module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 63
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : realMixedSpace K\nw : { w // w.IsReal }\n⊢ ((homeoRealMixedSpacePolarSpace K) x).1 ↑w = x.1 w",
"usedConstants": [
"Real",
"Pi.topologicalSpace",
"congrArg",
"NumberField.InfinitePlace.IsComplex",
"PseudoMetricSpace.toUniformSpace",
... | simp_rw [homeoRealMixedSpacePolarSpace_apply, dif_pos w.prop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 215,
"column": 75
} | {
"line": 215,
"column": 92
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw₁ w₂ : InfinitePlace K\ne₁ : { w // w ≠ w₁ } ≃ Fin (rank K)\ne₂ : { w // w ≠ w₂ } ≃ Fin (rank K)\nf : Fin (rank K + 1) ≃ InfinitePlace K := (finSuccEquiv (rank K)).trans ↑((Equiv.optionSubtype w₁).symm e₁... | prod_eq_abs_norm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 387,
"column": 2
} | {
"line": 388,
"column": 75
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\n⊢ (∃ u, (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b) ↔ ∃ ζ, ↑ζ • ↑a = ↑b",
"usedConstants": [
"Units.val",
"Real",
"instHSMul",
"MulEquiv.instEquivLike",
"HMul.hMul",
"_pri... | refine ⟨fun ⟨u, h⟩ ↦ ⟨⟨unitsNonZeroDivisorsEquiv u, ?_⟩, by simpa using h⟩,
fun ⟨⟨u, _⟩, h⟩ ↦ ⟨unitsNonZeroDivisorsEquiv.symm u, by simpa using h⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 559,
"column": 6
} | {
"line": 560,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nc : ℝ\nJ : ↥(Ideal (𝓞 K))⁰\nn : ℕ\n⊢ { a // ↑(preimageOfMemIntegerSet ↑a) ∈ ↑J } ≃ { I // ↑J ∣ ↑↑I.1 }",
"usedConstants": [
"NumberField.mixedEmbedding.fundamentalCone.preimageOfMemIntegerSet",
"Eq.mpr",
"... | convert! Equiv.subtypeEquivOfSubtype (p := fun I ↦ J.1 ∣ I.1) (integerSetEquivNorm K n)
rw [integerSetEquivNorm_apply_fst, dvd_span_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 559,
"column": 6
} | {
"line": 560,
"column": 60
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nc : ℝ\nJ : ↥(Ideal (𝓞 K))⁰\nn : ℕ\n⊢ { a // ↑(preimageOfMemIntegerSet ↑a) ∈ ↑J } ≃ { I // ↑J ∣ ↑↑I.1 }",
"usedConstants": [
"NumberField.mixedEmbedding.fundamentalCone.preimageOfMemIntegerSet",
"Eq.mpr",
"... | convert! Equiv.subtypeEquivOfSubtype (p := fun I ↦ J.1 ∣ I.1) (integerSetEquivNorm K n)
rw [integerSetEquivNorm_apply_fst, dvd_span_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 580,
"column": 4
} | {
"line": 601,
"column": 82
} | [
{
"pp": "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 }",
"usedConstants": [
"Iff.mpr",
"NumberField.mi... | simp_rw [← intNorm_idealSetEquiv_apply, ← Nat.le_floor_iff hs]
rw [torsionOrder, ← Nat.card_eq_fintype_card, ← Nat.card_prod]
refine Nat.card_congr <| @Equiv.ofFiberEquiv _ (γ := Finset.Iic ⌊s⌋₊) _
(fun I ↦ ⟨absNorm I.1.val.1, Finset.mem_Iic.mpr I.1.prop.2.2⟩)
(fun a ↦ ⟨intNorm (idealSetEquiv K J a.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 580,
"column": 4
} | {
"line": 601,
"column": 82
} | [
{
"pp": "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 }",
"usedConstants": [
"Iff.mpr",
"NumberField.mi... | simp_rw [← intNorm_idealSetEquiv_apply, ← Nat.le_floor_iff hs]
rw [torsionOrder, ← Nat.card_eq_fintype_card, ← Nat.card_prod]
refine Nat.card_congr <| @Equiv.ofFiberEquiv _ (γ := Finset.Iic ⌊s⌋₊) _
(fun I ↦ ⟨absNorm I.1.val.1, Finset.mem_Iic.mpr I.1.prop.2.2⟩)
(fun a ↦ ⟨intNorm (idealSetEquiv K J a.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 62,
"column": 4
} | {
"line": 64,
"column": 26
} | [
{
"pp": "case inl\nK : 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 : IsUnramified K w\nhv : v.IsReal\nthis : ComplexEmbedding.LiesOver (extensionEmbedding w) (extensionEmbedding v)\n⊢ Module.finrank v.Com... | rw [Algebra.finrank_eq_of_equiv_equiv (ringEquivRealOfIsReal hv) (ringEquivRealOfIsReal
(h.liesOver_isReal_over _ _ hv)) (RingHom.ext fun _ ↦ Complex.ofReal_inj.1 <| by simp),
Module.finrank_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 116,
"column": 36
} | {
"line": 117,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : NumberField K\n⊢ Units.rank ↥K⁺ = Units.rank K",
"usedConstants": [
"Eq.mpr",
"NumberField.maximalRealSubfield",
"congrArg",
"NumberField.Units.rank",
"HSub.hSub",
"Membership.mem"... | by
rw [Units.rank, Units.rank, card_infinitePlace_eq_card_infinitePlace K] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 517,
"column": 47
} | {
"line": 517,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ Real.exp (x w₀) ^ finrank ℚ K *\n ∏ x_1, (∏ i, i ((algebraMap (𝓞 K) K) ↑(fundSystem K (equivFinRank.symm x_1))) ^ i.mult) ^ x ↑x_1 =\n Real.exp (x w₀) ^ finrank ℚ K",
"usedConstants": [
"NumberField.InfinitePla... | prod_eq_abs_norm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 61
} | [
{
"pp": "case e_a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ ∏ x, (↑x.mult)⁻¹ = 2⁻¹ ^ nrComplexPlaces K",
"usedConstants": [
