module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Convex.Slope | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 13
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Slope | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 13
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 35
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"PartialOrder.toPr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 37
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"i... | · 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
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 201,
"column": 8
} | {
"line": 202,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 288,
"column": 2
} | {
"line": 289,
"column": 80
} | [
{
"pp": "w₁ w₂ p₁ p₂ : ℝ≥0\n⊢ w₁ + w₂ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ ≤ w₁ * p₁ + w₂ * p₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 295,
"column": 2
} | {
"line": 297,
"column": 20
} | [
{
"pp": "w₁ w₂ w₃ p₁ p₂ p₃ : ℝ≥0\n⊢ w₁ + w₂ + w₃ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ * p₃ ^ ↑w₃ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃",
"usedConstants": [
"Eq.mpr",
"NNReal.instCommSemiring",
"Semigroup.toMul",
"Real",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 303,
"column": 2
} | {
"line": 305,
"column": 20
} | [
{
"pp": "w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ≥0\n⊢ w₁ + w₂ + w₃ + w₄ = 1 → p₁ ^ ↑w₁ * p₂ ^ ↑w₂ * p₃ ^ ↑w₃ * p₄ ^ ↑w₄ ≤ w₁ * p₁ + w₂ * p₂ + w₃ * p₃ + w₄ * p₄",
"usedConstants": [
"Eq.mpr",
"NNReal.instCommSemiring",
"Semigroup.toMul",
"Real",
"HMul.hMul",
"CommSemiring.toNonUnitalC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 88
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toIn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 421,
"column": 2
} | {
"line": 421,
"column": 62
} | [
{
"pp": "a b : ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ a * b ≤ a ^ p / p.toNNReal + b ^ q / q.toNNReal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 35
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"HMul.hMul"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 37
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"instNeZeroNatHAdd_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
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Real",
"ENNReal.ofNNReal",
"ENNReal.instPowReal",
"id",
"LE.le",
"instHAdd",
"HPo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 341,
"column": 34
} | {
"line": 341,
"column": 45
} | [
{
"pp": "p : ℝ≥0∞\nh : p ∈ Set.Ioo 0 1\n⊢ p⁻¹ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ENNReal.inv_eq_top._simp_1",
"id",
"Ne",
"Inv.inv",
"ENNReal",
"Zero.toOfNat0",
"ENNReal.instInv",
"ENNReal.instTop",
"ENNReal.instZero",
"Top... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 255,
"column": 6
} | {
"line": 255,
"column": 17
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 507,
"column": 2
} | {
"line": 507,
"column": 84
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"congrArg",
"Finset",
"Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 261,
"column": 6
} | {
"line": 261,
"column": 17
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 27
} | [
{
"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",
"usedConstants": [
"ENNReal.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 572,
"column": 2
} | {
"line": 572,
"column": 13
} | [
{
"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)",
"usedConstants": [
"NNReal.instTopolog... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 618,
"column": 4
} | {
"line": 618,
"column": 24
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 619,
"column": 4
} | {
"line": 619,
"column": 53
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 634,
"column": 2
} | {
"line": 635,
"column": 56
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 655,
"column": 6
} | {
"line": 655,
"column": 36
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
"GroupWithZero.toMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 658,
"column": 4
} | {
"line": 658,
"column": 30
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 306,
"column": 25
} | {
"line": 306,
"column": 35
} | [
{
"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... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 306,
"column": 36
} | {
"line": 306,
"column": 46
} | [
{
"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... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 154,
"column": 48
} | {
"line": 154,
"column": 79
} | [
{
