module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 874,
"column": 2
} | {
"line": 874,
"column": 69
} | {
"line": 875,
"column": 4
} | [
{
"pp": "α : Type u\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\n⊢ MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) (Measure.pi fun x ↦ μ) (μ.prod μ)",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\n⊢ MeasurePreserving (⇑MeasurableEquiv.finTwoArrow) (Measure.pi fun x ↦ μ) (μ.prod μ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Holder | {
"line": 78,
"column": 56
} | {
"line": 78,
"column": 67
} | {
"line": 78,
"column": 68
} | [
{
"pp": "p q r : ℝ≥0∞\ninst✝ : p.HolderTriple q r\n⊢ 1 / p + 1 / q = 1 / r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"id",
"HDiv.hDiv",
"instHAdd",
"Inv.in... | [
"p q r : ℝ≥0∞\ninst✝ : p.HolderTriple q r\n⊢ p⁻¹ + q⁻¹ = r⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Holder | {
"line": 99,
"column": 16
} | {
"line": 99,
"column": 27
} | {
"line": 99,
"column": 28
} | [
{
"pp": "p q r : ℝ≥0∞\ninst✝ : p.HolderTriple q r\nhr : r ≠ 0\n⊢ r⁻¹ < ∞",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"PartialOrder.toPreorder",
"ENNReal.inv_lt_top._simp_1",
"id",
"Inv.inv",
"LT.lt",
"ENNReal",
... | [
"p q r : ℝ≥0∞\ninst✝ : p.HolderTriple q r\nhr : r ≠ 0\n⊢ 0 < r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 109,
"column": 27
} | {
"line": 109,
"column": 40
} | {
"line": 109,
"column": 40
} | [
{
"pp": "case inr.inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs' : s ≠ 0\np : ℝ\nhp : 1 < p\nhp' : 0 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\n⊢ 1 + p * s < rexp (log (1 + s) * p)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"R... | [
"case inr.inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs' : s ≠ 0\np : ℝ\nhp : 1 < p\nhp' : 0 < p\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\n⊢ rexp (log (1 + p * s)) < rexp (log (1 + s) * p)"
] | ← exp_log hs2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 603,
"column": 33
} | {
"line": 603,
"column": 58
} | {
"line": 603,
"column": 59
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\ns : Set α\nf : α → ε\nhsf : Function.support f ⊆ s\nhp0 : ¬p = 0\nhp_top : ¬p = ∞\n⊢ ¬p.toReal ≤ 0",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\ns : Set α\nf : α → ε\nhsf : Function.support f ⊆ s\nhp0 : ¬p = 0\nhp_top : ¬p = ∞\n⊢ 0 < p.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 604,
"column": 4
} | {
"line": 604,
"column": 22
} | {
"line": 604,
"column": 23
} | [
{
"pp": "case neg.e_a\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\ns : Set α\nf : α → ε\nhsf : Function.support f ⊆ s\nhp0 : ¬p = 0\nhp_top : ¬p = ∞\nthis : ¬p.toReal ≤ 0\n⊢ (Function.support fun x ↦ ‖f x‖ₑ ^ p.toReal) ⊆ s... | [
"case neg.e_a\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\ns : Set α\nf : α → ε\nhsf : Function.support f ⊆ s\nhp0 : ¬p = 0\nhp_top : ¬p = ∞\nthis : ¬p.toReal ≤ 0\n⊢ ∀ (x : α), ¬f x = 0 → x ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 150,
"column": 27
} | {
"line": 150,
"column": 40
} | {
"line": 150,
"column": 40
} | [
{
"pp": "case inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs' : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\n⊢ rexp (log (1 + s) * p) < 1 + p * s",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
... | [
"case inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs' : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\n⊢ rexp (log (1 + s) * p) < rexp (log (1 + p * s))"
] | ← exp_log hs2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.ConjExponents | {
"line": 116,
"column": 56
} | {
"line": 116,
"column": 67
} | {
"line": 116,
"column": 68
} | [
{
"pp": "p q r : ℝ\nh : p.HolderTriple q r\n⊢ 1 / p + 1 / q = 1 / r",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"id",
"HDiv.hDiv",
"Real.instAd... | [
"p q r : ℝ\nh : p.HolderTriple q r\n⊢ p⁻¹ + q⁻¹ = r⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 124,
"column": 23
} | {
"line": 124,
"column": 34
} | {
"line": 124,
"column": 35
} | [
{
"pp": "p q r : ℝ\nh : p.HolderTriple q r\n⊢ r < p",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q r : ℝ\nh : p.HolderTriple q r\n⊢ r < p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 161,
"column": 41
} | {
"line": 161,
"column": 52
} | {
"line": 161,
"column": 53
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ p⁻¹ - 1 = -q⁻¹",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℝ\nh : p.HolderConjugate q\n⊢ p⁻¹ - 1 = -q⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 56
} | {
"line": 167,
"column": 57
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ p * q = p + q",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℝ\nh : p.HolderConjugate q\n⊢ p * q = p + q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 837,
"column": 37
} | {
"line": 837,
"column": 62
} | {
"line": 837,
"column": 62
} | [
{
"pp": "case pos\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\nf : α → ε\nhf : AEStronglyMeasurable f μ\nh0 : p ≠ 0\nh_top : p = ∞\n⊢ eLpNormEssSup f μ = 0 ↔ f =ᵐ[μ] 0",
"ppTerm": "?pos✝",
"assigned": true,
"us... | [
"case pos\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ENormedAddMonoid ε\nf : α → ε\nhf : AEStronglyMeasurable f μ\nh0 : p ≠ 0\nh_top : p = ∞\n⊢ f =ᵐ[μ] 0 ↔ f =ᵐ[μ] 0"
] | eLpNormEssSup_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.ConjExponents | {
"line": 189,
"column": 27
} | {
"line": 189,
"column": 38
