module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.CMField | {
"line": 209,
"column": 2
} | {
"line": 217,
"column": 27
} | {
"line": 219,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : Algebra.IsIntegral ℚ K\nx : 𝓞 K\n⊢ (complexConj K) ↑x = ↑x ↔ ∃ y, (algebraMap (𝓞 ↥K⁺) K) y = ↑x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
... | [] | rw [complexConj_eq_self_iff]
refine ⟨fun h ↦ ?_, fun ⟨y, hy⟩ ↦ ?_⟩
· have : IsIntegral ℤ (⟨x, h⟩ : K⁺) :=
(isIntegral_algebraMap_iff (FaithfulSMul.algebraMap_injective K⁺ K)).mp x.isIntegral_coe
refine ⟨⟨⟨x, h⟩, this⟩, ?_⟩
rw [IsScalarTower.algebraMap_apply (𝓞 K⁺) K⁺, RingOfIntegers.map_mk]
rfl
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 32
} | {
"line": 579,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\n⊢ Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s } * torsionOrder K =\n Nat.card { a // mixedEmbedding.norm ↑a ≤ s }",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Nat.inst... | [
"case inl\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nhs : 0 ≤ s\n⊢ Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s } * torsionOrder K =\n Nat.card { a // mixedEmbedding.norm ↑a ≤ s }",
"case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nJ : ↥... | obtain hs | hs := le_or_gt 0 s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 48
} | {
"line": 281,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nx : realSpace K\nx✝ : InfinitePlace K\n⊢ ↑expMap (c • x) x✝ = (↑expMap x ^ c) x✝",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instPow",
... | [] | simp [mul_comm c _, ← mul_assoc, Real.exp_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 300,
"column": 52
} | {
"line": 300,
"column": 55
} | {
"line": 300,
"column": 56
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : mixedEmbedding.norm (mixedSpaceOfRealSpace (↑expMap x)) = 1\nx✝ : { w // w ≠ w₀ }\n⊢ x ↑x✝ - ↑(↑x✝).mult * (Real.log (mixedEmbedding.norm (mixedSpaceOfRealSpace (↑expMap x))) * (↑(finrank ℚ K))⁻¹) =\n x ↑x✝",
"ppTerm": ... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : mixedEmbedding.norm (mixedSpaceOfRealSpace (↑expMap x)) = 1\nx✝ : { w // w ≠ w₀ }\n⊢ x ↑x✝ - ↑(↑x✝).mult * (Real.log 1 * (↑(finrank ℚ K))⁻¹) = x ↑x✝"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 50,
"column": 2
} | {
"line": 55,
"column": 33
} | {
"line": 57,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\ninst✝ : (↑w).LiesOver ↑v\nh : IsRamified K w\n⊢ Module.finrank v.Completion w.Completion = 2",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | have H := NumberField.InfinitePlace.isRamified_iff.mp h
rw [NumberField.InfinitePlace.LiesOver.comap_eq w v] at H
have := LiesOver.extensionEmbedding_liesOver_of_isReal w H.2
rw [Algebra.finrank_eq_of_equiv_equiv (ringEquivRealOfIsReal H.2)
(ringEquivComplexOfIsComplex H.1) (by ext; simp),
Complex.finra... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 50,
"column": 2
} | {
"line": 55,
"column": 33
} | {
"line": 57,
"column": 0
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nv : InfinitePlace K\nw : InfinitePlace L\ninst✝ : (↑w).LiesOver ↑v\nh : IsRamified K w\n⊢ Module.finrank v.Completion w.Completion = 2",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | have H := NumberField.InfinitePlace.isRamified_iff.mp h
rw [NumberField.InfinitePlace.LiesOver.comap_eq w v] at H
have := LiesOver.extensionEmbedding_liesOver_of_isReal w H.2
rw [Algebra.finrank_eq_of_equiv_equiv (ringEquivRealOfIsReal H.2)
(ringEquivComplexOfIsComplex H.1) (by ext; simp),
Complex.finra... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 307,
"column": 43
} | {
"line": 307,
"column": 61
} | {
"line": 307,
"column": 61
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw' : InfinitePlace K\ne : { w // w ≠ w' } ≃ Fin (rank K)\n⊢ regOfFamily (fundSystem K) =\n |(Matrix.of fun i w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(fundSystem K (e i))))).det|",
"ppTerm": "?m.47",
"assigned": true,
"u... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw' : InfinitePlace K\ne : { w // w ≠ w' } ≃ Fin (rank K)\n⊢ |(Matrix.of fun i w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(fundSystem K (?e i))))).det| =\n |(Matrix.of fun i w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(fundSystem K (e i)... | regOfFamily_eq_det | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 51,
"column": 6
} | {
"line": 51,
"column": 9
} | {
"line": 51,
"column": 9
} | [
{
"pp": "n : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : ζ ^ n = 1\nhζ1 : ζ ≠ 1\nkey : ↑n = ∑ i ∈ range n, (1 - ζ ^ i)\n⊢ ζ - 1 ∣ ↑n",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"AddGr... | [
"n : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : ζ ^ n = 1\nhζ1 : ζ ≠ 1\nkey : ↑n = ∑ i ∈ range n, (1 - ζ ^ i)\n⊢ ζ - 1 ∣ ∑ i ∈ range n, (1 - ζ ^ i)"
] | key | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Ideal.Basic | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 71
} | {
"line": 98,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nI : Ideal (𝓞 K)\ninst✝¹ : NumberField K\nn : ℕ\ninst✝ : NeZero n\nζ : 𝓞 K\nhζ : IsPrimitiveRoot ζ n\nhI₁ : absNorm I ≠ 1\nhI₂ : (absNorm I).Coprime n\nh : IsPrimitiveRoot hζ.toRootsOfUnity n\n⊢ IsPrimitiveRoot ((Ideal.Quotient.mk I) ζ) n",
"ppTerm": "?m.52",
"a... | [] | exact IsPrimitiveRoot.coe_units_iff.mpr <|
h.map_of_injective <| Ideal.rootsOfUnityMapQuot_injective n hI₁ hI₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 73,
"column": 75
} | {
"line": 73,
"column": 78
} | {
"line": 73,
"column": 79
} | [
{
"pp": "n : ℕ\ninst✝² : NeZero n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\nσ : Gal(K/ℚ)\nx : 𝓞 K\nhx : x ^ n = 1\n⊢ ↑(x ^ n) = 1",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Number... | [
"n : ℕ\ninst✝² : NeZero n\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\nσ : Gal(K/ℚ)\nx : 𝓞 K\nhx : x ^ n = 1\n⊢ ↑1 = 1"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 23
} | {
"line": 96,
"column": 2
} | [
{
"pp": "n : ℕ\ninst✝⁷ : NeZero n\nK : Type u_1\ninst✝⁶ : Field K\ninst✝⁵ : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\nm : ℕ\ninst✝⁴ : NeZero m\nF : Type u_2\ninst✝³ : Field F\ninst✝² : NumberField F\nhF : IsCyclotomicExtension {m} ℚ F\ninst✝¹ : Algebra F K\ninst✝ : IsGalois ℚ F\nh : m ∣ n\nσ : Gal(K/ℚ)... | [] | rwa [Units.ext_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.NumberField.DedekindZeta | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 93
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case e_a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\n⊢ Nat.card { I // absNorm I ≤ n } = ∑ x ∈ Finset.Ioc 0 n, Nat.card { I // absNorm I = x } + 1",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Semiring.to... | [
"case e_a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\n⊢ Nat.card { I // absNorm I ≤ n } =\n ∑ x ∈ Finset.Ioc 0 n, Nat.card { I // absNorm I = x } + Nat.card { I // absNorm I = 0 }"
