\nh : Gᶜ.IsUniversal v\nx : V\nhx : G.Adj v x\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"False",
"SimpleGraph.Adj.ne",
"congrArg",
"Compl.compl",
"SimpleGraph.Adj",
"False.eli... | [] | simpa [hx] using h hx.ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 1116,
"column": 4
} | {
"line": 1116,
"column": 28
} | {
"line": 1117,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nv : V\nh : Gᶜ.IsUniversal v\nx : V\nhx : G.Adj v x\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"False",
"SimpleGraph.Adj.ne",
"congrArg",
"Compl.compl",
"SimpleGraph.Adj",
"False.eli... | [] | simpa [hx] using h hx.ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 40,
"column": 27
} | {
"line": 40,
"column": 32
} | {
"line": 40,
"column": 33
} | [
{
"pp": "case insert\nR : Type u_1\nV : Type u_2\nι : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : TopologicalSpace V\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ns : Set V\nw : ι → R\nz : ι → V\nhs : StrictConvex R s\ni : ι\nt : Finset ι\nhi : i ∉ t\nht :\n (∀ ... | [
"case insert\nR : Type u_1\nV : Type u_2\nι : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : TopologicalSpace V\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ns : Set V\nw : ι → R\nz : ι → V\nhs : StrictConvex R s\ni : ι\nt : Finset ι\nhi : i ∉ t\nht :\n (∀ i ∈ t, 0 ≤ w... | hi'j' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Side | {
"line": 578,
"column": 8
} | {
"line": 578,
"column": 21
} | {
"line": 578,
"column": 21
} | [
{
"pp": "case mp\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y : P\nhs : s.WSameSide x y\nho : s.WOppSide x y\n⊢ x ∈ s ∨ y ∈ s",
"ppTerm": "... | [
"case mp\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y : P\nhs : s.WSameSide x y\nho : s.WOppSide y x\n⊢ x ∈ s ∨ y ∈ s"
] | wOppSide_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Side | {
"line": 939,
"column": 2
} | {
"line": 940,
"column": 74
} | {
"line": 941,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : Fin (n + 1)\... | refine ⟨Finset.univ.affineCombination R s.points w₃,
(s.affineCombination_mem_affineSpan_faceOpposite_iff hw₃1).2 hw₃i, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Side | {
"line": 1037,
"column": 6
} | {
"line": 1037,
"column": 100
} | {
"line": 1038,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j... | [] | exact wSameSide_of_left_mem _ ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).2 h0) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Side | {
"line": 1037,
"column": 6
} | {
"line": 1037,
"column": 100
} | {
"line": 1038,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j... | [] | exact wSameSide_of_left_mem _ ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).2 h0) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Side | {
"line": 1037,
"column": 6
} | {
"line": 1037,
"column": 100
} | {
"line": 1038,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j... | [] | exact wSameSide_of_left_mem _ ((s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).2 h0) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Side | {
"line": 1074,
"column": 2
} | {
"line": 1075,
"column": 85
} | {
"line": 1076,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw : Fin (n + 1) → R\nhw : ∑ j, w j = 1\ni : Fin (n + 1)\n⊢ (affineSpan ... | [
"R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw : Fin (n + 1) → R\nhw : ∑ j, w j = 1\ni : Fin (n + 1)\n⊢ SignType.sign (w i) = Si... | rw [← Finset.univ.affineCombination_piSingle R s.points (Finset.mem_univ i),
s.sSameSide_affineSpan_faceOpposite_iff (Fintype.sum_pi_single' _ _) hw, eq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 288,
"column": 4
} | {
"line": 288,
"column": 39
} | {