} | {
"line": 189,
"column": 39
} | [
{
"pp": "a b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
... | [
"a b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a + b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 210,
"column": 42
} | {
"line": 210,
"column": 53
} | {
"line": 210,
"column": 54
} | [
{
"pp": "a b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ (1 / a).HolderConjugate (1 / b)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"id",
"HDiv.... | [
"a b : ℝ\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a⁻¹.HolderConjugate b⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 321,
"column": 27
} | {
"line": 321,
"column": 38
} | {
"line": 321,
"column": 39
} | [
{
"pp": "⊢ 2⁻¹ + 2⁻¹ = 1⁻¹",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
... | [
"⊢ 2⁻¹ + 2⁻¹ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 71
} | {
"line": 349,
"column": 72
} | [
{
"pp": "p q : ℝ≥0\nh : p.HolderConjugate q\n⊢ p * q = p + q",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℝ≥0\nh : p.HolderConjugate q\n⊢ p * q = p + q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 364,
"column": 27
} | {
"line": 364,
"column": 38
} | {
"line": 364,
"column": 39
} | [
{
"pp": "a b : ℝ≥0\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg... | [
"a b : ℝ≥0\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a + b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 369,
"column": 58
} | {
"line": 369,
"column": 69
} | {
"line": 369,
"column": 70
} | [
{
"pp": "a : ℝ≥0\nha₀ : 0 < a\nha₁ : a < 1\n⊢ a⁻¹⁻¹ + (1 - a)⁻¹⁻¹ = 1",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"congrArg",
"HSub.hSub",
"NNReal.instInv",
"id",
"Distrib.toAdd",
"NNReal",... | [
"a : ℝ≥0\nha₀ : 0 < a\nha₁ : a < 1\n⊢ a + (1 - a) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 387,
"column": 42
} | {
"line": 387,
"column": 53
} | {
"line": 387,
"column": 54
} | [
{
"pp": "a b : ℝ≥0\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ (1 / a).HolderConjugate (1 / b)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"congrArg",
"DivisionSemiring.toGro... | [
"a b : ℝ≥0\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\n⊢ a⁻¹.HolderConjugate b⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 13
} | {
"line": 397,
"column": 14
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ p.toNNReal.HolderConjugate q.toNNReal",
"ppTerm": "?m.1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℝ\nh : p.HolderConjugate q\n⊢ p.toNNReal.HolderConjugate q.toNNReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 445,
"column": 2
} | {
"line": 446,
"column": 24
} | {
"line": 446,
"column": 25
} | [
{
"pp": "p q r : ℝ\nh : p.HolderTriple q r\n⊢ (ENNReal.ofReal p).HolderTriple (ENNReal.ofReal q) (ENNReal.ofReal r)",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"_private.Mathlib.Data.Real.ConjExponents.0.Real.HolderTriple.ennrealOfReal._sim... | [
"p q r : ℝ\nh : p.HolderTriple q r\n⊢ (ENNReal.ofReal p)⁻¹ + (ENNReal.ofReal q)⁻¹ = (ENNReal.ofReal r)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 13
} | {
"line": 450,
"column": 14
} | [
{
"pp": "p q : ℝ\nh : p.HolderConjugate q\n⊢ (ENNReal.ofReal p).HolderConjugate (ENNReal.ofReal q)",
"ppTerm": "?m.1",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℝ\nh : p.HolderConjugate q\n⊢ (ENNReal.ofReal p).HolderConjugate (ENNReal.ofReal q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 41
} | {
"line": 457,
"column": 42
} | [
{
"pp": "p q r : ℝ≥0∞\nh : p.toReal.HolderTriple q.toReal r.toReal\nhp : 0 < p ∧ p < ∞\nhq : 0 < q ∧ q < ∞\nhr : 0 < r ∧ r < ∞\n⊢ p.HolderTriple q r",
"ppTerm": "?m.43",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q r : ℝ≥0∞\nh : p.toReal.HolderTriple q.toReal r.toReal\nhp : 0 < p ∧ p < ∞\nhq : 0 < q ∧ q < ∞\nhr : 0 < r ∧ r < ∞\n⊢ p.HolderTriple q r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 464,
"column": 2
} | {
"line": 465,
"column": 9
} | {
"line": 465,
"column": 10
} | [
{
"pp": "p q r : ℝ≥0∞\nhp : 0 < p ∧ p < ∞\nhq : 0 < q ∧ q < ∞\nh : p.HolderTriple q r\n⊢ p.toReal⁻¹ + q.toReal⁻¹ = r.toReal⁻¹",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q r : ℝ≥0∞\nhp : 0 < p ∧ p < ∞\nhq : 0 < q ∧ q < ∞\nh : p.HolderTriple q r\n⊢ p.toReal⁻¹ + q.toReal⁻¹ = r.toReal⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 474,
"column": 19
} | {
"line": 474,
"column": 60
} | {
"line": 474,
"column": 61
} | [
{
"pp": "p q r : ℝ≥0∞\nh : p.toNNReal.HolderTriple q.toNNReal r.toNNReal\n⊢ p.toReal.HolderTriple q.toReal r.toReal",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q r : ℝ≥0∞\nh : p.toNNReal.HolderTriple q.toNNReal r.toNNReal\n⊢ p.toReal.HolderTriple q.toReal r.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 504,
"column": 2
} | {
"line": 504,
"column": 13
} | {
"line": 504,
"column": 14
} | [
{
"pp": "p q : ℝ≥0∞\nhp : 1 < p.toReal\nh : p.HolderConjugate q\nhq : 0 < q.toReal\n⊢ p.toReal.HolderConjugate q.toReal",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℝ≥0∞\nhp : 1 < p.toReal\nh : p.HolderConjugate q\nhq : 0 < q.toReal\n⊢ p.toReal.HolderConjugate q.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 512,
"column": 19
} | {
"line": 512,
"column": 60
} | {
"line": 512,
"column": 61
} | [
{
"pp": "p q : ℝ≥0∞\nh : p.toNNReal.HolderConjugate q.toNNReal\n⊢ p.toReal.HolderConjugate q.toReal",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℝ≥0∞\nh : p.toNNReal.HolderConjugate q.toNNReal\n⊢ p.toReal.HolderConjugate q.toReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 557,