"NumberField.InfinitePlace.mult_isComplex",
"MulOne.toOne",
"Real",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
... | · simp [prod_eq_prod_mul_prod, mult_isReal, mult_isComplex] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 42
} | [
{
"pp": "𝒪 : Type u_2\ninst✝⁴ : CommRing 𝒪\ninst✝³ : Module.Finite ℤ 𝒪\ninst✝² : Algebra.Unramified ℤ 𝒪\ninst✝¹ : IsDomain 𝒪\ninst✝ : FaithfulSMul ℤ 𝒪\nthis : IsDedekindDomain 𝒪\nK : Type u_2 := FractionRing 𝒪\n⊢ Function.Bijective ⇑(algebraMap ℤ 𝒪)",
"usedConstants": [
"CommSemiring.toSemiri... | let : Algebra ℤ K := Ring.toIntAlgebra K | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 341,
"column": 43
} | {
"line": 341,
"column": 75
} | [
{
"pp": "n m p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhn : n = p ^ (k + 1) * m\nhm : ¬p ∣ m\nthis✝³ : IsAbelianGalois ℚ K\nthis✝² : NeZero m\nthis✝¹ : NeZero n\nhp' : 𝒑 ≠ ⊥\nζ : K := zeta n ℚ K\nhζ : IsPrimitiveRoot (zeta n... | rwa [hp.out.coprime_iff_not_dvd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 341,
"column": 43
} | {
"line": 341,
"column": 75
} | [
{
"pp": "n m p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhn : n = p ^ (k + 1) * m\nhm : ¬p ∣ m\nthis✝³ : IsAbelianGalois ℚ K\nthis✝² : NeZero m\nthis✝¹ : NeZero n\nhp' : 𝒑 ≠ ⊥\nζ : K := zeta n ℚ K\nhζ : IsPrimitiveRoot (zeta n... | rwa [hp.out.coprime_iff_not_dvd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 341,
"column": 43
} | {
"line": 341,
"column": 75
} | [
{
"pp": "n m p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhn : n = p ^ (k + 1) * m\nhm : ¬p ∣ m\nthis✝³ : IsAbelianGalois ℚ K\nthis✝² : NeZero m\nthis✝¹ : NeZero n\nhp' : 𝒑 ≠ ⊥\nζ : K := zeta n ℚ K\nhζ : IsPrimitiveRoot (zeta n... | rwa [hp.out.coprime_iff_not_dvd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : Fintype α\ninst✝² : Fintype β\nA : Matrix α β ℤ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nv : β → ℤ\nhv : 0 ≤ v ∧ v ≤ B'\n⊢ N ≤ A *ᵥ v ∧ A *ᵥ v ≤ P",
"usedConstants": [
"Int.instAddCommMonoid",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | have mulVec_def : A.mulVec v =
fun i ↦ Finset.sum univ fun j : β ↦ A i j * v j := rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\nhn : m < n\nhm : 0 < m\nx : β → ℤ\nhxT : x ∈ T\ny : β → ℤ\nhyT : y ∈ T\nhneq : x ≠ y\nhfeq : A *ᵥ x = A *ᵥ y\nn_mul_norm_A_pow_e_nonneg : 0 ≤ (↑n * max 1 ‖A‖) ^ e\ni : β\nj : Unit\n⊢ |x i - y i| ≤ ↑⌊(↑n * max 1 ‖A‖) ^ ... | rw [abs_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 21
} | [
{
"pp": "G : Type u_2\ninst✝³ : CommGroup G\ninst✝² : UniformSpace G\ninst✝¹ : IsUniformGroup G\ninst✝ : NonarchimedeanGroup G\nf : ℕ → G\nhf : Tendsto (fun n ↦ f (n + 1) / f n) atTop (𝓝 1)\ns : Set G\nhs : s ∈ 𝓝 1\nt : OpenSubgroup G\nht : ↑t ⊆ s\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → f (b + 1) / f b ∈ t\n⊢ ∃ a, ∀ ... | refine ⟨(N, N), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Ostrowski | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 33
} | [
{
"pp": "f g : AbsoluteValue ℚ ℝ\n⊢ (∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = g ↑n) ↔ f.IsEquiv g",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"Real",
"Real.instZero",
"congrArg",
"Rat",
"Real.instLT",
"Exists",
"Real.semiring",
... | rw [isEquiv_iff_exists_rpow_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Ostrowski | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 27
} | [
{
"pp": "case inr\nf : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\nhf_nontriv : ∀ (n : ℕ), n ≠ 0 → f ↑n = 1\nn : ℕ\nhn : n ≠ 0\n⊢ f ↑n = AbsoluteValue.trivial ↑n",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | · simp [hf_nontriv, hn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Ostrowski | {
"line": 473,
"column": 2
} | {
"line": 473,
"column": 33
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ ¬real.IsEquiv (padic p)",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"Real",
"Real.instZero",
"congrArg",
"Rat",
"Rat.AbsoluteValue.real",
"Real.instLT",
"Exists",
"Real.sem... | rw [isEquiv_iff_exists_rpow_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Pell | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 20
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nh : IsFundamental a\nn : ℤ\n⊢ a ^ n = 1 ↔ n = 0",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"zp... | rw [← zpow_zero a] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Pell | {
"line": 634,
"column": 2
} | {
"line": 634,
"column": 26
} | [
{
"pp": "d : ℤ\nh₀ : 0 < d\nhd : ¬IsSquare d\na₁ : Solution₁ d\nha₁ : IsFundamental a₁\na : Solution₁ d\nHx : 1 < a.x\nHy : 0 < a.y\nH : ∀ (a_1 : Solution₁ d), ∃ n, a_1 = a ^ n ∨ a_1 = -a ^ n\n⊢ a = a₁",
"usedConstants": []
}
] | obtain ⟨n₁, hn₁⟩ := H a₁ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Padics.HeightOneSpectrum | {
"line": 69,
"column": 2
} | {
"line": 72,
"column": 96
