module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.Modular | {
"line": 881,
"column": 59
} | {
"line": 881,
"column": 61
} | {
"line": 882,
"column": 4
} | [
{
"pp": "x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : Filter.Tendsto (fun t ↦ ↑t * ↑x) (𝓝 1) (𝓝 ↑x)\na : ℝ\n⊢ 0 < a → ↑a * ↑x = ↑(↑ofComplex (↑a * ↑x))",
"ppTerm": "?m.375",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"PartialOr... | [
"x : ℍ\nhxnorm : 1 < ‖↑x‖\nhxre : |x.re| ≤ 1 / 2\nthis : Filter.Tendsto (fun t ↦ ↑t * ↑x) (𝓝 1) (𝓝 ↑x)\na : ℝ\nha : 0 < a\n⊢ ↑a * ↑x = ↑(↑ofComplex (↑a * ↑x))"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 114,
"column": 13
} | {
"line": 114,
"column": 37
} | {
"line": 114,
"column": 38
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : 𝒢.IsFiniteRelIndex ℋ\ninst✝¹ : ℋ.HasDetPlusMinusOne\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\n⊢ IsBoundedAtImInfty ((SlashInvariantForm.norm ℋ f).toFun ∣[k * ↑(Nat.... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : 𝒢.IsFiniteRelIndex ℋ\ninst✝¹ : ℋ.HasDetPlusMinusOne\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\n⊢ IsBoundedAtImInfty ((∏ q, quotientFunc f q) ∣[k * ↑(Nat.card (↥ℋ ⧸ 𝒢.subgroupOf... | SlashInvariantForm.norm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 162,
"column": 25
} | {
"line": 162,
"column": 47
} | {
"line": 162,
"column": 48
} | [
{
"pp": "𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝¹ : 𝒢.IsArithmetic\ninst✝ : 𝒢.HasDetOne\nf : ModularForm 𝒢 0\nthis✝ : ModularFormClass\n (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range\n (0 *\n ↑(Nat.card\n (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinear... | [
"𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝¹ : 𝒢.IsArithmetic\ninst✝ : 𝒢.HasDetOne\nf : ModularForm 𝒢 0\nthis✝ : ModularFormClass\n (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range\n (0 *\n ↑(Nat.card\n (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearGroup.mapGL ... | ModularForm.coe_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 53
} | {
"line": 42,
"column": 2
} | [
{
"pp": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\nx y p : R\nh : p ∣ x - y\n⊢ p ∣ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) ↔ p ∣ ↑n * y ^ (n - 1)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"Semiring.toModule",
"AddGroupWithOne.toAddGroup",
"CommSemirin... | [
"R : Type u_1\nn : ℕ\ninst✝ : CommRing R\nx y p : R\nh : (Ideal.Quotient.mk (span {p})) x = (Ideal.Quotient.mk (span {p})) y\n⊢ p ∣ ∑ i ∈ range n, x ^ i * y ^ (n - 1 - i) ↔ p ∣ ↑n * y ^ (n - 1)"
] | rw [← mem_span_singleton, ← Ideal.Quotient.eq] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.LFunction | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 67
} | {
"line": 115,
"column": 6
} | [
{
"pp": "case e'_6.e'_6.hfanalytic\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : ModularFormClass F Γ k\nr : ℝ\nhpos : 0 < s.re\nhs : r + 1 < s.re\nhΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀... | [
"case e'_6.e'_6.hfper\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : ModularFormClass F Γ k\nr : ℝ\nhpos : 0 < s.re\nhs : r + 1 < s.re\nhΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s\nhcoeff : (fu... | · exact ModularFormClass.analyticAt_cuspFunction_zero f hh hΓ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 38
} | {
"line": 285,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nhv : v.IsComplex\n⊢ Function.Surjective ⇑(extensionEmbedding v)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"RingHom",
"NormedField.toField",
"Field.toDivisionRing",
"... | [
"K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nhv : v.IsComplex\n⊢ (extensionEmbedding v).fieldRange = ⊤"
] | rw [← RingHom.fieldRange_eq_top_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ModularForms.LFunction | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 10
} | {
"line": 192,
"column": 2
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ninst✝ : CuspFormClass F Γ k\n⊢ Differentiable ℂ (L hk f)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Differentiable",
"NormedCommRing.toSeminormed... | [
"Γ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ninst✝ : CuspFormClass F Γ k\n⊢ Differentiable ℂ fun s ↦ Λ hk f s * (2 / s.Gammaℂ)"
] | unfold L | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 24
} | {
"line": 121,
"column": 25
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nthis :\n (Subgroup.map (QuotientGroup.mk' (torsion K)) (Subgroup.closure (Set.range u))).index =\n (Subgroup.closure (Set.range u) ⊔ torsion K).index\n⊢ (AddSubgroup.map (↑(logEmbeddingEquiv K).toAddEquiv)\n ... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nthis :\n (Subgroup.map (QuotientGroup.mk' (torsion K)) (Subgroup.closure (Set.range u))).index =\n (Subgroup.closure (Set.range u) ⊔ torsion K).index\n⊢ (AddSubgroup.map (↑(logEmbeddingEquiv K).toAddEquiv)\n (Subgroup.t... | map_toAddSubgroup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 132,
"column": 47
} | {
"line": 132,
"column": 63
} | {
"line": 132,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nh₁ :\n (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔\n Finite\n (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))\nh₂ : DiscreteTopology ↥(... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nh₁ :\n (Subgroup.closure (Set.range u) ⊔ torsion K).index ≠ 0 ↔\n Finite\n (↥(unitLattice K) ⧸ span ℤ (Set.range (⇑(logEmbeddingEquiv K) ∘ ⇑Additive.toMul.symm ∘ QuotientGroup.mk ∘ u)))\nh₂ : DiscreteTopology ↥(span ℤ (Set.... | finiteIndex_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 14
