module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Polynomial.Basic | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 29
} | {
"line": 281,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh : Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)\n⊢ Q.comp (-X) ≠ 0",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Nor... | [
"𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhQ : Q ≠ 0\nh : Tendsto (fun x ↦ eval x P / eval x Q) atBot (𝓝 0)\n⊢ ¬(Q = 0 ∨ eval ((-X).coeff 0) Q = 0 ∧ -X = C ((-X).coeff 0))"
] | rw [Ne, comp_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 29
} | {
"line": 296,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree\nhQ : Q ≠ 0\n⊢ Q.comp (-X) ≠ 0",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"NormedC... | [
"𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nhdeg : (Q.comp (-X)).degree < (P.comp (-X)).degree\nhQ : Q ≠ 0\n⊢ ¬(Q = 0 ∨ eval ((-X).coeff 0) Q = 0 ∧ -X = C ((-X).coeff 0))"
] | rw [Ne, comp_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 452,
"column": 2
} | {
"line": 468,
"column": 64
} | {
"line": 470,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\ninst✝ : CompleteSpace E\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ L',\n ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunS... | [] | obtain ⟨m, C, h⟩ := exists_contractingWith_iterate_next hf
let L' := (∑ i ∈ Finset.range m, (K * max (tmax - t₀) (t₀ - tmin)) ^ i / i !) * (1 - C)⁻¹
have hL' : 0 ≤ L' := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
positivity
refine ⟨.mk L' hL', fun x y hx hy α... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 452,
"column": 2
} | {
"line": 468,
"column": 64
} | {
"line": 470,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℝ → E → E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L K : ℝ≥0\ninst✝ : CompleteSpace E\nhf : IsPicardLindelof f t₀ x₀ a r L K\n⊢ ∃ L',\n ∀ (x y : E) (hx : x ∈ closedBall x₀ ↑r) (hy : y ∈ closedBall x₀ ↑r) (α β : FunS... | [] | obtain ⟨m, C, h⟩ := exists_contractingWith_iterate_next hf
let L' := (∑ i ∈ Finset.range m, (K * max (tmax - t₀) (t₀ - tmin)) ^ i / i !) * (1 - C)⁻¹
have hL' : 0 ≤ L' := by
have : 0 ≤ max (tmax - t₀) (t₀ - tmin) := le_max_of_le_left <| sub_nonneg_of_le t₀.2.2
positivity
refine ⟨.mk L' hL', fun x y hx hy α... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.Basic | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 20
} | {
"line": 318,
"column": 21
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : P.degree ≤ Q.degree\nhp : P = 0\n⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
... | [
"case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : P.degree ≤ Q.degree\nhp : P = 0\n⊢ (fun x ↦ 0) =O[atTop] fun x ↦ eval x Q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 27
} | {
"line": 90,
"column": 28
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ p.supNorm ∈ Set.range fun x ↦ ‖p.coeff x‖",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real",
"SeminormedRing.toRing",
"Membership.mem",
... | [
"A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ ∃ y, ‖p.coeff y‖ = p.supNorm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Norm | {
"line": 90,
"column": 52
} | {
"line": 90,
"column": 81
} | {
"line": 90,
"column": 82
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ p.supNorm ∈ upperBounds (Set.range fun x ↦ ‖p.coeff x‖)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"SeminormedRing.toNorm",
"Real.instLE",
"Real",
"Preorder.toLE",
... | [
"A : Type u_1\ninst✝ : SeminormedRing A\np : A[X]\n⊢ ∀ (a : ℕ), ‖p.coeff a‖ ≤ p.supNorm"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 12
} | {
"line": 557,
"column": 13
} | [
{
"pp": "case coe\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\nt₀ tmin tmax : ℝ\nht₀ : t₀ ∈ Icc tmin tmax\nhα : ContinuousOn α (Icc tmin tmax)\nhmem : ∀ t ∈ Icc tmin tmax, α t ∈ u\nx₀ : E\nheqon : ∀ t ∈ Icc tmin tmax, α t =... | [] | | coe n => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 43
} | {
"line": 149,
"column": 44
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝ : Semiring R\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nvMul : ∀ (a b : R), v (a * b) ≤ v a * v b\nvna : IsNonarchimedean v\nvZero : v 0 = 0\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nt : σ →₀ ℕ\nk : (σ →₀ ℕ) ×... | [