"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 =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Convex.Mul | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nG : Type u_4\ninst✝¹¹ : CommRing 𝕜\ninst✝¹⁰ : LinearOrder 𝕜\ninst✝⁹ : IsStrictOrderedRing 𝕜\ninst✝⁸ : CommRing E\ninst✝⁷ : LinearOrder E\ninst✝⁶ : IsStrictOrderedRing E\ninst✝⁵ : AddCommGroup G\ninst✝⁴ : Module 𝕜 G\ninst✝³ : Module 𝕜 E\ns : Set G\ninst✝² : IsOrderedMod... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 725,
"column": 4
} | {
"line": 725,
"column": 24
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 726,
"column": 4
} | {
"line": 726,
"column": 45
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 13
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf g : ι → ℝ\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)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 747,
"column": 48
} | {
"line": 747,
"column": 59
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 17
} | [
{
"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",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 17
} | [
{
"pp": "case h₁.h₁\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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 42
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 56
} | [
{
"pp": "case bc\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 using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 48
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"ENN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 39
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 236,
"column": 23
} | {
"line": 236,
"column": 34
} | [
{
"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≥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 240,
"column": 23
} | {
"line": 240,
"column": 34
} | [
{
"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≥... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 246,
"column": 31
} | {
"line": 246,
"column": 42
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 246,
"column": 56
} | {
"line": 246,
"column": 67
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 842,
"column": 2
} | {
"line": 842,
"column": 13
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nf : α → E\np : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nφ : α → 𝕜\n⊢ eLpNorm (φ • f) p μ ≤ eLpNorm φ ∞ μ * eL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 286,
"column": 28
} | {
"line": 286,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nf : α → E\np : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nφ : α → 𝕜\nx✝ : α\n⊢ ‖(fun x1 x2 ↦ x1 • x2) (φ x✝) (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\np : ℝ≥0∞\nf : α → E\nφ : α → 𝕜\nhφ : AEStronglyMeasurable φ μ\n⊢ eLpNorm (φ • f) p μ ≤ eLpNorm φ p μ * eL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 291,
"column": 28
} | {
"line": 291,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\np : ℝ≥0∞\nf : α → E\nφ : α → 𝕜\nhφ : AEStronglyMeasurable φ μ\nx✝ : α\n⊢ ‖(fun x1 x2 ↦ x1 • x2) (φ x✝) (f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\np q r : ℝ\nf : α → E\nhf : AEStronglyMeasurable f μ\nφ : α → 𝕜\nhφ : AEStronglyMeasurable φ μ\nhp0_lt : 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 297,
"column": 28
} | {
"line": 297,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\np q r : ℝ\nf : α → E\nhf : AEStronglyMeasurable f μ\nφ : α → 𝕜\nhφ : AEStronglyMeasurable φ μ\nhp0_lt : 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\np q r : ℝ≥0∞\nf : α → E\nhf : AEStronglyMeasurable f μ\nφ : α → 𝕜\nhφ : AEStronglyMeasurable φ μ\nhpqr : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 305,
"column": 30
} | {
"line": 305,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\nE : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedRing 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MulActionWithZero 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\np q r : ℝ≥0∞\nf : α → E\nhf : AEStronglyMeasurable f μ\nφ : α → 𝕜\nhφ : AEStronglyMeasurable φ μ\nhpqr : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 30
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 981,
"column": 6
} | {
"line": 982,
"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\n⊢ (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 988,
"column": 4
} | {
"line": 989,
"column": 39
} | [
{
"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 ≠ ∞",
"usedConstants": [
"Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1006,
"column": 6
} | {
"line": 1006,
"column": 79
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1012,
"column": 4
} | {
"line": 1012,
"column": 87
} | [
{
"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 ≠ ∞",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1023,
"column": 4
} | {
"line": 1023,
"column": 60
} | [
{
"pp": "case h.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 ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1040,