} | {
"line": 382,
"column": 4
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\nx : realSpace K\nx✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.I... | [
"case refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\nx : realSpace K\nx✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w.IsReal then S... | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 45
} | {
"line": 226,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nc : ℝ\nhc : c ≠ 0\nh : c • x ∈ fundamentalCone K\n⊢ x ∈ fundamentalCone K",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivI... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nc : ℝ\nhc : c ≠ 0\nh : c • x ∈ fundamentalCone K\n⊢ x = c⁻¹ • c • x"
] | convert! smul_mem_of_mem h (inv_ne_zero hc) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 91,
"column": 28
} | {
"line": 91,
"column": 31
} | {
"line": 91,
"column": 32
} | [
{
"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\nhv : v = w.comap (algebraMap K L)\nh : IsUnramified K w\n⊢ v.mult * 1 = w.mult",
"ppTerm": "?inl",
"assigned": true,
"usedConstants... | [
"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\nhv : v = w.comap (algebraMap K L)\nh : IsUnramified K w\n⊢ (w.comap (algebraMap K L)).mult * 1 = w.mult"
] | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 92,
"column": 28
} | {
"line": 92,
"column": 31
} | {
"line": 92,
"column": 32
} | [
{
"pp": "case inr\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\nhv : v = w.comap (algebraMap K L)\nh : IsRamified K w\n⊢ v.mult * 2 = w.mult",
"ppTerm": "?inr",
"assigned": true,
"usedConstants":... | [
"case inr\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\nhv : v = w.comap (algebraMap K L)\nh : IsRamified K w\n⊢ (w.comap (algebraMap K L)).mult * 2 = w.mult"
] | hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 14
} | {
"line": 387,
"column": 6
} | [
{
"pp": "case mem\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx x✝ : realSpace K\nh : x✝ ∈ Set.range fun w ↦ completeFamily K ↑w\n⊢ ∑ w, x✝ w = 0",
"ppTerm": "?mem",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Units.val",
"Eq.mpr",
"Real",
"Number... | [] | | mem _ h => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.NumberField.Ideal.KummerDedekind | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 64
} | {
"line": 221,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\nthis : (span {↑p, (aeval θ) Q}).IsMaximal\n⊢ (↑((primesOverSpanEquivMonicFactorsMod hp).symm ⟨Poly... | [
"K : Type u_1\ninst✝² : Field K\nθ : 𝓞 K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : NumberField K\nhp : ¬p ∣ exponent θ\nQ : ℤ[X]\nhQ : Polynomial.map (Int.castRingHom (ZMod p)) Q ∈ monicFactorsMod θ p\nthis✝ : (span {↑p, (aeval θ) Q}).IsMaximal\nthis : (span {↑p, (aeval θ) Q}).LiesOver (span {↑p})\n⊢ (↑((primes... | have := liesOver_primesOverSpanEquivMonicFactorsMod_symm hp hQ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 73,
"column": 46
} | {
"line": 73,
"column": 99
} | {
"line": 75,
"column": 0
} | [
{
"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.54",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NonAssoc... | [] | by rw [← Subalgebra.coe_pow, hx, OneMemClass.coe_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 94,
"column": 38
} | {
"line": 94,
"column": 55
} | {
"line": 94,
"column": 56
} | [
{
"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)... | [
"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)\nhζ : IsPri... | ZMod.natCast_val, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 94,
"column": 56
} | {
"line": 94,
"column": 73
} | {
"line": 94,
"column": 74
} | [
{
"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)... | [
"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)\nhζ : IsPri... | ZMod.natCast_val, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 75
} | {
"line": 422,
"column": 0
} | [
{
"pp": "n m p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhn : n = p ^ (k + 1) * m\nhm : ¬p ∣ m\nthis : IsGalois ℚ K\n⊢ P.inertiaDeg ℤ = orderOf ↑p",
"ppTerm": "?m.57",... | [] | rw [← inertiaDegIn_eq_inertiaDeg 𝒑 P Gal(K/ℚ), inertiaDegIn_eq n K hn hm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.House | {
"line": 81,
"column": 7
} | {
"line": 81,
"column": 38
} | {
"line": 81,
"column": 38
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : 𝓞 K\nhα0 : α ≠ 0\nw : InfinitePlace K\nhw : 1 ≤ w ↑α\n⊢ 1 ≤ ‖w.embedding ↑α‖",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.InfinitePlace.instFunLikeReal",
"Eq.mpr",
... | [
"case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : 𝓞 K\nhα0 : α ≠ 0\nw : InfinitePlace K\nhw : 1 ≤ w ↑α\n⊢ 1 ≤ w ↑α"
] | InfinitePlace.norm_embedding_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 33
} | {
"line": 205,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\nha : a ≠ 0\nk' : α\nhs : ∀ (b : K →+* ℂ), (fun l ↦ ((newBasis K).repr (a k' l.1 * (newBasis K) l.2)) b) = 0 (k', b)\nl : β\nb : K →+* ℂ\nthis : a k' l * (newBasis K) b = 0\n⊢ a k' l = 0 k' l",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : Type u_2\nβ : Type u_3\na : Matrix α β (𝓞 K)\nha : a ≠ 0\nk' : α\nhs : ∀ (b : K →+* ℂ), (fun l ↦ ((newBasis K).repr (a k' l.1 * (newBasis K) l.2)) b) = 0 (k', b)\nl : β\nb : K →+* ℂ\nthis : a k' l = 0 ∨ (newBasis K) b = 0\n⊢ a k' l = 0 k' l"
] | simp only [mul_eq_zero] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Padics.AddChar | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 24
