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.Real.Hyperreal | {
"line": 346,
"column": 2
} | {
"line": 346,
"column": 31
} | [
{
"pp": "x : ℝ*\nhx : Germ.Tendsto x atTop\nthis : 0 < x\nn : ℕ\n⊢ n • |ArchimedeanOrder.val (ArchimedeanOrder.of 1)| < |ArchimedeanOrder.val (ArchimedeanOrder.of x)|",
"usedConstants": [
"ArchimedeanOrder.of",
"Hyperreal.instField",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 39
} | [
{
"pp": "x : ℝ*\nhx : Germ.Tendsto x atBot\nthis : x < 0\nn : ℕ\n⊢ n • |ArchimedeanOrder.val (ArchimedeanOrder.of 1)| < |ArchimedeanOrder.val (ArchimedeanOrder.of x)|",
"usedConstants": [
"ArchimedeanOrder.of",
"AddGroup.toSubtractionMonoid",
"Hyperreal.instField",
"Eq.mpr",
"N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 383,
"column": 6
} | {
"line": 383,
"column": 17
} | [
{
"pp": "case refine_2\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s < r\n⊢ coeRingHom s ≤ x",
"usedConstants": [
"Hyperreal.instField",
"Real",
"PartialOrder.toPreorder",
"OrderRingHom.instFunLike",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"Real.semiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 384,
"column": 6
} | {
"line": 384,
"column": 17
} | [
{
"pp": "case refine_3\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s > r\n⊢ x ≤ coeRingHom s",
"usedConstants": [
"Hyperreal.instField",
"Real",
"PartialOrder.toPreorder",
"OrderRingHom.instFunLike",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"Real.semiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 414,
"column": 32
} | {
"line": 414,
"column": 68
} | [
{
"pp": "x : ℝ\nhx : Irrational x\na b : ℤ\np_nonzero : C a * X + C b ≠ 0\nx_is_root : (aeval x) (C a * X + C b) = 0\n⊢ ↑a * x = -↑b",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddZeroClass",
"AddZeroC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 434,
"column": 4
} | {
"line": 434,
"column": 31
} | [
{
"pp": "case refine_1\nx : ℝ*\nh : x.InfinitePos\nhx : 0 < x\nn : ℕ\n⊢ n • |ArchimedeanOrder.val (ArchimedeanOrder.of 1)| < |ArchimedeanOrder.val (ArchimedeanOrder.of x)|",
"usedConstants": [
"ArchimedeanOrder.of",
"Hyperreal.instField",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 39
} | [
{
"pp": "case refine_1\nx : ℝ*\nh : x.InfiniteNeg\nhx : x < 0\nn : ℕ\n⊢ n • |ArchimedeanOrder.val (ArchimedeanOrder.of 1)| < |ArchimedeanOrder.val (ArchimedeanOrder.of x)|",
"usedConstants": [
"ArchimedeanOrder.of",
"AddGroup.toSubtractionMonoid",
"Hyperreal.instField",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 961,
"column": 32
} | {
"line": 961,
"column": 59
} | [
{
"pp": "x y : ℝ*\nhx : x.Infinitesimal\nhy : y.Infinitesimal\n⊢ (x + y).Infinitesimal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 966,
"column": 2
} | {
"line": 966,
"column": 29
} | [
{
"pp": "x : ℝ*\nhx : x.Infinitesimal\n⊢ (-x).Infinitesimal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 976,
"column": 32
} | {
"line": 976,
"column": 59
} | [
{
"pp": "x y : ℝ*\nhx : x.Infinitesimal\nhy : y.Infinitesimal\n⊢ (x * y).Infinitesimal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 998,
"column": 2
} | {
"line": 998,
"column": 29
} | [
{
"pp": "x : ℝ*\nr : ℝ\nhxr : x.IsSt r\n⊢ (x - ↑r).Infinitesimal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 86,
"column": 4
} | {
"line": 86,
"column": 47
} | [
{
"pp": "case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → dist (sqrt (x * y)) ((x + y) / 2) ≤ dist x y / 2\nh : ¬x ≤ y\n⊢ dist (sqrt (x * y)) ((x + y) / 2) ≤ dist x y / 2",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instLE",
"Real",
"instHDi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 126,
"column": 2
} | {
"line": 129,
"column": 55
} | [
{
"pp": "x y : ℝ≥0\nh : x ≠ y\nn m : ℕ\n⊢ (x.agmSequences y n).1 < (x.agmSequences y m).2",
"usedConstants": [
