module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.House | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 22
} | {
"line": 311,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\np q : ℕ\nhpq : p < q\nx : β × (K →+* ℂ) → ℤ\ninst✝² : Fintype β\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝ : Fintype α\nApos... | [
"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\np q : ℕ\nhpq : p < q\nx : β × (K →+* ℂ) → ℤ\ninst✝² : Fintype β\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝ : Fintype α\nApos : 0 ≤ A\nhx... | let h := finrank ℚ K | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.Ostrowski | {
"line": 171,
"column": 8
} | {
"line": 171,
"column": 21
} | {
"line": 171,
"column": 22
} | [
{
"pp": "case isUnit_or_isUnit\nf : AbsoluteValue ℚ ℝ\na b : ℕ\nhp0 : 0 < f ↑(a * b)\nhp1 : f ↑(a * b) < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → a * b ≤ m\nha₁ : a ≠ 1\nhb₁ : b ≠ 1\nha₀ : a ≠ 0\nhb₀ : b ≠ 0\nhap : a < a * b\nhbp : b < a * b\nha : 1 ≤ f ↑a\nhb : 1 ≤ f ↑b\n⊢ False",
"ppTerm": "?isUnit_or_i... | [
"case isUnit_or_isUnit\nf : AbsoluteValue ℚ ℝ\na b : ℕ\nhp0 : 0 < f ↑(a * b)\nhp1 : f (↑a * ↑b) < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → a * b ≤ m\nha₁ : a ≠ 1\nhb₁ : b ≠ 1\nha₀ : a ≠ 0\nhb₀ : b ≠ 0\nhap : a < a * b\nhbp : b < a * b\nha : 1 ≤ f ↑a\nhb : 1 ≤ f ↑b\n⊢ False"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 314,
"column": 7
} | {
"line": 321,
"column": 63
} | {
"line": 322,
"column": 2
} | [] | [
"case calc_1\nK : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\np q : ℕ\nhpq : p < q\nx : β × (K →+* ℂ) → ℤ\ninst✝² : Fintype β\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝ : Fintype α\nApo... | _ = house (algebraMap (𝓞 K) K (∑ r, (x (l, r)) * ((newBasis K) r))) := rfl
_ ≤ ∑ r, house (((algebraMap (𝓞 K) K) (x (l, r))) *
((algebraMap (𝓞 K) K) ((newBasis K) r))) := ?_
_ ≤ ∑ r, ‖x (l, r)‖ * house ((algebraMap (𝓞 K) K) ((newBasis K) r)) := ?_
_ ≤ ∑ r, ‖x (l, r)‖ * (supOfBasis K) :=... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.NumberTheory.NumberField.House | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 22
} | {
"line": 348,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\nha : a ≠ 0\np q : ℕ\nh0p : 0 < p\nhpq : p < q\ninst✝² : Fintype β\ncardβ : Fintype.card β = q\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →... | [
"K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\nha : a ≠ 0\np q : ℕ\nh0p : 0 < p\nhpq : p < q\ninst✝² : Fintype β\ncardβ : Fintype.card β = q\nA : ℝ\nhabs : ∀ (k : α) (l : β), house ((algebraMap (𝓞 K) K) (a k l)) ≤ A\ninst✝¹ : DecidableEq (K →+* ℂ)\ninst✝... | let h := finrank ℚ K | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 70,
"column": 70
} | {
"line": 70,
"column": 83
} | {
"line": 70,
"column": 84
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\nx : ℤ_[p]\nf : ℤ_[p] → ℤ_[p] := fun x ↦ Polynomial.eval x (ascPochhammer ℤ_[p] k)\nhC : ↑k.factorial ≠ 0\nhf : ContinuousAt f x\nn : ℕ\nhn : ‖f x - f ↑n‖ ≤ ‖↑k.factorial‖\n⊢ ‖↑(k.factorial * Ring.multichoose n k)‖ ≤ ‖↑k.factorial‖",
"ppTerm": "?m.165",
"as... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\nx : ℤ_[p]\nf : ℤ_[p] → ℤ_[p] := fun x ↦ Polynomial.eval x (ascPochhammer ℤ_[p] k)\nhC : ↑k.factorial ≠ 0\nhf : ContinuousAt f x\nn : ℕ\nhn : ‖f x - f ↑n‖ ≤ ‖↑k.factorial‖\n⊢ ‖↑k.factorial * ↑(Ring.multichoose n k)‖ ≤ ‖↑k.factorial‖"
] | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 21
} | {
"line": 138,
"column": 22
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoidWithOne M\ninst✝ : AddCommGroup G\nf : M → G\nn R : ℕ\nhR : 1 ≤ R\naux : Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0\n⊢ ∑ j ∈ range (R - 1), R.choose (j + 1) • Δ_[1]^[n + (j + 1)] f 0 + R.choose (R - 1 + 1) • Δ_[1]^[n + (R - 1 + 1)] f 0... | [
"M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoidWithOne M\ninst✝ : AddCommGroup G\nf : M → G\nn R : ℕ\nhR : 1 ≤ R\naux : Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0\n⊢ ∑ x ∈ range (R - 1 + 1), R.choose (x + 1) • Δ_[1]^[n + (x + 1)] f 0 =\n ∑ k ∈ range (n + 1), ((-1) ^ (n - k) * ↑(n.choose k))... | ← sum_range_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.Complex | {
"line": 206,
"column": 61
} | {
"line": 215,
"column": 33
} | {
"line": 217,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ (fun x ↦ ‖x‖) = Valued.v.norm",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"UniformContinuous",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"PadicComplex",
"L... | [] | by
apply UniformSpace.Completion.extension_unique (f := @norm (PadicAlgCl p) _) (g := Valued.v.norm)
· exact uniformContinuous_norm
· letI S := (Valued.toNormedField ℂ_[p] NNReal).toNormedCommRing.toNormedRing.toSeminormedRing
letI := S.toNonUnitalSeminormedRing.toSeminormedAddCommGroup.toSeminormedAddGroup
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Separation.DisjointCover | {
"line": 129,
"column": 62
} | {
"line": 134,
"column": 91
} | {
"line": 136,
"column": 0
} | [
{
"pp": "X : Type u_1\nV : Type u_2\ninst✝⁴ : TopologicalSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : TotallyDisconnectedSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nS : Set (V × V)\nf : C(X, V)\nhS : S ∈ 𝓝ˢ (diagonal V)\n⊢ ∃ n D,\n (∀ (i : Fin n), D i ≠ ⊥) ∧\n (∀ (i : Fin n), ∀ y ∈ D i, ∀ z ... | [] | by
have : (f.prodMap f) ⁻¹' S ∈ nhdsSet (diagonal X) := by
rw [mem_nhdsSet_iff_forall] at hS ⊢
rintro ⟨x, y⟩ (rfl : x = y)
exact (map_continuous _).continuousAt.preimage_mem_nhds (hS _ rfl)
exact exists_finite_disjoint_nonempty_clopen_cover_of_mem_nhds_diagonal_of_profinite this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Pell | {
"line": 532,
"column": 36
} | {
"line": 537,
"column": 20
} | {
"line": 540,
"column": 0
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ a₁.y ≤ a.y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_ze... | [] | by
have H : d * (a₁.y ^ 2 - a.y ^ 2) = a₁.x ^ 2 - a.x ^ 2 := by rw [a.prop_x, a₁.prop_x]; ring
rw [← abs_of_pos hay, ← abs_of_pos h.2.1, ← sq_le_sq, ← mul_le_mul_iff_right₀ h.d_pos,
← sub_nonpos, ← mul_sub, H, sub_nonpos, sq_le_sq, abs_of_pos (zero_lt_one.trans h.1),
abs_of_pos (zero_lt_one.trans hax)]
ex... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Pell | {
"line": 542,
"column": 63
} | {
"line": 542,
"column": 84
} | {
"line": 542,
"column": 85
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ |a.x| * a₁.y ≤ |a.y| * a₁.x",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"abs",
"congrArg",
"Pell.Solution₁.x",
"id",
... | [
"d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ |a.x| * a₁.y ≤ |a.y| * |a₁.x|"
] | ← abs_of_pos h.x_pos, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Valuation | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 17
} | {
"line": 55,
"column": 18
} | [
{