"R : Type u_1\nσ : Type u_2\nv : R → ℝ\nc : σ → ℝ\ninst✝ : Semiring R\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nvMul : ∀ (a b : R), v (a * b) ≤ v a * v b\nvna : IsNonarchimedean v\nvZero : v 0 = 0\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\nt : σ →₀ ℕ\nk : (σ →₀ ℕ) × (σ →₀ ℕ)\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Order | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 17
} | {
"line": 98,
"column": 18
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Even P.natDegree\nhlc' : 0 ≤ (P.comp (-X)).leadingCoeff\nthis : 0 < eval (-x) (P.comp (-X))\n⊢ 0 < eval x P",
"ppTerm": "?m.187",
"assigned": true,
"u... | [
"P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Even P.natDegree\nhlc' : 0 ≤ (P.comp (-X)).leadingCoeff\nthis : 0 < eval (-x) (P.comp (-X))\n⊢ 0 < eval x P"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.Order | {
"line": 105,
"column": 28
} | {
"line": 105,
"column": 39
} | {
"line": 105,
"column": 40
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Odd P.natDegree\nhlc' : 0 ≤ -(P.comp (-X)).leadingCoeff\nthis : eval (-x) (P.comp (-X)) < 0\n⊢ eval x P < 0",
"ppTerm": "?m.249",
"assigned": false,
"... | [
"P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\nh : Odd P.natDegree\nhlc' : 0 ≤ -(P.comp (-X)).leadingCoeff\nthis : eval (-x) (P.comp (-X)) < 0\n⊢ eval x P < 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 36
} | {
"line": 187,
"column": 37
} | [
{
"pp": "case inr\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nH : ∀ {a b : α}, f a ≠ f b → f a > f b → f (a + b) = max (f a) (f b)\nhab : ¬f a > f b\n⊢ f (a + b) = max (f a) (f b)",
... | [
"case inr\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nH : ∀ {a b : α}, f a ≠ f b → f a > f b → f (a + b) = max (f a) (f b)\nhab : ¬f a > f b\n⊢ f (a + b) = max (f a) (f b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 44
} | {
"line": 190,
"column": 45
} | [
{
"pp": "case inl\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (a + b)\n⊢ max (f a) (f b) ≤ f (a + b)",
"ppTerm": "?inl",
"assigned": true,... | [
"case inl\nα : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (a + b)\n⊢ f a ≤ f (a + b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 191,
"column": 35
} | {
"line": 191,
"column": 54
} | {
"line": 191,
"column": 55
} | [
{
"pp": "α : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (-b)\n⊢ f (-b) < f (a + b + -b)",
"ppTerm": "?m.119",
"assigned": true,
"usedConst... | [
"α : Type u_3\nS : Type u_4\ninst✝¹ : LinearOrder S\ninst✝ : AddCommGroup α\nf : α → S\nna : IsNonarchimedean f\nNeg : ∀ (a : α), f a = f (-a)\na b : α\nhne : f a ≠ f b\nhab : f a > f b\nh : f (a + b + -b) ≤ f (-b)\n⊢ f (-b) < f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 183,
"column": 53
} | {
"line": 186,
"column": 37
} | {
"line": 187,
"column": 2
} | [
{
"pp": "R : Type u_1\nF : Type u_2\ninst✝³ : Semiring R\ninst✝² : FunLike F R ℝ\nv : F\ninst✝¹ : ZeroHomClass F R ℝ\ninst✝ : NonnegHomClass F R ℝ\nhna : IsNonarchimedean ⇑v\nc : ℝ\nhc : 0 ≤ c\np q : R[X]\nh✝¹ : p ≠ 0\nh✝ : q ≠ 0\nhpq : p + q ≠ 0\ni : ℕ\na✝ : i ∈ (p + q).support\n⊢ v ((p + q).coeff i) * c ^ i ≤... | [] | by
rw [coeff_add]
gcongr
exact hna (p.coeff i) (q.coeff i) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 41
} | {
"line": 129,
"column": 42
} | [
{
"pp": "case pos\np q : ℂ[X]\nhpq : p * q = 0\n⊢ (p * q).mahlerMeasure = p.mahlerMeasure * q.mahlerMeasure",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real",
"HMul.hMul",
"Polynomial.mahlerMeasure_eq_zero_iff._sim... | [
"case pos\np q : ℂ[X]\nhpq : p * q = 0\n⊢ p = 0 ∨ q = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.ContinuousMap | {
"line": 39,
"column": 28
} | {
"line": 39,
"column": 39
} | {
"line": 39,
"column": 40
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : RCLike 𝕜\nf g : C(X, ℝ)\nhfg : realToRCLike 𝕜 f = realToRCLike 𝕜 g\nx : X\n⊢ f x = g x",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\n𝕜 : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : RCLike 𝕜\nf g : C(X, ℝ)\nhfg : realToRCLike 𝕜 f = realToRCLike 𝕜 g\nx : X\n⊢ f x = g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Rat.NatSqrt.Real | {
"line": 25,
"column": 2
} | {
"line": 25,
"column": 36
} | {
"line": 25,
"column": 37
} | [
{
"pp": "x prec : ℕ\nh : 0 < prec\nthis✝¹ : x.ratSqrt prec ^ 2 ≤ ↑x\nthis✝ : ↑(x.ratSqrt prec) ^ 2 ≤ ↑x\nthis : √(↑(x.ratSqrt prec) ^ 2) ≤ √↑x\n⊢ 0 ≤ ↑(x.ratSqrt prec)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Real.instLE",