"NNReal.agmSequences",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal",
"NNReal.agmSequences_fst_monotone",
"le_total",
"LE.le"... | suffices ∀ {k}, (agmSequences x y k).1 < (agmSequences x y k).2 by
obtain h | h := le_total n m
· exact (agmSequences_fst_monotone h).trans_lt this
· exact this.trans_le (agmSequences_snd_antitone h) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 122,
"column": 81
} | {
"line": 138,
"column": 91
} | [
{
"pp": "z : ℂ\nhz : ‖z‖ < 1\n⊢ HasSum (fun n ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)) z.arctan",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Even.add_one._simp_1",
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"Group... | by
have := ((hasSum_taylorSeries_log (z := z * I) (by simpa)).add
(hasSum_taylorSeries_neg_log (z := z * I) (by simpa))).mul_left (-I / 2)
simp_rw [← add_div, ← add_one_mul, hasSum_arctan_aux hz] at this
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
dsimp [Function.comp_def] at this
simp_rw [← m... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 36
} | [
{
"pp": "case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → x.agm y ≤ max x y\nh : ¬x ≤ y\n⊢ x.agm y ≤ max x y",
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.agm",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"NNReal.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 36
} | [
{
"pp": "case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → min x y ≤ x.agm y\nh : ¬x ≤ y\n⊢ min x y ≤ x.agm y",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.agm",
"id",
"NNReal",
"NNReal.instSemi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 13
} | [
{
"pp": "x y : ℝ≥0\n⊢ x.agm y = (sqrt (x * y)).agm ((x + y) / 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 36
} | [
{
"pp": "case inr\nx y : ℝ≥0\nhx : 0 < x\nhy : 0 < y\nhn : x ≠ y\nthis : ∀ {x y : ℝ≥0}, 0 < x → 0 < y → x ≠ y → x < y → min x y < x.agm y\nh : ¬x < y\n⊢ min x y < x.agm y",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"PartialOrder.toPreorder",
"NNR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 176,
"column": 6
} | {
"line": 176,
"column": 30
} | [
{
"pp": "case refine_2\nn : ℕ\n⊢ ((monomial 2) (-4)).natDegree + (sinPoly n).natDegree ≤ n + 2",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"Semiring.toModule",
"Polynomial.instNeg",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 36
} | [
{
"pp": "case inr\nx y : ℝ≥0\nhn : x ≠ y\nthis : ∀ {x y : ℝ≥0}, x ≠ y → x < y → x.agm y < max x y\nh : ¬x < y\n⊢ x.agm y < max x y",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"lt_sup_iff._simp_3",
"PartialOrder.toPreorder",
"NNReal.agm",
"SemilatticeInf.toPartialOrder... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 23
} | [
{
"pp": "x✝ y✝ x y : ℝ≥0\nhn : x ≠ y\nh : x < y\n⊢ x.agm y < max x y",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"NNReal.agm",
"SemilatticeSup.toMax",
"NNReal.instSemilatticeSup",
"id",
"LT.lt.le",
"NNReal... | max_eq_right h.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 145,
"column": 10
} | {
"line": 145,
"column": 21
} | [
{
"pp": "p : ℝ\nh : p ≠ 2⁻¹\nthis : ∀ {p : ℝ}, p ≠ 2⁻¹ → p < 2⁻¹ → binEntropy p < log 2\nhp : ¬p < 2⁻¹\n⊢ 1 - p < 2⁻¹",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"sub_lt_comm",
"congrArg",
"Real.instInv",
"instIsLeftCancelAddOfAddLeftReflectLE",
... | sub_lt_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Choose | {
"line": 34,
"column": 12
} | {
"line": 34,
"column": 23
} | [
{
"pp": "case zero\n⊢ (fun n ↦ ↑(n.descFactorial 0)) ~[atTop] fun n ↦ ↑n ^ 0",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Choose | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 15
} | [
{