"column": 2
} | {
"line": 558,
"column": 39
} | {
"line": 558,
"column": 40
} | [
{
"pp": "case inr.inr\np q : ℝ≥0∞\nh : p.HolderConjugate q\nhp : p ≠ ∞\nhq : q ≠ ∞\n⊢ p * q = p + q",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"congrArg",
"CommSemiring.toSemi... | [
"case inr.inr\np q : ℝ≥0∞\nh : p.HolderConjugate q\nhp : p ≠ ∞\nhq : q ≠ ∞\n⊢ p * q = q + p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 571,
"column": 27
} | {
"line": 571,
"column": 48
} | {
"line": 571,
"column": 49
} | [
{
"pp": "a b : ℝ≥0∞\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"id",
"semiOutParam",
... | [
"a b : ℝ≥0∞\nhab : a + b = 1\n⊢ a + b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.ConjExponents | {
"line": 571,
"column": 24
} | {
"line": 571,
"column": 52
} | {
"line": 573,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nhab : a + b = 1\n⊢ a⁻¹⁻¹ + b⁻¹⁻¹ = 1⁻¹",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"id",
"semiOutParam",
... | [] | by simpa [inv_inv] using hab | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.ConjExponents | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 13
} | {
"line": 577,
"column": 14
} | [
{
"pp": "a : ℝ≥0∞\nha : 1 ≤ a\n⊢ a.HolderConjugate (1 - a⁻¹)⁻¹",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a : ℝ≥0∞\nha : 1 ≤ a\n⊢ a.HolderConjugate (1 - a⁻¹)⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Slope | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 33
} | {
"line": 288,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhy : y ∈ s\nhxy : x < y\nhxy' : f y < f x\nthis : StrictMonoOn (f ∘ ⇑(-AffineMap.id 𝕜 𝕜)) (⇑(-AffineMap.id 𝕜 𝕜) ⁻¹' s ∩ Set.Ici (-x))\n⊢ StrictAntiOn f... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConvexOn 𝕜 s f\nx y : 𝕜\nhy : y ∈ s\nhxy : x < y\nhxy' : f y < f x\nthis : StrictMonoOn (f ∘ ⇑(-AffineMap.id 𝕜 𝕜)) (⇑(-AffineMap.id 𝕜 𝕜) ⁻¹' s ∩ Set.Ici (-x))\n⊢ StrictAntiOn f (s ∩ Set.Ii... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Slope | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 13
} | {
"line": 294,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y : 𝕜\nhy : y ∈ s\nhxy : x < y\nhxy' : f x < f y\n⊢ StrictMonoOn f (s ∩ Set.Iic x)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y : 𝕜\nhy : y ∈ s\nhxy : x < y\nhxy' : f x < f y\n⊢ StrictMonoOn f (s ∩ Set.Iic x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Slope | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 13
} | {
"line": 300,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f y < f x\n⊢ StrictAntiOn f (s ∩ Set.Ici y)",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y : 𝕜\nhx : x ∈ s\nhxy : x < y\nhxy' : f y < f x\n⊢ StrictAntiOn f (s ∩ Set.Ici y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monovary | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 38
} | {
"line": 99,
"column": 39
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\ns : Set ι\nf : ι → α\ng : ι → β\n⊢ MonovaryOn f g⁻¹ s ↔ AntivaryOn f g s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instIsRightC... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\ns : Set ι\nf : ι → α\ng : ι → β\n⊢ (∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → g j < g i → f i ≤ f j) ↔\n ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → g i < g j → f j ≤ f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monovary | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 38
} | {
"line": 103,
"column": 39
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\ns : Set ι\nf : ι → α\ng : ι → β\n⊢ AntivaryOn f g⁻¹ s ↔ MonovaryOn f g s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instIsRightC... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\ns : Set ι\nf : ι → α\ng : ι → β\n⊢ (∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → g j < g i → f j ≤ f i) ↔\n ∀ ⦃i : ι⦄, i ∈ s → ∀ ⦃j : ι⦄, j ∈ s → g i < g j → f i ≤ f j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monovary | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 34
} | {
"line": 106,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\nf : ι → α\ng : ι → β\n⊢ Monovary f g⁻¹ ↔ Antivary f g",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instIsRightCancelMulOfMulRightR... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\nf : ι → α\ng : ι → β\n⊢ (∀ ⦃i j : ι⦄, g j < g i → f i ≤ f j) ↔ ∀ ⦃i j : ι⦄, g i < g j → f j ≤ f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Monovary | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 34
} | {
"line": 109,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\nf : ι → α\ng : ι → β\n⊢ Antivary f g⁻¹ ↔ Monovary f g",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instIsRightCancelMulOfMulRightR... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α\ninst✝² : CommGroup β\ninst✝¹ : PartialOrder β\ninst✝ : IsOrderedMonoid β\nf : ι → α\ng : ι → β\n⊢ (∀ ⦃i j : ι⦄, g j < g i → f j ≤ f i) ↔ ∀ ⦃i j : ι⦄, g i < g j → f i ≤ f j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 35
} | {
"line": 113,
"column": 36
} | [
{
"pp": "case refine_2\nw₁ w₂ z₁ z₂ : ℝ≥0\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\nh : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p\n⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"R... | [