"line": 289,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 <... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\nH2 : ∀ s > 0, 0 < ∫ (t : ℝ) i... | rw [← restrict_Ioo_eq_restrict_Ioc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 37
} | {
"line": 421,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t\nG : ℝ → ℝ\nG_mble : Measurable G\nG_nn : 0 ≤ G\ng_eq_G : g =ᵐ[volume.restr... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t\nG : ℝ → ℝ\nG_mble : Measurable G\nG_nn : 0 ≤ G\ng_eq_G : g =ᵐ[volume.restrict (Ioi 0)]... | filter_upwards [g_eq_G] with t ht | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 424,
"column": 35
} | {
"line": 424,
"column": 37
} | {
"line": 424,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t\nG : ℝ → ℝ\nG_mble : Measurable G\nG_nn : 0 ≤ G\ng_eq_G : g =ᵐ[volume.restr... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ᵐ[μ] f\nf_mble : AEMeasurable f μ\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t\nG : ℝ → ℝ\nG_mble : Measurable G\nG_nn : 0 ≤ G\ng_eq_G : g =ᵐ[volume.restrict (Ioi 0)]... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 558,
"column": 34
} | {
"line": 558,
"column": 36
} | {
"line": 558,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nM : ℝ\nf_intble : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nf_bdd : f ≤ᵐ[μ] fun x ↦ M\nt : ℝ\nht : t ∈ Ioi 0 \\ Ioc 0 M\nhtM : M < t\na : α\n⊢ f a ≤ M → ¬M < f a",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"Real... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nM : ℝ\nf_intble : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nf_bdd : f ≤ᵐ[μ] fun x ↦ M\nt : ℝ\nht : t ∈ Ioi 0 \\ Ioc 0 M\nhtM : M < t\na : α\nha : f a ≤ M\n⊢ ¬M < f a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 559,
"column": 2
} | {
"line": 559,
"column": 50
} | {
"line": 560,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nM : ℝ\nf_intble : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nf_bdd : f ≤ᵐ[μ] fun x ↦ M\nt : ℝ\nht : t ∈ Ioi 0 \\ Ioc 0 M\nhtM : M < t\nobs : μ {a | M < f a} = 0\n⊢ μ.real {a | t ≤ f a} = 0",
"ppTerm": "?m.103",
"assigned": true,
"use... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nM : ℝ\nf_intble : Integrable f μ\nf_nn : 0 ≤ᵐ[μ] f\nf_bdd : f ≤ᵐ[μ] fun x ↦ M\nt : ℝ\nht : t ∈ Ioi 0 \\ Ioc 0 M\nhtM : M < t\nobs : μ {a | M < f a} = 0\n⊢ μ {a | t ≤ f a} = 0 ∨ μ {a | t ≤ f a} = ∞"
] | rw [measureReal_def, ENNReal.toReal_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 47
} | {
"line": 377,
"column": 2
} | [
{
"pp": "case h₂\nH : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := ⋯\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤ ↑k\n... | [
"case h₂\nH : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℝ H\nr : ℝ\nt : Set ℝ := {y | 1 / 2 < y}\nht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t\nhdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t\nhunique : UniqueDiffOn ℝ t\nN k : ℕ\nhk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k\nhk₁ : r ≤ ↑k... | have hx' : 1 / 2 < x := by simpa [t] using hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 58
} | {
"line": 153,
"column": 4
} | [
{
"pp": "case mp\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : σ = sort f\n⊢ StrictMono ⇑(Equiv.trans σ (graphEquiv₁ f))",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Equiv.instEquivLike",
"Strict... | [
"case mp\nn : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\nh : σ = sort f\n⊢ StrictMono ⇑((graphEquiv₂ f).trans (Equiv.refl ↥(graph f)))"
] | rw [h, sort, Equiv.trans_assoc, Equiv.symm_trans_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 18
} | {
"line": 163,
"column": 18