] | show 1 = Nat.card {I : Ideal (𝓞 K) // absNorm I = 0} by simp [Ideal.absNorm_eq_zero_iff], | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 91,
"column": 2
} | {
"line": 92,
"column": 67
} | {
"line": 93,
"column": 2
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : NumberField K\ninst✝³ : CommRing 𝒪\ninst✝² : Algebra 𝒪 K\ninst✝¹ : IsIntegralClosure 𝒪 ℤ K\ninst✝ : IsGalois ℚ K\nH : 1 < Module.finrank ℚ K\nthis✝⁶ : IsDomain 𝒪\nthis✝⁵ : IsDedekindDomain 𝒪\nthis✝⁴ : IsFractionRing 𝒪 K\nthis✝³ : Module.Fini... | [
"K : Type u_1\n𝒪 : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : NumberField K\ninst✝³ : CommRing 𝒪\ninst✝² : Algebra 𝒪 K\ninst✝¹ : IsIntegralClosure 𝒪 ℤ K\ninst✝ : IsGalois ℚ K\nH : 1 < Module.finrank ℚ K\nthis✝⁷ : IsDomain 𝒪\nthis✝⁶ : IsDedekindDomain 𝒪\nthis✝⁵ : IsFractionRing 𝒪 K\nthis✝⁴ : Module.Finite ℤ 𝒪\nthi... | have : .span {p} = Ideal.under ℤ Q :=
((Ideal.liesOver_span_iff Ideal.IsPrime.ne_top' this).mpr hQ).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 229,
"column": 61
} | {
"line": 229,
"column": 74
} | {
"line": 229,
"column": 75
} | [
{
"pp": "n : ℕ\ninst✝⁸ : NeZero n\nK : Type u_1\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)\ninst✝³ : IsAbelianGalois ℚ K\nF : IntermediateField ℚ K\nm : ℕ\ninst✝² : NeZero m\ninst✝... | [
"n : ℕ\ninst✝⁸ : NeZero n\nK : Type u_1\ninst✝⁷ : Field K\ninst✝⁶ : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : HasEnoughRootsOfUnity R (Monoid.exponent (ZMod n)ˣ)\ninst✝³ : IsAbelianGalois ℚ K\nF : IntermediateField ℚ K\nm : ℕ\ninst✝² : NeZero m\ninst✝¹ : IsGalois... | toUnitHom_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 119,
"column": 8
} | {
"line": 120,
"column": 74
} | {
"line": 120,
"column": 74
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nthis✝ : IsGalois ℚ K\nthis : IsGalois (FractionRing ℤ) (FractionRing (𝓞 K))\nσ : 𝓞 K ≃ₐ[ℤ] 𝓞 K\n| span {σ (hζ.toInteger ... | [
"p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nthis✝ : IsGalois ℚ K\nthis : IsGalois (FractionRing ℤ) (FractionRing (𝓞 K))\nσ : 𝓞 K ≃ₐ[ℤ] 𝓞 K\n| span {hζ.toInteger - 1}"
] | span_singleton_eq_span_singleton.mpr
(hζ.toInteger_isPrimitiveRoot.associated_sub_one_map_sub_one σ).symm | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 67,
"column": 2
} | {
"line": 68,
"column": 33
} | {
"line": 69,
"column": 2
} | [
{
"pp": "case mp\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nx✝ : mixedSpace K\n⊢ (∃ x,\n (x ∈ {x | (toMixed K) x ∈ fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} ∧\n ∃ y ∈ ↑↑↑J, (mixedEmbedding K) y = ↑↑(toMixed K) x) ∧... | [
"case mpr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nx✝ : mixedSpace K\n⊢ (x✝ ∈ fundamentalCone K ∧ ∃ a ∈ ↑J, (mixedEmbedding K) ↑a = x✝) ∧ mixedEmbedding.norm x✝ ≤ s →\n ∃ x,\n (x ∈ {x | (toMixed K) x ∈ fundamentalCone K ∧ mixedEmbedding.norm... | · rintro ⟨_, ⟨⟨hx₁, hx₂⟩, _, ⟨x, hx₃, rfl⟩, h⟩, rfl⟩
exact ⟨⟨hx₁, x, hx₃, h⟩, hx₂⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 231,
"column": 86
} | {
"line": 233,
"column": 43
} | {
"line": 235,
"column": 0
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ p\n⊢ (span {hζ.toInteger - 1}).inertiaDeg ℤ = 1",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"congrArg",
"Nat.ins... | [] | by
rw [← pow_one p] at hK hζ
exact inertiaDeg_span_zeta_sub_one p 0 hζ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 342,
"column": 6
} | {
"line": 342,
"column": 46
} | {
"line": 342,
"column": 47
} | [
{
"pp": "m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : ¬p ∣ m\nthis : IsGalois ℚ K\nP : Ideal (𝓞 K)\nleft✝ : P.IsPrime\nright✝ : P.LiesOver 𝒑\n⊢ 𝒑.inertiaDegIn (𝓞 K) = orderOf ↑p",
"ppTerm": "?m.87",
... | [
"m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : ¬p ∣ m\nthis : IsGalois ℚ K\nP : Ideal (𝓞 K)\nleft✝ : P.IsPrime\nright✝ : P.LiesOver 𝒑\n⊢ P.inertiaDeg ℤ = orderOf ↑p"
] | inertiaDegIn_eq_inertiaDeg 𝒑 P Gal(K/ℚ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 31
} | {
"line": 130,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\nhA : A ≠ 0\nh : ‖A‖ < 1\ni : α\nj : β\n⊢ A i j = 0 i j",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Matrix",
"id",
"Int",
"Zero.toOfNat0",
"OfNat.ofNat",
"E... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\nhA : A ≠ 0\nh : ‖A‖ < 1\ni : α\nj : β\n⊢ A i j = 0"
] | simp only [Matrix.zero_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.House | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 24
} | {
"line": 274,
"column": 24
} | [
{
"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)\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\nthis : 0 ≤ c K\n⊢ ‖asiegel K a‖ ≤ c... | [
"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)\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\nthis : 0 ≤ c K\n⊢ ∀ (i : α × (K →+* ℂ)) (j : β ... | Matrix.norm_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 61,
"column": 11
} | {
"line": 61,
"column": 25
} | {
"line": 61,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nT : Type u_2\nl : List T\ny x : R\n⊢ (List.mapIdx (fun i x_1 ↦ x * y ^ i) l).sum = x * (List.mapIdx (fun i x ↦ y ^ i) l).sum",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"List.mapIdx",
"HMul.hMul",
"List.sum",
"id",
... | [
"R : Type u_1\ninst✝ : Semiring R\nT : Type u_2\nl : List T\ny x : R\n⊢ (List.mapIdx (fun i x_1 ↦ x • y ^ i) l).sum = x • (List.mapIdx (fun i x ↦ y ^ i) l).sum"
] | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 48
} | {
"line": 185,
"column": 2
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\nhm : f ↑m < 1\nM : ℝ := max (f ↑p) (f ↑m)\nhM : M = max (f ↑p) (f ↑m)\n⊢ False",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants":... | [
"f : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\nhm : f ↑m < 1\nM : ℝ := max (f ↑p) (f ↑m)\nhM : M = max (f ↑p) (f ↑m)\nk : ℕ := ⌈logb M (1 / 2)⌉₊ + 1\nhk : k = ⌈logb M (1 / 2)⌉₊ + 1\n⊢ False"
] | set k := Nat.ceil (M.logb (1 / 2)) + 1 with hk | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.NumberTheory.Ostrowski | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 14
} | {
"line": 225,
"column": 2
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\npprime : Nat.Prime p\n⊢ f ↑p = ↑p ^ (- -logb (↑p) (f ↑p))",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
... | [
"f : AbsoluteValue ℚ ℝ\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\npprime : Nat.Prime p\n⊢ f ↑p = ↑p ^ logb (↑p) (f ↑p)"
] | rw [neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 173,
"column": 32
} | {
"line": 173,
"column": 42
} | {
"line": 173,
"column": 42
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn : ℕ\n⊢ ↑‖∑ j ∈ range (p... | [
"case refine_1\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn : ℕ\n⊢ ‖∑ j ∈ range (p ^ t - 1), (p... | coe_le_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.Complex | {