"pp": "case succ\nk : ℕ\nh : (fun n ↦ ↑(n.descFactorial k)) ~[atTop] fun n ↦ ↑n ^ k\nhz : ∀ᶠ (x : ℕ) in atTop, ↑x ≠ 0\n⊢ Tendsto (fun n ↦ ((fun n ↦ ↑(n - k)) / Nat.cast) (n + k)) atTop (𝓝 1)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 44
} | [
{
"pp": "case inr.inl\nq : ℕ\nhp₀✝ : 0 ≤ 1\nhp₁ : 1 ≤ 1\nhp₀ : 0 < 1\n⊢ 0 ≤ qaryEntropy q 1",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.qaryEntropy",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"AddMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 69
} | [
{
"pp": "case hf\n⊢ Tendsto (fun x ↦ log (1 - x)) (𝓝[<] 1) atBot",
"usedConstants": [
"Real",
"Set.Ioi",
"Real.instZero",
"nhdsWithin",
"PseudoMetricSpace.toUniformSpace",
"Real.tendsto_log_nhdsGT_zero",
"Real.log",
"Filter.Tendsto",
"Filter.atBot",
... | have : Tendsto log (𝓝[>] 0) atBot := Real.tendsto_log_nhdsGT_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 31
} | [
{
"pp": "l : Filter ℂ\nhre : Tendsto re l atTop\nhim : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l fun z ↦ |z.im|\n⊢ im =O[l] fun z ↦ z.re ^ 0",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"Monoid.toMulOneClass",
"congrArg",
"Complex.im",
"Asymptotics.IsBigO",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 13
} | [
{
"pp": "l : Filter ℂ\nhl : IsExpCmpFilter l\nn : ℕ\n⊢ (fun z ↦ |z.im| ^ n) ≤ᶠ[l] fun z ↦ Real.exp z.re",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 358,
"column": 8
} | {
"line": 358,
"column": 98
} | [
{
"pp": "case neg.inl\nq : ℕ\np : ℝ\nis_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)\nh : DifferentiableAt ℝ (deriv (qaryEntropy q)) p\ncontAt : ContinuousAt (deriv (qaryEntropy q)) p\nh✝ : p = 0\n⊢ False",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"False",
"Real.qaryEntropy",
... | simp_all [not_continuousAt_deriv_qaryEntropy_zero, not_continuousAt_deriv_qaryEntropy_one] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 358,
"column": 8
} | {
"line": 358,
"column": 98
} | [
{
"pp": "case neg.inr\nq : ℕ\np : ℝ\nis_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)\nh : DifferentiableAt ℝ (deriv (qaryEntropy q)) p\ncontAt : ContinuousAt (deriv (qaryEntropy q)) p\nh✝ : p = 1\n⊢ False",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"False",
"Real.qaryEntropy",
... | simp_all [not_continuousAt_deriv_qaryEntropy_zero, not_continuousAt_deriv_qaryEntropy_one] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 188,
"column": 54
} | {
"line": 188,
"column": 65
} | [
{
"pp": "l : Filter ℂ\nhl : IsExpCmpFilter l\na : ℂ\nb : ℝ\nhb : b < 0\n⊢ (fun z ↦ cexp (↑b * z)) =o[l] fun z ↦ z ^ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 33
} | [
{
"pp": "l : Filter ℂ\nb₁ b₂ : ℝ\nhl : IsExpCmpFilter l\nhb : b₁ < b₂\nm n : ℕ\n⊢ (fun z ↦ z ^ m * cexp (↑b₁ * z)) =o[l] fun z ↦ z ^ n * cexp (↑b₂ * z)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 33
} | [
{
"pp": "l : Filter ℂ\nb₁ b₂ : ℝ\nhl : IsExpCmpFilter l\nhb : b₁ < b₂\nm n : ℤ\n⊢ (fun z ↦ z ^ m * cexp (↑b₁ * z)) =o[l] fun z ↦ z ^ n * cexp (↑b₂ * z)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 20
} | [
{
"pp": "case pos\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsStrictlyPositive a\n⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)\n (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a els... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.EulerMascheroni | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 15
} | [
{
"pp": "this : Tendsto (fun n ↦ eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0)\n⊢ Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 30
} | [
{
"pp": "⊢ HasDerivAt Gamma (-γ) 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 30
} | [
{
"pp": "⊢ HasDerivAt Gamma (-↑γ) 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 23
} | [
{
"pp": "case refine_2\nthis : HasDerivAt Gamma ↑(-√π * (γ + 2 * Real.log 2)) ↑(1 / 2)\n⊢ HasDerivAt Gamma (-↑√π * (↑γ + 2 * log 2)) (1 / 2)",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Complex.log",