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : Algebra R ℚ\ninst✝¹ : IsIntegralClosure R ℤ ℚ\ninst✝ : IsFractionRing R ℚ\n⊢ Function.Surjective ⇑(algebraMap ℤ R)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddCommGroup.intIsScalarTower",
"IsDedekindDomain.isDedekindDomainDvr... | intro x
obtain ⟨y, hy⟩ := IsIntegrallyClosed.isIntegral_iff.1 <|
IsIntegral.algebraMap (B := ℚ) (IsIntegralClosure.isIntegral ℤ ℚ x)
exact ⟨y, IsFractionRing.injective R ℚ <| by simp only [← IsScalarTower.algebraMap_apply, hy]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.HeightOneSpectrum | {
"line": 69,
"column": 2
} | {
"line": 72,
"column": 96
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : Algebra R ℚ\ninst✝¹ : IsIntegralClosure R ℤ ℚ\ninst✝ : IsFractionRing R ℚ\n⊢ Function.Surjective ⇑(algebraMap ℤ R)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddCommGroup.intIsScalarTower",
"IsDedekindDomain.isDedekindDomainDvr... | intro x
obtain ⟨y, hy⟩ := IsIntegrallyClosed.isIntegral_iff.1 <|
IsIntegral.algebraMap (B := ℚ) (IsIntegralClosure.isIntegral ℤ ℚ x)
exact ⟨y, IsFractionRing.injective R ℚ <| by simp only [← IsScalarTower.algebraMap_apply, hy]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.HeightOneSpectrum | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 79
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁷ : CommRing R✝\ninst✝⁶ : Algebra R✝ ℚ\ninst✝⁵ : IsIntegralClosure R✝ ℤ ℚ\nR : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Algebra R ℚ\ninst✝² : IsIntegralClosure R ℤ ℚ\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsFractionRing R ℚ\np : Nat.Primes\n⊢ (fun v ↦ ⟨natGenerator v, ⋯⟩)\n ((fun... | simp only [Ideal.map_symm, natGenerator, HeightOneSpectrum.ofPrime_asIdeal] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Rayleigh | {
"line": 83,
"column": 87
} | {
"line": 90,
"column": 20
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\n⊢ ¬∃ j k m, ↑k < ↑j / r ∧ (↑j + 1) / r ≤ ↑k + 1 ∧ ↑m ≤ ↑j / s ∧ (↑j + 1) / s < ↑m + 1",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Int.cast",
"False",
"Real.partialOrder",
"Real.instLE",
"add_lt_... | by
intro ⟨j, k, m, h₁₁, h₁₂, h₂₁, h₂₂⟩
have h₃ := add_lt_add_of_lt_of_le h₁₁ h₂₁
have h₄ := add_lt_add_of_le_of_lt h₁₂ h₂₂
simp_rw [div_eq_inv_mul, ← right_distrib, hrs.inv_add_inv_eq_one, one_mul] at h₃ h₄
rw [← Int.cast_one, ← add_assoc, add_lt_add_iff_right, add_right_comm] at h₄
simp_rw [← Int.cast_add,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Rayleigh | {
"line": 158,
"column": 24
} | {
"line": 158,
"column": 41
} | [
{
"pp": "case a\nr s : ℝ\nhrs : r.HolderConjugate s\nj : ℤ\nhj : 0 < j\nhb₁ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq s k = x} ↔ j ∈ {x | ∃ k, beattySeq s k = x}\nhb₂ : ∀ s ≥ 0, j ∈ {x | ∃ k > 0, beattySeq' s k = x} ↔ j ∈ {x | ∃ k, beattySeq' s k = x}\n⊢ j ∈ {x | ∃ k > 0, beattySeq r k = x} ∧ j ∉ {x | ∃ k > 0, bea... | hb₁ _ hrs.nonneg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.HeightOneSpectrum | {
"line": 242,
"column": 2
} | {
"line": 248,
"column": 20
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Algebra R ℚ\ninst✝¹ : IsFractionRing R ℚ\ninst✝ : IsIntegralClosure R ℤ ℚ\np : Nat.Primes\nx : ℤ_[↑p]\n⊢ ↑((adicCompletionIntegersEquiv R p) x) = (adicCompletionEquiv R p) ↑x",
"usedConstants": [
"Int.instAddCommGroup",
... | simp only [adicCompletionIntegersEquiv, ContinuousAlgEquiv.trans_apply,
adicCompletionIntegers.coe_padicIntEquiv_symm_apply,
adicCompletionEquiv, ContinuousAlgEquiv.trans_apply, ContinuousAlgEquiv.cast_apply,
EmbeddingLike.apply_eq_iff_eq, Equiv.cast_apply, eq_cast_iff_heq]
rw [← Subtype.heq_iff_coe_heq (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.HeightOneSpectrum | {
"line": 242,
"column": 2
} | {
"line": 248,
"column": 20
} | [
{
"pp": "R : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\ninst✝² : Algebra R ℚ\ninst✝¹ : IsFractionRing R ℚ\ninst✝ : IsIntegralClosure R ℤ ℚ\np : Nat.Primes\nx : ℤ_[↑p]\n⊢ ↑((adicCompletionIntegersEquiv R p) x) = (adicCompletionEquiv R p) ↑x",
"usedConstants": [
"Int.instAddCommGroup",
... | simp only [adicCompletionIntegersEquiv, ContinuousAlgEquiv.trans_apply,
adicCompletionIntegers.coe_padicIntEquiv_symm_apply,
adicCompletionEquiv, ContinuousAlgEquiv.trans_apply, ContinuousAlgEquiv.cast_apply,
EmbeddingLike.apply_eq_iff_eq, Equiv.cast_apply, eq_cast_iff_heq]
rw [← Subtype.heq_iff_coe_heq (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Rayleigh | {
"line": 168,
"column": 58
} | {
"line": 173,
"column": 53
} | [
{
"pp": "r : ℝ\nhr : Irrational r\n⊢ {x | ∃ k > 0, beattySeq' r k = x} = {x | ∃ k > 0, beattySeq r k = x}",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"Preorder.toLT",
"HMul.hMul",
"instConditionallyCompleteLinearOrder",
"AddMonoid.toAddSemigroup",
"Int.... | by
dsimp only [beattySeq, beattySeq']
congr! 4; rename_i k; rw [and_congr_right_iff]; intro hk; congr!