"line": 214,
"column": 4
} | {
"line": 215,
"column": 33
} | {
"line": 217,
"column": 0
} | [
{
"pp": "case h\np : ℕ\nhp : Fact (Nat.Prime p)\nx : PadicAlgCl p\n⊢ ‖x‖ = ↑(embedding (Valued.v.restrict ↑x))",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"PadicComplex",
"LinearOrderedCommGroupWithZero... | [] | rw [embedding_restrict (PadicComplex.valued p).v x, valuation_extends,
← PadicAlgCl.valuation_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Separation.DisjointCover | {
"line": 130,
"column": 2
} | {
"line": 133,
"column": 70
} | {
"line": 134,
"column": 2
} | [
{
"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 ... | [
"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)\nthis : ⇑(f.prodMap f) ⁻¹' S ∈ 𝓝ˢ (diagonal X)\n⊢ ∃ n D,\n (∀ (i : Fin n), D i ≠ ⊥) ∧\n ... | 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) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Padics.Measure.Basic | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 18
} | {
"line": 196,
"column": 19
} | [
{
"pp": "case e_6\nX : Type u_1\nY : Type u_2\nR : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : IsTopologicalRing R\nμ : AbstractMeasure X R R\nν : AbstractMeasure Y R R\ninst✝¹ : LocallyCompactSpace X\ninst✝ : LocallyCompactSpace... | [
"case e_6\nX : Type u_1\nY : Type u_2\nR : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : IsTopologicalRing R\nμ : AbstractMeasure X R R\nν : AbstractMeasure Y R R\ninst✝¹ : LocallyCompactSpace X\ninst✝ : LocallyCompactSpace Y\nf : C(X,... | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Padics.Measure.Basic | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 18
} | {
"line": 217,
"column": 19
} | [
{
"pp": "case e_6\nX : Type u_1\nY : Type u_2\nR : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : IsTopologicalRing R\nμ : AbstractMeasure X R R\nν : AbstractMeasure Y R R\ninst✝¹ : LocallyCompactSpace X\ninst✝ : LocallyCompactSpace... | [
"case e_6\nX : Type u_1\nY : Type u_2\nR : Type u_3\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : IsTopologicalRing R\nμ : AbstractMeasure X R R\nν : AbstractMeasure Y R R\ninst✝¹ : LocallyCompactSpace X\ninst✝ : LocallyCompactSpace Y\nf : C(X,... | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Pell | {
"line": 197,
"column": 16
} | {
"line": 197,
"column": 19
} | {
"line": 197,
"column": 20
} | [
{
"pp": "d : ℤ\nh₀ : 0 ≤ d\na : Solution₁ d\nhx : a.x = 0\nh : 0 ≤ a.x ^ 2 - 1\n⊢ False",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"congrArg",
"Pell.Solution₁.x",
"HSub.hSub",
"Eq.mp",
"instOfNatNat",
"Int",
"LE.le",
"NPow.toPow",
... | [
"d : ℤ\nh₀ : 0 ≤ d\na : Solution₁ d\nhx : a.x = 0\nh : 0 ≤ 0 ^ 2 - 1\n⊢ False"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 21
} | {
"line": 204,
"column": 2
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nha : 1 < a.x\nhy : a.y = 0\n⊢ False",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Pell.Solution₁.x",
"HSub.hSub",
"Pell.Solution₁.prop",
"instOfNatNat",
"Int",
"NPow.toPow",
"Int.instMonoid",
... | [
"d : ℤ\na : Solution₁ d\nha : 1 < a.x\nhy : a.y = 0\nprop : a.x ^ 2 - d * a.y ^ 2 = 1\n⊢ False"
] | have prop := a.prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Pell | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 21
} | {
"line": 230,
"column": 2
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nH : a.y = 0\n⊢ a = 1 ∨ a = -1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Pell.Solution₁.x",
"HSub.hSub",
"Pell.Solution₁.prop",
"instOfNatNat",
"Int",
"NPow.toPow",
"Int.instMonoid",
"... | [
"d : ℤ\na : Solution₁ d\nH : a.y = 0\nprop : a.x ^ 2 - d * a.y ^ 2 = 1\n⊢ a = 1 ∨ a = -1"
] | have prop := a.prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Pell | {
"line": 495,
"column": 2
} | {
"line": 495,
"column": 34
} | {
"line": 496,
"column": 2
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nh : IsFundamental a\nH : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y\nn : ℤ\n⊢ (a ^ n).y < (a ^ (n + 1)).y",
"ppTerm": "?m.192",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Int.instLinearOrder",
"PartialOrder.toPreorder",
"DivInv... | [
"case inl\nd : ℤ\na : Solution₁ d\nh : IsFundamental a\nH : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y\nn : ℤ\nhn : 0 ≤ n\n⊢ (a ^ n).y < (a ^ (n + 1)).y",
"case inr\nd : ℤ\na : Solution₁ d\nh : IsFundamental a\nH : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y\nn : ℤ\nhn : n < 0\n⊢ (a ^ n).y < (a ^ (n + 1)).... | rcases le_or_gt 0 n with hn | hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 78
} | {
"line": 88,
"column": 4
} | [
{
"pp": "m n : ℕ\nhc : m.Coprime n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\n⊢ IsSquare (-1)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"Prod.instMul",
"Add... | [
"m n : ℕ\nhc : m.Coprime n\nhm : IsSquare (-1)\nhn : IsSquare (-1)\n⊢ IsSquare (-1, -1)"
] | rw [show (-1 : ZMod m × ZMod n) = ((-1 : ZMod m), (-1 : ZMod n)) from rfl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 90,
"column": 8
} | {
"line": 90,
"column": 11
} | {
"line": 90,
"column": 12
} | [
{
"pp": "m n : ℕ\nhc : m.Coprime n\nx : ZMod m\nhx : -1 = x * x\ny : ZMod n\nhy : -1 = y * y\n⊢ IsSquare (-1, -1)",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSe... | [
"m n : ℕ\nhc : m.Coprime n\nx : ZMod m\nhx : -1 = x * x\ny : ZMod n\nhy : -1 = y * y\n⊢ IsSquare (x * x, -1)"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.Basic | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 74
} | {
"line": 70,
"column": 2
} | [
{
"pp": "a : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : a.natAbs.Coprime b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na1 : |a * ↑q - ↑b * p| * ↑q ^ (b + 1) < ↑b * ↑q\na0 : a * ↑q - ↑b * p ≠ 0\nap : 0 < |a * ↑q - ↑b * p|\n⊢ False",
"ppTerm": "?m.399",
"assigned... | [
"a : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : a.natAbs.Coprime b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na0 : a * ↑q - ↑b * p ≠ 0\ne : ℕ\nhe : ↑e = |a * ↑q - ↑b * p|\na1 : ↑e * ↑q ^ (b + 1) < ↑b * ↑q\nap : 0 < ↑e\n⊢ False"
] | lift |a * ↑q - ↑b * p| to ℕ using abs_nonneg (a * ↑q - ↑b * p) with e he | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.NumberTheory.Transcendental.Liouville.Basic | {
"line": 144,
"column": 2
} | {
"line": 148,
"column": 71
} | {
"line": 150,
"column": 2
} | [
{
"pp": "α : ℝ\nha : Irrational α\nf : ℤ[X]\nf0 : f ≠ 0\nfR : ℝ[X] := map (algebraMap ℤ ℝ) f\nfa : eval α fR = 0\nfR0 : fR ≠ 0\nar : α ∈ ↑fR.roots.toFinset\nζ : ℝ\nz0 : ζ > 0\nU : closedBall α ζ ∩ ↑fR.roots.toFinset = {α}\n⊢ ∃ A, 0 < A ∧ ∀ (a : ℤ) (b : ℕ), 1 ≤ (↑b + 1) ^ f.natDegree * (|α - ↑a / (↑b + 1)| * A)"... | [
"α : ℝ\nha : Irrational α\nf : ℤ[X]\nf0 : f ≠ 0\nfR : ℝ[X] := map (algebraMap ℤ ℝ) f\nfa : eval α fR = 0\nfR0 : fR ≠ 0\nar : α ∈ ↑fR.roots.toFinset\nζ : ℝ\nz0 : ζ > 0\nU : closedBall α ζ ∩ ↑fR.roots.toFinset = {α}\nxm : ℝ\nhM : IsMaxOn (fun x ↦ |eval x (derivative fR)|) (Icc (α - ζ) (α + ζ)) xm\n⊢ ∃ A, 0 < A ∧ ∀ (a... | obtain ⟨xm, -, hM⟩ : ∃ xm : ℝ, xm ∈ Icc (α - ζ) (α + ζ) ∧