"case refine_2\nw₁ w₂ z₁ z₂ : ℝ≥0\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\nh : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p\n⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 37
} | {
"line": 114,
"column": 2
} | [
{
"pp": "case refine_2\nw₁ w₂ z₁ z₂ : ℝ≥0\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\nh : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p\n⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"N... | [
"case refine_1\nw₁ w₂ z₁ z₂ : ℝ≥0\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\n⊢ ∑ i, ![w₁, w₂] i = 1"
] | · simpa [Fin.sum_univ_succ] using h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 50
} | {
"line": 186,
"column": 51
} | [
{
"pp": "case inr.inr\nα : Type u_2\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\np q : ℝ\nhp✝ : 0 ≤ p\nhq✝ : 0 ≤ q\nhpq : p + q = 1\nhp : 0 < p\nhq : 0 < q\nh2p : 1 < 1 / p\nh2pq : (1 / p)⁻¹ + (1 / q)⁻¹ = 1\nthis :\n ∫⁻ (a : α), ((fun x ↦ f x ^ p) * f... | [
"case inr.inr\nα : Type u_2\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\np q : ℝ\nhp✝ : 0 ≤ p\nhq✝ : 0 ≤ q\nhpq : p + q = 1\nhp : 0 < p\nhq : 0 < q\nh2p : 1 < 1 / p\nh2pq : (1 / p)⁻¹ + (1 / q)⁻¹ = 1\nthis :\n ∫⁻ (a : α), ((fun x ↦ f x ^ p) * fun x ↦ g x ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 35
} | {
"line": 268,
"column": 36
} | [
{
"pp": "case refine_2\nw₁ w₂ z₁ z₂ : ℝ≥0∞\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\nh : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p\n⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case refine_2\nw₁ w₂ z₁ z₂ : ℝ≥0∞\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\nh : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p\n⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 37
} | {
"line": 269,
"column": 2
} | [
{
"pp": "case refine_2\nw₁ w₂ z₁ z₂ : ℝ≥0∞\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\nh : (∑ i, ![w₁, w₂] i * ![z₁, z₂] i) ^ p ≤ ∑ i, ![w₁, w₂] i * ![z₁, z₂] i ^ p\n⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case refine_1\nw₁ w₂ z₁ z₂ : ℝ≥0∞\nhw' : w₁ + w₂ = 1\np : ℝ\nhp : 1 ≤ p\n⊢ ∑ i, ![w₁, w₂] i = 1"
] | · simpa [Fin.sum_univ_succ] using h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 54
} | {
"line": 292,
"column": 4
} | [
{
"pp": "case neg\np : ℝ\nhp1 : 1 ≤ p\nhp_pos : 0 < p\na b : ℝ≥0\nh_top : ¬↑a + ↑b = ∞\n⊢ ↑a ^ p + ↑b ^ p ≤ (↑a + ↑b) ^ p",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"ENNReal.ofNNReal",
"ENNReal.instPowReal",
"id"... | [
"case neg\np : ℝ\nhp1 : 1 ≤ p\nhp_pos : 0 < p\na b : ℝ≥0\nh_top : ¬↑a + ↑b = ∞\n⊢ ↑a ^ p + ↑b ^ p ≤ (↑a + ↑b) ^ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 201,
"column": 8
} | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 16
} | [
{
"pp": "α : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ι → α → ℝ≥0∞\ni₀ : ι\ns : Finset ι\nhi₀ : i₀ ∉ s\nih :\n (∀ i ∈ s, AEMeasurable (f i) μ) →\n ∀ {p : ι → ℝ},\n ∑ i ∈ s, p i = 1 →\n (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f... | [
"α : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : ι → α → ℝ≥0∞\ni₀ : ι\ns : Finset ι\nhi₀ : i₀ ∉ s\nih :\n (∀ i ∈ s, AEMeasurable (f i) μ) →\n ∀ {p : ι → ℝ},\n ∑ i ∈ s, p i = 1 →\n (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 341,
"column": 34
} | {
"line": 341,
"column": 45
} | {
"line": 341,
"column": 46
} | [
{
"pp": "p : ℝ≥0∞\nh : p ∈ Set.Ioo 0 1\n⊢ p⁻¹ ≠ ∞",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"ENNReal.inv_eq_top._simp_1",
"id",
"Ne",
"Inv.inv",
"ENNReal",
"Zero.toOfNat0",
"ENNReal.instInv",
"ENNReal.i... | [
"p : ℝ≥0∞\nh : p ∈ Set.Ioo 0 1\n⊢ ¬p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 17
} | {
"line": 73,
"column": 18
} | [
{
"pp": "case inr.inl.convert_3\nα : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p < 1\n⊢ p.toReal ≤ 1",
"ppTerm": "?inr.inl.c... | [
"case inr.inl.convert_3\nα : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : p < 1\n⊢ p.toReal ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 42
} | {
"line": 74,
"column": 43
} | [
{
"pp": "case inr.inr\nα : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : 1 ≤ p\n⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p ... | [
"case inr.inr\nα : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\nhp : p ≠ 0\nh'p : 1 ≤ p\n⊢ eLpNorm (f + g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 48
} | {
"line": 105,
"column": 49
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\n⊢ eLpNorm (f - g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)",
"ppTerm": "?m.40",
"assigned": true,
... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\n⊢ eLpNorm (f + -g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 39
} | {
"line": 109,
"column": 40
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhp : 1 ≤ p\n⊢ eLpNorm (f - g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ",
"ppTerm": "?m.40",
"assigned": false,
"use... | [
"α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhp : 1 ≤ p\n⊢ eLpNorm (f - g) p μ ≤ eLpNorm f p μ + eLpNorm g p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 15
} | {
"line": 250,
"column": 16
} | [
{
"pp": "α : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Finset ι\ng : α → ℝ≥0∞\nf : ι → α → ℝ≥0∞\nhg : AEMeasurable g μ\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\nq : ℝ\np : ι → ℝ\nhpq : q + ∑ i ∈ s, p i = 1\nhq : 0 ≤ q\nhp : ∀ i ∈ s, 0 ≤ p i\nthis :\n ∫⁻ (t : α), ∏ j ∈ insertNone s, (j.el... | [