"DivInvMonoid.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 57,
"column": 25
} | {
"line": 57,
"column": 36
} | [
{
"pp": "s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ s ≠ 0",
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 34
} | [
{
"pp": "a : ℝ\nha : a < 0\nb s : ℝ\nthis : (fun x ↦ rexp (a * x ^ 2 + b * x)) =o[atTop] fun x ↦ rexp (-x)\n⊢ (fun x ↦ rexp (-x)) =o[atTop] fun x ↦ x ^ s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 79,
"column": 12
} | {
"line": 79,
"column": 23
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStrictlyPositive x⁻¹ʳ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | [
{
"pp": "case e_a\np t x : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nhx : 0 ≤ x\n⊢ x ^ (p - 1) * x = x ^ p",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 162,
"column": 14
} | {
"line": 162,
"column": 46
} | [
{
"pp": "p t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 ≤ t\nx : ℝ\nhx : x ∈ Ici 0\ny : ℝ\nhy : y ∈ Ici 0\nhxy : x ≤ y\nh : x = 0\n⊢ p.rpowIntegrand₀₁ t x ≤ p.rpowIntegrand₀₁ t y",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"sub_self",
"Real.instZero",
"Re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 169,
"column": 2
} | {
"line": 177,
"column": 87
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\ns : Set ℝ\nhs : s ⊆ Ici 0\n⊢ ContinuousOn (Function.uncurry p.rpowIntegrand₀₁) (Ioi 0 ×ˢ s)",
"usedConstants": [
"Set.instSProd",
"Real.instPow",
"Real",
"Set.Ioi",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.... | let g : ℝ × ℝ → ℝ := fun q => q.1 ^ (p - 1) * q.2 / (q.1 + q.2)
refine ContinuousOn.congr (f := g) ?_ fun q => ?_
· simp only [g]
refine ContinuousOn.mul ?_ ?_
· refine ContinuousOn.mul ?_ (by fun_prop)
exact ContinuousOn.rpow_const (by fun_prop) (by grind)
· exact ContinuousOn.inv₀ (by fun_prop) ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 169,
"column": 2
} | {
"line": 177,
"column": 87
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\ns : Set ℝ\nhs : s ⊆ Ici 0\n⊢ ContinuousOn (Function.uncurry p.rpowIntegrand₀₁) (Ioi 0 ×ˢ s)",
"usedConstants": [
"Set.instSProd",
"Real.instPow",
"Real",
"Set.Ioi",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.... | let g : ℝ × ℝ → ℝ := fun q => q.1 ^ (p - 1) * q.2 / (q.1 + q.2)
refine ContinuousOn.congr (f := g) ?_ fun q => ?_
· simp only [g]
refine ContinuousOn.mul ?_ ?_
· refine ContinuousOn.mul ?_ (by fun_prop)
exact ContinuousOn.rpow_const (by fun_prop) (by grind)
· exact ContinuousOn.inv₀ (by fun_prop) ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 49
} | [
{
"pp": "a : ℂ\nha : 0 < a.re\n⊢ ∑' (n : ℤ), cexp (-↑π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), cexp (-↑π / a * ↑n ^ 2)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 131,
"column": 2
} | {
"line": 133,
"column": 19
} | [
{
"pp": "a : ℝ\nha : 0 < a\n⊢ ∑' (n : ℤ), rexp (-π * a * ↑n ^ 2) = 1 / a ^ (1 / 2) * ∑' (n : ℤ), rexp (-π / a * ↑n ^ 2)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 239,
"column": 63
} | {
"line": 240,
"column": 43
} | [
{
"pp": "b x : ℝ\nhb : 1 < b\nhx : 0 < x\n⊢ logb b x ≤ 0 ↔ x ≤ 1",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Real.instLT",
"Preorder.toLE",
"id",
... | by
rw [← not_lt, logb_pos_iff hb hx, not_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 17
} | [
{
"pp": "case hff'\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ ∀ x_1 ∈ Ioi 0, HasDerivWithinAt (fun x_2 ↦ x * x_2) x (Ioi x_1) x_1",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 45
} | [
{
"pp": "case hg_cont\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ ContinuousOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ((fun x_1 ↦ x * x_1) '' Ioi 0)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 45