"Real",
... | [
"x prec : ℕ\nh : 0 < prec\nthis✝¹ : x.ratSqrt prec ^ 2 ≤ ↑x\nthis✝ : ↑(x.ratSqrt prec) ^ 2 ≤ ↑x\nthis : √(↑(x.ratSqrt prec) ^ 2) ≤ √↑x\n⊢ 0 ≤ x.ratSqrt prec"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Rat.NatSqrt.Real | {
"line": 36,
"column": 25
} | {
"line": 36,
"column": 36
} | {
"line": 36,
"column": 37
} | [
{
"pp": "x prec : ℕ\nh : 0 < prec\nthis✝¹ : ↑x < (x.ratSqrt prec + 1 / ↑prec) ^ 2\nthis✝ : ↑x < ↑((x.ratSqrt prec + 1 / ↑prec) ^ 2)\nthis : √↑x < √(↑(x.ratSqrt prec + 1 / ↑prec) ^ 2)\n⊢ 0 ≤ ↑(x.ratSqrt prec)",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Eq.m... | [
"x prec : ℕ\nh : 0 < prec\nthis✝¹ : ↑x < (x.ratSqrt prec + 1 / ↑prec) ^ 2\nthis✝ : ↑x < ↑((x.ratSqrt prec + 1 / ↑prec) ^ 2)\nthis : √↑x < √(↑(x.ratSqrt prec + 1 / ↑prec) ^ 2)\n⊢ 0 ≤ x.ratSqrt prec"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 339,
"column": 8
} | {
"line": 339,
"column": 20
} | {
"line": 339,
"column": 20
} | [
{
"pp": "p : ℂ[X]\nthis✝¹ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis✝ : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\nthis : ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), 0 < ‖eval (circleMap 0 1 θ) p‖\nhlogAe :\n ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), rexp (log ‖eval (circleMap 0 1 θ) p... | [
"p : ℂ[X]\nthis✝¹ : IsFiniteMeasure (volume.restrict (uIoc 0 (2 * π)))\nthis✝ : NeZero (volume (uIoc 0 (2 * π)))\nhp : p ≠ 0\nthis : ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), 0 < ‖eval (circleMap 0 1 θ) p‖\nhlogAe :\n ∀ᵐ (θ : ℝ) ∂volume.restrict (uIoc 0 (2 * π)), rexp (log ‖eval (circleMap 0 1 θ) p‖) = ‖eval (... | rw [sqrt_sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Real.Irrational | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 13
} | {
"line": 146,
"column": 14
} | [
{
"pp": "⊢ Irrational √2",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ Irrational √2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 194,
"column": 48
} | {
"line": 194,
"column": 79
} | {
"line": 194,
"column": 80
} | [
{
"pp": "x : ℝ\nh : Irrational x\n⊢ x ≠ 1",
"ppTerm": "?m.4",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nh : Irrational x\n⊢ x ≠ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 45
} | {
"line": 270,
"column": 46
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (x - ↑q)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"AddMonoid.toAddZeroClass",
"Real.instRatCast",
"sub_eq_add_neg",
"HSub.hSub",
"... | [
"q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (x + -↑q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 35
} | {
"line": 272,
"column": 36
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (↑q - x)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"AddMonoid.toAddZeroClass",
"Real.instRatCast",
"sub_eq_add_neg",
"HSub.hSub",
"... | [
"q : ℚ\nx : ℝ\nh : Irrational x\n⊢ Irrational (↑q + -x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 274,
"column": 28
} | {
"line": 274,
"column": 71
} | {
"line": 274,
"column": 72
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational (x - ↑q)\n⊢ Irrational (x + ↑(-q))",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Real.instRatCast",
"Rat",
"id",
"Rat.... | [
"q : ℚ\nx : ℝ\nh : Irrational (x - ↑q)\n⊢ Irrational (x + -↑q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 276,
"column": 31
} | {
"line": 276,
"column": 64
} | {
"line": 276,
"column": 65
} | [
{
"pp": "q : ℚ\nx : ℝ\nh : Irrational (↑q - x)\n⊢ Irrational (↑q + -x)",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"q : ℚ\nx : ℝ\nh : Irrational (↑q - x)\n⊢ Irrational (↑q + -x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 37
} | {
"line": 278,
"column": 38
} | [
{
"pp": "x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (x - ↑m)",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (x - ↑m)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 37
} | {
"line": 280,
"column": 38
} | [
{
"pp": "x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (↑m - x)",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nh : Irrational x\nm : ℤ\n⊢ Irrational (↑m - x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Real.Irrational | {
"line": 416,
"column": 32
} | {
"line": 416,
"column": 68
} | {
"line": 416,
"column": 69
} | [
{
"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",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"Real",