} | {
"line": 63,
"column": 25
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝⁴ : NormedRing R\ninst✝³ : Algebra ℤ_[p] R\ninst✝² : IsBoundedSMul ℤ_[p] R\ninst✝¹ : IsUltrametricDist R\ninst✝ : CompleteSpace R\nr : R\nhr : Tendsto (fun x ↦ r ^ x) atTop (𝓝 0)\n⊢ (mahlerSeries fun x ↦ r ^ x) 0 = 1",
"ppTerm": "?m.62",
"... | [
"p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝⁴ : NormedRing R\ninst✝³ : Algebra ℤ_[p] R\ninst✝² : IsBoundedSMul ℤ_[p] R\ninst✝¹ : IsUltrametricDist R\ninst✝ : CompleteSpace R\nr : R\nhr : Tendsto (fun x ↦ r ^ x) atTop (𝓝 0)\n⊢ (mahlerSeries fun x ↦ r ^ x) ↑0 = 1"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 84,
"column": 76
} | {
"line": 84,
"column": 92
} | {
"line": 85,
"column": 4
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nk : ℕ\n⊢ ↑(Polynomial.eval x (ascPochhammer ℤ_[p] k)) = ↑((ascPochhammer ℕ k).smeval x)",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Polynomial.eval",
"Real.instLE",
"Real",
"... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nk : ℕ\n⊢ Polynomial.eval x (ascPochhammer ℤ_[p] k) = (ascPochhammer ℕ k).smeval x"
] | Subtype.coe_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 83
} | {
"line": 65,
"column": 4
} | [
{
"pp": "case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp0' : 0 < ↑p\nhp0 : 0 < (↑p)⁻¹\nhp1' : 1 < ↑p\nhp1 : (↑p)⁻¹ < 1\nn : ℕ\nhn : Valued.v (↑p ^ n) = exp (-↑n)\nx y : WithVal (Rat.padicValuation p)\nx' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x\nhx : x' = (WithVal.equiv (Rat.padicValuation p)) x\ny' : ℚ :=... | [
"case h\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp0' : 0 < ↑p\nhp0 : 0 < (↑p)⁻¹\nhp1' : 1 < ↑p\nhp1 : (↑p)⁻¹ < 1\nn : ℕ\nhn : Valued.v (↑p ^ n) = exp (-↑n)\nx y : WithVal (Rat.padicValuation p)\nx' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x\nhx : x' = (WithVal.equiv (Rat.padicValuation p)) x\ny' : ℚ := (WithVal.eq... | rw [← Nat.cast_pow, ← Rat.cast_natCast, ← Rat.cast_inv_of_ne_zero, Rat.cast_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 239,
"column": 32
} | {
"line": 239,
"column": 94
} | {
"line": 239,
"column": 94
} | [
{
"pp": "p : 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)\nε : ℝ\nhε : ε > 0\nthis✝ : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nhf : f ≠ 0\nthis : 0 ... | [] | simpa only [dist_eq_norm_sub] using (hδf (hxy.trans_lt ht)).le | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 239,
"column": 32
} | {
"line": 239,
"column": 94
} | {
"line": 239,
"column": 94
} | [
{
"pp": "p : 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)\nε : ℝ\nhε : ε > 0\nthis✝ : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nhf : f ≠ 0\nthis : 0 ... | [] | simpa only [dist_eq_norm_sub] using (hδf (hxy.trans_lt ht)).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 239,
"column": 32
} | {
"line": 239,
"column": 94
} | {
"line": 239,
"column": 94
} | [
{
"pp": "p : 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)\nε : ℝ\nhε : ε > 0\nthis✝ : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nhf : f ≠ 0\nthis : 0 ... | [] | simpa only [dist_eq_norm_sub] using (hδf (hxy.trans_lt ht)).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.Measure.Basic | {
"line": 195,
"column": 30
} | {
"line": 195,
"column": 55
} | {
"line": 195,
"column": 56
} | [
{
"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,... | ContinuousMap.smul_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Padics.Measure.Basic | {
"line": 195,
"column": 2
} | {
"line": 196,
"column": 30
} | {
"line": 197,
"column": 2
} | [
{
"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,... | simp_rw [contractFst_apply, ContinuousMap.smul_apply, smul_eq_mul, mul_comm (μ f) (g y),
← smul_eq_mul, ← map_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.Padics.Measure.Basic | {
"line": 198,
"column": 11
} | {
"line": 198,
"column": 36
} | {
"line": 198,
"column": 37
} | [
{
"pp": "case e_6.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✝ : LocallyCompactS... | [
"case e_6.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 : ... | ContinuousMap.smul_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Padics.Measure.Basic | {
"line": 216,
"column": 11
} | {
"line": 216,
"column": 36
} | {
"line": 216,
"column": 37
} | [
{
"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,... | ContinuousMap.smul_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Pell | {
"line": 543,
"column": 84
} | {
"line": 543,
"column": 94
} | {
"line": 543,
"column": 95
} | [
{
"pp": "d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ (1 + d * a.y ^ 2) * a₁.y ^ 2 ≤ a.y ^ 2 * a₁.x ^ 2",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"Int.instCommMonoid",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneC... | [
"d : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ (1 + d * a.y ^ 2) * a₁.y ^ 2 ≤ a.y ^ 2 * (1 + d * a₁.y ^ 2)"
] | a₁.prop_x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 28
} | {
"line": 141,
"column": 2
} | [
{
"pp": "case calc_3\nR : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\nins... | [] | · rw [A_smul, smul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Pell | {
"line": 577,
"column": 23
} | {
"line": 577,
"column": 33
} | {
"line": 577,
"column": 34
} | [
{
"pp": "case calc.step\nd : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ a.y * a₁.x ^ 2 < a₁.y * (d * (a.y * a₁.y) + a.x)",
"ppTerm": "?«calc».step",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Eq.mpr",