"pp": "case inr\nA : Type u_1\nB : Type u_4\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDedekindDomain A\ninst✝⁴ : CommRing B\ninst✝³ : IsDedekindDomain B\ninst✝² : Algebra A B\ninst✝¹ : Module.IsTorsionFree A B\nv : HeightOneSpectrum A\nw : HeightOneSpectrum B\ninst✝ : w.asIdeal.LiesOver v.asIdeal\nx : A\nhx : Ideal.sp... | [
"case inr\nA : Type u_1\nB : Type u_4\ninst✝⁶ : CommRing A\ninst✝⁵ : IsDedekindDomain A\ninst✝⁴ : CommRing B\ninst✝³ : IsDedekindDomain B\ninst✝² : Algebra A B\ninst✝¹ : Module.IsTorsionFree A B\nv : HeightOneSpectrum A\nw : HeightOneSpectrum B\ninst✝ : w.asIdeal.LiesOver v.asIdeal\nx : A\nhx : Ideal.span {x} ≠ ⊥\n... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 106,
"column": 12
} | {
"line": 106,
"column": 33
} | {
"line": 106,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra... | [
"R : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra K L\ninst✝⁵... | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 605,
"column": 4
} | {
"line": 605,
"column": 52
} | {
"line": 606,
"column": 4
} | [
{
"pp": "case h.inr\nd : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\nx : ℕ\nih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n\na : Solution₁ d\nhay : 0 ≤ a.y\nhax' : ↑x = a.x\nhax : 0 < ↑x\nhy : 0 < a.y\nhx₁ : 1 < a.x\nhxx₁ : 0 < (a * a₁⁻¹).x\nhxx₂ : (a * a₁⁻¹).x < a.x\nhyy : 0 ≤ (... | [
"case h.inr\nd : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\nx : ℕ\nih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n\na : Solution₁ d\nhay : 0 ≤ a.y\nhax' : ↑x = a.x\nhax : 0 < ↑x\nhy : 0 < a.y\nhx₁ : 1 < a.x\nhyy : 0 ≤ (a * a₁⁻¹).y\nx' : ℕ\nhx' : ↑x' = (a * a₁⁻¹).x\nhxx₁ : 0 < ↑x'\nh... | lift (a * a₁⁻¹).x to ℕ using hxx₁.le with x' hx' | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.NumberTheory.Pell | {
"line": 662,
"column": 2
} | {
"line": 662,
"column": 67
} | {
"line": 664,
"column": 0
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nh : 1 < a.x ∧ 0 < a.y ∧ ∀ (b : Solution₁ d), ∃ n, b = a ^ n ∨ b = -a ^ n\nh₀ : 0 < d\nhd : ¬IsSquare d\na₁ : Solution₁ d\nha₁ : IsFundamental a₁\nb : Solution₁ d\nhb₂ : ∀ (y : Solution₁ d), (fun a₁ ↦ 1 < a₁.x ∧ 0 < a₁.y ∧ ∀ (a : Solution₁ d), ∃ n, a = a₁ ^ n ∨ a = -a₁ ^ n) y → y... | [] | rwa [hb₂ a h, ← hb₂ a₁ ⟨ha₁.1, ha₁.2.1, ha₁.eq_zpow_or_neg_zpow⟩] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.RamificationInertia.Valuation | {
"line": 111,
"column": 32
} | {
"line": 111,
"column": 82
} | {
"line": 111,
"column": 82
} | [
{
"pp": "A : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝¹⁶ : CommRing A\ninst✝¹⁵ : IsDedekindDomain A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : IsDedekindDomain B\ninst✝¹² : Algebra A B\ninst✝¹¹ : Module.IsTorsionFree A B\ninst✝¹⁰ : Field K\ninst✝⁹ : Field L\ninst✝⁸ : Algebra K L\ninst✝⁷ : Algebra A K\nins... | [] | by simp [EmbeddingLike.map_eq_zero_iff (f := σwV)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 175,
"column": 15
} | {
"line": 175,
"column": 39
} | {
"line": 175,
"column": 40
} | [
{
"pp": "x y : ℤ[i]\n⊢ (toComplex { re := round (↑(x * star y).re / ↑(norm y)), im := round (↑(x * star y).im / ↑(norm y)) }).im =\n ↑(round (toComplex x / toComplex y).im)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"G... | [
"x y : ℤ[i]\n⊢ (toComplex { re := round ↑(↑(x * star y).re / ↑(norm y)), im := round (↑(x * star y).im / ↑(norm y)) }).im =\n ↑(round (toComplex x / toComplex y).im)"
] | ← @Rat.round_cast ℝ _ _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity | {
"line": 46,
"column": 72
} | {
"line": 46,
"column": 85
} | {
"line": 46,
"column": 86
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : Prime ↑p\nhp1 : p % 2 = 1\nhp3 : p % 4 ≠ 3\nk : ℕ\nk_lt_p : k < p\nhk : -1 = ↑k * ↑k\n⊢ ↑(k * k) + ↑1 = 0",
"ppTerm": "?m.150",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : Prime ↑p\nhp1 : p % 2 = 1\nhp3 : p % 4 ≠ 3\nk : ℕ\nk_lt_p : k < p\nhk : -1 = ↑k * ↑k\n⊢ ↑k * ↑k + ↑1 = 0"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 22
} | {
"line": 221,
"column": 23
} | [
{
"pp": "x y : ℤ[i]\nhy : y ≠ 0\n⊢ (norm x).natAbs ≤ (norm (x * y)).natAbs",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Zsqrtd.instMul",
"Eq.mpr",
"GaussianInt",
"HMul.hMul",
"congrArg",
"Zsqrtd.norm_mul",
"id",
"Int.instNegInt",
"I... | [
"x y : ℤ[i]\nhy : y ≠ 0\n⊢ (norm x).natAbs ≤ (norm x * norm y).natAbs"
] | Zsqrtd.norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 176,
"column": 51
} | {
"line": 178,
"column": 73
} | {
"line": 180,
"column": 0
} | [
{
"pp": "n x y : ℕ\nh : n = x ^ 2 + y ^ 2\nhc : x.Coprime y\n⊢ IsSquare (-1)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Int.ofNat_inj._simp_2",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.instMonoid",
"Ad... | [] | by
zify at h
exact ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime h hc.isCoprime | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.SumFourSquares | {
"line": 178,
"column": 80
} | {
"line": 178,
"column": 93
} | {
"line": 178,
"column": 94
} | [
{
"pp": "p : ℕ\nhp : Prime p\nthis✝ : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 ... | [
"p : ℕ\nhp : Prime p\nthis✝ : Fact (Prime p)\nnatAbs_iff :\n ∀ {a b c d : ℤ} {k : ℕ},\n a.natAbs ^ 2 + b.natAbs ^ 2 + c.natAbs ^ 2 + d.natAbs ^ 2 = k ↔ a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = ↑k\nhm✝ : ∃ m < p, 0 < m ∧ ∃ a b c d, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = m * p\nm : ℕ\nhmin : ∀ m_1 < m, ¬(m_1 < p ∧ 0 < m_1 ∧ ∃ a ... | Nat.cast_mul, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.NumberTheory.Transcendental.Liouville.Basic | {
"line": 197,
"column": 2
} | {
"line": 197,
"column": 95
} | {
"line": 200,
"column": 2
} | [
{
"pp": "x : ℝ\nlx : Liouville x\nf : ℤ[X]\nf0 : f ≠ 0\nef0 : eval x (map (algebraMap ℤ ℝ) f) = 0\nA : ℝ\nhA : 0 < A\nh : ∀ (a : ℤ) (b : ℕ), 1 ≤ (↑b + 1) ^ f.natDegree * (|x - ↑a / (↑b + 1)| * A)\nr : ℕ\nhn : A < (1 + 1) ^ r\na b : ℤ\nb1 : 1 < b\na1 : |x - ↑a / ↑b| < 1 / ↑b ^ (r + f.natDegree)\n⊢ False",
"p... | [
"x : ℝ\nlx : Liouville x\nf : ℤ[X]\nf0 : f ≠ 0\nef0 : eval x (map (algebraMap ℤ ℝ) f) = 0\nA : ℝ\nhA : 0 < A\nh : ∀ (a : ℤ) (b : ℕ), 1 ≤ (↑b + 1) ^ f.natDegree * (|x - ↑a / (↑b + 1)| * A)\nr : ℕ\nhn : A < (1 + 1) ^ r\na b : ℤ\nb1 : 1 < b\na1 : |x - ↑a / ↑b| < 1 / ↑b ^ (r + f.natDegree)\nb0 : 0 < ↑b\n⊢ False"
] | have b0 : (0 : ℝ) < b := zero_lt_one.trans (by rw [← Int.cast_one]; exact Int.cast_lt.mpr b1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 418,
"column": 2
} | {
"line": 418,
"column": 54
} | {
"line": 419,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ninst✝¹ : IsDedekindDomain S\ninst✝ : P.IsPrime\nhP : P ≠ ⊥\ni : ℕ\nhi : ... | [
"case refine_2\nR : Type u\ninst✝⁴ : CommRing R\nS : Type v\ninst✝³ : CommRing S\ninst✝² : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ninst✝¹ : IsDedekindDomain S\ninst✝ : P.IsPrime\nhP : P ≠ ⊥\ni : ℕ\nhi : failed to pr... | · exact quotientToQuotientRangePowQuotSucc p P a_mem | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 480,