"HMul.hMul",
... | [
"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 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 380,
"column": 6
} | {
"line": 380,
"column": 17
} | {
"line": 380,
"column": 18
} | [
{
"pp": "case refine_2\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s < r\n⊢ coeRingHom s ≤ x",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"Real",
"congrArg",
"PartialOrder.toPreorder",
"OrderRingHom.instFunLike",
... | [
"case refine_2\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s < r\n⊢ ↑s ≤ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 381,
"column": 6
} | {
"line": 381,
"column": 17
} | {
"line": 381,
"column": 18
} | [
{
"pp": "case refine_3\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s > r\n⊢ x ≤ coeRingHom s",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"Hyperreal.instField",
"Eq.mpr",
"Real",
"congrArg",
"PartialOrder.toPreorder",
"OrderRingHom.instFunLike",
... | [
"case refine_3\nx : ℝ*\nr : ℝ\nh : x.IsSt r\ns : ℝ\nhs : s > r\n⊢ x ≤ ↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Hyperreal | {
"line": 424,
"column": 2
} | {
"line": 426,
"column": 36
} | {
"line": 427,
"column": 2
} | [
{
"pp": "case refine_1\nx : ℝ*\nh : x.InfinitePos\n⊢ 0 < x ∧ ArchimedeanClass.mk x < 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"ArchimedeanOrder.of",
"Hyperreal.instField",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"instHSM... | [
"case refine_2\nx : ℝ*\nx✝ : 0 < x ∧ ArchimedeanClass.mk x < 0\nr : ℝ\nhx : 0 < x\nhx' : ArchimedeanClass.mk x < 0\n⊢ ↑r < x"
] | · have hx : 0 < x := h 0
refine ⟨h 0, fun n ↦ ?_⟩
simpa [abs_of_pos hx] using! h n | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 123,
"column": 81
} | {
"line": 139,
"column": 91
} | {
"line": 141,
"column": 0
} | [
{
"pp": "z : ℂ\nhz : ‖z‖ < 1\n⊢ HasSum (fun n ↦ (-1) ^ n * z ^ (2 * n + 1) / ↑(2 * n + 1)) z.arctan",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Even.add_one._simp_1",
"Norm.norm",
"Mathlib.Tactic.Ring.C... | [] | 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": 86,
"column": 4
} | {
"line": 86,
"column": 47
} | {
"line": 86,
"column": 48
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"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 ((x + y) / 2) (sqrt (x * y)) ≤ dist x y / 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 126,
"column": 2
} | {
"line": 129,
"column": 55
} | {
"line": 130,
"column": 2
} | [
{
"pp": "x y : ℝ≥0\nh : x ≠ y\nn m : ℕ\n⊢ (x.agmSequences y n).1 < (x.agmSequences y m).2",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNReal.agmSequences",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal",
"NNReal.agmSequences_fs... | [
"x y : ℝ≥0\nh : x ≠ y\nn m : ℕ\n⊢ ∀ {k : ℕ}, (x.agmSequences y k).1 < (x.agmSequences y k).2"
] | 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.ArithmeticGeometricMean | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 36
} | {
"line": 180,
"column": 37
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.agm",
"SemilatticeInf.toPartialOrder",
... | [
"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 ≤ x ∨ x.agm y ≤ y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Choose | {
"line": 34,
"column": 12
} | {
"line": 34,
"column": 23
} | {
"line": 34,
"column": 24
} | [
{
"pp": "case zero\n⊢ (fun n ↦ ↑(n.descFactorial 0)) ~[atTop] fun n ↦ ↑n ^ 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"Monoid.toMulOneClass",
"congrArg",
"AddGroupWithO... | [
"case zero\n⊢ (fun n ↦ 1) ~[atTop] fun n ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 36
} | {
"line": 212,
"column": 37
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.agm",
... | [
"case inr\nx y : ℝ≥0\nthis : ∀ {x y : ℝ≥0}, x ≤ y → min x y ≤ x.agm y\nh : ¬x ≤ y\n⊢ x ≤ x.agm y ∨ y ≤ x.agm y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Choose | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 15
} | {
"line": 41,
"column": 16
} | [
{
"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)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminorm... | [
"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 ↦ ↑n / (↑n + ↑k)) atTop (𝓝 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 13
} | {
"line": 250,
"column": 14
} | [
{
"pp": "x y : ℝ≥0\n⊢ x.agm y = (sqrt (x * y)).agm ((x + y) / 2)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℝ≥0\n⊢ x.agm y = (sqrt (x * y)).agm ((x + y) / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 36