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\n⊢ σ = sort f ↔ Monotone (f ∘ ⇑σ) ∧ ∀ (i j : Fin n), i < j → f (σ i) = f (σ j) → σ i < σ j",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"StrictMo... | [
"n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nσ : Equiv.Perm (Fin n)\n⊢ StrictMono ⇑(Equiv.trans σ (graphEquiv₁ f)) ↔\n Monotone (f ∘ ⇑σ) ∧ ∀ (i j : Fin n), i < j → f (σ i) = f (σ j) → σ i < σ j"
] | eq_sort_iff' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 12
} | {
"line": 90,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ : 𝕜\nv : E\nhv : v ∈ (eigenspace T μ)ᗮ\n⊢ T v ∈ (eigenspace T μ)ᗮ",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"InnerProd... | [
"𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ : 𝕜\nv : E\nhv : v ∈ (eigenspace T μ)ᗮ\nw : E\nhw : w ∈ eigenspace T μ\n⊢ ⟪w, T v⟫ = 0"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Matrix.Order | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 68
} | {
"line": 171,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\n⊢ A.PosSemidef ↔ A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Matrix.PosSemidef.isHermitian",
"NonUnitalCommRi... | [
"case refine_1\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nh : A.PosSemidef\na : 𝕜\n⊢ a ∈ spectrum 𝕜 A → a ∈ {a | 0 ≤ a}",
"case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n ... | refine ⟨fun h => ⟨h.isHermitian, fun a => ?_⟩, fun ⟨h1, h2⟩ => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Matrix.Order | {
"line": 189,
"column": 38
} | {
"line": 189,
"column": 41
} | {
"line": 189,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → 𝕜\ny : Matrix n n 𝕜\nhx : (star y * y).PosSemidef\nh : IsUnit (star y * y)\nhv : y *ᵥ v = 0\n⊢ yᴴ *ᵥ y *ᵥ v = 0",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → 𝕜\ny : Matrix n n 𝕜\nhx : (star y * y).PosSemidef\nh : IsUnit (star y * y)\nhv : y *ᵥ v = 0\n⊢ yᴴ *ᵥ 0 = 0"
] | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.SmoothApprox | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 63
} | {
"line": 95,
"column": 2
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : Chart... | [
"E : Type u_1\nF : Type u_2\nH : Type u_3\nM : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : FiniteDimensional ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\... | apply exists_contMDiffMap_forall_mem_convex_of_local I t_conv | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 203,
"column": 8
} | {
"line": 203,
"column": 41
} | {
"line": 204,
"column": 6
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : N... | [
"case inr\n𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : NonUnitalCont... | cfcₙ_apply_of_not_map_zero _ hf0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 869,
"column": 2
} | {
"line": 869,
"column": 26
} | {
"line": 870,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n}_{K}(E, F... | [
"case pos\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n}_{K}(E, F)\... | by_cases! hk : k + 1 ≤ n | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 296,
"column": 54
} | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 61
} | [
{
"pp": "A : Type u_2\ninst✝¹⁰ : NonUnitalNormedRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : ContinuousStar A\ninst✝⁷ : NormedSpace ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRi... | [] | cfc_tac | _aux_Mathlib_Tactic_ContinuousFunctionalCalculus___macroRules_cfcTac_1 | cfcTac |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 296,
"column": 54
} | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 61
} | [
{
"pp": "A : Type u_2\ninst✝¹⁰ : NonUnitalNormedRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : ContinuousStar A\ninst✝⁷ : NormedSpace ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRi... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 296,