"Semigroup.to... | [
"case calc.step\nd : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\na : Solution₁ d\nhax : 1 < a.x\nhay : 0 < a.y\n⊢ a.y * (1 + d * a₁.y ^ 2) < a₁.y * (d * (a.y * a₁.y) + a.x)"
] | a₁.prop_x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 288,
"column": 2
} | {
"line": 289,
"column": 33
} | {
"line": 291,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nn : ℕ\n⊢ ‖newton_seq (n + 1) - newton_seq n‖ =\n ‖(Polynomial.aeval (newton_seq n)... | [] | rw [newton_seq_gen, newton_seq_gen, newton_seq_aux, ih_n]
simp [sub_eq_add_neg, add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 288,
"column": 2
} | {
"line": 289,
"column": 33
} | {
"line": 291,
"column": 0
} | [
{
"pp": "p : ℕ\ninst✝² : Fact (Nat.Prime p)\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R ℤ_[p]\nF : Polynomial R\na : ℤ_[p]\nhnorm : ‖(Polynomial.aeval a) F‖ < ‖(Polynomial.aeval a) (Polynomial.derivative F)‖ ^ 2\nn : ℕ\n⊢ ‖newton_seq (n + 1) - newton_seq n‖ =\n ‖(Polynomial.aeval (newton_seq n)... | [] | rw [newton_seq_gen, newton_seq_gen, newton_seq_aux, ih_n]
simp [sub_eq_add_neg, add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.RamificationInertia.HilbertTheory | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 60
} | {
"line": 298,
"column": 60
} | [
{
"pp": "case refine_4\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝³⁹ : Field K\ninst✝³⁸ : Field L\ninst✝³⁷ : Algebra K L\ninst✝³⁶ : CommRing A\ninst✝³⁵ : CommRing B\ninst✝³⁴ : Algebra A B\np : Ideal A\nP : Ideal B\ninst✝³³ : P.LiesOver p\ninst✝³² : Algebra A K\ninst✝³¹ : IsFractionRing A K\ni... | [
"case refine_4\nA : Type u_1\nK : Type u_2\nL : Type u_3\nB : Type u_4\ninst✝³⁹ : Field K\ninst✝³⁸ : Field L\ninst✝³⁷ : Algebra K L\ninst✝³⁶ : CommRing A\ninst✝³⁵ : CommRing B\ninst✝³⁴ : Algebra A B\np : Ideal A\nP : Ideal B\ninst✝³³ : P.LiesOver p\ninst✝³² : Algebra A K\ninst✝³¹ : IsFractionRing A K\ninst✝³⁰ : Alg... | inertiaDegIn_eq_inertiaDeg _ P (stabilizer Gal(L/K) P) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Rayleigh | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 22
} | {
"line": 148,
"column": 2
} | [
{
"pp": "case a\nr s : ℝ\nhrs : r.HolderConjugate s\n⊢ {n | 0 < n} ⊆ {x | ∃ k > 0, beattySeq r k = x} ∆ {x | ∃ k > 0, beattySeq' s k = x}",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Set.ofPred",
"Membership.mem",
"Int",
"Int.instLTInt",
"instOfNat",
"... | [
"case a\nr s : ℝ\nhrs : r.HolderConjugate s\nj : ℤ\nhj : 0 < j\n⊢ j ∈ {x | ∃ k > 0, beattySeq r k = x} ∆ {x | ∃ k > 0, beattySeq' s k = x}"
] | intro j (hj : 0 < j) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.SelbergSieve | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 13
} | {
"line": 291,
"column": 13
} | [
{
"pp": "s : BoundingSieve\nl : ℕ\nhl : l ∣ s.prodPrimes\np : ℕ\nhp : p ∈ l.primeFactors\n⊢ 0 < (1 - s.nu p)⁻¹",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Grou... | [
"s : BoundingSieve\nl : ℕ\nhl : l ∣ s.prodPrimes\np : ℕ\nhp : p ∈ l.primeFactors\n⊢ 0 < 1 - s.nu p"
] | inv_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 63
} | {
"line": 78,
"column": 2
} | [
{
"pp": "case hbc.convert_10\nf : ℕ → ℂ[X]\ns : ℂ\nc : ℝ\nhc : ∀ (p : ℕ), ∀ x ∈ Set.Ioc 0 1, ‖eval (x • s) (f p)‖ ≤ c ^ p\np : ℕ\n⊢ MeasureTheory.volume (Set.Ioc 0 1) < ⊤",
"ppTerm": "?hbc.convert_10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"Measure... | [] | rw [Real.volume_Ioc, sub_zero]; exact ENNReal.ofReal_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 63
} | {
"line": 78,
"column": 2
} | [
{
"pp": "case hbc.convert_10\nf : ℕ → ℂ[X]\ns : ℂ\nc : ℝ\nhc : ∀ (p : ℕ), ∀ x ∈ Set.Ioc 0 1, ‖eval (x • s) (f p)‖ ≤ c ^ p\np : ℕ\n⊢ MeasureTheory.volume (Set.Ioc 0 1) < ⊤",
"ppTerm": "?hbc.convert_10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"Measure... | [] | rw [Real.volume_Ioc, sub_zero]; exact ENNReal.ofReal_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SumFourSquares | {
"line": 190,
"column": 6
} | {
"line": 192,
"column": 35
} | {
"line": 193,
"column": 6
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Prime p\nthis✝¹ : 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... | [
"case refine_2\np : ℕ\nhp : Prime p\nthis✝¹ : 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 ... | suffices ((a : ZMod m) ^ 2 + (b : ZMod m) ^ 2 + (c : ZMod m) ^ 2 + (d : ZMod m) ^ 2) = 0 by
simpa [← ZMod.intCast_zmod_eq_zero_iff_dvd, hf_mod, sq, add_comm, add_assoc,
add_left_comm] using this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.Transcendental.Liouville.Residual | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 80
} | {
"line": 56,
"column": 4
} | [
{
"pp": "case refine_2\n⊢ {x | Irrational x} ∩ ⋂ n, ⋃ a, ⋃ b, ⋃ (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n) ⊆ {x | Liouville x}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Set.ofPred",
"Se... | [
"case refine_2\n⊢ ⋂ i, ⋃ i_1, ⋃ i_2, ⋃ (_ : 1 < i_2), {x | Irrational x} ∩ ball (↑i_1 / ↑i_2) (1 / ↑i_2 ^ i) ⊆\n ⋂ n, ⋃ a, ⋃ b, ⋃ (_ : 1 < b), ball (↑a / ↑b) (1 / ↑b ^ n) \\ {↑a / ↑b}"
] | simp only [inter_iInter, inter_iUnion, setOfPred_liouville_eq_iInter_iUnion] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 51,
"column": 23
} | {
"line": 51,
"column": 26
} | {
"line": 51,
"column": 27
} | [
{
"pp": "p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\n⊢ LiouvilleWith p x →\n x ∈ Ico 0 1 → ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))",
"ppTerm": "?m.251",
"assigned": true,
"usedConstants": [
"LiouvilleWith"
],
"usedFVars... | [