} | [
{
"pp": "case hg1\np x : ℝ\nhp : p ∈ Ioo 0 1\nhx✝ : 0 ≤ x\nhx : 0 < x\nthis :\n ∫ (t : ℝ) in Ioi 0, ((fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ∘ fun x_1 ↦ x * x_1) t * x =\n x ^ p * ∫ (t : ℝ) in Ioi 0, p.rpowIntegrand₀₁ t 1\n⊢ IntegrableOn (fun x_1 ↦ p.rpowIntegrand₀₁ x_1 x) ((fun x_1 ↦ x * x_1) '' Ici 0) volume"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 53
} | [
{
"pp": "case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"PartialOrder.toPreorder",
"AddGroup... | have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 53
} | [
{
"pp": "case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\n⊢ ⌈logb (↑b) r⌉ = Int.clog b r",
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Real.instRCLike",
"Real.instZeroLEOneClass",
"PartialOrder.toPreorder",
"AddGrou... | have hb1' : 1 < (b : ℝ) := Nat.one_lt_cast.mpr hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 508,
"column": 13
} | {
"line": 508,
"column": 34
} | [
{
"pp": "case inl\nb a c : ℝ\nn : ℕ\nha : a ≠ 0\nh : log b = 0\n⊢ Tendsto (fun x ↦ logb b x ^ n / (a * x + c)) atTop (𝓝 0)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Real.ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 514,
"column": 4
} | {
"line": 514,
"column": 15
} | [
{
"pp": "b : ℝ\nn : ℕ\n⊢ Tendsto (fun x ↦ logb b x ^ n / id x) atTop (𝓝 0)",
"usedConstants": [
"Real",
"instHDiv",
"NormedDivisionRing.toNormedRing",
"PseudoMetricSpace.toUniformSpace",
"NormedDivisionRing.toDivisionRing",
"nhds",
"DivisionRing.toDivisionSemiring"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 529,
"column": 14
} | {
"line": 529,
"column": 51
} | [
{
"pp": "case pos.inl\n⊢ logb (-1) =O[⊤] log",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Asymptotics.IsBigO",
"Real.logb_neg_base_eq_logb",
"id",
"Pi.instZero",
"Real.log",
"Real.logb_one_left_eq_zero",
"Real.instOne"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 529,
"column": 14
} | {
"line": 529,
"column": 51
} | [
{
"pp": "case pos.inr.inl\n⊢ logb 0 =O[⊤] log",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Asymptotics.IsBigO",
"Real.logb_zero_left_eq_zero",
"id",
"Pi.instZero",
"Real.log",
"Real.logb",
"Zero.toOfNat0",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 529,
"column": 14
} | {
"line": 529,
"column": 51
} | [
{
"pp": "case pos.inr.inr\n⊢ logb 1 =O[⊤] log",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Asymptotics.IsBigO",
"id",
"Pi.instZero",
"Real.log",
"Real.logb_one_left_eq_zero",
"Real.instOne",
"Real.logb",
"One.toO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 530,
"column": 4
} | {
"line": 531,
"column": 11
} | [
{
"pp": "case neg\nb : ℝ\nh : b ≠ -1 ∧ b ≠ 0 ∧ b ≠ 1\n⊢ logb b =O[⊤] log",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 535,
"column": 4
} | {
"line": 535,
"column": 15
} | [
{
"pp": "case inl\n⊢ (fun x ↦ log (0 * x)) =O[atTop] log",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"MulZeroClass.zero_mul",
"Asymptotics.IsBigO",
"Real.semiring",
"id",
"Real.log",
"Filter.atTop",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 24
} | [
{
"pp": "c : ℝ\n⊢ (fun x ↦ log (x * c)) =O[atTop] log",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Asymptotics.IsBigO",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 546,
"column": 2
} | {
"line": 547,
"column": 9
} | [
{
"pp": "b c : ℝ\n⊢ (fun x ↦ logb b (c * x)) =O[atTop] log",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Mono... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 24
} | [
{
"pp": "b c : ℝ\n⊢ (fun x ↦ logb b (x * c)) =O[atTop] log",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Asymptotics.IsBigO",
"id",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 598,
"column": 2
} | {
"line": 598,
"column": 23
} | [
{