rw [sub_eq_iff_eq_add, Int.ceil_eq_iff, Int.cast_add, Int.cast_one, add_sub_cancel_right]
refine ⟨(Int.floor_le _).lt_of_ne fun h ↦ ?_, (Int.lt_floor_add_one _).le⟩
exact (hr.intCast_mul hk.ne').ne_int ⌊k * ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 20
} | [
{
"pp": "x : ℤ[i]\n⊢ norm x = |x.re| * |x.re| + |x.im| * |x.im|",
"usedConstants": [
"Int.instAddCommGroup",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Zsqrtd.re",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"AddMonoid.toAddSemigroup",
"abs",
"congrArg... | simp [Zsqrtd.norm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.SumFourSquares | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 67
} | [
{
"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 < ... | rcases (Nat.one_le_iff_ne_zero.2 hm₀.ne').eq_or_lt with rfl | hm₁ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 69
} | [
{
"pp": "f : ℤ[X]\ns : ℂ\nthis : Bornology.IsBounded ((fun x ↦ max (x * ‖s‖) 1 * ‖(aeval (↑x * s)) f‖) '' Set.Ioc 0 1)\nc : ℝ\np : ℕ\nx : ℝ\nhx : x ∈ Set.Ioc 0 1\nh : |max (x * ‖s‖) 1 * ‖(aeval (↑x * s)) f‖| ≤ c\n⊢ x ^ (p - 1) * ‖s‖ ^ (p - 1) * ‖(aeval (↑x * s)) f‖ ^ p ≤ |max (x * ‖s‖) 1| ^ p * ‖(aeval (↑x * s)... | refine mul_le_mul_of_nonneg_right ?_ (pow_nonneg (norm_nonneg _) _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Transcendental.Liouville.Residual | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 24
} | [
{
"pp": "case refine_2.refine_1\nr : ℚ\nn : ℕ\n⊢ 1 < ↑r.den * 2",
"usedConstants": [
"_private.Mathlib.NumberTheory.Transcendental.Liouville.Residual.0.eventually_residual_liouville._proof_1_3",
"Rat.pos",
"Rat.den",
"instOfNatNat",
"Nat",
"LT.lt",
"instLTNat",
... | · have := r.pos; lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 40
} | [
{
"pp": "r : ℝ\nhr : 2 < r\nB : ℤ → ℕ → Set ℝ := fun a b ↦ ball (↑a / ↑b) (1 / ↑b ^ r)\nhB : ∀ (a : ℤ) (b : ℕ), volume (B a b) = ↑(2 / ↑b ^ r)\nthis : ∀ (b : ℕ), volume (⋃ a ∈ Finset.Icc 0 ↑b, B a b) ≤ ↑(2 * (↑b ^ (1 - r) + ↑b ^ (-r)))\n⊢ Summable fun a ↦ 2 * (↑a ^ (1 - r) + ↑a ^ (-r))",
"usedConstants": [
... | refine (Summable.add ?_ ?_).mul_left _ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Wilson | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 76
} | [
{
"pp": "case inr\nn : ℕ\nh : ↑(n - 1)! = -1\nh0 : n ≠ 0\nh1 : 1 < n\nh2 : ¬Prime n\nm : ℕ\nhm1 : m ∣ n\nhm2 : 1 < m\nhm3 : m < n\nhm : m ∣ (n - 1)!\n⊢ False",
"usedConstants": [
"Nat.dvd_add_right",
"Dvd.dvd",
"Nat.dvd_one",
"HSub.hSub",
"instSubNat",
"instOfNatNat",
... | refine hm2.ne' (Nat.dvd_one.mp ((Nat.dvd_add_right hm).mp (hm1.trans ?_))) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 156,
"column": 26
} | {
"line": 156,
"column": 61
} | [
{
"pp": "p x : ℝ\nn : ℕ\nhn : n ≠ 0\n⊢ LiouvilleWith p (x * ↑↑n) ↔ LiouvilleWith p x",
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real",
"NormedRing.toRing",
"HMul.hMul",
"DivisionRing.toRatCast",
"congrArg",
"AddMonoid.toAdd... | mul_rat_iff (Nat.cast_ne_zero.2 hn) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Bounded | {
"line": 316,
"column": 2
} | {
"line": 317,
"column": 19
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a ≤ b}) ↔ Bounded (fun x1 x2 ↦ x1 < x2) s",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Set.Bounded",
"HEq.refl",
"Set.bounded_lt_inter... | convert! @bounded_lt_inter_not_lt _ s _ a
exact not_lt.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Bounded | {
"line": 316,
"column": 2
} | {
"line": 317,
"column": 19
} | [
{
"pp": "α : Type u_1\ns : Set α\ninst✝ : LinearOrder α\na : α\n⊢ Bounded (fun x1 x2 ↦ x1 < x2) (s ∩ {b | a ≤ b}) ↔ Bounded (fun x1 x2 ↦ x1 < x2) s",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"Set.Bounded",
"HEq.refl",
"Set.bounded_lt_inter... | convert! @bounded_lt_inter_not_lt _ s _ a
exact not_lt.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 307,
"column": 4
} | {
"line": 309,
"column": 78
} | [
{
"pp": "x : ℝ\nhx : Liouville x\nn N : ℕ\nhN : ∀ b ≥ N, ∀ (a : ℤ), x ≠ ↑a / ↑b → 1 / ↑b ^ n ≤ |x - ↑a / ↑b|\nb : ℕ\nhb : 1 < ↑b\n⊢ ∀ᶠ (m : ℕ) in atTop, ∀ (a : ℤ), 1 / ↑b ^ m ≤ |x - ↑a / ↑b|",
"usedConstants": [
"tendsto_inv_atTop_zero",
"Eq.mpr",
"MulOne.toOne",
"Real",
"DivIn... | have H : Tendsto (fun m => 1 / (b : ℝ) ^ m : ℕ → ℝ) atTop (𝓝 0) := by
simp only [one_div]
exact tendsto_inv_atTop_zero.comp (tendsto_pow_atTop_atTop_of_one_lt hb) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Concept | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 30
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nc d : Concept α β r\nh : c.extent = d.extent\n⊢ c = d",
"usedConstants": []
}
] | obtain ⟨s₁, t₁, rfl, _⟩ := c | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Concept | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 34
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → β → Prop\nc : Concept α β r\nx : α\ny : β\nhx : x ∈ c.extent\nhy : y ∈ c.intent\n⊢ r x y",
"usedConstants": [
"congrArg",
"Membership.mem",
"Concept.intent",
"Eq.mp",
"Concept.extent",
"upperPolar",
"Eq.symm",
"Set.... | rw [← c.upperPolar_extent] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Extension.Linear | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 41
} | [
{