"column": 2
} | {
"line": 1041,
"column": 85
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1054,
"column": 4
} | {
"line": 1054,
"column": 85
} | [
{
"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 ≠ ∞",
"usedConstants": [
"Finset",
"Membership.mem",
"id",
"Ne",
"And",
"Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.MeanInequalities | {
"line": 1053,
"column": 2
} | {
"line": 1054,
"column": 88
} | [
{
"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)",
"usedConstants": [
"Real.inst... | 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.LpSpace.Complete | {
"line": 288,
"column": 49
} | {
"line": 288,
"column": 60
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 79
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 68
} | [
{
"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.ConvergenceInMeasure | {
"line": 419,
"column": 2
} | {
"line": 419,
"column": 41
} | [
{
"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 : ... | rw [ENNReal.tendsto_nhds_zero] at hfg ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 452,
"column": 2
} | {
"line": 452,
"column": 41
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_5\ninst✝ : SeminormedAddCommGroup E\nf : ι → α → E\ng : α → E\nl : Filter ι\nδ : ℝ≥0∞\nhδ : 0 < δ\nhδ_top : δ ≠ ∞\nhfg : Tendsto (fun n ↦ essSup (fun x ↦ ‖(f n - g) x‖ₑ) μ) l (𝓝 0)\n⊢ Tendsto (fun i ↦ μ {x | δ ≤ edist (f i x)... | rw [ENNReal.tendsto_nhds_zero] at hfg ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Multilinear.Bounded | {
"line": 54,
"column": 45
} | {
"line": 54,
"column": 56
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nF : Type u_3\nE : ι → Type u_4\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : (i : ι) → AddCommGroup (E i)\ninst✝⁵ : (i : ι) → Module 𝕜 (E i)\ninst✝⁴ : (i : ι) → TopologicalSpace (E i)\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\ninst✝ : Nonempty ι\ns :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 383,
"column": 12
} | {
"line": 383,
"column": 86
} | [
{
"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✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nhp : Fact (1 ≤ p)\nf g : ↥(Lp E p μ)\nthis : ‖f + g‖ₑ ≤ ‖f‖ₑ + ‖g‖ₑ\n⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 412,
"column": 48
} | {
"line": 412,
"column": 59
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 437,
"column": 4
} | {
"line": 438,
"column": 11
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 475,
"column": 4
} | {
"line": 475,
"column": 62
} | [
{
"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 < ∞",
"usedConstants": [
"Eq... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Multilinear.Bounded | {
"line": 80,
"column": 14
} | {
"line": 80,
"column": 37
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nF : Type u_3\nE : ι → Type u_4\ninst✝⁷ : NormedField 𝕜\ninst✝⁶ : (i : ι) → AddCommGroup (E i)\ninst✝⁵ : (i : ι) → Module 𝕜 (E i)\ninst✝⁴ : (i : ι) → TopologicalSpace (E i)\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\ninst✝ : Nonempty ι\ns :... | rw [norm_mul, norm_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 501,
"column": 2
} | {
"line": 509,
"column": 14
} | [
{
"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",
"usedConstants": [
... | 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": 501,
"column": 2
} | {
"line": 509,
"column": 14
} | [
{
"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",
"usedConstants": [
... | 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.MeasureTheory.Function.LpSpace.Basic | {
"line": 647,
"column": 4
} | {
"line": 647,
"column": 20
} | [
{
"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‖",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 42
} | [
{
"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‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Units | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : Rˣ\n⊢ (fun t ↦ (↑x + t)⁻¹ʳ - ↑x⁻¹) =O[𝓝 0] fun t ↦ ‖t‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Units | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : Rˣ\nh_is_o : (fun t ↦ (↑x + t)⁻¹ʳ - ↑x⁻¹) =o[𝓝 0] fun x ↦ 1\nh_lim : Tendsto (fun y ↦ y - ↑x) (𝓝 ↑x) (𝓝 0)\n⊢ Tendsto (fun e ↦ ‖e⁻¹ʳ - ↑x⁻¹‖) (𝓝 ↑x) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Units | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nI : Ideal R\nx : R\nhxI : x ∈ I\nhx : ‖1 - x‖ < 1\nu : Rˣ := Units.oneSub (1 - x) hx\n⊢ 1 ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Ring.Units | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nI : Ideal R\nhI : I ≠ ⊤\nh : closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑I ⊆ nonunits R\n⊢ I.closure ≠ ⊤",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semiring.toMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 944,