} | {
"line": 276,
"column": 37
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",... | [
"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⊢ x < x.agm y ∨ y < x.agm y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 36
} | {
"line": 284,
"column": 37
} | [
{
"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",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"lt_sup_iff._simp_3",
"PartialOrder.toPreorder",
"NNRea... | [
"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 < x ∨ x.agm y < y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 285,
"column": 6
} | {
"line": 285,
"column": 23
} | {
"line": 285,
"column": 23
} | [
{
"pp": "x✝ y✝ x y : ℝ≥0\nhn : x ≠ y\nh : x < y\n⊢ x.agm y < max x y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"NNReal.agm",
"SemilatticeSup.toMax",
"NNReal.instSemilatticeSup"... | [
"x✝ y✝ x y : ℝ≥0\nhn : x ≠ y\nh : x < y\n⊢ x.agm y < y"
] | max_eq_right h.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 31
} | {
"line": 76,
"column": 32
} | [
{
"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",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real",
"Monoid.toMulOneClass",
"congrArg",
... | [
"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 ↦ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 28
} | {
"line": 176,
"column": 29
} | [
{
"pp": "case refine_2\nn : ℕ\n⊢ ((monomial 2) (-4)).natDegree + (sinPoly n).natDegree ≤ n + 2",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Nat.instIsOrderedAddMonoid",
"Semiring.toModule",
"Polynomial.instNeg",
"AddLef... | [
"case refine_2\nn : ℕ\n⊢ (sinPoly n).natDegree ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 145,
"column": 10
} | {
"line": 145,
"column": 21
} | {
"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",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"sub_lt_comm",
"congrArg",
"Real.instInv",
... | [
"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 - 2⁻¹ < p"
] | sub_lt_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 44
} | {
"line": 227,
"column": 45
} | [
{
"pp": "case inr.inl\nq : ℕ\nhp₀✝ : 0 ≤ 1\nhp₁ : 1 ≤ 1\nhp₀ : 0 < 1\n⊢ 0 ≤ qaryEntropy q 1",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.qaryEntropy",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Rea... | [
"case inr.inl\nq : ℕ\nhp₀✝ : 0 ≤ 1\nhp₁ : 1 ≤ 1\nhp₀ : 0 < 1\n⊢ 0 ≤ log ↑(↑q - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 69
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case hf\n⊢ Tendsto (fun x ↦ log (1 - x)) (𝓝[<] 1) atBot",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ioi",
"Real.instZero",
"nhdsWithin",
"PseudoMetricSpace.toUniformSpace",
"Real.tendsto_log_nhdsGT_zero",
"Real.log",
... | [
"case hf\nthis : Tendsto log (𝓝[>] 0) atBot\n⊢ Tendsto (fun x ↦ log (1 - x)) (𝓝[<] 1) 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": 188,
"column": 54
} | {
"line": 188,
"column": 65
} | {
"line": 188,
"column": 66
} | [
{
"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",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 33
} | {
"line": 195,
"column": 34
} | [
{
"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)",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.CompareExp | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 33
} | {
"line": 202,
"column": 34
} | [
{
"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)",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 358,
"column": 8
} | {
"line": 358,
"column": 98
} | {
"line": 360,
"column": 0
} | [
{
"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",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing"... | [] | 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
} | {
"line": 360,
"column": 0
} | [
{
"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",
"ppTerm": "?neg.inr✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing"... | [] | simp_all [not_continuousAt_deriv_qaryEntropy_zero, not_continuousAt_deriv_qaryEntropy_one] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 20
} | {
"line": 60,
"column": 21
} | [
{
"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... | [
"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 ↦ cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a) (𝓝[>] 0) (𝓝 (log a))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.EulerMascheroni | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 16