"pp": "case refine_1.inl.inl\nα : Type u\nr : α → α → Prop\ninst✝ : IsPartialOrder α r\nS : Set (α → α → Prop) := {s | IsPartialOrder α s}\nhS : ∀ c ⊆ S, IsChain (fun x1 x2 ↦ x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ S, ∀ z ∈ c, z ≤ ub\ns : α → α → Prop\nhrs : r ≤ s\nhs : Maximal (fun x ↦ x ∈ S) s\nthis : IsPartialOrder α... | · exact Or.inl (_root_.trans ab bc) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Interval.Set.SurjOn | {
"line": 73,
"column": 24
} | {
"line": 73,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na : α\n⊢ SurjOn f (Ioi a ∪ {a}) (Ici (f a))",
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"Set.Ici",
"congrArg",
"PartialOrder.toPreorder",
... | ← Ioi_union_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Lattice.Congruence | {
"line": 75,
"column": 6
} | {
"line": 78,
"column": 41
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ r (x ⊓ y ⊓ z) (y ⊓ z)",
"usedC... | suffices r (x ⊓ y ⊓ (y ⊓ z)) ((x ⊔ y) ⊓ (y ⊓ z)) by
rw [inf_comm x, inf_assoc]
simpa [inf_comm x, ← inf_inf_distrib_left] using this
exact (h₄ inf_le_sup (h₂.mp hxy)).1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Lattice.Congruence | {
"line": 75,
"column": 6
} | {
"line": 78,
"column": 41
} | [
{
"pp": "α : Type u_2\ninst✝ : Lattice α\nr : α → α → Prop\nh₂ : ∀ ⦃x y : α⦄, r x y ↔ r (x ⊓ y) (x ⊔ y)\nh₃ : ∀ ⦃x y z : α⦄, x ≤ y → y ≤ z → r x y → r y z → r x z\nh₄ : ∀ ⦃x y t : α⦄, x ≤ y → r x y → r (x ⊓ t) (y ⊓ t) ∧ r (x ⊔ t) (y ⊔ t)\nx y z : α\nhxy : r x y\nhyz : r y z\n⊢ r (x ⊓ y ⊓ z) (y ⊓ z)",
"usedC... | suffices r (x ⊓ y ⊓ (y ⊓ z)) ((x ⊔ y) ⊓ (y ⊓ z)) by
rw [inf_comm x, inf_assoc]
simpa [inf_comm x, ← inf_inf_distrib_left] using this
exact (h₄ inf_le_sup (h₂.mp hxy)).1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.PrimeIdeal | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 23
} | [
{
"pp": "P : Type u_1\ninst✝ : BooleanAlgebra P\nx : P\nI : Ideal P\nhI : I.IsPrime\n⊢ x ⊓ xᶜ ∈ I",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Compl.compl",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.me... | rw [inf_compl_eq_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Partition.Basic | {
"line": 296,
"column": 2
} | {
"line": 299,
"column": 76
} | [
{
"pp": "α : Type u_1\nx : α\nu : Set α\nP : Partition u\n⊢ x ∈ u ↔ ∃! t, t ∈ P ∧ x ∈ t",
"usedConstants": [
"Partition.subset_of_mem",
"congrArg",
"Partition.eq_of_mem_of_mem",
"Set.sUnion",
"Membership.mem",
"Exists",
"Eq.mp",
"Partition.sUnion_eq",
"A... | refine ⟨fun hx ↦ ?_, fun ⟨_, ⟨htP, hxt⟩, _⟩ ↦ subset_of_mem htP hxt⟩
rw [← P.sUnion_eq, mem_sUnion] at hx
obtain ⟨t, ht, hxt⟩ := hx
exact ⟨t, ⟨ht, hxt⟩, fun s ⟨hsP, hxs⟩ ↦ P.eq_of_mem_of_mem hsP ht hxs hxt⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Partition.Basic | {
"line": 296,
"column": 2
} | {
"line": 299,
"column": 76
} | [
{
"pp": "α : Type u_1\nx : α\nu : Set α\nP : Partition u\n⊢ x ∈ u ↔ ∃! t, t ∈ P ∧ x ∈ t",
"usedConstants": [
"Partition.subset_of_mem",
"congrArg",
"Partition.eq_of_mem_of_mem",
"Set.sUnion",
"Membership.mem",
"Exists",
"Eq.mp",
"Partition.sUnion_eq",
"A... | refine ⟨fun hx ↦ ?_, fun ⟨_, ⟨htP, hxt⟩, _⟩ ↦ subset_of_mem htP hxt⟩
rw [← P.sUnion_eq, mem_sUnion] at hx
obtain ⟨t, ht, hxt⟩ := hx
exact ⟨t, ⟨ht, hxt⟩, fun s ⟨hsP, hxs⟩ ↦ P.eq_of_mem_of_mem hsP ht hxs hxt⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Sublocale | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 74
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\n⊢ InfClosed ↑S",
"usedConstants": [
"Eq.mpr",
"Set.pair_subset",
"sInf_pair",
"CompleteLattice.toLattice",
"congrArg",
"Sublocale.sInf_mem",
"Membership.mem",
"Sublocale.instSetLike",
"Comple... | rintro a ha b hb; rw [← sInf_pair]; exact S.sInf_mem (pair_subset ha hb) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Sublocale | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 74
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\n⊢ InfClosed ↑S",
"usedConstants": [
"Eq.mpr",
"Set.pair_subset",
"sInf_pair",
"CompleteLattice.toLattice",
"congrArg",
"Sublocale.sInf_mem",
"Membership.mem",
"Sublocale.instSetLike",
"Comple... | rintro a ha b hb; rw [← sInf_pair]; exact S.sInf_mem (pair_subset ha hb) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Predictable | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 50
} | [
{
"pp": "case h\nΩ : Type u_1\nm✝ : MeasurableSpace Ω\n𝓕 : Filtration ℕ m✝\nn : ℕ\ns : Set Ω\nhs : MeasurableSet s\nm : ℕ\n⊢ m ∈ {n + 1} ↔ m ∈ Set.Ioc n (n + 1)",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Preorder.toLT",
"congrArg",
"_private.Mathlib.Probability.Process.Predict... | simp only [Set.mem_singleton_iff, Set.mem_Ioc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Martingale.Basic | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 72