"column": 26
} | {
"line": 480,
"column": 39
} | {
"line": 480,
"column": 40
} | [
{
"pp": "case pos\nR : Type u\ninst✝⁵ : CommRing R\nS : Type v\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\nP : Ideal S\ninst✝² : IsDedekindDomain S\nhP0 : P ≠ ⊥\ninst✝¹ : p.IsMaximal\ninst✝ : P.IsPrime\nhe : e ≠ 0\nthis✝¹ : NeZero e := { out := he }\nthis✝ : Algebra (R ⧸ p) (S ⧸ P) := Quotient.alge... | [
"case pos\nR : Type u\ninst✝⁵ : CommRing R\nS : Type v\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\np : Ideal R\nP : Ideal S\ninst✝² : IsDedekindDomain S\nhP0 : P ≠ ⊥\ninst✝¹ : p.IsMaximal\ninst✝ : P.IsPrime\nhe : e ≠ 0\nthis✝¹ : NeZero e := { out := he }\nthis✝ : Algebra (R ⧸ p) (S ⧸ P) := Quotient.algebraQuotientO... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 609,
"column": 2
} | {
"line": 614,
"column": 94
} | {
"line": 616,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹⁷ : CommRing R\nS : Type v\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\ninst✝¹⁴ : IsDedekindDomain S\nK : Type u_1\nL : Type u_2\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : IsDedekindDomain R\ninst✝¹⁰ : Algebra R K\ninst✝⁹ : IsFractionRing R K\ninst✝⁸ : Algebra S L\ninst✝⁷ : IsF... | [] | classical
have hP : P ∈ IsDedekindDomain.primesOverFinset p S :=
(IsDedekindDomain.mem_primesOverFinset_iff hp0 _).mpr ⟨hP₁, hP₂⟩
rw [← sum_ramification_inertia S K L hp0, ← Finset.add_sum_erase _ _ hP]
refine le_trans (Nat.le_mul_of_pos_left _ ?_) (Nat.le_add_right _ _)
exact Nat.pos_iff_ne_zero.mpr <| IsD... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 609,
"column": 2
} | {
"line": 614,
"column": 94
} | {
"line": 616,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹⁷ : CommRing R\nS : Type v\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\ninst✝¹⁴ : IsDedekindDomain S\nK : Type u_1\nL : Type u_2\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : IsDedekindDomain R\ninst✝¹⁰ : Algebra R K\ninst✝⁹ : IsFractionRing R K\ninst✝⁸ : Algebra S L\ninst✝⁷ : IsF... | [] | classical
have hP : P ∈ IsDedekindDomain.primesOverFinset p S :=
(IsDedekindDomain.mem_primesOverFinset_iff hp0 _).mpr ⟨hP₁, hP₂⟩
rw [← sum_ramification_inertia S K L hp0, ← Finset.add_sum_erase _ _ hP]
refine le_trans (Nat.le_mul_of_pos_left _ ?_) (Nat.le_add_right _ _)
exact Nat.pos_iff_ne_zero.mpr <| IsD... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 609,
"column": 2
} | {
"line": 614,
"column": 94
} | {
"line": 616,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹⁷ : CommRing R\nS : Type v\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : Algebra R S\ninst✝¹⁴ : IsDedekindDomain S\nK : Type u_1\nL : Type u_2\ninst✝¹³ : Field K\ninst✝¹² : Field L\ninst✝¹¹ : IsDedekindDomain R\ninst✝¹⁰ : Algebra R K\ninst✝⁹ : IsFractionRing R K\ninst✝⁸ : Algebra S L\ninst✝⁷ : IsF... | [] | classical
have hP : P ∈ IsDedekindDomain.primesOverFinset p S :=
(IsDedekindDomain.mem_primesOverFinset_iff hp0 _).mpr ⟨hP₁, hP₂⟩
rw [← sum_ramification_inertia S K L hp0, ← Finset.add_sum_erase _ _ hP]
refine le_trans (Nat.le_mul_of_pos_left _ ?_) (Nat.le_add_right _ _)
exact Nat.pos_iff_ne_zero.mpr <| IsD... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 65
} | {
"line": 206,
"column": 2
} | [
{
"pp": "case h.right.refine_2\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^... | [
"case h.right.refine_2\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^ p)) = (p - ... | have h : ((f.aroots ℂ).map c').toFinset.Nonempty := ⟨c' r, aux⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.WellApproximable | {
"line": 254,
"column": 94
} | {
"line": 258,
"column": 38
} | {
"line": 259,
"column": 4
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nδ : ℕ → ℝ\nhδ : Tendsto δ atTop (𝓝 0)\nthis✝ : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p\nμ : Measure 𝕊 := volume\nu : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)\nhu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p\nhu : Tendsto (addOrderOf ∘ u)... | [] | by
apply (ergodic_nsmul hp.one_lt).ae_empty_or_univ_of_image_ae_le (hA₀ p).nullMeasurableSet
apply (LE.le.eventuallyLE this).congr EventuallyEq.rfl
exact blimsup_thickening_mul_ae_eq μ (fun n => 0 < n ∧ p∤n) (fun n => {y | addOrderOf y = n})
(Nat.cast_pos.mpr hp.pos) _ hδ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Completion | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 18
} | {
"line": 183,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : PartialOrder β\nf : β ↪o α\nx : β\n⊢ f x ≤ sSup (⇑f '' (principal x).left)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : PartialOrder β\nf : β ↪o α\nx : β\n⊢ ∀ b ∈ upperBounds (⇑f '' (principal x).left), f x ≤ b"
] | rw [le_sSup_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.CompleteSublattice | {
"line": 101,
"column": 29
} | {
"line": 101,
"column": 39
} | {
"line": 101,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\nS : Set ↥L\n⊢ sInf (Subtype.val '' S) = ⨅ N ∈ S, ↑N",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"congrArg",
"sInf_image",
"Membership.mem",
"CompleteLattice... | [
"α : Type u_1\ninst✝ : CompleteLattice α\nL : CompleteSublattice α\nS : Set ↥L\n⊢ ⨅ a ∈ S, ↑a = ⨅ N ∈ S, ↑N"
] | sInf_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.WellApproximable | {
"line": 314,
"column": 6
} | {
"line": 316,
"column": 45
} | {
"line": 318,
"column": 0
} | [
{
"pp": "case neg.inr\nT : ℝ\nhT : Fact (0 < T)\nδ : ℕ → ℝ\nhδ : Tendsto δ atTop (𝓝 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p\nμ : Measure 𝕊 := volume\nu : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)\nhu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p\nhu : Tendsto (ad... | [] | rcases hB p with _ | h; · contradiction
simp only [μ, h, union_ae_eq_univ_of_ae_eq_univ_left,
union_ae_eq_univ_of_ae_eq_univ_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.WellApproximable | {
"line": 314,
"column": 6
} | {
"line": 316,
"column": 45
} | {
"line": 318,
"column": 0
} | [
{
"pp": "case neg.inr\nT : ℝ\nhT : Fact (0 < T)\nδ : ℕ → ℝ\nhδ : Tendsto δ atTop (𝓝 0)\nthis : SemilatticeSup Nat.Primes := Nat.Subtype.semilatticeSup fun p ↦ Nat.Prime p\nμ : Measure 𝕊 := volume\nu : Nat.Primes → 𝕊 := fun p ↦ ↑(↑1 / ↑↑p * T)\nhu₀ : ∀ (p : Nat.Primes), addOrderOf (u p) = ↑p\nhu : Tendsto (ad... | [] | rcases hB p with _ | h; · contradiction
simp only [μ, h, union_ae_eq_univ_of_ae_eq_univ_left,
union_ae_eq_univ_of_ae_eq_univ_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Nat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 55
} | {
"line": 34,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"HSub.hSub",
"Nat.instLocallyFiniteOrder",
"Eq.mp",
"Finset.coe_Ioo",
"Nat.card_Ioo",
"instSubNat",