"α : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Finset ι\ng : α → ℝ≥0∞\nf : ι → α → ℝ≥0∞\nhg : AEMeasurable g μ\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\nq : ℝ\np : ι → ℝ\nhpq : q + ∑ i ∈ s, p i = 1\nhq : 0 ≤ q\nhp : ∀ i ∈ s, 0 ≤ p i\nthis :\n ∫⁻ (t : α), ∏ j ∈ insertNone s, (j.elim (g t) fun... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 255,
"column": 6
} | {
"line": 255,
"column": 17
} | {
"line": 255,
"column": 18
} | [
{
"pp": "case refine_1.some\nα : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Finset ι\ng : α → ℝ≥0∞\nf : ι → α → ℝ≥0∞\nhg : AEMeasurable g μ\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\nq : ℝ\np : ι → ℝ\nhpq : q + ∑ i ∈ s, p i = 1\nhq : 0 ≤ q\nhp : ∀ i ∈ s, 0 ≤ p i\ni : ι\nhi : some i ∈ insert... | [
"case refine_1.some\nα : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Finset ι\ng : α → ℝ≥0∞\nf : ι → α → ℝ≥0∞\nhg : AEMeasurable g μ\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\nq : ℝ\np : ι → ℝ\nhpq : q + ∑ i ∈ s, p i = 1\nhq : 0 ≤ q\nhp : ∀ i ∈ s, 0 ≤ p i\ni : ι\nhi : some i ∈ insertNone s\n⊢ i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 261,
"column": 6
} | {
"line": 261,
"column": 17
} | {
"line": 261,
"column": 18
} | [
{
"pp": "case refine_3.some\nα : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Finset ι\ng : α → ℝ≥0∞\nf : ι → α → ℝ≥0∞\nhg : AEMeasurable g μ\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\nq : ℝ\np : ι → ℝ\nhpq : q + ∑ i ∈ s, p i = 1\nhq : 0 ≤ q\nhp : ∀ i ∈ s, 0 ≤ p i\ni : ι\nhi : some i ∈ insert... | [
"case refine_3.some\nα : Type u_2\nι : Type u_3\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Finset ι\ng : α → ℝ≥0∞\nf : ι → α → ℝ≥0∞\nhg : AEMeasurable g μ\nhf : ∀ i ∈ s, AEMeasurable (f i) μ\nq : ℝ\np : ι → ℝ\nhpq : q + ∑ i ∈ s, p i = 1\nhq : 0 ≤ q\nhp : ∀ i ∈ s, 0 ≤ p i\ni : ι\nhi : some i ∈ insertNone s\n⊢ i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 27
} | {
"line": 270,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞\nhg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞\nhp1 : 1 ≤ p\na : α\n⊢ (f a + g a) ^ p ≤ 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p",
"ppTerm": "?m.122",
"assigned":... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞\nhg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞\nhp1 : 1 ≤ p\na : α\n⊢ (f a + g a) ^ p ≤ 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 81
} | {
"line": 503,
"column": 4
} | [
{
"pp": "a b p q : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhpq : p.HolderConjugate q\n⊢ a * b ≤ a ^ p / p + b ^ q / q",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.t... | [
"a b p q : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhpq : p.HolderConjugate q\n⊢ a * b ≤ p⁻¹ * a ^ p + q⁻¹ * b ^ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 518,
"column": 2
} | {
"line": 518,
"column": 81
} | {
"line": 519,
"column": 4
} | [
{
"pp": "a b p q : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhpq : p.HolderConjugate q\n⊢ a * b = a ^ p / p + b ^ q / q ↔ a ^ p = b ^ q",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivi... | [
"a b p q : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhpq : p.HolderConjugate q\n⊢ a * b = p⁻¹ * a ^ p + q⁻¹ * b ^ q ↔ a ^ p = b ^ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 535,
"column": 2
} | {
"line": 535,
"column": 43
} | {
"line": 535,
"column": 44
} | [
{
"pp": "a b : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ a * b ≤ a ^ p / p.toNNReal + b ^ q / q.toNNReal",
"ppTerm": "?m.27",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ a * b ≤ a ^ p / p.toNNReal + b ^ q / q.toNNReal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 43
} | {
"line": 545,
"column": 44
} | [
{
"pp": "a b : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ a * b = a ^ p / p.toNNReal + b ^ q / q.toNNReal ↔ a ^ p = b ^ q",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ a * b = a ^ p / p.toNNReal + b ^ q / q.toNNReal ↔ a ^ p = b ^ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 153,
"column": 48
} | {
"line": 153,
"column": 79
} | {
"line": 153,
"column": 80
} | [
{
"pp": "α : Type u_1\nε' : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\np q : ℝ≥0∞\nf : α → ε'\ns : Set α\nhfq : MemLp f q (μ.restrict (toMeasurable μ s))\nhf : ∀ x ∉ s, f x = 0\nhs : μ s ≠ ∞\nhpq : p ≤ q\nthis : (toMeasurable μ s).indicator f =... | [
"α : Type u_1\nε' : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\np q : ℝ≥0∞\nf : α → ε'\ns : Set α\nhfq : MemLp f q (μ.restrict (toMeasurable μ s))\nhf : ∀ x ∉ s, f x = 0\nhs : μ s ≠ ∞\nhpq : p ≤ q\nthis : (toMeasurable μ s).indicator f = f\n⊢ ¬μ s =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 306,
"column": 25
} | {
"line": 306,
"column": 35
} | {
"line": 306,
"column": 36
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\n... | [
"α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\nq2 : ℝ := p2... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 306,
"column": 36
} | {
"line": 306,
"column": 46
} | {
"line": 306,
"column": 47
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\n... | [
"α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\nq2 : ℝ := p2... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 223,
"column": 6
} | {
"line": 224,
"column": 81
} | {
"line": 226,
"column": 0
} | [
{
"pp": "case h₂\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np q r : ℝ\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G... | [] | simpa only [eLpNorm', enorm_mul, enorm_norm] using!