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\ninst✝¹ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ι m0\ninst✝ : SigmaFiniteFiltration μ ℱ\nf : ι → Ω → ℝ\nhadp : StronglyAdapted ℱ f\nhint : ∀ (i : ι), Integrable (f i) μ\ni j : ι\nhij : i ≤ j\ns : Set Ω\nhs : MeasurableSet s\nx✝ : (μ.trim ⋯) s < ∞\n... | integral_sub' integrable_condExp.integrableOn (hint i).integrableOn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Basic | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\ninst✝⁶ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ι m0\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : PartialOrder F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\ninst✝¹ : IsOrderedModule ℝ F\ninst✝ : IsOrderedAddMonoid F\nf : ι → Ω → ... | exact (hf.smul_nonneg <| neg_nonneg.2 hc).neg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Martingale.Basic | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\ninst✝⁶ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ι m0\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : PartialOrder F\ninst✝³ : IsOrderedAddMonoid F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : CompleteSpace F\ninst✝ : IsOrderedModule ℝ F\nf : ι → Ω → ... | exact (hf.smul_nonneg <| neg_nonneg.2 hc).neg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.HittingTime | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 55
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\ns : Set β\nn : ι\nω : Ω\nm : ι\nh_exists : ∃ j ∈ Set.Icc n m, u j ω ∈ s\nh : ¬n ≤ m\n⊢ False",
"usedConstants": [
"not_le",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreor... | rw [Set.Icc_eq_empty_of_lt (not_le.mp h)] at h_exists | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Process.HittingTime | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 24
} | [
{
"pp": "case mp\nΩ : 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\ni' : ι\nhittingAfter_le_iff : ∀ [WellFoundedLT ι], ↑i' ≤ ↑i ↔ ∃ j ∈ Set.Icc n i, u j ω ∈ s\nh' : ↑i' ≤ ↑i\nh_le : ↑n ≤ ↑i'\n⊢ (↑i').untopA ∈... | norm_cast at h' h_le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Probability.Process.HittingTime | {
"line": 481,
"column": 2
} | {
"line": 487,
"column": 56
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : CompleteLattice ι\nu : ι → Ω → β\ns : Set β\nω : Ω\n⊢ hittingBtwn u s ⊥ ⊤ ω = sInf {i | u i ω ∈ s}",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"ChainCompletePartialOrder.instOfCompleteLattice",
"_private.Mathli... | simp only [hittingBtwn, Set.Icc_bot,
Set.Iic_top, Set.univ_inter, ite_eq_left_iff, not_exists]
intro h_notMem_s
symm
rw [sInf_eq_top]
simp only [Set.mem_univ, true_and] at h_notMem_s
exact fun i hi_mem_s => absurd hi_mem_s (h_notMem_s i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.HittingTime | {
"line": 481,
"column": 2
} | {
"line": 487,
"column": 56
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : CompleteLattice ι\nu : ι → Ω → β\ns : Set β\nω : Ω\n⊢ hittingBtwn u s ⊥ ⊤ ω = sInf {i | u i ω ∈ s}",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"ChainCompletePartialOrder.instOfCompleteLattice",
"_private.Mathli... | simp only [hittingBtwn, Set.Icc_bot,
Set.Iic_top, Set.univ_inter, ite_eq_left_iff, not_exists]
intro h_notMem_s
symm
rw [sInf_eq_top]
simp only [Set.mem_univ, true_and] at h_notMem_s
exact fun i hi_mem_s => absurd hi_mem_s (h_notMem_s i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 634,
"column": 2
} | {
"line": 670,
"column": 9
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nhab : a < b\n⊢ upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N n = lowerCrossingTime a b f N n",
"usedConstants": [
"IsRightCancelAdd.addRightStr... | have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine ⟨fun h => ?_, fun h => ?_⟩
· rwa [← sub_le_sub_iff_right a, ←
posPart_eq_of_posPart_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [posPart_eq_self.2 (le... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 634,
"column": 2
} | {
"line": 670,
"column": 9
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nhab : a < b\n⊢ upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N n = upperCrossingTime a b f N n ∧\n lowerCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N n = lowerCrossingTime a b f N n",
"usedConstants": [
"IsRightCancelAdd.addRightStr... | have hab' : 0 < b - a := sub_pos.2 hab
have hf : ∀ ω i, b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω := by
intro i ω
refine ⟨fun h => ?_, fun h => ?_⟩
· rwa [← sub_le_sub_iff_right a, ←
posPart_eq_of_posPart_pos (lt_of_lt_of_le hab' h)]
· rw [← sub_le_sub_iff_right a] at h
rwa [posPart_eq_self.2 (le... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Stopping | {
"line": 489,
"column": 2
} | {
"line": 495,
"column": 87
} | [
{
"pp": "case h.mpr\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nf : Filtration ι m\ni : ι\ns : Set Ω\nh : MeasurableSet s\n⊢ MeasurableSet s ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | ↑i ≤ ↑i_1})",
"usedConstants": [
"Eq.mpr",
"False",
"MeasurableSet",
"eq_fal... | · refine ⟨f.le i _ h, fun j ↦ ?_⟩
by_cases hij : i ≤ j
· norm_cast
simp only [hij, Set.setOf_true, Set.inter_univ]
exact f.mono hij _ h
· norm_cast
simp only [hij, Set.setOf_false, Set.inter_empty, @MeasurableSet.empty _ (f.1 j)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 119,
"column": 96
} | {
"line": 136,
"column": 25
} | [
{
"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\nhf0 : f 0 = 0\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\n⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) → ∃ c, Tendsto (fun n ↦ ... | by
have ht : ∀ᵐ ω ∂μ, ∀ i : ℕ, ∃ c, Tendsto (fun n => stoppedAbove f i n ω) atTop (𝓝 c) := by
rw [ae_all_iff]
exact fun i ↦ Submartingale.exists_ae_tendsto_of_bdd (hf.stoppedAbove i)
(hf.eLpNorm_stoppedAbove_le' i.cast_nonneg hf0 hbdd)
filter_upwards [ht] with ω hω hωb
rw [BddAbove] at hωb