"column": 54
} | {
"line": 296,
"column": 61
} | {
"line": 296,
"column": 61
} | [
{
"pp": "A : Type u_2\ninst✝¹⁰ : NonUnitalNormedRing A\ninst✝⁹ : StarRing A\ninst✝⁸ : ContinuousStar A\ninst✝⁷ : NormedSpace ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRi... | [] | cfc_tac | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 20
} | {
"line": 180,
"column": 20
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf f' g : ↥(Lp E 2 μ)\n⊢ ∫ (a : α), ⟪↑↑(f + f') a, ↑↑g a⟫ ∂μ = ∫ (a : α), ⟪↑↑f a, ↑↑g a⟫ + ⟪↑↑f' a, ↑↑g a⟫ ∂μ",
"ppTerm": "?m.36",
"... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝² : RCLike 𝕜\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf f' g : ↥(Lp E 2 μ)\n⊢ ∫ (a : α), ⟪↑↑(f + f') a, ↑↑g a⟫ ∂μ = ∫ (a : α), ⟪↑↑f a + ↑↑f' a, ↑↑g a⟫ ∂μ"
] | ← inner_add_left | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 1004,
"column": 4
} | {
"line": 1004,
"column": 86
} | {
"line": 1005,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\nn : ℕ∞\nK : Compacts E\nm : MeasurableSpace E\ninst✝¹⁰ : OpensMeasurableSpace E\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : ... | [
"case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\nn : ℕ∞\nK : Compacts E\nm : MeasurableSpace E\ninst✝¹⁰ : OpensMeasurableSpace E\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedSpace ... | apply le_trans (norm_integral_le_of_norm_le ((hφ.norm.mul_const _).mul_const _) h) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 22
} | {
"line": 113,
"column": 0
} | [
{
"pp": "case h'\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ... | [] | exact hg₄.coeFn_toLp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 224,
"column": 2
} | {
"line": 227,
"column": 50
} | {
"line": 228,
"column": 2
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nβ : Type u_3\nb : UnconditionalSchauderBasis β 𝕜 X\ninst✝ : CompleteSpace X\nx : X\nA₀ : Finset β\nhA₀ : ∀ (t : Finset β), Disjoint t A₀ → ‖∑ i ∈ t, (b.coord i) x • ↑b i‖... | [
"case h\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nβ : Type u_3\nb : UnconditionalSchauderBasis β 𝕜 X\ninst✝ : CompleteSpace X\nx : X\nA₀ : Finset β\nhA₀ : ∀ (t : Finset β), Disjoint t A₀ → ‖∑ i ∈ t, (b.coord i) x • ↑b i‖ < 1\nA : Fi... | have hdecomp : b.proj A x = b.proj (A ∩ A₀) x + b.proj (A \ A₀) x := by
simp only [GeneralSchauderBasis.proj_apply]
rw [← Finset.sum_union (Finset.disjoint_sdiff_inter A A₀).symm,
Finset.union_comm, Finset.sdiff_union_inter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 174,
"column": 4
} | {
"line": 180,
"column": 27
} | {
"line": 181,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\nhp : p ≠ ∞\n... | [
"case e'_3\nα : Type u_1\ninst✝⁸ : TopologicalSpace α\ninst✝⁷ : NormalSpace α\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : BorelSpace α\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝³ : NormedSpace ℝ E\ninst✝² : R1Space α\ninst✝¹ : WeaklyLocallyCompactSpace α\ninst✝ : μ.Regular\nhp : p ≠ ∞\nf... | convert!
(hδ _ _
(f_mem.aestronglyMeasurable.sub
(aestronglyMeasurable_const.indicator s_closed.measurableSet))
((aestronglyMeasurable_const.indicator s_closed.measurableSet).sub
(aestronglyMeasurable_const.indicator ht))
I2 I1).le using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 271,
"column": 4
} | {
"line": 277,
"column": 27
} | {
"line": 278,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\nhp : p ≠ ∞\nf✝ : α → E\nhf : MemLp f✝ p μ\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠... | [
"case e'_3\nα : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.WeaklyRegular\nhp : p ≠ ∞\nf✝ : α → E\nhf : MemLp f✝ p μ\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ ... | convert!