"instOfNatNat",
... | [] | simpa [← Set.ncard_coe_finset] using Nat.card_Ioo a b | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Interval.Set.Nat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 55
} | {
"line": 34,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"HSub.hSub",
"Nat.instLocallyFiniteOrder",
"Eq.mp",
"Finset.coe_Ioo",
"Nat.card_Ioo",
"instSubNat",
"instOfNatNat",
... | [] | simpa [← Set.ncard_coe_finset] using Nat.card_Ioo a b | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.Nat | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 55
} | {
"line": 34,
"column": 0
} | [
{
"pp": "a b : ℕ\n⊢ (Ioo a b).ncard = b - a - 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"HSub.hSub",
"Nat.instLocallyFiniteOrder",
"Eq.mp",
"Finset.coe_Ioo",
"Nat.card_Ioo",
"instSubNat",
"instOfNatNat",
... | [] | simpa [← Set.ncard_coe_finset] using Nat.card_Ioo a b | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Nucleus | {
"line": 247,
"column": 23
} | {
"line": 247,
"column": 54
} | {
"line": 249,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m : Nucleus X\nx✝ y : X\nn : Nucleus X\nx : X\nhx : ↑(rangeFactorization (⇑n) x) ≤ x\n⊢ ⟨x, ⋯⟩ = rangeFactorization (⇑n) x",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CompleteLattice.toLattice",
"Nucleus",
"... | [] | ext; exact le_apply.antisymm hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Nucleus | {
"line": 247,
"column": 23
} | {
"line": 247,
"column": 54
} | {
"line": 249,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : Frame X\nn✝ m : Nucleus X\nx✝ y : X\nn : Nucleus X\nx : X\nhx : ↑(rangeFactorization (⇑n) x) ≤ x\n⊢ ⟨x, ⋯⟩ = rangeFactorization (⇑n) x",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CompleteLattice.toLattice",
"Nucleus",
"... | [] | ext; exact le_apply.antisymm hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Partition.Basic | {
"line": 367,
"column": 46
} | {
"line": 369,
"column": 29
} | {
"line": 371,
"column": 0
} | [
{
"pp": "α : Type u_1\nx y : α\nu : Set α\nP : Partition u\nh : P.Rel x y\n⊢ x ∈ u",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Partition.subset_of_mem",
"Membership.mem",
"And.casesOn",
"And",
"Partition",
"Exists.casesOn",
"CompleteBooleanAlg... | [] | by
obtain ⟨t, htP, hxt, -⟩ := h
exact subset_of_mem htP hxt | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Sublocale | {
"line": 147,
"column": 8
} | {
"line": 149,
"column": 53
} | {
"line": 150,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nι : Sort u_2\nS✝ T : Sublocale X\ns✝ : Set X\nf : ι → X\na✝ b✝ : X\nS : Sublocale X\na b : X\ns : ↥S\n⊢ ↑(Sublocale.restrictAux✝ S b) ≤ a ⇨ ↑s ↔ b ≤ a ⇨ ↑s",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.coe_prop"... | [] | set c : S := ⟨a ⇨ s, S.himp_mem s.coe_prop⟩
change Sublocale.restrictAux S b ≤ c.val ↔ b ≤ c
rw [S.giAux.u_le_u_iff, S.giAux.gc.le_iff_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Sublocale | {
"line": 147,
"column": 8
} | {
"line": 149,
"column": 53
} | {
"line": 150,
"column": 6
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nι : Sort u_2\nS✝ T : Sublocale X\ns✝ : Set X\nf : ι → X\na✝ b✝ : X\nS : Sublocale X\na b : X\ns : ↥S\n⊢ ↑(Sublocale.restrictAux✝ S b) ≤ a ⇨ ↑s ↔ b ≤ a ⇨ ↑s",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.coe_prop"... | [] | set c : S := ⟨a ⇨ s, S.himp_mem s.coe_prop⟩
change Sublocale.restrictAux S b ≤ c.val ↔ b ≤ c
rw [S.giAux.u_le_u_iff, S.giAux.gc.le_iff_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Types.Arithmetic | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 58
} | {
"line": 137,
"column": 0
} | [
{
"pp": "a b c : OrderType.{u_1}\nx✝⁵ : Type u_1\nx✝⁴ : LinearOrder x✝⁵\nx✝³ : Type u_1\nx✝² : LinearOrder x✝³\nx✝¹ : Type u_1\nx✝ : LinearOrder x✝¹\n⊢ type (Lex ((x✝³ ⊕ₗ x✝¹) × x✝⁵)) = type (Lex (x✝³ × x✝⁵) ⊕ₗ Lex (x✝¹ × x✝⁵))",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Lex",
... | [] | exact (Prod.Lex.sumLexProdLexDistrib _ _ _).type_congr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Martingale.Centering | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 43
} | {
"line": 157,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝ : SigmaFiniteFiltration μ ℱ\nhf : IsStronglyPredictable ℱ f\nhfint : ∀ (n : ℕ), Integrable (f n) μ\nn : ℕ\n⊢ martingalePart f ℱ μ n =ᵐ[μ]... | [] | filter_upwards [hf.predictablePart_eq (μ := μ) hfint n] with ω hω
simp [martingalePart, hω, sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Centering | {
"line": 154,
"column": 2
} | {
"line": 155,
"column": 43
} | {
"line": 157,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝ : SigmaFiniteFiltration μ ℱ\nhf : IsStronglyPredictable ℱ f\nhfint : ∀ (n : ℕ), Integrable (f n) μ\nn : ℕ\n⊢ martingalePart f ℱ μ n =ᵐ[μ]... | [] | filter_upwards [hf.predictablePart_eq (μ := μ) hfint n] with ω hω
simp [martingalePart, hω, sub_eq_add_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.HittingTime | {
"line": 228,
"column": 2
} | {
"line": 238,
"column": 73
} | {
"line": 240,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\ns : Set β\nn i : ι\nω : Ω\ninst✝ : WellFoundedLT ι\nm : ι\nh_exists : ∃ j ∈ Set.Icc n m, u j ω ∈ s\n⊢ hittingBtwn u s n m ω ≤ i ↔ ∃ j ∈ Set.Icc n i, u j ω ∈ s",
"ppTerm": "?m.29",
"assigned": t... | [] | constructor <;> intro h'
· exact ⟨hittingBtwn u s n m ω, ⟨le_hittingBtwn_of_exists h_exists, h'⟩,
hittingBtwn_mem_set h_exists⟩
· have h'' : ∃ k ∈ Set.Icc n (min m i), u k ω ∈ s := by
obtain ⟨k₁, hk₁_mem, hk₁_s⟩ := h_exists
obtain ⟨k₂, hk₂_mem, hk₂_s⟩ := h'
refine ⟨min k₁ k₂, ⟨le_min hk₁_mem... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.HittingTime | {
"line": 228,
"column": 2
} | {
"line": 238,
"column": 73
} | {
"line": 240,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\ns : Set β\nn i : ι\nω : Ω\ninst✝ : WellFoundedLT ι\nm : ι\nh_exists : ∃ j ∈ Set.Icc n m, u j ω ∈ s\n⊢ hittingBtwn u s n m ω ≤ i ↔ ∃ j ∈ Set.Icc n i, u j ω ∈ s",
"ppTerm": "?m.29",
"assigned": t... | [] | constructor <;> intro h'
· exact ⟨hittingBtwn u s n m ω, ⟨le_hittingBtwn_of_exists h_exists, h'⟩,
hittingBtwn_mem_set h_exists⟩
· have h'' : ∃ k ∈ Set.Icc n (min m i), u k ω ∈ s := by
obtain ⟨k₁, hk₁_mem, hk₁_s⟩ := h_exists
obtain ⟨k₂, hk₂_mem, hk₂_s⟩ := h'
refine ⟨min k₁ k₂, ⟨le_min hk₁_mem... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Convergence | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 29
} | {
"line": 316,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhunif : UniformIntegrable f 1 μ\n⊢ Tendsto (fun n ↦ eLpNorm (f n - limitProcess f ℱ μ) 1 μ) atTop (𝓝 0)",
"ppTerm": "?m.47",
"assigned": true,
"used... | [
"Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhunif : UniformIntegrable f 1 μ\nR : ℝ≥0\nhR : ∀ (i : ℕ), eLpNorm (f i) 1 μ ≤ ↑R\n⊢ Tendsto (fun n ↦ eLpNorm (f n - limitProcess f ℱ μ) 1 μ) atTop (𝓝 0)"