obtain... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 148,
"column": 96
} | {
"line": 168,
"column": 36
} | [
{
"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\n⊢ ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n ↦ f n ω) ↔ ∃ c, Tendsto (fun n ↦ f n ω) atTop (�... | by
set g : ℕ → Ω → ℝ := fun n ω => f n ω - f 0 ω
have hg : Submartingale g ℱ μ :=
hf.sub_martingale (martingale_const_fun _ _ (hf.stronglyAdapted 0) (hf.integrable 0))
have hg0 : g 0 = 0 := by
ext ω
simp only [g, sub_self, Pi.zero_apply]
have hgbdd : ∀ᵐ ω ∂μ, ∀ i : ℕ, |g (i + 1) ω - g i ω| ≤ ↑R := b... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.BorelCantelli | {
"line": 52,
"column": 80
} | {
"line": 55,
"column": 87
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nβ : Type u_3\ninst✝⁵ : LinearOrder ι\nmβ : MeasurableSpace β\ninst✝⁴ : NormedAddCommGroup β\ninst✝³ : BorelSpace β\nf : ι → Ω → β\ni j : ι\ninst✝² : SecondCountableTopology β\ninst✝¹ : CompleteSpace β\ninst✝ : NormedSpace ℝ β\nhf : ∀ (i... | by
have : IsProbabilityMeasure μ := hfi.isProbabilityMeasure
exact condExp_indep_eq (hf j).measurable.comap_le (Filtration.le _ _)
(comap_measurable <| f j).stronglyMeasurable (hfi.indep_comap_natural_of_lt hf hij) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Process.Stopping | {
"line": 896,
"column": 6
} | {
"line": 901,
"column": 29
} | [
{
"pp": "case refine_1.coe\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁷ : Nonempty ι\ninst✝⁶ : LinearOrder ι\nτ : Ω → WithTop 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✝ :... | have h_set_eq : (fun x : s => τ (x : Set.Iic i × Ω).snd) ⁻¹' Set.Iic j =
(fun x : s => (x : Set.Iic i × Ω).snd) ⁻¹' {ω | τ ω ≤ min i j} := by
ext1 ω
simp only [Set.mem_preimage, Set.mem_Iic, coe_min, le_inf_iff,
Set.preimage_setOf_eq, Set.mem_setOf_eq, iff_and_self]
exact fun... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Process.Stopping | {
"line": 1364,
"column": 2
} | {
"line": 1364,
"column": 68
} | [
{
"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... | rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Process.Stopping | {
"line": 1393,
"column": 2
} | {
"line": 1393,
"column": 68
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁸ : LinearOrder ι\nμ : Measure Ω\nℱ : Filtration ι m\nτ : Ω → WithTop ι\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : CompleteSpace E\nf : Ω → E\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : Fi... | rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 498,
"column": 4
} | {
"line": 498,
"column": 84
} | [
{
"pp": "case h_directed\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 → StieltjesFunction ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsCondKernelCDF f κ ν\na : α\ns : Set β\nhs : MeasurableSet s\nh_dir : Directed (fun x y ↦ x ⊆ y) fun q ↦ Ii... | refine Monotone.directed_le fun i j hij t ↦ measure_mono (Iic_subset_Iic.mpr ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 154,
"column": 25
} | {
"line": 154,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\n⊢ Integrable (fun a ↦ rexp ((v - t) * X a)) μ",
"usedConstants": []
}
] | simpa using ht_int_neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 154,
"column": 25
} | {
"line": 154,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\n⊢ Integrable (fun a ↦ rexp ((v - t) * X a)) μ",
"usedConstants": []
}
] | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 154,
"column": 25
} | {
"line": 154,
"column": 47
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\n⊢ Integrable (fun a ↦ rexp ((v - t) * X a)) μ",
"usedConstants": []
}
] | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 38
} | [
{
"pp": "case refine_4\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... | refine ae_of_all _ fun ω ε hε ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSe... | simpa using ht_int_neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSe... | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSe... | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 38
} | [
{
"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... | refine ae_of_all _ fun ω ε hε ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 245,
"column": 2
} | {
"line": 276,
"column": 8
} | [
{
"pp": "case neg\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv : ℝ\nh : v ∈ interior (integrableExpSet X μ)\nhμ : ¬μ = 0\n⊢ (∫ (x : Ω), (fun ω ↦ X ω ^ 2 * rexp (v * X ω)) x ∂μ) / mgf X μ v - deriv (cgf X μ) v ^ 2 =\n (∫ (x : Ω), (fun ω ↦ (X ω - deriv (cgf X μ) v) ^ 2 * rexp (v * X ω)) x ... | calc (∫ ω, (X ω) ^ 2 * exp (v * X ω) ∂μ) / mgf X μ v - deriv (cgf X μ) v ^ 2
_ = (∫ ω, (X ω) ^ 2 * exp (v * X ω) ∂μ - 2 * (∫ ω, X ω * exp (v * X ω) ∂μ) * deriv (cgf X μ) v
+ deriv (cgf X μ) v ^ 2 * mgf X μ v) / mgf X μ v := by
rw [add_div, sub_div, sub_add]
congr 1
rw [mul_div_cancel_right₀, deriv_c... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 270,
"column": 8
} | {
"line": 270,
"column": 39
} | [
{
"pp": "case refine_1\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\n⊢ v + t ≠ v - t",
"usedConstants"... | · rw [← sub_ne_zero]; simp [ht] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 68
} | [
{
"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 :\n ... | rwa [integral_map hX (by fun_prop), integral_map hY (by fun_prop)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 346,