IsMaxOn (|fR.derivative.eval ·|) (Icc (α - ζ) (α + ζ)) xm :=
IsCompact.exists_isMaxOn isCompact_Icc
⟨α, (sub_lt_self α z0).le, (lt_add_of_pos_right α z0).le⟩
(continuous_abs.comp fR.derivative.continuous_aeval).continuousOn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 21
} | {
"line": 268,
"column": 4
} | [
{
"pp": "case refine_1.convert_1\nR : Type u\ninst✝¹⁴ : CommRing R\nS : Type v\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : Algebra R K\nL : Type u_2\ninst✝⁹ : Field L\ninst✝⁸ : Algebra S L\ninst✝⁷ : IsFractionRing S L\nhRK : IsFractionRing R K\ninst✝⁶ : ... | [] | exact hp.ne_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 348,
"column": 4
} | {
"line": 348,
"column": 60
} | {
"line": 348,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ni : ℕ\na : S\na_mem : a ∈ P ^ i\nx y : S ⧸ P\n⊢ p.quotientToQuotientRangePowQuotSuccAux ... | [
"case h\nR : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ni : ℕ\na : S\na_mem : a ∈ P ^ i\nx y : S\n⊢ p.quotientToQuotientRangePowQuotSuccAux P a_mem ... | induction x, y using Quotient.inductionOn₂' with | _ x y
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 60
} | {
"line": 352,
"column": 60
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ni : ℕ\na : S\na_mem : a ∈ P ^ i\nx : R ⧸ p\ny : S ⧸ P\n⊢ p.quotientToQuotientRangePowQuo... | [
"case h\nR : Type u\ninst✝² : CommRing R\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : Ideal R\nP : Ideal S\nhfp : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ni : ℕ\na : S\na_mem : a ∈ P ^ i\nx : R\ny : S\n⊢ p.quotientToQuotientRangePowQuotSuccAux P a... | induction x, y using Quotient.inductionOn₂' with | _ x y
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleNumber | {
"line": 141,
"column": 6
} | {
"line": 145,
"column": 26
} | {
"line": 147,
"column": 0
} | [
{
"pp": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 / m ^ (n ! * (n + 1)) ≤ 1 / (m ^ n !) ^ n",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Distrib.leftDistribClass",
"Real.instIsOrderedRing",
"Eq.mpr",
"Grou... | [] | rw [← pow_mul, div_le_div_iff₀, one_mul, mul_add_one, pow_add, mul_comm 2]
· gcongr
-- `2 ≤ m ^ n!` is a consequence of monotonicity of exponentiation at `2 ≤ m`.
exact hm.trans <| le_self_pow₀ (one_le_two.trans hm) <| by positivity
all_goals positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleNumber | {
"line": 141,
"column": 6
} | {
"line": 145,
"column": 26
} | {
"line": 147,
"column": 0
} | [
{
"pp": "n : ℕ\nm : ℝ\nhm : 2 ≤ m\n⊢ 2 / m ^ (n ! * (n + 1)) ≤ 1 / (m ^ n !) ^ n",
"ppTerm": "?m.207",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Distrib.leftDistribClass",
"Real.instIsOrderedRing",
"Eq.mpr",
"Grou... | [] | rw [← pow_mul, div_le_div_iff₀, one_mul, mul_add_one, pow_add, mul_comm 2]
· gcongr
-- `2 ≤ m ^ n!` is a consequence of monotonicity of exponentiation at `2 ≤ m`.
exact hm.trans <| le_self_pow₀ (one_le_two.trans hm) <| by positivity
all_goals positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 63,
"column": 20
} | {
"line": 63,
"column": 48
} | {
"line": 63,
"column": 49
} | [
{
"pp": "case h\nx : ℝ\nn : ℕ\nhn : 0 < n\nhn' : 0 < ↑n\nthis : x < ↑(⌊x * ↑n⌋ + 1) / ↑n\n⊢ |↑(⌊x * ↑n⌋ + 1) / ↑n - x| < 2 / ↑n ^ 1",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Iff.mpr",
"sub_pos",
"AddGroup.to... | [
"case h\nx : ℝ\nn : ℕ\nhn : 0 < n\nhn' : 0 < ↑n\nthis : x < ↑(⌊x * ↑n⌋ + 1) / ↑n\n⊢ ↑(⌊x * ↑n⌋ + 1) / ↑n - x < 2 / ↑n ^ 1"
] | abs_of_pos (sub_pos.2 this), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 20
} | {
"line": 282,
"column": 0
} | [
{
"pp": "case hab\np : ℝ\nhp : 1 < p\nM : ℤ\nh : LiouvilleWith p ↑M\nn : ℕ\nhn : 0 < n\nm : ℤ\nhne : ¬↑M * ↑n = ↑m\nhlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)\nhn' : 0 < ↑n\n⊢ M * ↑n ≠ m",
"ppTerm": "?hab",
"assigned": true,
"usedConstants": [
"Int.cast",
"GroupWithZero.toMonoidWithZero",
"I... | [] | exact mod_cast hne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Comparable | {
"line": 254,
"column": 72
} | {
"line": 255,
"column": 36
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u_1\na b : α\nr : α → α → Prop\n⊢ ¬SymmGen r a b ↔ IncompRel r a b",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"congrArg",
"iff_self",
"And",
"Iff",
"not_or._simp_1",
"True",
"of_eq_true",
"congrFun'",
"Relation.Sym... | [] | by
simp [Relation.SymmGen, IncompRel] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.WellApproximable | {
"line": 264,
"column": 4
} | {
"line": 264,
"column": 22
} | {
"line": 265,
"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) ... | [
"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) atTop atTop\... | let x := u ⟨p, hp⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Order.Concept | {
"line": 98,
"column": 11
} | {
"line": 98,
"column": 30
} | {
"line": 98,
"column": 31
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\na : α\nb : β\n⊢ a ∈ lowerPolar r {b} ↔ r a b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"Set.instSingletonSet",
"id",
"Iff",
"_private.Mathlib.Order.... | [
"α : Type u_2\nβ : Type u_3\nr : α → β → Prop\na : α\nb : β\n⊢ (∀ ⦃b_1 : β⦄, b_1 ∈ {b} → r a b_1) ↔ r a b"
] | mem_lowerPolar_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Nucleus | {
"line": 192,
"column": 26
} | {
"line": 192,
"column": 43
} | {
"line": 194,
"column": 0
} | [
{
"pp": "case a\nX : Type u_1\ninst✝ : Frame X\nn : Nucleus X\nx y : X\n⊢ y ⊓ x ≤ y",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"CompleteLattice.toLattice",
"Order.Frame.toCompleteLattice",
"inf_le_left",
"Lattice.toSemilatticeInf"
],
"usedFVars": [
... | [] | exact inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Partition.Basic | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 29
} | {
"line": 184,
"column": 0
} | [
{
"pp": "α : Type u_1\ns x : α\ninst✝ : CompleteLattice α\nP : Partition s\nhxP : x ∈ P\n⊢ s ≠ ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"OrderBot.toBot",
"PartialOrder.toPreorder",
... | [] | rintro rfl
exact P.notMem_of_bot _ hxP | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Partition.Basic | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 29
} | {
"line": 184,
"column": 0
} | [
{
"pp": "α : Type u_1\ns x : α\ninst✝ : CompleteLattice α\nP : Partition s\nhxP : x ∈ P\n⊢ s ≠ ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"False",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"OrderBot.toBot",
"PartialOrder.toPreorder",
... | [] | rintro rfl
exact P.notMem_of_bot _ hxP | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.PrimeSeparator | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 55
} | {