"column": 7
} | {
"line": 944,
"column": 49
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Mul | {
"line": 42,
"column": 52
} | {
"line": 42,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝³ : NonUnitalSeminormedRing R\ninst✝² : NormedSpace 𝕜 R\ninst✝¹ : IsScalarTower 𝕜 R R\ninst✝ : SMulCommClass 𝕜 R R\nx y : R\n⊢ ‖((LinearMap.mul 𝕜 R) x) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Mul | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : SeminormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁴ : SeminormedRing R\ninst✝³ : NormedAlgebra 𝕜 R\ninst✝² : Module R E\ninst✝¹ : IsBoundedSMul R E\ninst✝ : IsScalarTower 𝕜 R E\nc : R\nx : E\n⊢ ‖((Algebra... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Mul | {
"line": 206,
"column": 15
} | {
"line": 206,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : SeminormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁴ : SeminormedRing R\ninst✝³ : NormedAlgebra 𝕜 R\ninst✝² : Module R E\ninst✝¹ : IsBoundedSMul R E\ninst✝ : IsScalarTower 𝕜 R E\nx y : E\nh : (lsmul 𝕜 R).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Mul | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 13
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁵ : NormedDivisionRing R\ninst✝⁴ : NormedAlgebra 𝕜 R\ninst✝³ : Module R E\ninst✝² : NormSMulClass R E\ninst✝¹ : IsScalarTower 𝕜 R E\ninst✝ : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 13
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np q : M →L[R] M\nhp : IsIdempotentElem p\nhq : IsIdempotentElem q\n⊢ p = q ↔ (↑p).range = (↑q).range ∧ (↑p).ker = (↑q).ker",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf T : M →L[R] M\nhf : IsIdempotentElem f\n⊢ (↑f).range ∈ Module.End.invtSubmodule ↑T ↔ f ∘SL T ∘SL f = T ∘SL f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 46
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf T : M →L[R] M\nhf : IsIdempotentElem f\n⊢ (↑f).ker ∈ Module.End.invtSubmodule ↑T ↔ f ∘SL T ∘SL f = f ∘SL T",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 71
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf T : M →L[R] M\nhf : IsIdempotentElem f\n⊢ Commute f T ↔ (↑f).range ∈ Module.End.invtSubmodule ↑T ∧ (↑f).ker ∈ Module.End.invtSubmodule ↑T",
"usedConstants": [
"Sublattice",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 91
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsTopologicalAddGroup M\nf : M →L[R] M\nhf : IsIdempotentElem f\nT : (M →L[R] M)ˣ\nthis : IsUnit (toLinearMapRingHom ↑T)\n⊢ Commute f ↑T ↔ Submodule.map (↑↑T) (↑f).range = (↑f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 135,
"column": 30
} | {
"line": 135,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : Module 𝕜 F\nf : E → F\nhf : IsBoundedLinearMap 𝕜 f\n⊢ ∃ M, 0 < M ∧ ∀ (x : E), ‖-f x‖ ≤ M * ‖x‖",
"usedConstants": [
"Norm.norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 147,
"column": 50
} | {
"line": 147,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : Module 𝕜 F\nf g : E → F\nhf : IsBoundedLinearMap 𝕜 f\nhg : IsBoundedLinearMap 𝕜 g\n⊢ IsBoundedLinearMap 𝕜 fun e ↦ f e - g e",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 237,
"column": 49
} | {
"line": 237,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : Module 𝕜 G\nf : E × F → G\nh : IsBoundedBilinearMap 𝕜 f\nC : ℝ\nl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 254,
"column": 4
} | {
"line": 254,
"column": 84
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : Module 𝕜 G\nf : E × F → G\nh : IsBoundedBilinearMap 𝕜 f\nx : E × ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Maps.Strict.Basic | {
"line": 115,
"column": 20
} | {
"line": 115,
"column": 31
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nf_quot : IsQuotientMap f\nΦ : ↑(range (g ∘ f)) ≃ₜ ↑(range g) := Homeomorph.setCongr ⋯\nkey : rangeFactorization g ∘ f = ⇑Φ ∘ rangeFactorization (g ∘ f)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Maps.Strict.Basic | {
"line": 132,
"column": 15
} | {
"line": 132,
"column": 39
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\ng_emb : IsEmbedding g\nΦ : Quotient (ker (g ∘ f)) ≃ₜ Quotient (ker f) := Homeomorph.Quotient.congrRight ⋯\nkey : g ∘ kerLift f ∘ ⇑Φ = kerLift (g ∘ f)\nH ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Complement | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nthis : IsIdempotentElem 0\n⊢ IsTopCompl ⊥ ⊤",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Complement | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝³ : Ring R\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nthis : IsIdempotentElem (ContinuousLinearMap.id R M)\n⊢ IsTopCompl ⊤ ⊥",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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