"p : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\n⊢ x ∈ Ico 0 1 → ∃ᶠ (b : ℕ) in atTop, ∃ a ∈ Finset.Icc 0 ↑b, |x - ↑a / ↑b| < 1 / ↑b ^ (2 + 1 / ↑(n + 1))"
] | hxp | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Wilson | {
"line": 55,
"column": 12
} | {
"line": 55,
"column": 14
} | {
"line": 56,
"column": 4
} | [
{
"pp": "case refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : 0 < p\na : (ZMod p)ˣ\n⊢ a ∈ univ → (↑a).val ∈ Ico 1 (p - 1).succ",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Finset.univ",
... | [
"case refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp : 0 < p\na : (ZMod p)ˣ\nha : a ∈ univ\n⊢ (↑a).val ∈ Ico 1 (p - 1).succ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 72,
"column": 4
} | {
"line": 73,
"column": 48
} | {
"line": 75,
"column": 0
} | [
{
"pp": "case refine_2\np : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\na : ℤ\nhb : 1 ≤ ↑b\nhb0 : 0 < ↑b\nhlt : x * ↑b - 1 < ↑a ∧ ↑a < 1 + x * ↑b\n⊢ 1 + x * ↑b ≤ 1 + ↑b",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | rw [add_le_add_iff_left]
exact mul_le_of_le_one_left hb0.le hx01.2.le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Transcendental.Liouville.Measure | {
"line": 72,
"column": 4
} | {
"line": 73,
"column": 48
} | {
"line": 75,
"column": 0
} | [
{
"pp": "case refine_2\np : ℝ\nhp : p > 2\nn : ℕ\nhn : 2 + 1 / (↑n + 1) < p\nx : ℝ\nhxp : LiouvilleWith p x\nhx01 : x ∈ Ico 0 1\nb : ℕ\na : ℤ\nhb : 1 ≤ ↑b\nhb0 : 0 < ↑b\nhlt : x * ↑b - 1 < ↑a ∧ ↑a < 1 + x * ↑b\n⊢ 1 + x * ↑b ≤ 1 + ↑b",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | rw [add_le_add_iff_left]
exact mul_le_of_le_one_left hb0.le hx01.2.le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 22
} | {
"line": 274,
"column": 2
} | [
{
"pp": "p : ℝ\nhp : 1 < p\nM : ℤ\nh : LiouvilleWith p ↑M\nn : ℕ\nhn : 0 < n\nm : ℤ\nhne : ↑M ≠ ↑m / ↑n\nhlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)\n⊢ False",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real.instPow",
"Real",
"instHDiv",
"Real.lattice",
... | [
"p : ℝ\nhp : 1 < p\nM : ℤ\nh : LiouvilleWith p ↑M\nn : ℕ\nhn : 0 < n\nm : ℤ\nhne : ↑M ≠ ↑m / ↑n\nhlt : |↑M - ↑m / ↑n| < ↑n ^ (-1)\n⊢ ↑n ^ (-1) ≤ |↑M - ↑m / ↑n|"
] | refine hlt.not_ge ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.WellApproximable | {
"line": 118,
"column": 23
} | {
"line": 118,
"column": 45
} | {
"line": 118,
"column": 46
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedCommGroup A\nm : ℕ\nδ : ℝ\nn : ℕ\nhm : 0 < m\na : A\nha : a ∈ approxOrderOf A (n * m) δ\nb : A\nhb : orderOf b = n * m\nhab : a ∈ ball b δ\n⊢ orderOf (b ^ m) = n",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Eq.mpr",
... | [
"A : Type u_1\ninst✝ : SeminormedCommGroup A\nm : ℕ\nδ : ℝ\nn : ℕ\nhm : 0 < m\na : A\nha : a ∈ approxOrderOf A (n * m) δ\nb : A\nhb : orderOf b = n * m\nhab : a ∈ ball b δ\n⊢ orderOf b / (orderOf b).gcd m = n"
] | orderOf_pow' b hm.ne', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Comparable | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 32
} | {
"line": 271,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Total r\na b : α\n⊢ ¬IncompRel r a b",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"IncompRel",
"propext",
"Eq",
"Relation.SymmGen",
"Not",
"not_incompRel... | [
"α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Total r\na b : α\n⊢ SymmGen r a b"
] | rw [not_incompRel_iff_symmGen] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 24
} | {
"line": 58,
"column": 4
} | [
{
"pp": "case refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nhe : e ∈ s\nhs : e < sInf s\n⊢ False",
"ppTerm": "?refine_1",
"assigned":... | [
"case refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nhe : e ∈ s\na : ι\nha : (∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j a → e j = sInf s j) ∧ (fun {i} x1... | obtain ⟨a, ha⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 24
} | {
"line": 61,
"column": 4
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nh : e ∈ lowerBounds s\nhs : sInf s < e\n⊢ False",
"ppTerm": "?refine_2",
... | [
"case refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nh : e ∈ lowerBounds s\na : ι\nha : (∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j a → sInf s j = e j) ∧ ... | obtain ⟨a, ha⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 24
} | {
"line": 86,
"column": 4
} | [
{
"pp": "case refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nhe : e ∈ s\nhs : sSup s < e\n⊢ False",
"ppTerm": "?refine_1",
"assigned":... | [
"case refine_1\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nhe : e ∈ s\na : ι\nha : (∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j a → sSup s j = e j) ∧ (fun {i} x1... | obtain ⟨a, ha⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 24
} | {
"line": 89,
"column": 4
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nh : e ∈ upperBounds s\nhs : e < sSup s\n⊢ False",
"ppTerm": "?refine_2",
... | [
"case refine_2\nι : Type u_1\nα : ι → Type u_2\ninst✝² : LinearOrder ι\ninst✝¹ : (i : ι) → CompleteLinearOrder (α i)\ninst✝ : WellFoundedLT ι\ns : Set (Lex ((i : ι) → (fun i ↦ α i) i))\ne : Lex ((i : ι) → (fun i ↦ α i) i)\nh : e ∈ upperBounds s\na : ι\nha : (∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j a → e j = sSup s j) ∧ ... | obtain ⟨a, ha⟩ := hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Concept | {
"line": 475,
"column": 15
} | {
"line": 475,
"column": 17
} | {
"line": 476,
"column": 2
} | [
{