"pp": "b y : ℝ\n⊢ Tendsto (fun x ↦ logb b (x + y) - logb b x) atTop (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 627,
"column": 12
} | {
"line": 627,
"column": 23
} | [
{
"pp": "case zero\nP : ℝ → Prop\nx₀ r : ℝ\nhr : 1 < r\nhx₀ : 0 < x₀\nbase : ∀ x ∈ Ico x₀ (r * x₀), P x\nstep : ∀ n ≥ 1, (∀ z ∈ Ico x₀ (r ^ n * x₀), P z) → ∀ z ∈ Ico (r ^ n * x₀) (r ^ (n + 1) * x₀), P z\n⊢ ∀ x ∈ Ico x₀ (r ^ (0 + 1) * x₀), P x",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 35,
"column": 33
} | {
"line": 35,
"column": 44
} | [
{
"pp": "x : ℝ\nhx : x ∈ interior (Ici (rexp (-1)))\n⊢ rexp (-1) < x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 43,
"column": 42
} | {
"line": 43,
"column": 53
} | [
{
"pp": "⊢ HasDerivAt ?m.53 1 ?m.55",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 35
} | [
{
"pp": "H : ContinuousAt (fun x ↦ (log x)⁻¹) (-1)\n⊢ ContinuousAt (fun x ↦ (log x)⁻¹) 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 15
} | [
{
"pp": "case inl\n⊢ deriv (fun x ↦ (log x)⁻¹) 0 = -0⁻¹ / log 0 ^ 2",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"instHDiv",
"Semiring.toModule",
"Real.denselyNormedField",
"Real.instZero",
"congrArg",
"Real.ins... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 15
} | [
{
"pp": "case inr.inl\nh0 : 1 ≠ 0\n⊢ deriv (fun x ↦ (log x)⁻¹) 1 = -1⁻¹ / log 1 ^ 2",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"instHDiv",
"Semiring.toModule",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.InvLog | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 15
} | [
{
"pp": "case inr.inr.inl\nh0 : -1 ≠ 0\nh1 : -1 ≠ 1\n⊢ deriv (fun x ↦ (log x)⁻¹) (-1) = -(-1)⁻¹ / log (-1) ^ 2",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 469,
"column": 10
} | {
"line": 472,
"column": 18
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint... | refine cfcₙ_smul (R := ℝ) (t ^ ((p : ℝ) - 1)) _ a ?_
refine ContinuousOn.mono ?_ hspec
have := continuousOn_rpowIntegrand₀₁_Ici hp zero_lt_one
fun_prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 469,
"column": 10
} | {
"line": 472,
"column": 18
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint... | refine cfcₙ_smul (R := ℝ) (t ^ ((p : ℝ) - 1)) _ a ?_
refine ContinuousOn.mono ?_ hspec
have := continuousOn_rpowIntegrand₀₁_Ici hp zero_lt_one
fun_prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 459,
"column": 89
} | {
"line": 477,
"column": 51
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalNormedRing A\ninst✝⁷ : StarRing A\ninst✝⁶ : NormedSpace ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : PartialOrder A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonnegSpectrumClass ℝ A\ninst✝ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint... | by
have hspec : quasispectrum ℝ a ⊆ Ici 0 := by grind
have h_mapsTo : MapsTo (t⁻¹ • · : ℝ → ℝ) (Ici 0) (Ici 0) := by
intro x hx
simp only [mem_Ici, smul_eq_mul] at hx ⊢
positivity
calc _ = cfcₙ (fun x => t ^ ((p : ℝ) - 1) * (rpowIntegrand₀₁ p 1 (t⁻¹ • x))) a := by
refine cfcₙ_congr ?_
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.RegularityCompacts | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 61
} | [
{
"pp": "case mpr.refine_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace α\ninst✝ : R1Space α\nh : μ.InnerRegularWRT IsCompact IsClosed\nA : Set α\nhA : IsClosed A\nr : ℝ≥0∞\nhr : r < μ A\nK : Set α\nhK1 : K ⊆ A\nhK2 : IsCompact K\nhK3 : r < μ K\n⊢ (IsCompact ∘ closure) (cl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 70,
"column": 47
} | {
"line": 70,
"column": 80
} | [
{
"pp": "X : Type u_1\ninst✝² : MeasurableSpace X\ninst✝¹ : TopologicalSpace X\ninst✝ : OpensMeasurableSpace X\nf : X →ᵇ ℝ≥0\nμ : Measure X\n⊢ AEStronglyMeasurable (fun x ↦ ↑(f x)) μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 154,
"column": 59
} | {
"line": 154,
"column": 70