] | obtain ⟨R, hR⟩ := hunif.2.2 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Probability.Martingale.Convergence | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 29
} | {
"line": 367,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\ng : Ω → ℝ\nhg : Integrable g μ\nhgmeas : StronglyMeasurable g\nhle : ⨆ n, ↑ℱ n ≤ m0\nhunif : UniformIntegrable (fun n ↦ μ[g | ↑ℱ n]) 1 μ\n⊢ ∀ᵐ (x : Ω) ∂μ, Tendsto (fun n ↦ μ[g | ↑ℱ n] x) atTop (𝓝 (g x)... | [
"Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\ng : Ω → ℝ\nhg : Integrable g μ\nhgmeas : StronglyMeasurable g\nhle : ⨆ n, ↑ℱ n ≤ m0\nhunif : UniformIntegrable (fun n ↦ μ[g | ↑ℱ n]) 1 μ\nR : ℝ≥0\nhR : ∀ (i : ℕ), eLpNorm ((fun n ↦ μ[g | ↑ℱ n]) i) 1 μ ≤ ↑R\n⊢ ∀ᵐ (x... | obtain ⟨R, hR⟩ := hunif.2.2 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Probability.Process.Stopping | {
"line": 425,
"column": 32
} | {
"line": 436,
"column": 98
} | {
"line": 438,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁵ : Add ι\ninst✝⁴ : LinearOrder ι\ninst✝³ : CanonicallyOrderedAdd ι\ninst✝² : Countable ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\n⊢ IsStoppin... | [] | by
intro j
have h : {ω | (τ + π) ω ≤ j} = ⋃ k : Set.Iic j, {ω | π ω = k} ∩ {ω | τ ω + k ≤ j} := by
ext ω
simp only [Pi.add_apply, Set.mem_setOf_eq, Set.mem_iUnion, Set.mem_inter_iff]
cases τ ω with
| top => simp
| coe a =>
cases π ω with
| top => simp
| coe b => norm_cast; simp... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Process.Stopping | {
"line": 561,
"column": 55
} | {
"line": 564,
"column": 38
} | {
"line": 566,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : LinearOrder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhτ : IsStoppingTime f τ\ni : ι\n⊢ MeasurableSet {ω | ↑i < τ ω}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Preorder.toLT",
... | [] | by
have : {ω : Ω | i < τ ω} = {ω : Ω | τ ω ≤ i}ᶜ := by ext1 ω; simp
rw [this]
exact (hτ.measurableSet_le' i).compl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 42
} | {
"line": 155,
"column": 2
} | [
{
"pp": "Ω : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\nR : ℝ≥0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\ng : ℕ → Ω → ℝ := fun n ω ↦ f n ω - f 0 ω\nhg : Submartingale g ℱ μ\nω : Ω\n⊢ g 0 ω = 0 ω",
... | [] | simp only [g, sub_self, Pi.zero_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Martingale.OptionalStopping | {
"line": 163,
"column": 30
} | {
"line": 181,
"column": 74
} | {
"line": 182,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\nthis :\n ε • μ {ω | ↑ε ≤ (range (n + 1)).sup' ⋯ fun k ↦ f k ω} +\n ENNReal.ofReal (∫ (ω : Ω) in {ω | ((range (n + 1)).... | [] | by
have hadd : ENNReal.ofReal (∫ ω, f n ω ∂μ) =
ENNReal.ofReal
(∫ ω in {ω | ε ≤ (range (n + 1)).sup' nonempty_range_add_one fun k => f k ω}, f n ω ∂μ) +
ENNReal.ofReal
(∫ ω in {ω | ((range (n + 1)).sup' nonempty_range_add_one fun k => f k ω) < ε},
f n ω ∂μ) := by
rw [← EN... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.OptionalStopping | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 46
} | {
"line": 192,
"column": 6
} | [
{
"pp": "case hf\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\n⊢ IntegrableOn (fun ω ↦ f n ω) {ω | ((range (n + 1)).sup' ⋯ fun k ↦ f k ω) < ↑ε} μ",
"ppTerm": "?hf",
"assi... | [
"case hg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ninst✝ : IsFiniteMeasure μ\nhsub : Submartingale f 𝒢 μ\nhnonneg : 0 ≤ f\nε : ℝ≥0\nn : ℕ\n⊢ IntegrableOn (fun ω ↦ stoppedValue f (fun ω ↦ ↑(hittingBtwn f {y | ↑ε ≤ y} 0 n ω)) ω)\n {ω | ((range (n + 1)).sup' ⋯ fun k... | · exact (hsub.integrable n).integrableOn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Process.Stopping | {
"line": 815,
"column": 7
} | {
"line": 815,
"column": 45
} | {
"line": 817,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : Nonempty ι\nu : ι → Ω → β\ninst✝ : Bot ι\nx✝ : Ω\n⊢ stoppedValue u (fun x ↦ ⊥) x✝ = u ⊥ x✝",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Bot.bot",
"eq_self",
"of_eq_true",
"Eq"
],
"usedFVars": [
... | [] | simp [stoppedValue, ← WithTop.coe_bot] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ConditionalExpectation | {
"line": 59,
"column": 22
} | {
"line": 59,
"column": 27
} | {
"line": 59,
"column": 28
} | [
{
"pp": "case pos.refine_3\nΩ : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : Indep m₁ m₂ μ\nhfi... | [
"case pos.refine_3\nΩ : Type u_1\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nm₁ m₂ m : MeasurableSpace Ω\nμ : Measure Ω\nf : Ω → E\nhle₁ : m₁ ≤ m\nhle₂ : m₂ ≤ m\ninst✝ : SigmaFinite (μ.trim hle₂)\nhf : StronglyMeasurable f\nhindp : Indep m₁ m₂ μ\nhfint : MemLp f... | hvint | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 29
} | {
"line": 270,
"column": 0
} | [
{
"pp": "Ω : Type u_2\ns : ℕ → Set Ω\nω : Ω\nn : ℕ\nx✝¹ : Ω\nx✝ : x✝¹ ∈ s (n + 1)\n⊢ ‖1 x✝¹‖ ≤ 1",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"le_refl",
"Real.i... | [] | rw [Pi.one_apply, norm_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.BorelCantelli | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 31
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ns : ℕ → Set Ω\nhsm : ∀ (n : ℕ), MeasurableSet (s n)\nhs : iIndepSet s μ\nhs' : ∑' (n : ℕ), μ (s n) = ∞\nthis✝ : IsProbabilityMeasure μ\nthis :\n ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), μ[(s (n + 1)).indicator 1 | ↑(filtrationOfSet hsm) n] ω = (fun ... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Stopping | {
"line": 931,
"column": 76
} | {
"line": 973,
"column": 15
} | {
"line": 975,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁷ : Nonempty ι\nτ : Ω → WithTop ι\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : SecondCountableTopology ι\ninst✝¹ : BorelSpace ι\nf : Filtration ι m\ninst✝ : PseudoMetrizableSp... | [] | by
refine fun i ↦ (Measurable.untopA ?_).stronglyMeasurable
let m_prod : MeasurableSpace (Set.Iic i × Ω) := Subtype.instMeasurableSpace.prod (f i)
let m_set : ∀ t : Set (Set.Iic i × Ω), MeasurableSpace t := fun _ =>
@Subtype.instMeasurableSpace (Set.Iic i × Ω) _ m_prod
let s := {p : Set.Iic i × Ω | τ p.2 ≤ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 76,
"column": 6
} | {
"line": 76,
"column": 17
} | {
"line": 76,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhX : AEMeasurable X μ\nh : ¬Integrable (fun ω ↦ rexp (z.re * X ω)) μ\n⊢ complexMGF X μ z = 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"Inner... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhX : AEMeasurable X μ\nh : ¬Integrable (fun ω ↦ rexp (z.re * X ω)) μ\n⊢ ∫ (ω : Ω), cexp (z * ↑(X ω)) ∂μ = 0"
] | complexMGF, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 17
} | {
"line": 82,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhX : AEMeasurable X μ\nt : ℂ\n⊢ complexMGF id (Measure.map X μ) t = complexMGF X μ t",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.to... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nhX : AEMeasurable X μ\nt : ℂ\n⊢ ∫ (ω : ℝ), cexp (t * ↑(id ω)) ∂Measure.map X μ = complexMGF X μ t"