} | [
{
"pp": "this : Tendsto (fun n ↦ eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0)\n⊢ Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant)",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"this : Tendsto (fun n ↦ eulerMascheroniSeq' n - eulerMascheroniSeq n) atTop (𝓝 0)\n⊢ Tendsto eulerMascheroniSeq' atTop (𝓝 eulerMascheroniConstant)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 30
} | {
"line": 98,
"column": 31
} | [
{
"pp": "⊢ HasDerivAt Gamma (-γ) 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ HasDerivAt Gamma (-γ) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 93,
"column": 12
} | {
"line": 93,
"column": 23
} | {
"line": 93,
"column": 24
} | [
{
"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... | [
"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\nhsp : IsSt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Digamma | {
"line": 57,
"column": 25
} | {
"line": 57,
"column": 36
} | {
"line": 57,
"column": 37
} | [
{
"pp": "s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ s ≠ 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Complex.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"Complex",
"OfNat.ofNat"
],
"usedFVars": [
"s"
],
"usedGoals": [
{
"new"... | [
"s : ℂ\nhs : ∀ (m : ℕ), s ≠ -↑m\n⊢ ¬s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 30
} | {
"line": 182,
"column": 31
} | [
{
"pp": "⊢ HasDerivAt Gamma (-↑γ) 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ HasDerivAt Gamma (-↑γ) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 187,
"column": 4
} | {
"line": 188,
"column": 23
} | {
"line": 188,
"column": 24
} | [
{
"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)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"NormedCommRing.toSeminormedComm... | [
"case refine_2\nthis : HasDerivAt Gamma ↑(-√π * (γ + 2 * Real.log 2)) ↑(1 / 2)\n⊢ HasDerivAt Gamma (-(↑√π * (↑γ + 2 * log 2))) 2⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 34
} | {
"line": 42,
"column": 35
} | [
{
"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",
"ppTerm": "?m.114",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | {
"line": 140,
"column": 14
} | [
{
"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",
"ppTerm": "?e_a✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 162,
"column": 14
} | {
"line": 162,
"column": 46
} | {
"line": 162,
"column": 47
} | [
{
"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",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
... | [
"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⊢ 0 ≤ t ^ p * (t⁻¹ - (t + y)⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 169,
"column": 2
} | {
"line": 177,
"column": 87
} | {
"line": 179,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Real.instPow",
"Real",
"Set.Ioi",
"instHDiv",
"NonUnitalCommRi... | [] | 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
} | {
"line": 179,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Real.instPow",
"Real",
"Set.Ioi",
"instHDiv",
"NonUnitalCommRi... | [] | 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": 121,
"column": 2
} | {
"line": 121,
"column": 49
} | {
"line": 121,
"column": 50
} | [
{
"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)",
"ppTerm": "?m.100",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation | {
"line": 126,
"column": 2
} | {
"line": 128,
"column": 19
} | {
"line": 128,
"column": 20
} | [
{
"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)",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
... | [
"a : ℝ\nha : 0 < a\n⊢ ∑' (a_1 : ℤ), cexp (-↑π * ↑a * ↑a_1 ^ 2) = 1 / ↑a ^ (1 / 2) * ∑' (a_1 : ℤ), cexp (-↑π / ↑a * ↑a_1 ^ 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 17
} | {
"line": 294,
"column": 18
} | [
{
"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",
"ppTerm": "?... | [
"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 : ℝ), 0 < x_1 → HasDerivWithinAt (fun x_2 ↦ x * x_2) x (Ici x_1) x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 295,
"column": 6
} | {
"line": 295,
"column": 45
} | {
"line": 295,
"column": 46
} | [
{
"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)",
... | [
"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) (Ioi 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 45
} | {
"line": 296,
"column": 46
} | [
{
"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"... | [