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → limsup (fun i ↦ ∫ (x : X), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 155,
"column": 59
} | {
"line": 155,
"column": 70
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → limsup (fun i ↦ ∫ (x : X), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 176,
"column": 59
} | {
"line": 176,
"column": 70
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 177,
"column": 59
} | {
"line": 177,
"column": 70
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : MeasurableSpace X\ninst✝² : OpensMeasurableSpace X\nι : Type u_2\nL : Filter ι\nμ : Measure X\ninst✝¹ : IsProbabilityMeasure μ\nμs : ι → Measure X\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\nh : ∀ (f : X →ᵇ ℝ), 0 ≤ f → ∫ (x : X), f x ∂μ ≤ liminf ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 610,
"column": 6
} | {
"line": 615,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 < t\na : A\nha : 0 ≤ a\nb : A\nhb : 0 ≤ b\nhab : a ≤ b\n⊢ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • a) ≤ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b)",
"... | gcongr
unfold rpowIntegrand₀₁
simp only [Real.one_rpow, one_mul, inv_one]
refine CFC.monotoneOn_one_sub_one_add_inv_real
(?_ : 0 ≤ t⁻¹ • a) (?_ : 0 ≤ t⁻¹ • b) (by gcongr)
all_goals positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 610,
"column": 6
} | {
"line": 615,
"column": 26
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\np t : ℝ\nhp : p ∈ Ioo 0 1\nht : 0 < t\na : A\nha : 0 ≤ a\nb : A\nhb : 0 ≤ b\nhab : a ≤ b\n⊢ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • a) ≤ t ^ (p - 1) • cfcₙ (p.rpowIntegrand₀₁ 1) (t⁻¹ • b)",
"... | gcongr
unfold rpowIntegrand₀₁
simp only [Real.one_rpow, one_mul, inv_one]
refine CFC.monotoneOn_one_sub_one_add_inv_real
(?_ : 0 ≤ t⁻¹ • a) (?_ : 0 ≤ t⁻¹ • b) (by gcongr)
all_goals positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 34
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ Polynomial.eval (↑n - 1) (descPochhammer ℝ n) = 0",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Real",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"congrArg",
"descPochhammer",
"Real.instSub",
"AddGroupWithOne.toAddM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 68
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nι : Type u_2\nt : Finset ι\np : ι → ℕ\nw : ι → ℝ\nh₀ : ∀ i ∈ t, 0 ≤ w i\nh₁ : ∑ i ∈ t, w i = 1\nh_avg : ↑n - 1 ≤ ∑ i ∈ t, w i * ↑(p i)\nf : ℝ → ℝ := (Set.Ici (↑n - 1)).piecewise (fun x ↦ Polynomial.eval x (descPochhammer ℝ n)) 0\nh_jensen : f (∑ i ∈ t, w i • ↑(p i)) ≤ ∑ i ∈ t, w i • ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 113,
"column": 31
} | {
"line": 113,
"column": 67
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nι : Type u_2\nt : Finset ι\np : ι → ℕ\nw : ι → ℝ\nh₀ : ∀ i ∈ t, 0 ≤ w i\nh₁ : ∑ i ∈ t, w i = 1\nh_avg : ↑n - 1 ≤ ∑ i ∈ t, w i * ↑(p i)\n⊢ Polynomial.eval (∑ i ∈ t, w i * ↑(p i)) (descPochhammer ℝ n) / ↑n.factorial ≤\n (∑ i ∈ t, w i * Polynomial.eval (↑(p i)) (descPochhammer ℝ n)) ... | descPochhammer_eval_eq_descFactorial | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 25
} | [
{
"pp": "case succ\nn a : ℕ\nH : ∃ c, a < (c + 1) ^ (n + 1)\nk : ℕ\nhc : k + 1 = Nat.find H\n⊢ (k + 1) ^ (n + 1) ≤ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 54
} | [
{
"pp": "case h_lim.hp\nE : Type u_1\ninst✝³ : TopologicalSpace E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nP : Measure E\ninst✝ : IsFiniteMeasure P\nε : ℝ\ng : E →ᵇ ℝ\nhε : 0 < ε\nx : E\n⊢ Tendsto (fun x_1 ↦ ((1 + (↑x_1)⁻¹ • -(ε • g * g)) ^ x_1) x) atTop (𝓝 (rexp (-(ε * g x * g x))))",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 177,
"column": 35
} | {
"line": 177,
"column": 68
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy : dist 0 ((1 - ε / 4) • x) < ε / 4\n⊢ 1 - ε / 4 < ε / 4",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 17