"line": 92,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DistribLattice α\nF : PFilter α\nI : Ideal α\nhFI : Disjoint ↑F ↑I\nS : Set (Set α) := ⋯\nJset : Set α\nleft✝ : ↑I ⊆ Jset\nhmax : Maximal (fun x ↦ x ∈ S) Jset\nJidl : IsIdeal Jset\nIJ : ↑I ⊆ Jset\nJF : Disjoint (↑F) Jset\nJ : Ideal α := ⋯\nIJ' : I ≤ J\nJpr : J.IsProper\na₁ a₂ : α\... | [
"α : Type u_1\ninst✝ : DistribLattice α\nF : PFilter α\nI : Ideal α\nhFI : Disjoint ↑F ↑I\nS : Set (Set α) := {J | IsIdeal J ∧ ↑I ⊆ J ∧ Disjoint (↑F) J}\nJset : Set α\nleft✝ : ↑I ⊆ Jset\nhmax : Maximal (fun x ↦ x ∈ S) Jset\nJidl : IsIdeal Jset\nIJ : ↑I ⊆ Jset\nJF : Disjoint (↑F) Jset\nJ : Ideal α := Jidl.toIdeal\nI... | simp only [mem_setOf_eq, SetLike.coe_subset_coe, S] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.PrimeSeparator | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 55
} | {
"line": 97,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DistribLattice α\nF : PFilter α\nI : Ideal α\nhFI : Disjoint ↑F ↑I\nS : Set (Set α) := ⋯\nJset : Set α\nleft✝ : ↑I ⊆ Jset\nhmax : Maximal (fun x ↦ x ∈ S) Jset\nJidl : IsIdeal Jset\nIJ : ↑I ⊆ Jset\nJF : Disjoint (↑F) Jset\nJ : Ideal α := ⋯\nIJ' : I ≤ J\nJpr : J.IsProper\na₁ a₂ : α\... | [
"α : Type u_1\ninst✝ : DistribLattice α\nF : PFilter α\nI : Ideal α\nhFI : Disjoint ↑F ↑I\nS : Set (Set α) := {J | IsIdeal J ∧ ↑I ⊆ J ∧ Disjoint (↑F) J}\nJset : Set α\nleft✝ : ↑I ⊆ Jset\nhmax : Maximal (fun x ↦ x ∈ S) Jset\nJidl : IsIdeal Jset\nIJ : ↑I ⊆ Jset\nJF : Disjoint (↑F) Jset\nJ : Ideal α := Jidl.toIdeal\nI... | simp only [mem_setOf_eq, SetLike.coe_subset_coe, S] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Partition.Basic | {
"line": 473,
"column": 2
} | {
"line": 473,
"column": 35
} | {
"line": 474,
"column": 2
} | [
{
"pp": "α : Type u_1\nu : Set α\nP : Partition u\nf : α → α\na b : α\nhf : P.IsRepFun f\nha : a ∈ u\nhab : f a = f b\nhb : b ∉ u\n⊢ False",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"Partition.IsRepFun.apply_of_notMem",
... | [
"α : Type u_1\nu : Set α\nP : Partition u\nf : α → α\na b : α\nhf : P.IsRepFun f\nha : a ∈ u\nhab : f a = b\nhb : b ∉ u\n⊢ False"
] | rw [hf.apply_of_notMem hb] at hab | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.SuccPred.Tree | {
"line": 54,
"column": 2
} | {
"line": 58,
"column": 15
} | {
"line": 59,
"column": 2
} | [
{
"pp": "case zero\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nh : Nat.find ⋯ = 0\n⊢ Order.pred (Order.pred^[0 - 1] r) = ⊥",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case succ\nα : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\nn✝ : ℕ\nh : Nat.find ⋯ = n✝ + 1\n⊢ Order.pred (Order.pred^[n✝ + 1 - 1] r) = ⊥"
] | · have : Order.pred^[0] r = ⊥ := by
rw [← h]
apply Nat.find_spec (bot_le (a := r)).exists_pred_iterate
simp only [Function.iterate_zero, id_eq] at this
simp [this] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Centering | {
"line": 197,
"column": 37
} | {
"line": 197,
"column": 40
} | {
"line": 197,
"column": 41
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝¹ : CompleteSpace E\nhf : StronglyAdapted ℱ f\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\ninst✝ : SigmaFiniteFiltration μ ℱ\ni j : ℕ\nhij : i... | [
"Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝¹ : CompleteSpace E\nhf : StronglyAdapted ℱ f\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\ninst✝ : SigmaFiniteFiltration μ ℱ\ni j : ℕ\nhij : i ≤ j\nh_eq_s... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 199,
"column": 69
} | {
"line": 200,
"column": 70
} | {
"line": 202,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot ι\na b : ℝ\nf : ι → Ω → ℝ\nN : ι\nn : ℕ\nω : Ω\n⊢ upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Real",
"ConditionallyCompleteLinearO... | [] | by
simp only [lowerCrossingTime, le_hittingBtwn upperCrossingTime_le ω] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 76
} | {
"line": 216,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot ι\na b : ℝ\nf : ι → Ω → ℝ\nN : ι\nn m : ℕ\nω : Ω\nhnm : n ≤ m\n⊢ upperCrossingTime a b f N n ω ≤ upperCrossingTime a b f N m ω",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
... | [
"Ω : Type u_1\nι : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot ι\na b : ℝ\nf : ι → Ω → ℝ\nN : ι\nn m : ℕ\nω : Ω\nhnm : n ≤ m\n⊢ Monotone fun n ↦ upperCrossingTime a b f N n ω"
] | suffices Monotone fun n => upperCrossingTime a b f N n ω by exact this hnm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 66
} | {
"line": 290,
"column": 2
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nω : Ω\nhab : a < b\nk : ℕ\nhk : upperCrossingTime a b f N k ω = N\n⊢ BddAbove {n | upperCrossingTime a b f N n ω < N}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"setOf",
"Membership.mem",
"MeasureTheory.upperCross... | [
"Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nω : Ω\nhab : a < b\nk : ℕ\nhk : upperCrossingTime a b f N k ω = N\nn : ℕ\nhn : upperCrossingTime a b f N n ω < N\n⊢ n ≤ k"
] | refine ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Process.Stopping | {
"line": 260,
"column": 93
} | {
"line": 270,
"column": 26
} | {
"line": 272,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝¹ : Preorder ι\ninst✝ : Countable ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhτ : ∀ (i : ι), MeasurableSet {ω | τ ω = ↑i}\n⊢ IsStoppingTime f τ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | by
intro i
have h_eq_iUnion : {ω | τ ω ≤ i} = ⋃ k ≤ i, {ω | τ ω = k} := by
ext ω
simp only [Set.mem_setOf_eq, Set.mem_iUnion, exists_prop]
cases τ ω with
| top => simp
| coe a => norm_cast; simp
rw [h_eq_iUnion]
refine MeasurableSet.biUnion (Set.to_countable _) fun k hk => ?_
exact f.mono ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 517,
"column": 2
} | {
"line": 517,
"column": 10
} | {
"line": 518,
"column": 2
} | [
{
"pp": "case a\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nhab : a < b\nN M : ℕ\nhNM : N ≤ M\nω : Ω\nn : ℕ\n⊢ upperCrossingTime a b f N n ω < N → upperCrossingTime a b f M n ω < M",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"MeasureTheory.upperCrossingTime",
... | [
"case a\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nhab : a < b\nN M : ℕ\nhNM : N ≤ M\nω : Ω\nn : ℕ\nhn : upperCrossingTime a b f N n ω < N\n⊢ upperCrossingTime a b f M n ω < M"
] | intro hn | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 517,
"column": 2
} | {
"line": 517,
"column": 10
} | {
"line": 518,
"column": 2
} | [
{
"pp": "case a\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nhab : a < b\nN M : ℕ\nhNM : N ≤ M\nω : Ω\nn : ℕ\n⊢ upperCrossingTime a b f N n ω < N → upperCrossingTime a b f M n ω < M",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"MeasureTheory.upperCrossingTime",
... | [
"case a\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nhab : a < b\nN M : ℕ\nhNM : N ≤ M\nω : Ω\nn : ℕ\nhn : upperCrossingTime a b f N n ω < N\n⊢ upperCrossingTime a b f M n ω < M"