"pp": "α : Type u_6\ninst✝ : PartialOrder α\nc : Concept α α fun x1 x2 ↦ x1 < x2\na b : α\nhb : b ≤ a\n⊢ a ∈ c.extent → b ∈ c.extent",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Membership.mem",
"Concept.extent",
... | [
"α : Type u_6\ninst✝ : PartialOrder α\nc : Concept α α fun x1 x2 ↦ x1 < x2\na b : α\nhb : b ≤ a\nha : a ∈ c.extent\n⊢ b ∈ c.extent"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Concept | {
"line": 483,
"column": 15
} | {
"line": 483,
"column": 17
} | {
"line": 484,
"column": 2
} | [
{
"pp": "α : Type u_6\ninst✝ : PartialOrder α\nc : Concept α α fun x1 x2 ↦ x1 < x2\na b : α\nhb : a ≤ b\n⊢ a ∈ c.intent → b ∈ c.intent",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Membership.mem",
"Concept.intent",
... | [
"α : Type u_6\ninst✝ : PartialOrder α\nc : Concept α α fun x1 x2 ↦ x1 < x2\na b : α\nhb : a ≤ b\nha : a ∈ c.intent\n⊢ b ∈ c.intent"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.CountableSupClosed | {
"line": 252,
"column": 12
} | {
"line": 252,
"column": 14
} | {
"line": 252,
"column": 15
} | [
{
"pp": "α : Type u_2\ninst✝ : Preorder α\ns : Set α\na : α\n⊢ a ∈ upperBounds s → a ∈ upperBounds (countableSupClosure s)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Preorder.toLE",
"Membership.mem",
"upperBounds",
"Set.instMembership",
"Set"
],
... | [
"α : Type u_2\ninst✝ : Preorder α\ns : Set α\na : α\nha : a ∈ upperBounds s\n⊢ a ∈ upperBounds (countableSupClosure s)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Fin.InsertNth | {
"line": 33,
"column": 2
} | {
"line": 37,
"column": 20
} | {
"line": 39,
"column": 0
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : StrictMono f\nx : α\nhx : x < f 0\n⊢ StrictMono (insertNth 0 x f)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Preorder.toLT",
"Fin.cons_succ",
"St... | [] | rw [Fin.strictMono_iff_lt_succ] at hf ⊢
intro i
obtain rfl | ⟨i, rfl⟩ := i.eq_zero_or_eq_succ
· simpa
· simpa using hf i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Fin.InsertNth | {
"line": 33,
"column": 2
} | {
"line": 37,
"column": 20
} | {
"line": 39,
"column": 0
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : Preorder α\nf : Fin (n + 1) → α\nhf : StrictMono f\nx : α\nhx : x < f 0\n⊢ StrictMono (insertNth 0 x f)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Preorder.toLT",
"Fin.cons_succ",
"St... | [] | rw [Fin.strictMono_iff_lt_succ] at hf ⊢
intro i
obtain rfl | ⟨i, rfl⟩ := i.eq_zero_or_eq_succ
· simpa
· simpa using hf i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Height | {
"line": 107,
"column": 4
} | {
"line": 109,
"column": 21
} | {
"line": 110,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ns : Set α\nr : α → α → Prop\nh : s.chainHeight r = 0\n⊢ s = ∅",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Set.chainHeight",
"Set.ext",
"Eq.mpr",
"False",
"Set.encard",
"instCompleteLinearOrderENat",
"i... | [] | (Tactic.tacticSeq1Indented
[(Tactic.simp
"simp"
(Tactic.optConfig [])
[]
["only"]
["["
[(Tactic.simpLemma [] [] `chainHeight)
","
(Tactic.simpLemma [] [] `iSup_eq_zero)
","
(Tactic.simpLemma [] [] `encard_eq_zero)
","
(Tactic.simpLemma [] [] `Subtype.forall)
","
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Nucleus | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 85
} | {
"line": 173,
"column": 85
} | [
{
"pp": "X : Type u_1\ninst✝ : CompleteLattice X\nx✝ : Set (Nucleus X)\n⊢ sInf x✝ ∈ upperBounds (lowerBounds x✝)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"iInf",
"Nucleus.instInfSet",
"CompleteLattice.toLattice",
"lowerBounds",
"_private.Mathlib.Order.N... | [] | by simp +contextual [mem_lowerBounds, mem_upperBounds, ← coe_le_coe, Pi.le_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.PrimeIdeal | {
"line": 115,
"column": 17
} | {
"line": 122,
"column": 32
} | {
"line": 124,
"column": 0
} | [
{
"pp": "P : Type u_1\ninst✝¹ : SemilatticeInf P\nI : Ideal P\ninst✝ : I.IsProper\nhI : ∀ {x y : P}, x ⊓ y ∈ I → x ∈ I ∨ y ∈ I\n⊢ I.IsPrime",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Order.IsPFilter.of_def",
"Eq.mpr",
"False",
"Order.Ideal.isP... | [] | by
rw [isPrime_iff]
use ‹_›
refine .of_def ?_ ?_ ?_
· exact Set.nonempty_compl.2 (I.isProper_iff.1 ‹_›)
· intro x hx y hy
exact ⟨x ⊓ y, fun h => (hI h).elim hx hy, inf_le_left, inf_le_right⟩
· exact @mem_compl_of_ge _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Partition.Basic | {
"line": 247,
"column": 14
} | {
"line": 247,
"column": 16
} | {
"line": 247,
"column": 17
} | [
{
"pp": "α✝ : Type u_1\ns✝ t x y z : α✝\nS : Set α✝\ninst✝¹ : CompleteLattice α✝\nP✝ Q✝ : Partition s✝\nα : Type u_2\ninst✝ : Order.Frame α\ns : α\nP Q : Partition s\na : α\n⊢ a ∈ {a | ∃ p ∈ P, ∃ q ∈ Q, a = p ⊓ q} →\n ∀ ⦃y : α⦄, y ∈ {a | ∃ p ∈ P, ∃ q ∈ Q, a = p ⊓ q} → a ≠ y → Function.onFun Disjoint id a y",... | [
"α✝ : Type u_1\ns✝ t x y z : α✝\nS : Set α✝\ninst✝¹ : CompleteLattice α✝\nP✝ Q✝ : Partition s✝\nα : Type u_2\ninst✝ : Order.Frame α\ns : α\nP Q : Partition s\na : α\nha : a ∈ {a | ∃ p ∈ P, ∃ q ∈ Q, a = p ⊓ q}\n⊢ ∀ ⦃y : α⦄, y ∈ {a | ∃ p ∈ P, ∃ q ∈ Q, a = p ⊓ q} → a ≠ y → Function.onFun Disjoint id a y"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Sublocale | {
"line": 79,
"column": 24
} | {
"line": 79,
"column": 35
} | {
"line": 79,
"column": 35
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\na : X\nha : a ∈ ↑S\nb : X\nhb : b ∈ ↑S\n⊢ a ⊓ b ∈ ↑S",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"sInf_pair",
"CompleteLattice.toLattice",
"congrArg",
"Membership.mem",
"Subloca... | [
"X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\na : X\nha : a ∈ ↑S\nb : X\nhb : b ∈ ↑S\n⊢ sInf {a, b} ∈ ↑S"