] | complexMGF, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 17
} | {
"line": 92,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\n⊢ ‖complexMGF X μ z‖ ≤ mgf X μ z.re",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Re... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\n⊢ ‖∫ (ω : Ω), cexp (z * ↑(X ω)) ∂μ‖ ≤ mgf X μ z.re"
] | complexMGF, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 92,
"column": 2
} | {
"line": 96,
"column": 61
} | {
"line": 98,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\n⊢ ‖complexMGF X μ z‖ ≤ mgf X μ z.re",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Complex.mul_im",
"add_mul",
"Norm.norm",
"Eq.mpr",
"In... | [] | rw [complexMGF, ← re_add_im z]
simp_rw [add_mul, Complex.exp_add, re_add_im]
calc ‖∫ ω, cexp (z.re * X ω) * cexp (z.im * I * X ω) ∂μ‖
_ ≤ ∫ ω, ‖cexp (z.re * X ω) * cexp (z.im * I * X ω)‖ ∂μ := norm_integral_le_integral_norm _
_ = ∫ ω, rexp (z.re * X ω) ∂μ := by simp [Complex.norm_exp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 92,
"column": 2
} | {
"line": 96,
"column": 61
} | {
"line": 98,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\n⊢ ‖complexMGF X μ z‖ ≤ mgf X μ z.re",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Complex.mul_im",
"add_mul",
"Norm.norm",
"Eq.mpr",
"In... | [] | rw [complexMGF, ← re_add_im z]
simp_rw [add_mul, Complex.exp_add, re_add_im]
calc ‖∫ ω, cexp (z.re * X ω) * cexp (z.im * I * X ω) ∂μ‖
_ ≤ ∫ ω, ‖cexp (z.re * X ω) * cexp (z.im * I * X ω)‖ ∂μ := norm_integral_le_integral_norm _
_ = ∫ ω, rexp (z.re * X ω) ∂μ := by simp [Complex.norm_exp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 17
} | {
"line": 99,
"column": 18
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nx : ℝ\n⊢ complexMGF X μ ↑x = ↑(mgf X μ x)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"HMul.hMul... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nx : ℝ\n⊢ ∫ (ω : Ω), cexp (↑x * ↑(X ω)) ∂μ = ↑(mgf X μ x)"
] | complexMGF, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 1299,
"column": 2
} | {
"line": 1299,
"column": 66
} | {
"line": 1300,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nu : ℕ → Ω → β\nτ π : Ω → ℕ∞\ninst✝ : AddCommGroup β\nhle : τ ≤ π\nN : ℕ\nhbdd : ∀ (ω : Ω), π ω ≤ ↑N\nhπ_top : ∀ (ω : Ω), π ω ≠ ⊤\nhτ_top : ∀ (ω : Ω), τ ω ≠ ⊤\nω : Ω\n⊢ (∑ i ∈ Finset.Ico (untopA (τ ω)) (untopA (π ω)), (u (i + 1) - u i)) ω =\n (∑ i ∈ Finset.range (N + 1), {... | [
"Ω : Type u_1\nβ : Type u_2\nu : ℕ → Ω → β\nτ π : Ω → ℕ∞\ninst✝ : AddCommGroup β\nhle : τ ≤ π\nN : ℕ\nhbdd : ∀ (ω : Ω), π ω ≤ ↑N\nhπ_top : ∀ (ω : Ω), π ω ≠ ⊤\nhτ_top : ∀ (ω : Ω), τ ω ≠ ⊤\nω : Ω\n⊢ ∑ c ∈ Finset.Ico (untopA (τ ω)) (untopA (π ω)), (u (c + 1) - u c) ω =\n ∑ c ∈ Finset.range (N + 1) with ω ∈ {ω | τ ω... | simp only [Finset.sum_apply, Finset.sum_indicator_eq_sum_filter] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 329,
"column": 11
} | {
"line": 329,
"column": 22
} | {
"line": 329,
"column": 23
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nΩ' : Type u_3\nmΩ' : MeasurableSpace Ω'\nY : Ω' → ℝ\nμ' : Measure Ω'\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure μ'\nhX : AEMeasurable X μ\nhY : AEMeasurable Y μ'\ninner_ne_zero : ∀ (x : ℝ), x ≠ 0 → (innerₗ ℝ) x ≠ 0\nw : ℝ\nh : com... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nΩ' : Type u_3\nmΩ' : MeasurableSpace Ω'\nY : Ω' → ℝ\nμ' : Measure Ω'\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure μ'\nhX : AEMeasurable X μ\nhY : AEMeasurable Y μ'\ninner_ne_zero : ∀ (x : ℝ), x ≠ 0 → (innerₗ ℝ) x ≠ 0\nw : ℝ\nh :\n ∫ (ω : Ω), c... | complexMGF, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Process.Stopping | {
"line": 1409,
"column": 73
} | {
"line": 1413,
"column": 68
} | {
"line": 1415,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁵ : LinearOrder ι\nμ : Measure Ω\nℱ : Filtration ι m\nτ : Ω → WithTop ι\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : Ω → E\ninst✝¹ : SigmaFiniteFiltration μ ℱ\nhτ : IsStoppingTime ℱ τ\nh_cou... | [] | by
refine condExp_ae_eq_restrict_of_measurableSpace_eq_on
(hτ.measurableSpace_le) (ℱ.le i)
(hτ.measurableSet_eq_of_countable_range' h_countable i) fun t => ?_
rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 556,
"column": 64
} | {
"line": 557,
"column": 40
} | {
"line": 558,
"column": 2
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ ∫ (ω : ℝ), ω ^ 2 ∂gaussianReal 0 v = iteratedDeriv 2 (mgf (fun x ↦ x) (gaussianReal 0 v)) 0",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Real.denselyNormedField",
"Real.instZe... | [] | by
rw [iteratedDeriv_mgf_zero] <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 12
} | {
"line": 227,
"column": 13
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace F\ninst✝¹ : BorelSpace F\nμ : Measure E\ninst✝ : IsGaussian μ\nc : E\nL : StrongDual ... | [
"E : Type u_1\nF : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : MeasurableSpace F\ninst✝¹ : BorelSpace F\nμ : Measure E\ninst✝ : IsGaussian μ\nc : E\nL : StrongDual ℝ E\nhL_comp... | hL_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 265,
"column": 4
} | {
"line": 265,
"column": 82
} | {
"line": 266,
"column": 4
} | [
{
"pp": "case neg.inr.inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : NormedSpace ℝ E\nc : E := ∫ (x : E), x ∂μ\nthis✝ : id = fun x ↦ x - c + c\nhx : ¬c = 0\nI : Integrable (fun x ↦ ‖x‖) μ\ny : E\np : ℝ≥0\nh_Lp : MemLp (fun x ↦ x - c) (↑p) μ\nhp0 : ↑p ≠ 0\nth... | [
"case neg.inr.inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : NormedSpace ℝ E\nc : E := ∫ (x : E), x ∂μ\nthis✝¹ : id = fun x ↦ x - c + c\nhx : ¬c = 0\nI : Integrable (fun x ↦ ‖x‖) μ\ny : E\np : ℝ≥0\nh_Lp : MemLp (fun x ↦ x - c) (↑p) μ\nhp0 : ↑p ≠ 0\nthis✝ : Integ... | have : ‖c‖ ≤ ‖y‖ + ‖y - c‖ := Eq.trans_le (by abel_nf) (norm_sub_le y (y - c)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 90,
"column": 2
} | {
"line": 94,
"column": 8
} | {
"line": 95,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ\nhf : f.toBilinForm.Is... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ\nhf : f.toBilinForm.IsPosSemidef\n... | have h L : (n L : ℂ) = (L (∫ x, id x ∂μ) * I - covarianceBilinDual μ L L / 2 -
L m * I + f L L / 2) / (2 * π * I) := by
rw [hn L]
field_simp
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 75,
"column": 2
} | {
"line": 79,
"column": 37
} | {
"line": 80,
"column": 2
} | [
{
"pp": "case refine_1\nT : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : T → Type u_4\nX : (t : T) → S t → Ω → E\ninst✝ : NormedSpac... | [
"case refine_2\nT : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : T → Type u_4\nX : (t : T) → S t → Ω → E\ninst✝ : NormedSpace ℝ E\nhX : ... | · let L : (I.sigma (fun i ↦ if hi : i ∈ I then J ⟨i, hi⟩ else ∅) → E) →L[ℝ] (i : I) → J i → E :=