ENNReal.lintegral_Lp_mul_le_Lq_mul_Lr hro_lt hrp hpqr μ hf.enorm hg.enorm | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 235,
"column": 23
} | {
"line": 235,
"column": 34
} | {
"line": 235,
"column": 35
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\nq r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\nq r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥0\nh : ∀ᵐ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 239,
"column": 23
} | {
"line": 239,
"column": 34
} | {
"line": 239,
"column": 35
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥0\nh : ∀ᵐ (x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 245,
"column": 31
} | {
"line": 245,
"column": 42
} | {
"line": 245,
"column": 43
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np q r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np q r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥0\nh : ∀ᵐ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 84
} | {
"line": 645,
"column": 4
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\n⊢ ∑ i ∈ s, (f i * g i) ^ r ≤ (∑ i ∈ s, f i ^ p) ^ (r / p) * (∑ i ∈ s, g i ^ q) ^ (r / q)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul"... | [
"ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np q r : ℝ\nhpqr : p.HolderTriple q r\n⊢ ∑ x ∈ s, f x ^ r * g x ^ r ≤ (∑ i ∈ s, f i ^ p) ^ (r / p) * (∑ i ∈ s, g i ^ q) ^ (r / q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 245,
"column": 56
} | {
"line": 245,
"column": 67
} | {
"line": 245,
"column": 68
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np q r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np q r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥0\nh : ∀ᵐ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 709,
"column": 2
} | {
"line": 709,
"column": 13
} | {
"line": 709,
"column": 14
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ f i * g i) ∧\n ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)",
"ppTerm": "?m.84",
"assigned": true,
... | [
"ι : Type u\nf g : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : Summable fun i ↦ f i ^ p\nhg : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ f i * g i) ∧ ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ p⁻¹ * (∑' (i : ι), g i ^ q) ^ q⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 755,
"column": 4
} | {
"line": 755,
"column": 24
} | {
"line": 755,
"column": 25
} | [
{
"pp": "case refine_1\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ q) (B ^ q)\nH₁ : Summable fun i ↦ f i * g i\nH₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nhA : A ... | [
"case refine_1\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ q) (B ^ q)\nH₁ : Summable fun i ↦ f i * g i\nH₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nhA : A = (∑' (i : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 756,
"column": 4
} | {
"line": 756,
"column": 53
} | {
"line": 756,
"column": 54
} | [
{
"pp": "case refine_2\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ q) (B ^ q)\nH₁ : Summable fun i ↦ f i * g i\nH₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nhA : A ... | [
"case refine_2\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ q) (B ^ q)\nH₁ : Summable fun i ↦ f i * g i\nH₂ : ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1 / q)\nhA : A = (∑' (i : ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 771,
"column": 2
} | {
"line": 772,
"column": 56
} | {
"line": 773,
"column": 4
} | [
{
"pp": "case inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nq : ℝ := p / (p - 1)\nhpq : p.HolderConjugate q\nhp₁ : 1 / p * p = 1\nhq : 1 / q * p = p - 1\n⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": []... | [
"case inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nq : ℝ := p / (p - 1)\nhpq : p.HolderConjugate q\nhp₁ : 1 / p * p = 1\nhq : 1 / q * p = p - 1\n⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 792,
"column": 6
} | {
"line": 792,
"column": 36
} | {
"line": 792,
"column": 37
} | [
{
"pp": "case h.inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : 0 < ∑ i ∈ s, f i ^ p\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ (∑ i ∈ s, f i ^ p) ^ (1 / q) ≠ 0",
"ppTerm": "?h.inr✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case h.inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\nhf : 0 < ∑ i ∈ s, f i ^ p\nA : p + q - q ≠ 0\nB : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p\n⊢ ∃ x ∈ s, f x = 0 → p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 795,
"column": 4
} | {
"line": 795,
"column": 30
} | {
"line": 796,
"column": 6
} | [
{
"pp": "case right\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\ng : ι → ℝ≥0\nhg : g ∈ {g | ∑ i ∈ s, g i ^ q ≤ 1}\n⊢ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p)",
"ppTerm": "?right",
"assigned": false,
"usedConstants": [],... | [
"case right\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\ng : ι → ℝ≥0\nhg : g ∈ {g | ∑ i ∈ s, g i ^ q ≤ 1}\n⊢ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 30
} | {
"line": 350,
"column": 31
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\n𝕜 : Type u_3\nx✝ : MeasurableSpace α\ninst✝ : NormedCommRing 𝕜\nμ : Measure α\nf : ι → α → 𝕜\np : ι → ℝ≥0∞\ns : Finset ι\nhf : ∀ i ∈ s, MemLp (f i) (p i) μ\n⊢ MemLp (fun ω ↦ ∏ i ∈ s, f i ω) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ",
"ppTerm": "?m.32",
"assigned": false,
"usedCo... | [
"ι : Type u_1\nα : Type u_2\n𝕜 : Type u_3\nx✝ : MeasurableSpace α\ninst✝ : NormedCommRing 𝕜\nμ : Measure α\nf : ι → α → 𝕜\np : ι → ℝ≥0∞\ns : Finset ι\nhf : ∀ i ∈ s, MemLp (f i) (p i) μ\n⊢ MemLp (fun ω ↦ ∏ i ∈ s, f i ω) (∑ i ∈ s, (p i)⁻¹)⁻¹ μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 862,
"column": 4
} | {
"line": 862,
"column": 24
} | {
"line": 862,
"column": 25
} | [
{
"pp": "case refine_1\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ p) (B ^ p)\nhp' : p ≠ 0\nH₁ : Summable fun i ↦ (f i + g i) ^ p\nH₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p... | [
"case refine_1\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ p) (B ^ p)\nhp' : p ≠ 0\nH₁ : Summable fun i ↦ (f i + g i) ^ p\nH₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 863,
"column": 4
} | {
"line": 863,
"column": 45
} | {
"line": 863,
"column": 46
} | [
{
"pp": "case refine_2\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ p) (B ^ p)\nhp' : p ≠ 0\nH₁ : Summable fun i ↦ (f i + g i) ^ p\nH₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p... | [