] | ← sInf_pair | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.Tree | {
"line": 211,
"column": 10
} | {
"line": 211,
"column": 12
} | {
"line": 211,
"column": 13
} | [
{
"pp": "t : RootedTree\nt₁ : SubRootedTree t\nht₁ : t₁ ∈ t.subtrees\nt₂ : SubRootedTree t\nht₂ : t₂ ∈ t.subtrees\nh : t₁ ≠ t₂\na : ↑t\n⊢ a ∈ ↑t₁ → a ∉ ↑t₂",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"RootedTree.α",
"SubRootedTree",
"Membership.mem",
"instSetLik... | [
"t : RootedTree\nt₁ : SubRootedTree t\nht₁ : t₁ ∈ t.subtrees\nt₂ : SubRootedTree t\nht₂ : t₂ ∈ t.subtrees\nh : t₁ ≠ t₂\na : ↑t\nha : a ∈ ↑t₁\n⊢ a ∉ ↑t₂"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Types.Arithmetic | {
"line": 70,
"column": 29
} | {
"line": 72,
"column": 68
} | {
"line": 72,
"column": 68
} | [
{
"pp": "o : OrderType.{u}\nα : Type u\nx✝ : LinearOrder α\n⊢ type α + 0 = type α",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Lex",
"PartialOrder.toPreorder",
"OrderType.type",
"Preorder.toLE",
"OrderIso.sumLexEmpty",
... | [] | by
simp only [show 0 = type PEmpty by rfl, ← type_lex_sum]
exact (OrderIso.sumLexEmpty (β := PEmpty) (α := α)).type_congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.Centering | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 46
} | {
"line": 60,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn : ℕ\n⊢ predictablePart f ℱ μ (n + 1) = predictablePart f ℱ μ n + μ[f (n + 1) - f n | ↑ℱ n]",
"ppTerm": "?m.43",
"assigned": true,
... | [] | simp [predictablePart, Finset.sum_range_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Martingale.Centering | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 46
} | {
"line": 60,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn : ℕ\n⊢ predictablePart f ℱ μ (n + 1) = predictablePart f ℱ μ n + μ[f (n + 1) - f n | ↑ℱ n]",
"ppTerm": "?m.43",
"assigned": true,
... | [] | simp [predictablePart, Finset.sum_range_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Centering | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 46
} | {
"line": 60,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\nn : ℕ\n⊢ predictablePart f ℱ μ (n + 1) = predictablePart f ℱ μ n + μ[f (n + 1) - f n | ↑ℱ n]",
"ppTerm": "?m.43",
"assigned": true,
... | [] | simp [predictablePart, Finset.sum_range_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Basic | {
"line": 577,
"column": 4
} | {
"line": 579,
"column": 57
} | {
"line": 580,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\n𝒢 : Filtration ℕ m0\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsFinit... | [] | exact (hξ.stronglyMeasurable_le hi.le).smul
((hf.stronglyAdapted.stronglyMeasurable_le (Nat.succ_le_of_lt hi)).sub
(hf.stronglyAdapted.stronglyMeasurable_le hi.le)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Stopping | {
"line": 300,
"column": 2
} | {
"line": 301,
"column": 70
} | {
"line": 302,
"column": 2
} | [
{
"pp": "case neg\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : ConditionallyCompleteLinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : FirstCountableTopology ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhf : f.IsRightContinuous\nhτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑... | [
"case neg\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : ConditionallyCompleteLinearOrder ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : FirstCountableTopology ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhf : f.IsRightContinuous\nhτ1 : ∀ (i : ι), MeasurableSet {ω | τ ω < ↑i}\nhτ2 : ∀ ... | have h_exists_lt (u : ι) (hu : t < u) : ∃ i, s i < u :=
Eventually.exists (f := atTop) (hs_tendsto.eventually_lt_const hu) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Martingale.Convergence | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 22
} | {
"line": 119,
"column": 22
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nω : Ω\nhab : a < b\nhω : ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω < k\nh₁ : ∃ᶠ (n : ℕ) in atTop, f n ω < a\nh₂ : ∃ᶠ (n : ℕ) in atTop, b < f n ω\n⊢ False",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"... | [
"Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nω : Ω\nhab : a < b\nhω : ∃ k, ∀ (N : ℕ), upcrossingsBefore a b f N ω < k\nh₁ : ∀ (a_1 : ℕ), ∃ b, a_1 ≤ b ∧ f b ω < a\nh₂ : ∀ (a : ℕ), ∃ b_1, a ≤ b_1 ∧ b < f b_1 ω\n⊢ False"
] | frequently_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Basic | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 18
} | {
"line": 154,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\ninst✝ : IsFiniteMeasure μ\nhμ : μ ≠ 0\nc : ℝ\n⊢ log (μ.real Set.univ) + log (rexp (t * c)) = log (μ.real Set.univ) + t * c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.log_exp",... | [] | rw [log_exp _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Moments.Basic | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 18
} | {
"line": 154,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\ninst✝ : IsFiniteMeasure μ\nhμ : μ ≠ 0\nc : ℝ\n⊢ log (μ.real Set.univ) + log (rexp (t * c)) = log (μ.real Set.univ) + t * c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.log_exp",... | [] | rw [log_exp _] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.Basic | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 18
} | {
"line": 154,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\nt : ℝ\ninst✝ : IsFiniteMeasure μ\nhμ : μ ≠ 0\nc : ℝ\n⊢ log (μ.real Set.univ) + log (rexp (t * c)) = log (μ.real Set.univ) + t * c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.log_exp",... | [] | rw [log_exp _] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Stopping | {