] | intro hn | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 555,
"column": 8
} | {
"line": 555,
"column": 25
} | {
"line": 555,
"column": 25
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\nhab : a < b\nhn : upcrossingsBefore a b f N ω < n\n⊢ N ≤ upperCrossingTime a b f N n ω",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"setOf",
"Eq.mp",
"MeasureTheory.... | [
"Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nω : Ω\nhab : a < b\nhn : sSup {n | upperCrossingTime a b f N n ω < N} < n\n⊢ N ≤ upperCrossingTime a b f N n ω"
] | upcrossingsBefore | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 695,
"column": 4
} | {
"line": 695,
"column": 28
} | {
"line": 696,
"column": 4
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nN : ℕ\nhab : b ≤ a\n⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω ↦ (f N ω - a)⁺) x ∂μ",
"ppTerm": "?neg✝",... | [
"case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nN : ℕ\nhab : b - a ≤ 0\n⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤ ∫ (x : Ω), (fun ω ↦ (f N ω - a)⁺) x ∂μ"
] | rw [← sub_nonpos] at hab | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 826,
"column": 4
} | {
"line": 826,
"column": 28
} | {
"line": 827,
"column": 4
} | [
{
"pp": "case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : b ≤ a\n⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ",
"ppTerm": "... | [
"case neg\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : b - a ≤ 0\n⊢ ENNReal.ofReal (b - a) * ∫⁻ (ω : Ω), upcrossings a b f ω ∂μ ≤ ⨆ N, ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ"
] | rw [← sub_nonpos] at hab | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 285,
"column": 66
} | {
"line": 308,
"column": 51
} | {
"line": 310,
"column": 0
} | [
{
"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\nhfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω\nhf : StronglyAdapted ℱ f\nhint : ∀ (n : ℕ), Integrable (f n) μ\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑... | [] | by
simp only [← Real.norm_eq_abs] at hbdd
have h₀ := martingalePart_bdd_difference ℱ hbdd
simp only [Real.norm_eq_abs, ← NNReal.coe_ofNat, ← NNReal.coe_mul 2 R] at h₀
have h₁ := (martingale_martingalePart hf hint).ae_not_tendsto_atTop_atTop h₀
have h₂ := (martingale_martingalePart hf hint).ae_not_tendsto_atTo... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 89
} | {
"line": 337,
"column": 2
} | [
{
"pp": "Ω : Type u_2\nm0 : MeasurableSpace Ω\nℱ : Filtration ℕ m0\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\ns : ℕ → Set Ω\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ ∀ᵐ (ω : Ω) ∂μ,\n ω ∈ limsup s atTop ↔ Tendsto (fun n ↦ ∑ k ∈ Finset.range n, μ[(s (k + 1)).indicator 1 | ↑ℱ k] ω) atTop atTop",
"ppTerm": "?m... | [
"Ω : Type u_2\nm0 : MeasurableSpace Ω\nℱ : Filtration ℕ m0\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\ns : ℕ → Set Ω\nhs : ∀ (n : ℕ), MeasurableSet (s n)\n⊢ ∀ᵐ (ω : Ω) ∂μ,\n ω ∈ {ω | Tendsto (fun n ↦ ∑ k ∈ Finset.range n, (s (k + 1)).indicator (fun x ↦ 1) ω) atTop atTop} ↔\n Tendsto (fun n ↦ ∑ k ∈ Finset.ran... | rw [← limsup_nat_add s 1,
Set.limsup_eq_tendsto_sum_indicator_atTop (zero_lt_one (α := ℝ)) (fun n ↦ s (n + 1))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Process.Stopping | {
"line": 1124,
"column": 2
} | {
"line": 1140,
"column": 30
} | {
"line": 1142,
"column": 0
} | [
{
"pp": "case inr\nΩ : Type u_1\nι : Type u_3\ninst✝² : Nonempty ι\nτ : Ω → WithTop ι\nE : Type u_4\nu : ι → Ω → E\ninst✝¹ : LinearOrder ι\ninst✝ : AddCommMonoid E\ns : Finset ι\nn : ι\nhbdd : ∀ (ω : Ω), τ ω < ↑n → τ ω ∈ WithTop.some '' ↑s\nω : Ω\nh : τ ω < ↑n\n⊢ stoppedProcess u τ n ω = {a | ↑n ≤ τ a}.indicato... | [] | · rw [stoppedProcess_eq_of_ge (le_of_lt h)]
have h_top : τ ω ≠ ⊤ := fun h_contra ↦ by simp [h_contra] at h
specialize hbdd ω h
lift τ ω to ι using h_top with i hi
rw [Finset.sum_eq_single_of_mem i]
· simp only [untopD_coe]
rw [Set.indicator_of_notMem, zero_add, Set.indicator_of_mem] <;> rw [Se... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 163,
"column": 13
} | {
"line": 163,
"column": 27
} | {
"line": 163,
"column": 28
} | [
{
"pp": "case refine_6\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\nn : ℕ\nhX : AEMeasurable X μ\nl u : ℝ\nhlu : z.re ∈ Set.Ioo l u\nh_subset : Set.Ioo l u ⊆ integrableExpSet X μ\nt : ℝ := min (z.re - l) (u - z.re) / 2\nh_pos : 0 < min (z.re... | [
"case refine_6\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X μ)\nn : ℕ\nhX : AEMeasurable X μ\nl u : ℝ\nhlu : z.re ∈ Set.Ioo l u\nh_subset : Set.Ioo l u ⊆ integrableExpSet X μ\nt : ℝ := min (z.re - l) (u - z.re) / 2\nh_pos : 0 < min (z.re - l) (u - z... | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 249,
"column": 2
} | {
"line": 256,
"column": 60
} | {
"line": 258,
"column": 0
} | [
{
"pp": "case refine_2\nx t p : ℝ\nhp : 0 ≤ p\nht : t ≠ 0\n⊢ (p / |t|) ^ p * max (rexp (|t| * x)) (rexp (-|t| * x)) = (p / |t|) ^ p * rexp (|t| * |x|)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
... | [] | · congr
rcases le_total 0 x with hx | hx
· rw [abs_of_nonneg hx]
simp only [neg_mul, sup_eq_left, exp_le_exp, neg_le_self_iff]
positivity
· rw [abs_of_nonpos hx]
simp only [neg_mul, mul_neg, sup_eq_right, exp_le_exp, le_neg_self_iff]
exact mul_nonpos_of_nonneg_of_nonpos (abs_nonneg _... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 287,
"column": 4
} | {
"line": 288,
"column": 18
} | {
"line": 289,
"column": 2
} | [
{
"pp": "case hbc\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v x : ℝ\nh_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nh_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\nh_nonneg : 0 ≤ x\nhx : x < |t|\np : ℝ\nhp : 0 ≤ p\nht : t ≠ 0\nhX : AEMeasurable X μ\na : Ω\nthis : |t| =... | [] | · rw [sub_ne_zero]
exact hx.ne' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Process.Stopping | {
"line": 1428,
"column": 6
} | {
"line": 1428,
"column": 23
} | {
"line": 1429,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁴ : LinearOrder ι\nμ : Measure Ω\nℱ : Filtration ι m\nτ : Ω → WithTop ι\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : Ω → E\nhτ : IsStoppingTime ℱ τ\ni : ι\ninst✝ : SigmaFinite (μ.trim ⋯)\n⊢ ... | [] | exact inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 77,
"column": 66
} | {
"line": 81,
"column": 20
} | {
"line": 83,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommMonoid E\ninst✝¹ : Module ℝ E\nmE : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nμ : Measure E\nh : ∀ (L : E →L[ℝ] ℝ), IsGaussian (Measure.map (⇑L) μ)\n⊢ IsGaussian μ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [] | by
refine ⟨fun L ↦ ?_⟩
rw [(h L).eq_gaussianReal, integral_map, variance_map]
· simp
all_goals fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 420,
"column": 2
} | {
"line": 427,
"column": 20
} | {
"line": 429,
"column": 0
} | [
{