(hδ _ _
(f_mem.aestronglyMeasurable.sub
(aestronglyMeasurable_const.indicator s_closed.measurableSet))
((aestronglyMeasurable_const.indicator s_closed.measurableSet).sub
(aestronglyMeasurable_const.indicator ht))
I2 I1).le using 2 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 39
} | {
"line": 117,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\nG : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : ↥(lp G 2)\ni : ι\n⊢ ‖⟪↑f i, ↑g i⟫‖ ≤ ‖↑f i‖ * ‖↑g i‖",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants":... | [] | exact norm_inner_le_norm (𝕜 := 𝕜) _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 25
} | {
"line": 251,
"column": 6
} | [
{
"pp": "case refine_2.h\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[�... | [
"case refine_2.h\nι : Type u_1\n𝕜 : Type u_2\ninst✝⁶ : RCLike 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nG : ι → Type u_4\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\ninst✝¹ : CompleteSpace E\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : O... | rintro i x ⟨x, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 340,
"column": 2
} | {
"line": 341,
"column": 30
} | {
"line": 343,
"column": 0
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nc : ℂ\nn : ℤ\n⊢ fourierCoeff (c • f) n = c • fourierCoeff f n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCo... | [] | simp_rw [fourierCoeff, Pi.smul_apply, ← smul_assoc, smul_eq_mul, mul_comm, ← smul_eq_mul,
smul_assoc, integral_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 340,
"column": 2
} | {
"line": 341,
"column": 30
} | {
"line": 343,
"column": 0
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nc : ℂ\nn : ℤ\n⊢ fourierCoeff (c • f) n = c • fourierCoeff f n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCo... | [] | simp_rw [fourierCoeff, Pi.smul_apply, ← smul_assoc, smul_eq_mul, mul_comm, ← smul_eq_mul,
smul_assoc, integral_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 340,
"column": 2
} | {
"line": 341,
"column": 30
} | {
"line": 343,
"column": 0
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nc : ℂ\nn : ℤ\n⊢ fourierCoeff (c • f) n = c • fourierCoeff f n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCo... | [] | simp_rw [fourierCoeff, Pi.smul_apply, ← smul_assoc, smul_eq_mul, mul_comm, ← smul_eq_mul,
smul_assoc, integral_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 36
} | {
"line": 70,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\na : E\nhs : MeasurableSet s\nh'st : t ∈ 𝓝[s] x₀\nhlφ... | [] | exact hu.trans inter_subset_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 36
} | {
"line": 124,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\nhts : t ⊆... | [] | exact hu.trans inter_subset_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 176,
"column": 11
} | {
"line": 176,
"column": 28
} | {
"line": 176,
"column": 29
} | [
{
"pp": "case e'_12\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v... | [
"case e'_12\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v) w') (L v) ... | fourierSMulRight, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 23
} | {
"line": 185,
"column": 24
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\n⊢ ‖fourierSMulRight L f v‖ = 2 * π * ‖L v‖ * ‖f v‖... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\n⊢ ‖-(2 * ↑π * I) • (L v).smulRight (f v)‖ = 2 * π * ‖L v‖ * ‖f... | fourierSMulRight, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 53
} | {
"line": 156,
"column": 2
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : 𝓢(E, F)\ninst✝ : ProperSpace E\ns : ℝ\nk : ℕ := ⌈-s⌉₊\nhk : -↑k ≤ s\n⊢ ⇑f =O[cocompact E] fun x ↦ ‖x‖ ^ s",
"ppTerm": "?m.53",
"assigned": true,
... | [
"E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : 𝓢(E, F)\ninst✝ : ProperSpace E\ns : ℝ\nk : ℕ := ⌈-s⌉₊\nhk : -↑k ≤ s\n⊢ (fun x ↦ ‖x‖ ^ (-↑k)) =O[cocompact E] fun x ↦ ‖x‖ ^ s"
] | refine (isBigO_cocompact_zpow_neg_nat f k).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 313,
"column": 15
} | {
"line": 313,
"column": 17
} | {
"line": 313,
"column": 18
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCo... | [
"α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCompact s\nhμ ... | φ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 821,
"column": 4
} | {
"line": 822,
"column": 69
} | {
"line": 823,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nf : 𝓢(E, F)\nhg : Function.HasT... | [] | simpa [smulLeftCLM_apply hg] using
⟨tsupport_smul_subset_right g f, tsupport_smul_subset_left g f⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 821,
"column": 4
} | {
"line": 822,
"column": 69
} | {
"line": 823,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nf : 𝓢(E, F)\nhg : Function.HasT... | [] | simpa [smulLeftCLM_apply hg] using
⟨tsupport_smul_subset_right g f, tsupport_smul_subset_left g f⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 821,
"column": 4
} | {
"line": 822,
"column": 69
} | {
"line": 823,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nf : 𝓢(E, F)\nhg : Function.HasT... | [] | simpa [smulLeftCLM_apply hg] using
⟨tsupport_smul_subset_right g f, tsupport_smul_subset_left g f⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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