"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) (Ici 0) volume"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Monotone | {
"line": 35,
"column": 33
} | {
"line": 35,
"column": 44
} | {
"line": 35,
"column": 45
} | [
{
"pp": "x : ℝ\nhx : x ∈ interior (Ici (rexp (-1)))\n⊢ rexp (-1) < x",
"ppTerm": "?m.45",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x : ℝ\nhx : x ∈ interior (Ici (rexp (-1)))\n⊢ rexp (-1) < x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 239,
"column": 63
} | {
"line": 240,
"column": 43
} | {
"line": 242,
"column": 0
} | [
{
"pp": "b x : ℝ\nhb : 1 < b\nhx : 0 < x\n⊢ logb b x ≤ 0 ↔ x ≤ 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Real.... | [] | by
rw [← not_lt, logb_pos_iff hb hx, not_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 53
} | {
"line": 370,
"column": 4
} | [
{
"pp": "case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"FloorRing.toFloorSemiring",
"Real.instZeroLEOneC... | [
"case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\nhb1' : 1 < ↑b\n⊢ ⌊logb (↑b) r⌋ = Int.log b r"
] | 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.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 469,
"column": 10
} | {
"line": 472,
"column": 18
} | {
"line": 473,
"column": 4
} | [
{
"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
} | {
"line": 473,
"column": 4
} | [
{
"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
} | {
"line": 479,
"column": 0
} | [
{
"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.Analysis.SpecialFunctions.Log.Base | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 53
} | {
"line": 388,
"column": 4
} | [
{
"pp": "case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\n⊢ ⌈logb (↑b) r⌉ = Int.clog b r",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"FloorRing.toFloorSemiring",
"Real.instZeroLEOne... | [
"case pos\nb : ℕ\nr : ℝ\nhr✝ : 0 ≤ r\nhr : 0 < r\nhb : 1 < b\nhb1' : 1 < ↑b\n⊢ ⌈logb (↑b) r⌉ = Int.clog b r"
] | 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.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 36
} | {
"line": 39,
"column": 37
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nf : X →ᵇ ℝ≥0\nx : X\n⊢ ↑(f x) ≤ edist 0 f",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nf : X →ᵇ ℝ≥0\nx : X\n⊢ ↑(f x) ≤ edist 0 f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 514,
"column": 4
} | {
"line": 514,
"column": 15
} | {
"line": 514,
"column": 16
} | [
{
"pp": "b : ℝ\nn : ℕ\n⊢ Tendsto (fun x ↦ logb b x ^ n / id x) atTop (𝓝 0)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"NormedDivisionRing.toNormedRing",
"PseudoMetricSpace.toUniformSpace",
"NormedDivisionRing.toDivisionRing",
"n... | [
"b : ℝ\nn : ℕ\n⊢ Tendsto (fun x ↦ logb b x ^ n / x) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 535,
"column": 4
} | {
"line": 535,
"column": 15
} | {
"line": 535,
"column": 16
} | [
{
"pp": "case inl\n⊢ (fun x ↦ log (0 * x)) =O[atTop] log",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"MulZeroClass.zero_mul",
"Asymptotics.IsBigO",
"Real.semiring",
"id",
... | [
"case inl\n⊢ (fun x ↦ 0) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 24
} | {
"line": 543,
"column": 25
} | [
{
"pp": "c : ℝ\n⊢ (fun x ↦ log (x * c)) =O[atTop] log",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.t... | [
"c : ℝ\n⊢ (fun x ↦ log (c * x)) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 160,
"column": 59
} | {
"line": 160,
"column": 70
} | {
"line": 160,
"column": 71
} | [
{
"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), ... | [
"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), f x ∂μs i) L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 546,
"column": 2
} | {
"line": 547,
"column": 9
} | {
"line": 547,
"column": 10
} | [
{
"pp": "b c : ℝ\n⊢ (fun x ↦ logb b (c * x)) =O[atTop] log",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonA... | [
"b c : ℝ\n⊢ (fun x ↦ (log b)⁻¹ * log (c * x)) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 24
} | {
"line": 550,
"column": 25
} | [
{
"pp": "b c : ℝ\n⊢ (fun x ↦ logb b (x * c)) =O[atTop] log",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommR... | [
"b c : ℝ\n⊢ (fun x ↦ logb b (c * x)) =O[atTop] log"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 161,
"column": 59
} | {
"line": 161,
"column": 70
} | {
"line": 161,
"column": 71
} | [
{
"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), ... | [