"pp": "l u v : ℝ\nhv : v ∈ Set.Ioo l u\n⊢ v + min (v - l) (u - v) / 2 ∈ Set.Ioo l u",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Probability.Moments.IntegrableExpMul.0.ProbabilityTheory.add_half_inf_sub_mem_Ioo._abel_1_1",
"half_lt_self",
"IsRightCa... | [] | have h_pos : 0 < (v - l) ⊓ (u - v) := by simp [hv.1, hv.2]
constructor
· calc l < v := hv.1
_ ≤ v + ((v - l) ⊓ (u - v)) / 2 := le_add_of_nonneg_right (by positivity)
· calc v + ((v - l) ⊓ (u - v)) / 2
_ < v + ((v - l) ⊓ (u - v)) := by gcongr; exact half_lt_self (by positivity)
_ ≤ v + (u - v) := by gc... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 420,
"column": 2
} | {
"line": 427,
"column": 20
} | {
"line": 429,
"column": 0
} | [
{
"pp": "l u v : ℝ\nhv : v ∈ Set.Ioo l u\n⊢ v + min (v - l) (u - v) / 2 ∈ Set.Ioo l u",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Probability.Moments.IntegrableExpMul.0.ProbabilityTheory.add_half_inf_sub_mem_Ioo._abel_1_1",
"half_lt_self",
"IsRightCa... | [] | have h_pos : 0 < (v - l) ⊓ (u - v) := by simp [hv.1, hv.2]
constructor
· calc l < v := hv.1
_ ≤ v + ((v - l) ⊓ (u - v)) / 2 := le_add_of_nonneg_right (by positivity)
· calc v + ((v - l) ⊓ (u - v)) / 2
_ < v + ((v - l) ⊓ (u - v)) := by gcongr; exact half_lt_self (by positivity)
_ ≤ v + (u - v) := by gc... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 434,
"column": 46
} | {
"line": 434,
"column": 63
} | {
"line": 435,
"column": 4
} | [
{
"pp": "case h\nl u v : ℝ\nhv : v ∈ Set.Ioo l u\nh_pos : 0 < min (v - l) (u - v)\n⊢ min (v - l) (u - v) ≤ v - l",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Real",
"Real.instSub",
"HSub.hSub",
"Real.instSemilatticeInf",
"instHSub",
"inf_le_left"
],... | [] | exact inf_le_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 230,
"column": 39
} | {
"line": 230,
"column": 53
} | {
"line": 230,
"column": 54
} | [
{
"pp": "case e_a\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)\nh_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u\... | [
"case e_a\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)\nh_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u\nh_d_mgf : d... | mul_div_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 228,
"column": 4
} | {
"line": 233,
"column": 10
} | {
"line": 234,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)\nh_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u\nh_d_mgf :... | [] | rw [sub_div]
congr 1
· rw [pow_two, div_mul_eq_div_div, mul_div_assoc, div_self, mul_one]
exact (mgf_pos' hμ (interior_subset (s := integrableExpSet X μ) h)).ne'
· rw [deriv_cgf h]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 228,
"column": 4
} | {
"line": 233,
"column": 10
} | {
"line": 234,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\nh_mem : ∀ᶠ (y : ℝ) in 𝓝 v, y ∈ interior (integrableExpSet X μ)\nh_d_cgf : deriv (cgf X μ) =ᶠ[𝓝 v] fun u ↦ (∫ (x : Ω), (fun ω ↦ X ω * rexp (u * X ω)) x ∂μ) / mgf X μ u\nh_d_mgf :... | [] | rw [sub_div]
congr 1
· rw [pow_two, div_mul_eq_div_div, mul_div_assoc, div_self, mul_one]
exact (mgf_pos' hμ (interior_subset (s := integrableExpSet X μ) h)).ne'
· rw [deriv_cgf h]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Fernique | {
"line": 256,
"column": 8
} | {
"line": 256,
"column": 27
} | {
"line": 256,
"column": 28
} | [
{
"pp": "E : 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\na : ℝ\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\... | [
"E : 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\na : ℝ\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\nha_gt : 2⁻¹... | ENNReal.toReal_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Fernique | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 23
} | {
"line": 263,
"column": 24
} | [
{
"pp": "case e_a\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\na : ℝ\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) =... | [
"case e_a\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\na : ℝ\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\nh... | ENNReal.toReal_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 82,
"column": 2
} | {
"line": 83,
"column": 69
} | {
"line": 84,
"column": 2
} | [
{
"pp": "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✝ : 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 : ... | have h2 : L₂ ∘ (fun ω k ↦ X j k ω) = ∑ k : J j, (L₂ ∘L .single ℝ _ k) ∘ X j k := by
ext; simp [-ContinuousLinearMap.comp_apply, ← L₂.sum_comp_single] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Process.Basic | {
"line": 271,
"column": 20
} | {
"line": 271,
"column": 22
} | {
"line": 272,
"column": 2
} | [
{
"pp": "S : Type u_1\nΩ : Type u_3\nmΩ : MeasurableSpace Ω\nα : Type u_4\nmα : MeasurableSpace α\nκ : Kernel α Ω\nP : Measure α\nT : S → Type u_5\n𝓧 : (i : S) → T i → Type u_6\ninst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)\nX : (i : S) → (j : T i) → Ω → 𝓧 i j\nhX : ∀ (i : S) (j : T i), Measurable (X... | [
"S : Type u_1\nΩ : Type u_3\nmΩ : MeasurableSpace Ω\nα : Type u_4\nmα : MeasurableSpace α\nκ : Kernel α Ω\nP : Measure α\nT : S → Type u_5\n𝓧 : (i : S) → T i → Type u_6\ninst✝ : (i : S) → (j : T i) → MeasurableSpace (𝓧 i j)\nX : (i : S) → (j : T i) → Ω → 𝓧 i j\nhX : ∀ (i : S) (j : T i), Measurable (X i j)\nh : ∀... | hη | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 161,
"column": 33
} | {
"line": 161,
"column": 53
} | {
"line": 161,
"column": 54
} | [
{
"pp": "case hv.inr\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 ... | [
"case hv.inr\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 (↥I → ℝ)\n... | this j i (by grind), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 183,
"column": 8
} | {
"line": 183,
"column": 28
} | {
"line": 183,
"column": 29
} | [
{
"pp": "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\nh1 : i.succ ≤ j.succ\nh2 : i.castSucc ≤ j.succ\nh3 : i.castSucc ≤ j.castSucc\n⊢ ↑(min (t i... | [
"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\nh1 : i.succ ≤ j.succ\nh2 : i.castSucc ≤ j.succ\nh3 : i.castSucc ≤ j.castSucc\n⊢ ↑(t i.succ) - ↑(min (t... | min_eq_left (ht h1), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.BrownianMotion.Basic | {
"line": 171,
"column": 30
} | {
"line": 185,
"column": 72
} | {
"line": 187,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nB : ℝ≥0 → Ω → ℝ\nP : Measure Ω\nhB : IsPreBrownianReal B P\n⊢ HasIndepIncrements B P",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"Semiring.toModu... | [] | by
have : IsProbabilityMeasure P := hB.isGaussianProcess.isProbabilityMeasure
refine fun n t ht ↦ hB.isGaussianProcess.hasGaussianLaw_increments.iIndepFun_of_covariance_eq_zero
fun i j hij ↦ ?_
rw [covariance_fun_sub_fun_sub]
· simp_rw [hB.covariance_fun_eval]
wlog h : i < j generalizing i j
· simp_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 60
} | {
"line": 156,
"column": 4
} | [
{
"pp": "case neg.hf_int\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\nhρ_zero : ¬(ν a).restrict s = 0\nh :\n ∫⁻ (b : β... | [