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\ny : E\nhy : dist y ((1 - ε / 4) • x) < ε / 4\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy₀ : y ≠ 0\n⊢ ‖y‖ ≤ dist y ((1 - ε / 4) • x) + ‖(1 - ε / 4) • x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 40
} | [
{
"pp": "case neg\nε : ℝ\nE : Type u_2\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : PseudoEMetricSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : CompleteSpace E\ninst✝² : SecondCountableTopology E\nP P' : Measure E\ninst✝¹ : IsFiniteMeasure P\ninst✝ : IsFiniteMeasure P'\nf : E →ᵇ ℝ\nA : Subalgebra ℝ (E →ᵇ ℝ)\nhA : (Subalgebr... | have hgA : g ∈ A := hg'A.choose_spec.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 196,
"column": 4
} | {
"line": 197,
"column": 11
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 1\nε : ℝ\nhε : 0 < ε\nhε2 : ε ≤ 2\ny : E\nhy : dist y ((1 - ε / 4) • x) < ε / 4\nhabs : |1 - ε / 4| = 1 - ε / 4\nhy₀ : y ≠ 0\nhy₁ : ‖y‖ < 1\nu : E := ‖y‖⁻¹ • y\nhu₁ : ‖u‖ = 1\nhyx : dist y x < ε / 2\nH : u - y = (1 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 61
} | [
{
"pp": "⊢ sigmoid 0 = 2⁻¹",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"instHDiv",
"Mathlib.Meta.NormNum.IsNat.to_isNNRat",
"GroupW... | norm_num [sigmoid] | Mathlib.Tactic._aux_Mathlib_Tactic_NormNum_Core___elabRules_Mathlib_Tactic_normNum_1 | Mathlib.Tactic.normNum |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 61
} | [
{
"pp": "⊢ sigmoid 0 = 2⁻¹",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"instHDiv",
"Mathlib.Meta.NormNum.IsNat.to_isNNRat",
"GroupW... | norm_num [sigmoid] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 68,
"column": 43
} | {
"line": 68,
"column": 61
} | [
{
"pp": "⊢ sigmoid 0 = 2⁻¹",
"usedConstants": [
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"instHDiv",
"Mathlib.Meta.NormNum.IsNat.to_isNNRat",
"GroupW... | norm_num [sigmoid] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 224,
"column": 6
} | {
"line": 225,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\nμ : Measure E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nε : ℝ\nhε : 0 < ε\nx : ↑(sphere 0 1)\nthis✝ : Nontrivial E\nthis : ∀ {ε : ℝ}, 0 < ε → ε ≤ 2 → ↑(toSphereBa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Sigmoid | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 13
} | [
{
"pp": "⊢ Tendsto sigmoid atTop (𝓝 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 17
} | [
{
"pp": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : Nontrivial E\nμ : Measure E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nf : ℝ → F\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Stirling | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 27
} | [
{
"pp": "n : ℕ\nr : ℝ := (1 / (2 * (↑n + 1) + 1)) ^ 2\nhr : r = (1 / (2 * (↑n + 1) + 1)) ^ 2\nhr1 : r < 1\nthis : HasSum (fun j ↦ r ^ (j + 1) / 3) (1 / (12 * ↑(n + 1) * (↑(n + 1) + 1)))\nj : ℕ\n⊢ 1 / (2 * ↑(j + 1) + 1) * ((1 / (2 * ↑(n + 1) + 1)) ^ 2) ^ (j + 1) ≤ r ^ (j + 1) / 3",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 13
} | [
{
"pp": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis✝ : sin ∘ arcsin =ᶠ[𝓝[≥] (-1)] id\nthis : HasDerivWithinAt id (cos (arcsin (-1)) * derivWithin arcsin (Ici (-1)) (-1)) (Ici (-1)) (-1)\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.InverseDeriv | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 58
} | [
{
"pp": "x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)\nthis : DifferentiableWithinAt ℝ (fun i ↦ -(arcsin ∘ Neg.neg) i) (Ici (-x)) (-x)\n⊢ x ≠ 1",
"usedConstants": [
"Real",
"id",
"Ne",
"Real.instOne",
"One.toOfNat1",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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