"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), f x ∂μs i) L... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 627,
"column": 12
} | {
"line": 627,
"column": 23
} | {
"line": 627,
"column": 24
} | [
{
"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",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [... | [
"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 : ℝ), x₀ ≤ x → x < r * x₀ → P x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.RegularityCompacts | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 61
} | {
"line": 42,
"column": 62
} | [
{
"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... | [
"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 K)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 182,
"column": 59
} | {
"line": 182,
"column": 70
} | {
"line": 182,
"column": 71
} | [
{
"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 ... | [
"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 (fun i ↦ ∫ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.BoundedContinuousFunction | {
"line": 183,
"column": 59
} | {
"line": 183,
"column": 70
} | {
"line": 183,
"column": 71
} | [
{
"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 ... | [
"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 (fun i ↦ ∫ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 610,
"column": 6
} | {
"line": 615,
"column": 26
} | {
"line": 616,
"column": 4
} | [
{
"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
} | {
"line": 616,
"column": 4
} | [
{
"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.MulExpNegMulSqIntegral | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 54
} | {
"line": 117,
"column": 6
} | [
{
"pp": "case h_lim\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))))",
"ppTerm": "?... | [
"case h_lim\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 + -(ε * (g x * g x)) / ↑x_1) ^ x_1) atTop (𝓝 (rexp (-(ε * (g x * g x)))))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 34
} | {
"line": 92,
"column": 35
} | [
{
"pp": "n : ℕ\nhn : 0 < n\n⊢ Polynomial.eval (↑n - 1) (descPochhammer ℝ n) = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Real",
"Real.instZero",
"AddGroupWithOne.toAddGroup",
"congrArg",
"descPochhammer",
... | [
"n : ℕ\nhn : 0 < n\n⊢ Polynomial.eval (↑(n - 1)) (descPochhammer ℝ n) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 68
} | {
"line": 103,
"column": 69
} | [
{
"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 • ... | [
"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 • f ↑(p i)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pochhammer | {
"line": 113,
"column": 31
} | {
"line": 113,
"column": 67
} | {
"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)) ... | [
"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 (∑ x ∈ t, w x * ↑((p x).descFactorial n)) / ↑n.factorial"
] | descPochhammer_eval_eq_descFactorial | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.MulExpNegMulSqIntegral | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 40
} | {
"line": 197,
"column": 2
} | [
{
"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... | [
"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 : (Subalgebra.map (toCon... | 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": 176,
"column": 35
} | {
"line": 176,
"column": 68
} | {
"line": 176,
"column": 69
} | [
{
"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",
"ppTerm": "?m.179",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 180,
"column": 6
} | {
"line": 180,
"column": 17
} | {
"line": 180,
"column": 18
} | [
{
"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‖",
"ppTerm": "?m.261",
"assig... | [
"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‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas | {
"line": 89,
"column": 14
} | {
"line": 89,
"column": 25
} | {
"line": 89,
"column": 26
} | [
{
"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",
"ppTerm": "?succ",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.HaarToSphere | {
"line": 195,
"column": 4
} | {
"line": 196,
"column": 11
} | {
"line": 196,
"column": 12
} | [
{
"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 ... | [
"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 - ‖y‖) • u\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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