{ toFun x i j := x ⟨⟨i, j⟩, by simp⟩
map_add' x y := by ext; simp
map_smul' c x := by ext; simp }
exact (hX.hasGaussianLaw _).map L | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Fernique | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 84
} | {
"line": 559,
"column": 2
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : SecondCountableTopology E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod... | [] | exact exists_integrable_exp_sq_of_map_rotation_eq_self' h_rot ha_pos ha_gt ha_lt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 64
} | {
"line": 322,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nx y : ℝ\nhxy : x ≤ y\nthis : Nonempty { r' // y < ↑r' }\nr : { q' // y < ↑q' }\n⊢ BddBelow (range fun q ↦ f a ↑q)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Rea... | [
"case refine_2\nα : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nx y : ℝ\nhxy : x ≤ y\nthis : Nonempty { r' // y < ↑r' }\nr : { q' // y < ↑q' }\n⊢ { q' // x < ↑q' }",
"case refine_3\nα : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\n... | · refine ⟨0, fun z ↦ ?_⟩; rintro ⟨u, rfl⟩; exact hf.nonneg a _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | {
"line": 364,
"column": 2
} | {
"line": 364,
"column": 64
} | {
"line": 365,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nx : ℝ\nr : ℚ\nhrx : x < ↑r\n⊢ BddBelow (range fun r ↦ f a ↑r)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"lowerBounds",
... | [
"case refine_2\nα : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\nx : ℝ\nr : ℚ\nhrx : x < ↑r\n⊢ { r' // x < ↑r' }"
] | · refine ⟨0, fun z ↦ ?_⟩; rintro ⟨u, rfl⟩; exact hf.nonneg a _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 165,
"column": 8
} | {
"line": 165,
"column": 66
} | {
"line": 166,
"column": 6
} | [
{
"pp": "case hv.hZ\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nX : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nh1 : IsGaussianProcess X P\nh2 : ∀ (t : ℝ≥0), ∫ (x : Ω), X t x ∂P = 0\nh3 : ∀ (s t : ℝ≥0), s ≤ t → cov[X s, X t; P] = ↑s\nI : Finset ℝ≥0\nthis : IsGaussian (Measure.map (fun ω ↦ I.restrict fun x ↦ X x ω) P)\nx : WithLp 2 ... | [] | exact aemeasurable_pi_lambda _ (fun _ ↦ h1.aemeasurable _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 56
} | {
"line": 97,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDF f κ ν\na : α\nq : ℚ\n⊢ (fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a, b)) ↑q) =ᵐ[ν a] fun b ↦ f (a, b) q",
"ppTerm": "?m.32",
"assigned": tru... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\nhf : IsRatCondKernelCDF f κ ν\na✝ : α\nq : ℚ\na : β\nha : IsRatStieltjesPoint f (a✝, a)\n⊢ ↑(stieltjesOfMeasurableRat f ⋯ (a✝, a)) ↑q = f (a✝, a) q"
] | filter_upwards [hf.isRatStieltjesPoint_ae a] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 65
} | {
"line": 114,
"column": 4
} | [
{
"pp": "case hfi\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nq : ℚ\ns : Set β\nhs : MeasurableSet s\n⊢ Integrable (fun b ↦ ↑(stieltjesOfMeasurableRat f ⋯ (a,... | [
"case hfi\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nq : ℚ\ns : Set β\nhs : MeasurableSet s\n⊢ Integrable (fun b ↦ f (a, b) q) (ν a)"
] | rw [integrable_congr (stieltjesOfMeasurableRat_ae_eq hf a q)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 178,
"column": 4
} | {
"line": 179,
"column": 11
} | {
"line": 180,
"column": 4
} | [
{
"pp": "case inr\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nthis✝ : IsProbabilityMeasure P\nn : ℕ\nt : Fin (n + 1) → ℝ≥0\nht : Monotone t\ni j : Fin n\nhij : i ≠ j\nthis :\n ∀ (i j : Fin n),\n i ≠ j →\n i < j →\n ↑(min (t i.succ) (t j.succ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\nthis : IsProbabilityMeasure P\nn : ℕ\nt : Fin (n + 1) → ℝ≥0\nht : Monotone t\ni j : Fin n\nhij : i ≠ j\nh : i < j\n⊢ ↑(min (t i.succ) (t j.succ)) - ↑(min (t i.succ) (t j.castSucc)) - ↑(min (t i.castSucc) (t j.succ)) +... | · simp_rw [← this j i hij.symm (by grind), min_comm]
grind | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 31
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case hq\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ (κ a).real (s ×ˢ Iic x)",
"ppTerm": "?hq",
... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 31
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case hq\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ (κ a).real (s ×ˢ Iic x)",
"ppTerm": "?hq",
... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 31
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case hq\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ (κ a).real (s ×ˢ Iic x)",
"ppTerm": "?hq",
... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 69
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case f_nn\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\n⊢ 0 ≤ᵐ[(ν a).restrict s] fun b ↦ ↑(stieltjesOfMeasurabl... | [] | · exact ae_of_all _ (fun _ ↦ stieltjesOfMeasurableRat_nonneg _ _ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 44
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case hf\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ns : Set α\nhs : MeasurableSet s\ninst✝ : IsFiniteMeasure ρ\n⊢ ∀ᵐ (x : α) ∂ρ.fst.restrict s, preCDF ρ r x < ∞",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.preCDF",
"MeasureThe... | [] | refine ae_restrict_of_ae ?_
filter_upwards [preCDF_le_one ρ] with a ha
exact (ha r).trans_lt ENNReal.one_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 44
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case hf\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ns : Set α\nhs : MeasurableSet s\ninst✝ : IsFiniteMeasure ρ\n⊢ ∀ᵐ (x : α) ∂ρ.fst.restrict s, preCDF ρ r x < ∞",
"ppTerm": "?hf",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.preCDF",
"MeasureThe... | [] | refine ae_restrict_of_ae ?_
filter_upwards [preCDF_le_one ρ] with a ha
exact (ha r).trans_lt ENNReal.one_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.CentralLimitTheorem | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 82
} | {
"line": 63,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nX : Ω → ℝ\nhX : AEMeasurable X P\nh0 : ∫ (x : Ω), X x ∂P = 0\nh1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1\nt : ℝ\nthis :\n (fun n ↦ charFun (Measure.map X P) ((√↑n)⁻¹ * t) - (1 + -((↑(√↑n)⁻¹ * ↑t) ^ 2 / 2))) =o[atTop] fun n ↦\n ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nX : Ω → ℝ\nhX : AEMeasurable X P\nh0 : ∫ (x : Ω), X x ∂P = 0\nh1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1\nt : ℝ\nthis :\n (fun n ↦ charFun (Measure.map X P) ((√↑n)⁻¹ * t) - (1 + -((↑(√↑n)⁻¹ * ↑t) ^ 2 / 2))) =o[atTop] fun n ↦\n ((√↑n)⁻¹ ... | rw [← Asymptotics.isLittleO_norm_right, aux, Asymptotics.isLittleO_norm_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.Lemmas | {
"line": 100,
"column": 72