"case refine_2\nι : Type u\nf g : ι → ℝ≥0\nA B : ℝ≥0\np : ℝ\nhp : 1 ≤ p\nhf : HasSum (fun i ↦ f i ^ p) (A ^ p)\nhg : HasSum (fun i ↦ g i ^ p) (B ^ p)\nhp' : p ≠ 0\nH₁ : Summable fun i ↦ (f i + g i) ^ p\nH₂ : (∑' (i : ι), (f i + g i) ^ p) ^ (1 / p) ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) + (∑' (i : ι), g i ^ p) ^ (1 / p)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 884,
"column": 48
} | {
"line": 884,
"column": 59
} | {
"line": 884,
"column": 60
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ ∑ i ∈ s, ?m.71 i ≤ (∑ i ∈ s, |f i| ^ p) ^ (1 / p) * (∑ i ∈ s, |g i| ^ q) ^ (1 / q)",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.t... | [
"ι : Type u\ns : Finset ι\nf g : ι → ℝ\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ ∑ i ∈ s, ?m.71 i ≤ (∑ i ∈ s, |f i| ^ p) ^ p⁻¹ * (∑ i ∈ s, |g i| ^ q) ^ q⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 978,
"column": 2
} | {
"line": 978,
"column": 13
} | {
"line": 978,
"column": 14
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np q : ℝ\nhpq : p.HolderConjugate q\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ f i * g i) ∧\n ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ (1 / p) * (∑' (i : ι), g i ^ q) ^ (1... | [
"ι : Type u\nf g : ι → ℝ\np q : ℝ\nhpq : p.HolderConjugate q\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ f i * g i) ∧ ∑' (i : ι), f i * g i ≤ (∑' (i : ι), f i ^ p) ^ p⁻¹ * (∑' (i : ι), g i ^ q) ^ q⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 13
} | {
"line": 282,
"column": 14
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ edist (MemLp.toLp f hf) 0 = eLpNorm f p μ",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhf : MemLp f p μ\n⊢ edist (MemLp.toLp f hf) 0 = eLpNorm f p μ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1117,
"column": 6
} | {
"line": 1118,
"column": 40
} | {
"line": 1118,
"column": 41
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0\n⊢ (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0",
"ppTerm": "?m.126",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": ... | [
"ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0\n⊢ (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1124,
"column": 4
} | {
"line": 1125,
"column": 39
} | {
"line": 1125,
"column": 40
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0\nH' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = ∞)\n⊢ (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞",
"ppTerm": "?m.247",
"assi... | [
"ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0\nH' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = ∞)\n⊢ (∀ i ∈ s, ¬f i = ∞) ∧ ∀ i ∈ s, ¬g i = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 424,
"column": 48
} | {
"line": 424,
"column": 59
} | {
"line": 424,
"column": 60
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜' E\ninst✝¹ : IsBoundedSMul 𝕜 ... | [
"α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedRing 𝕜\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜' E\ninst✝¹ : IsBoundedSMul 𝕜 E\ninst✝ : I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 449,
"column": 4
} | {
"line": 450,
"column": 11
} | {
"line": 450,
"column": 12
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedRing 𝕜\ninst✝⁵ : NormedRing 𝕜'\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜' E\ninst✝² : IsBoundedSMul 𝕜 ... | [
"α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedRing 𝕜\ninst✝⁵ : NormedRing 𝕜'\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Module 𝕜' E\ninst✝² : IsBoundedSMul 𝕜 E\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 62
} | {
"line": 487,
"column": 63
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_6\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\nhf : MemLp f p μ\nq : ℝ≥0∞\nq_top : ¬q = ∞\nq_zero : q = 0\np_zero : ¬p = 0\n⊢ eLpNorm (fun x ↦ ‖f x‖ₑ ^ 0) ∞ μ < ∞",
"ppTerm": "?neg✝",
"assi... | [
"case neg\nα : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_6\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\nhf : MemLp f p μ\nq : ℝ≥0∞\nq_top : ¬q = ∞\nq_zero : q = 0\np_zero : ¬p = 0\n⊢ eLpNormEssSup (fun x ↦ 1) μ < ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1142,
"column": 6
} | {
"line": 1142,
"column": 79
} | {
"line": 1142,
"column": 80
} | [
{
"pp": "ι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) = 0 ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = 0\n⊢ (∀ i ∈ s, w i = 0) ∨ ∀ i ∈ s, w i = 0 ∨ f i = 0",
"ppTerm": "?m.176",
"assigned": false,
"usedConstants": [],
... | [
"ι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) = 0 ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = 0\n⊢ (∀ i ∈ s, w i = 0) ∨ ∀ i ∈ s, w i = 0 ∨ f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 513,
"column": 2
} | {
"line": 521,
"column": 14
} | {
"line": 523,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_6\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nq : ℝ≥0∞\nf : α → ε\nhf : AEStronglyMeasurable f μ\nq_zero : q ≠ 0\nq_top : q ≠ ∞\n⊢ MemLp (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ ↔ MemLp f p μ",
"ppTerm": "?m.35",
... | [] | refine ⟨fun h => ?_, fun h => h.enorm_rpow_div q⟩
apply (memLp_enorm_iff hf).1
convert! h.enorm_rpow_div q⁻¹ using 1
· ext x
have : q.toReal * q.toReal⁻¹ = 1 :=
CommGroupWithZero.mul_inv_cancel q.toReal <| ENNReal.toReal_ne_zero.mpr ⟨q_zero, q_top⟩
simp [← ENNReal.rpow_mul, this, ENNReal.rpow_one]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 513,
"column": 2
} | {
"line": 521,
"column": 14
} | {
"line": 523,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_6\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nq : ℝ≥0∞\nf : α → ε\nhf : AEStronglyMeasurable f μ\nq_zero : q ≠ 0\nq_top : q ≠ ∞\n⊢ MemLp (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ ↔ MemLp f p μ",
"ppTerm": "?m.35",
... | [] | refine ⟨fun h => ?_, fun h => h.enorm_rpow_div q⟩
apply (memLp_enorm_iff hf).1
convert! h.enorm_rpow_div q⁻¹ using 1
· ext x
have : q.toReal * q.toReal⁻¹ = 1 :=
CommGroupWithZero.mul_inv_cancel q.toReal <| ENNReal.toReal_ne_zero.mpr ⟨q_zero, q_top⟩
simp [← ENNReal.rpow_mul, this, ENNReal.rpow_one]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 1148,
"column": 4
} | {
"line": 1148,
"column": 87
} | {
"line": 1148,
"column": 88
} | [
{
"pp": "ι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0\nH' : ¬((∑ i ∈ s, w i) ^ (1 - p⁻¹) = ∞ ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = ∞)\n⊢ (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞",... | [
"ι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0\nH' : ¬((∑ i ∈ s, w i) ^ (1 - p⁻¹) = ∞ ∨ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ = ∞)\n⊢ (∀ i ∈ s, ¬w i = ∞) ∧ ∀ i ∈ s, ¬w i * f i ^ p = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 79
} | {
"line": 300,