"column": 8
} | {
"line": 346,
"column": 39
} | [
{
"pp": "case refine_1\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt v : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp ((v + t) * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp ((v - t) * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ v + t ≠ v - t",
"usedConstants": [
"NormedCommRing.toNorm... | · rw [← sub_ne_zero]; simp [ht] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using ht_int_neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using ht_int_neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 402,
"column": 4
} | {
"line": 402,
"column": 26
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nht : t ≠ 0\nht_int_pos : Integrable (fun ω ↦ rexp (t * X ω)) μ\nht_int_neg : Integrable (fun ω ↦ rexp (-t * X ω)) μ\np : ℝ\nhp : 0 ≤ p\n⊢ Integrable (fun ω ↦ rexp ((0 - t) * X ω)) μ",
"usedConstants": [
"Eq.m... | simpa using ht_int_neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 435,
"column": 6
} | {
"line": 435,
"column": 46
} | [
{
"pp": "case e_a\nμ : ℝ\nv : ℝ≥0\nz : ℂ\nhv : ¬v = 0\n⊢ (↑π * (2 * ↑↑v)) ^ 2⁻¹ = ↑((2 * π * ↑v) ^ 2⁻¹)",
"usedConstants": [
"zero_le",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real.partialOrder",
"Real",
... | rw [Complex.ofReal_cpow (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Disintegration.StandardBorel | {
"line": 279,
"column": 46
} | {
"line": 279,
"column": 63
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nΩ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nκ : Kernel α (β × Ω)\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) ℝ\ninst✝ : IsSFiniteKernel η\nhη : (κ.map (Prod.map id (embe... | comapRight_apply' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 81
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nf : β × Ω → F\nhY : AEMe... | rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prodMk₀ hY] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 81
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nf : β × Ω → F\nhY : AEMe... | rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prodMk₀ hY] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 81
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\nF : Type u_4\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : NormedAddCommGroup F\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nX : α → β\nY : α → Ω\nmβ : MeasurableSpace β\nf : β × Ω → F\nhY : AEMe... | rw [condDistrib, ← hf.ae_integrable_condKernel_iff, Measure.fst_map_prodMk₀ hY] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝⁵ : MeasurableSpace Ω\ninst✝⁴ : StandardBorelSpace Ω\ninst✝³ : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : IsFiniteMeasure μ\nY : α → Ω\nmβ : MeasurableSpace β\nΩ' : Type u_5\nmΩ' : MeasurableSpace Ω'\ninst✝¹ : StandardBorelSpace Ω'\ninst✝ ... | rw [AEMeasurable.map_map_of_aemeasurable (by fun_prop) (by fun_prop)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Integral | {
"line": 58,
"column": 27
} | {
"line": 58,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nf : β → E\na : α\ninst✝ : CompleteSpace E\ng : α → β\nhg : Measurable g\nhf : StronglyMeasurable f\n⊢ ∫ (x : β), f x ∂Measure.dirac (g a) = f (g a)",
"u... | integral_dirac' _ _ hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Integral | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\n⊢ ∫ (b : β), g b ∂(piecewise hs κ η)... | simp_rw [piecewise_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.Integral | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 27
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nη : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝ : DecidablePred fun x ↦ x ∈ s\na : α\ng : β → E\nt : Set β\n⊢ ∫ (b : β) in t, g b ∂(p... | simp_rw [piecewise_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.ProductMeasure | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 55
} | [
{
"pp": "case h.e_6.h.h\nX : ℕ → Type u_1\nmX : (n : ℕ) → MeasurableSpace (X n)\nμ : (n : ℕ) → Measure (X n)\ninst✝ : ∀ (n : ℕ), SigmaFinite (μ n)\nn : ℕ\ns : (i : ↥(Ioc n (n + 1))) → Set (X ↑i)\nhs : ∀ (i : ↥(Ioc n (n + 1))), MeasurableSet (s i)\nthis : Subsingleton ↥(Ioc n (n + 1))\nx : X (n + 1)\n⊢ (∀ (a : ℕ... | exact ⟨fun h ↦ h (n + 1) rfl, fun h a b ↦ b.symm ▸ h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.IonescuTulcea.PartialTraj | {
"line": 374,
"column": 33
} | {
"line": 374,
"column": 45
} | [
{
"pp": "case h.inr.inr\nX : 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\nf : ((n : ℕ) → X n) → ℝ≥0∞\nhf : Measurable f\nx₀ : (n : ℕ) → X n\nhab : a < b\nhbc : b < c\n⊢ ∫⁻ (z : (i : ↥(Iic b)) → X ↑i),\n ... | frestrictLe, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 384,
"column": 2
} | {
"line": 386,
"column": 62
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nA : ℕ → Set ((n : ℕ) → X n)\nA_mem : ∀ (n : ℕ), A n ∈ measurableCylinders X\nA_anti : Antitone A\nA_inter : ⋂ n, A n = ∅\np : ℕ\nx₀ : (i : ↥(Iic... | have lma_inv k M n (h : a n ≤ M) :
lmarginalPartialTraj κ k M (χ n) = lmarginalPartialTraj κ k (a n) (χ n) :=
(χ_dep n).lmarginalPartialTraj_const_right (mχ n) h le_rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
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