} | {
"line": 100,
"column": 89
} | {
"line": 100,
"column": 89
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\n⊢ (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))",
"ppTerm": "?m.70... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\n⊢ (⇑κ ∘ₘ μ).prod ν = (⇑κ ∘ₘ μ).prod ν"
] | Measure.prod_swap | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Lemmas | {
"line": 103,
"column": 65
} | {
"line": 103,
"column": 82
} | {
"line": 103,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(κ ∥ₖ Kernel.id... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(κ ∥ₖ Kernel.id) ∘ₘ μ.prod ... | Measure.prod_swap | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Lemmas | {
"line": 105,
"column": 10
} | {
"line": 105,
"column": 22
} | {
"line": 105,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(κ ∥ₖ Kernel.id... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(κ ∥ₖ Kernel.id) ∘ₘ map Pro... | Kernel.swap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Lemmas | {
"line": 109,
"column": 10
} | {
"line": 109,
"column": 22
} | {
"line": 109,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(Kernel.swap β ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\ninst✝² : SFinite μ\ninst✝¹ : SFinite ν\nκ : Kernel α γ\ninst✝ : IsSFiniteKernel κ\nh1 : (⇑κ ∘ₘ μ).prod ν = map Prod.swap (ν.prod (⇑κ ∘ₘ μ))\n⊢ ⇑(Kernel.deterministic Prod.... | Kernel.swap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Basic | {
"line": 171,
"column": 2
} | {
"line": 186,
"column": 14
} | {
"line": 189,
"column": 0
} | [
{
"pp": "case pos.right\nα : Type u_1\nβ : Type u_2\nΩ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\nκ : Kernel α (β × Ω)\nκCond : Kernel (α × β) Ω\ninst✝¹ : IsFiniteKernel κ.fst\ninst✝ : κ.IsCondKernel κCond\na : α\nh : κ.fst ⊗ₖ κCond = κ\nh_sfin : IsSFiniteKernel κCond\nh... | [] | · rw [ae_const_le_iff_forall_lt_measure_zero]
intro r hr
let s := {b | κCond (a, b) Set.univ ≤ r}
have hs : MeasurableSet s := h_meas measurableSet_Iic
have h_2_le : s.indicator (fun b ↦ (κCond (a, b)) Set.univ) ≤ s.indicator (fun _ ↦ r) := by
intro b
by_cases hbs : b ∈ s
· simpa [hbs]... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 585,
"column": 8
} | {
"line": 592,
"column": 35
} | {
"line": 593,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns t : Set (β × ℝ)\nht : MeasurableSet t\nht_eq : ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (... | [] | have h_le : (fun x ↦ hf.toKernel f (a, x) (Prod.mk x ⁻¹' t))
≤ᵐ[ν a] fun x ↦ hf.toKernel f (a, x) univ :=
Eventually.of_forall fun _ ↦ measure_mono (subset_univ _)
rw [lintegral_sub _ _ h_le]
· exact Kernel.measurable_kernel_prodMk_left' ht a
refine ((lintegral_mono_ae h_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 585,
"column": 8
} | {
"line": 592,
"column": 35
} | {
"line": 593,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns t : Set (β × ℝ)\nht : MeasurableSet t\nht_eq : ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (... | [] | have h_le : (fun x ↦ hf.toKernel f (a, x) (Prod.mk x ⁻¹' t))
≤ᵐ[ν a] fun x ↦ hf.toKernel f (a, x) univ :=
Eventually.of_forall fun _ ↦ measure_mono (subset_univ _)
rw [lintegral_sub _ _ h_le]
· exact Kernel.measurable_kernel_prodMk_left' ht a
refine ((lintegral_mono_ae h_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 453,
"column": 2
} | {
"line": 453,
"column": 64
} | {
"line": 455,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nx : γ\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable ((fun p ↦ κ.density ν p.1 p.2 s) ∘ fun a ↦ (a, x))",
"ppTerm": "... | [] | exact (measurable_density κ ν hs).comp measurable_prodMk_right | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 452,
"column": 2
} | {
"line": 453,
"column": 64
} | {
"line": 455,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nx : γ\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun a ↦ κ.density ν a x s",
"ppTerm": "?m.24",
"assigned": t... | [] | change Measurable ((fun (p : α × γ) ↦ density κ ν p.1 p.2 s) ∘ (fun a ↦ (a, x)))
exact (measurable_density κ ν hs).comp measurable_prodMk_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 452,
"column": 2
} | {
"line": 453,
"column": 64
} | {
"line": 455,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nν : Kernel α γ\nx : γ\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun a ↦ κ.density ν a x s",
"ppTerm": "?m.24",
"assigned": t... | [] | change Measurable ((fun (p : α × γ) ↦ density κ ν p.1 p.2 s) ∘ (fun a ↦ (a, x)))
exact (measurable_density κ ν hs).comp measurable_prodMk_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 209,
"column": 67
} | {
"line": 213,
"column": 9
} | {
"line": 215,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nmβ : MeasurableSpace β\nX : α → β\nc : Ω\n⊢ ⇑(condDistrib (fun x ↦ c) X μ) =ᵐ[Measure.map X μ] ⇑(Kernel.determinist... | [] | by
have : (fun _ : α ↦ c) = (fun _ : β ↦ c) ∘ X := rfl
rw [this]
filter_upwards [condDistrib_comp_self X (measurable_const (a := c))] with b hb
rw [hb] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 316,
"column": 2
} | {
"line": 317,
"column": 53
} | {
"line": 319,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nhX : Measurable X\nhY : AEMeasurable Y μ\nhs : MeasurableS... | [] | obtain ⟨t', ht', rfl⟩ := ht
rw [setLIntegral_preimage_condDistrib hX hY hs ht'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 316,
"column": 2
} | {
"line": 317,
"column": 53
} | {
"line": 319,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝³ : MeasurableSpace Ω\ninst✝² : StandardBorelSpace Ω\ninst✝¹ : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\ns : Set Ω\nhX : Measurable X\nhY : AEMeasurable Y μ\nhs : MeasurableS... | [] | obtain ⟨t', ht', rfl⟩ := ht
rw [setLIntegral_preimage_condDistrib hX hY hs ht'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 186,
"column": 66
} | {
"line": 190,
"column": 47
} | {
"line": 192,
"column": 0
} | [
{
"pp": "X : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\na b c : ℕ\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\nhab : a ≤ b\nhbc : b ≤ c\n⊢ partialTraj κ b c ∘ₖ partialTraj κ a b = partialTraj κ a c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
induction c, hbc using Nat.le_induction with
| base => simp
| succ k h hk => rw [partialTraj_succ_eq_comp h, comp_assoc, hk,
← partialTraj_succ_eq_comp (hab.trans h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 655,
"column": 4
} | {
"line": 655,
"column": 30
} | {
"line": 657,
"column": 0
} | [
{
"pp": "case neg.hI\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × β)\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nh : ¬(κ.fst a) (countablePartitionSet n x) = 0\nthis : countablePartitionSet... | [] | · exact measure_ne_top _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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