"column": 80
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k ↦ {x | 2⁻¹... | [
"α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : PseudoEMetricSpace E\nf : ℕ → α → E\ng : α → E\nhfg : TendstoInMeasure μ f atTop g\nh_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε\nns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg\nS : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 20
} | {
"line": 659,
"column": 21
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_6\nE : Type u_7\nF : Type u_8\nK : ℝ≥0\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : α → E\ng : E → F\nhg : LipschitzWith K g\ng0 : g 0 = 0\nhL : MemLp f p μ\nx : α\n⊢ ‖g (f x)‖ ≤ ↑K * ‖f x‖",
"ppTerm": "?m.41",
... | [
"p : ℝ≥0∞\nα : Type u_6\nE : Type u_7\nF : Type u_8\nK : ℝ≥0\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : α → E\ng : E → F\nhg : LipschitzWith K g\ng0 : g 0 = 0\nhL : MemLp f p μ\nx : α\n⊢ ‖g (f x)‖ ≤ ↑K * ‖f x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1160,
"column": 4
} | {
"line": 1160,
"column": 60
} | {
"line": 1160,
"column": 61
} | [
{
"pp": "case e'_3.a.inr\nι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0\nH' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞\nthis :\n ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑... | [
"case e'_3.a.inr\nι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0\nH' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞\nthis :\n ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ i ∈ s, ↑(w ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 717,
"column": 2
} | {
"line": 717,
"column": 42
} | {
"line": 717,
"column": 43
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf : ↥(Lp E p μ)\n⊢ ‖hg.compLp g0 f‖ ≤ ↑c * ‖f‖",
"ppTerm": "?m.35",
"assigned": false,... | [
"α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\nf : ↥(Lp E p μ)\n⊢ ‖hg.compLp g0 f‖ ≤ ↑c * ‖f‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1177,
"column": 2
} | {
"line": 1178,
"column": 85
} | {
"line": 1179,
"column": 4
} | [
{
"pp": "case inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0∞\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nq : ℝ := p / (p - 1)\nhpq : p.HolderConjugate q\nhp₁ : 1 / p * p = 1\nhq : 1 / q * p = p - 1\n⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [... | [
"case inr\nι : Type u\ns : Finset ι\nf : ι → ℝ≥0∞\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nq : ℝ := p / (p - 1)\nhpq : p.HolderConjugate q\nhp₁ : 1 / p * p = 1\nhq : 1 / q * p = p - 1\n⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1191,
"column": 4
} | {
"line": 1191,
"column": 85
} | {
"line": 1191,
"column": 86
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ p) ^ (1 / p) = ∞)\npos : 0 < p\n⊢ (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞",
"ppTerm": "?m.169",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
... | [
"ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ p) ^ (1 / p) = ∞)\npos : 0 < p\n⊢ (∀ i ∈ s, ¬f i = ∞) ∧ ∀ i ∈ s, ¬g i = ∞"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1190,
"column": 2
} | {
"line": 1191,
"column": 88
} | {
"line": 1192,
"column": 2
} | [
{
"pp": "case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : ¬((∑ i ∈ s, f i ^ p) ^ (1 / p) = ∞ ∨ (∑ i ∈ s, g i ^ p) ^ (1 / p) = ∞)\npos : 0 < p\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)",
"ppTerm": "?neg✝",
"assigned": ... | [
"case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\npos : 0 < p\nH' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)"
] | replace H' : (∀ i ∈ s, f i ≠ ⊤) ∧ ∀ i ∈ s, g i ≠ ⊤ := by
simpa [ENNReal.rpow_eq_top_iff, asymm pos, pos, ENNReal.sum_eq_top, not_or] using H' | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 68
} | {
"line": 330,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PseudoEMetricSpace E\nu : Filter ι\ninst✝¹ : u.NeBot\ninst✝ : u.IsCountablyGenerated\nf : ι → α → E\ng : α → E\nhfg : TendstoInMeasure μ f u g\nns : ℕ → ι\nh_tendsto_ns : Tendsto ns atTop u\n⊢ ∃ ns, Tendsto ns atTo... | [] | exact ⟨ns, h_tendsto_ns, fun ε hε => (hfg ε hε).comp h_tendsto_ns⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 913,
"column": 2
} | {
"line": 913,
"column": 28
} | {
"line": 914,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nν : Measure α\nc : ℝ≥0∞\nhc : c ≠ ∞\nh : μ ≤ c • ν\ninst✝ : Fact (1 ≤ p)\nf : ↥(Lp E p ν)\n⊢ ‖(LpToLpOfMeasureLeSMulₗ hc h) f‖ ≤ c.toReal ^ (1 / p).toReal * ‖f‖",
"pp... | [
"α : Type u_1\nE : Type u_4\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nν : Measure α\nc : ℝ≥0∞\nhc : c ≠ ∞\nh : μ ≤ c • ν\ninst✝ : Fact (1 ≤ p)\nf : ↥(Lp E p ν)\n⊢ ‖(LpToLpOfMeasureLeSMulₗ hc h) f‖ₑ.toReal ≤ c.toReal ^ (1 / p).toReal * ‖f‖ₑ.toReal"
] | simp only [← toReal_enorm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 1016,
"column": 7
} | {
"line": 1016,
"column": 49
} | {
"line": 1016,
"column": 50
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np✝ : ℝ≥0∞\nμ : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\nR : Type u_6\ninst✝² : NormedAddCommGroup R\ninst✝¹ : StarAddMonoid R\ninst✝ : NormedStarGroup R\np : ℝ≥0∞\nf : ↥(Lp R ... | [
"α : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np✝ : ℝ≥0∞\nμ : Measure α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\nR : Type u_6\ninst✝² : NormedAddCommGroup R\ninst✝¹ : StarAddMonoid R\ninst✝ : NormedStarGroup R\np : ℝ≥0∞\nf : ↥(Lp R p μ)\n⊢ eLpN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 288,
"column": 49
} | {
"line": 288,
"column": 60
} | {
"line": 288,
"column": 61
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : ℕ → α → E\np : ℝ\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nhB : ∑' (i : ℕ), B i ≠ ∞\nh_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : ℕ → α → E\np : ℝ\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nhB : ∑' (i : ℕ), B i ≠ ∞\nh_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm' (f n - f m_1) p μ < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 41
} | {
"line": 420,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\ninst✝ : SeminormedAddCommGroup E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ≥0∞\nhε : 0 < ε\nhε_top : ... | [
"α : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nf : ι → α → E\ng : α → E\ninst✝ : SeminormedAddCommGroup E\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nhf : ∀ (n : ι), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nl : Filter ι\nε : ℝ≥0∞\nhε : 0 < ε\nhε_top : ε ≠ ∞\nhfg :... | rw [ENNReal.tendsto_nhds_zero] at hfg ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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