"case neg.hf_int\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\nhρ_zero : ¬(ν a).restrict s = 0\nh :\n ∫⁻ (b : β) in s, ENNR... | rw [setLIntegral_stieltjesOfMeasurableRat_rat hf a _ hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 128,
"column": 6
} | {
"line": 129,
"column": 63
} | {
"line": 131,
"column": 0
} | [
{
"pp": "case h₂\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nL₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ\nL₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ... | [] | grw [le_opNorm₂]
gcongr <;> grw [opNorm_comp_le, norm_inr_le_one, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 128,
"column": 6
} | {
"line": 129,
"column": 63
} | {
"line": 131,
"column": 0
} | [
{
"pp": "case h₂\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nL₁✝ : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ\nL₂✝ : StrongDual ℝ F →L[ℝ] StrongDual ℝ F →L[ℝ] ℝ... | [] | grw [le_opNorm₂]
gcongr <;> grw [opNorm_comp_le, norm_inr_le_one, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 70
} | {
"line": 198,
"column": 4
} | [
{
"pp": "case refine_2\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nx : ℚ\nt : Set α\nx✝¹ : MeasurableSet t\nx✝ : ρ.fst t ≠ ∞\n⊢ ∫⁻ (x_1 : α) in t, ‖(preCDF ρ x x_1).toReal‖ₑ ∂ρ.fst ≤ ρ.fst univ",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [
"case refine_2\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\nx : ℚ\nt : Set α\nx✝¹ : MeasurableSet t\nx✝ : ρ.fst t ≠ ∞\n⊢ ∫⁻ (x_1 : α) in t, ENNReal.ofReal (preCDF ρ x x_1).toReal ∂ρ.fst ≤ ρ.fst univ"
] | simp_rw [← ofReal_norm, Real.norm_of_nonneg ENNReal.toReal_nonneg] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.CentralLimitTheorem | {
"line": 87,
"column": 4
} | {
"line": 90,
"column": 74
} | {
"line": 92,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nP : Measure Ω\nP' : Measure Ω'\nX : ℕ → Ω → ℝ\nY : Ω' → ℝ\ninst✝¹ : IsProbabilityMeasure P\ninst✝ : IsProbabilityMeasure P'\nhY : HasLaw Y (gaussianReal 0 1) P'\nh0 : ∫ (x : Ω), X 0 x ∂P = 0\nh1 : ∫ (x : Ω), (X 0 ^ 2) x ∂P =... | [] | refine ProbabilityMeasure.tendsto_iff_tendsto_charFun.2 fun t ↦ ?_
rw! [hY.map_eq]
simpa [charFun_inv_sqrt_mul_sum hindep hident, charFun_gaussianReal, neg_div] using
tendsto_charFun_inv_sqrt_mul_pow (hident 0).aemeasurable_fst h0 h1 t | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.CentralLimitTheorem | {
"line": 87,
"column": 4
} | {
"line": 90,
"column": 74
} | {
"line": 92,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nP : Measure Ω\nP' : Measure Ω'\nX : ℕ → Ω → ℝ\nY : Ω' → ℝ\ninst✝¹ : IsProbabilityMeasure P\ninst✝ : IsProbabilityMeasure P'\nhY : HasLaw Y (gaussianReal 0 1) P'\nh0 : ∫ (x : Ω), X 0 x ∂P = 0\nh1 : ∫ (x : Ω), (X 0 ^ 2) x ∂P =... | [] | refine ProbabilityMeasure.tendsto_iff_tendsto_charFun.2 fun t ↦ ?_
rw! [hY.map_eq]
simpa [charFun_inv_sqrt_mul_sum hindep hident, charFun_gaussianReal, neg_div] using
tendsto_charFun_inv_sqrt_mul_pow (hident 0).aemeasurable_fst h0 h1 t | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 94
} | {
"line": 505,
"column": 0
} | [
{
"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\n⊢ ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) univ ∂ν a = (κ a) univ",
"ppTerm": "... | [] | rw [← setLIntegral_univ, setLIntegral_toKernel_univ hf a MeasurableSet.univ, univ_prod_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 94
} | {
"line": 505,
"column": 0
} | [
{
"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\n⊢ ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) univ ∂ν a = (κ a) univ",
"ppTerm": "... | [] | rw [← setLIntegral_univ, setLIntegral_toKernel_univ hf a MeasurableSet.univ, univ_prod_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 94
} | {
"line": 505,
"column": 0
} | [
{
"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\n⊢ ∫⁻ (b : β), ((IsCondKernelCDF.toKernel f hf) (a, b)) univ ∂ν a = (κ a) univ",
"ppTerm": "... | [] | rw [← setLIntegral_univ, setLIntegral_toKernel_univ hf a MeasurableSet.univ, univ_prod_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 535,
"column": 10
} | {
"line": 535,
"column": 50
} | {
"line": 536,
"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 → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nt✝ t : Set ℝ\nht : MeasurableSet t\niht : ∫⁻ (b : β) in s, ((I... | [] | exact Or.inl ⟨subset_rfl, subset_univ t⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 118,
"column": 4
} | {
"line": 120,
"column": 75
} | {
"line": 121,
"column": 2
} | [
{
"pp": "case refine_2\nα : 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\nn : ℕ\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable fun p ↦ (ν p.1) ↑p.2",
"ppTerm": "?refine_2",
... | [] | · refine measurable_from_prod_countable_left ?_
rintro ⟨t, ht⟩
exact Kernel.measurable_coe _ (measurableSet_countablePartition _ ht) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 580,
"column": 8
} | {
"line": 582,
"column": 51
} | {
"line": 583,
"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) (... | [] | congr with x : 1
exact measure_compl (measurable_prodMk_left ht)
(measure_ne_top (hf.toKernel f (a, x)) _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 580,
"column": 8
} | {
"line": 582,
"column": 51
} | {
"line": 583,
"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) (... | [] | congr with x : 1
exact measure_compl (measurable_prodMk_left ht)
(measure_ne_top (hf.toKernel f (a, x)) _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 313,
"column": 12
} | {
"line": 313,
"column": 31
} | {
"line": 313,
"column": 32
} | [
{
"pp": "case pos\nα : 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\nκ' : Kernel α (γ × β)\nhκκ' : κ ≤ κ'\nhκ'ν : κ'.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns : Set β\nh0 : (ν a) (countableP... | [
"case pos\nα : 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\nκ' : Kernel α (γ × β)\nhκκ' : κ ≤ κ'\nhκ'ν : κ'.fst ≤ ν\nn : ℕ\na : α\nx : γ\ns : Set β\nh0 : (ν a) (countablePartitionSet ... | ENNReal.toReal_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 351,
"column": 4
} | {
"line": 353,
"column": 8
} | {
"line": 354,
"column": 2
} | [
{
"pp": "case neg.refine_1\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)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ ... | [] | rw [ne_eq, ENNReal.div_eq_top]
push Not
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 351,
"column": 4
} | {
"line": 353,
"column": 8
} | {
"line": 354,
"column": 2
} | [
{
"pp": "case neg.refine_1\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)\nν : Kernel α γ\ninst✝ : IsFiniteKernel κ\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Antitone seq\nhseq_iInter : ⋂ ... | [] | rw [ne_eq, ENNReal.div_eq_top]
push Not
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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