"line": 1068,
"column": 6
} | {
"line": 1068,
"column": 41
} | {
"line": 1069,
"column": 6
} | [
{
"pp": "case neg\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝¹⁰ : Nonempty ι\nu : ι → Ω → β\nτ : Ω → WithTop ι\ninst✝⁹ : LinearOrder ι\ninst✝⁸ : MeasurableSpace ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : OrderTopology ι\ninst✝⁵ : SecondCountableTopology ι\ninst✝⁴ : BorelSpace ι\ninst✝³... | [
"case neg\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝¹⁰ : Nonempty ι\nu : ι → Ω → β\nτ : Ω → WithTop ι\ninst✝⁹ : LinearOrder ι\ninst✝⁸ : MeasurableSpace ι\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : OrderTopology ι\ninst✝⁵ : SecondCountableTopology ι\ninst✝⁴ : BorelSpace ι\ninst✝³ : Topologic... | rw [tendsto_atTop] at h_seq_tendsto | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Moments.Basic | {
"line": 253,
"column": 11
} | {
"line": 253,
"column": 38
} | {
"line": 253,
"column": 39
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nν : Measure Ω\nhμ : Integrable (fun ω ↦ rexp (t * X ω)) μ\nhν : Integrable (fun ω ↦ rexp (t * X ω)) ν\n⊢ ∫ (x : Ω), (fun ω ↦ rexp (t * X ω)) x ∂(μ + ν) = mgf X μ t + mgf X ν t",
"ppTerm": "?m.44",
"assigned": true,
"usedC... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\nν : Measure Ω\nhμ : Integrable (fun ω ↦ rexp (t * X ω)) μ\nhν : Integrable (fun ω ↦ rexp (t * X ω)) ν\n⊢ ∫ (x : Ω), rexp (t * X x) ∂μ + ∫ (x : Ω), rexp (t * X x) ∂ν = mgf X μ t + mgf X ν t"
] | integral_add_measure hμ hν, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 1145,
"column": 8
} | {
"line": 1145,
"column": 21
} | {
"line": 1146,
"column": 6
} | [
{
"pp": "case inr.h\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\nω : Ω\ni : ι\nhi : ↑i = τ ω\nhbdd : ↑i ∈ WithTop.some '' ↑s\nh : ↑i < ↑n\n⊢ τ ω = ↑i",
"ppTerm": "?inr.h✝",
"assigne... | [] | exact hi.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Stopping | {
"line": 1145,
"column": 8
} | {
"line": 1145,
"column": 21
} | {
"line": 1146,
"column": 6
} | [
{
"pp": "case inr.h\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\nω : Ω\ni : ι\nhi : ↑i = τ ω\nhbdd : ↑i ∈ WithTop.some '' ↑s\nh : ↑i < ↑n\n⊢ τ ω = ↑i",
"ppTerm": "?inr.h✝",
"assigne... | [] | exact hi.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.Stopping | {
"line": 1145,
"column": 8
} | {
"line": 1145,
"column": 21
} | {
"line": 1146,
"column": 6
} | [
{
"pp": "case inr.h\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\nω : Ω\ni : ι\nhi : ↑i = τ ω\nhbdd : ↑i ∈ WithTop.some '' ↑s\nh : ↑i < ↑n\n⊢ τ ω = ↑i",
"ppTerm": "?inr.h✝",
"assigne... | [] | exact hi.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 299,
"column": 6
} | {
"line": 299,
"column": 69
} | {
"line": 300,
"column": 4
} | [
{
"pp": "case mp.refine_1\nΩ : 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 +... | [] | exact Finset.sum_mono_set_of_nonneg hω₃ (Finset.range_mono hnm) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.ComplexMGF | {
"line": 211,
"column": 4
} | {
"line": 216,
"column": 69
} | {
"line": 217,
"column": 4
} | [
{
"pp": "case succ\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nn : ℕ\nhn :\n ∀ {z : ℂ},\n z.re ∈ interior (integrableExpSet X μ) →\n HasDerivAt (iteratedDeriv n (complexMGF X μ)) (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ) z\nz : ℂ\nhz : z.re ∈ interior (integra... | [
"case succ\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nn : ℕ\nhn :\n ∀ {z : ℂ},\n z.re ∈ interior (integrableExpSet X μ) →\n HasDerivAt (iteratedDeriv n (complexMGF X μ)) (∫ (x : Ω), (fun ω ↦ ↑(X ω) ^ (n + 1) * cexp (z * ↑(X ω))) x ∂μ) z\nz : ℂ\nhz : z.re ∈ interior (integrableExpSet X ... | have : deriv (iteratedDeriv n (complexMGF X μ))
=ᶠ[𝓝 z] fun z ↦ μ[fun ω ↦ X ω ^ (n + 1) * cexp (z * X ω)] := by
have h_mem : ∀ᶠ y in 𝓝 z, y.re ∈ interior (integrableExpSet X μ) := by
refine IsOpen.eventually_mem ?_ hz
exact isOpen_interior.preimage Complex.continuous_re
filter_upwa... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 456,
"column": 4
} | {
"line": 457,
"column": 61
} | {
"line": 458,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nv l u : ℝ\nhvlu : v ∈ Set.Ioo l u\nh_subset : Set.Ioo l u ⊆ integrableExpSet X μ\nt : ℝ := min (v - l) (u - v) / 2\nh_pos : 0 < min (v - l) (u - v)\nht : 0 < t\nhvt : v + t = 0\nhvt' : v - t ≠ 0\n⊢ AEMeasurable X μ",
"ppTerm":... | [] | exact aemeasurable_of_aemeasurable_exp_mul hvt'
(h_subset (sub_half_inf_sub_mem_Ioo hvlu)).aemeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 550,
"column": 2
} | {
"line": 550,
"column": 45
} | {
"line": 551,
"column": 2
} | [
{
"pp": "case neg\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nh : 0 ∈ interior (integrableExpSet X μ)\np : ℝ≥0\nhX : AEMeasurable X μ\nhp_zero : ¬p = 0\n⊢ Integrable (fun x ↦ ‖X x‖ ^ (↑p).toReal) μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"case neg\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nh : 0 ∈ interior (integrableExpSet X μ)\np : ℝ≥0\nhX : AEMeasurable X μ\nhp_zero : ¬p = 0\n⊢ Integrable (fun x ↦ |X x| ^ ↑p) μ"
] | simp only [norm_eq_abs, ENNReal.coe_toReal] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 257,
"column": 4
} | {
"line": 263,
"column": 10
} | {
"line": 264,
"column": 4
} | [
{
"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\n⊢ ∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ - (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v +\n deriv (cgf X μ) v ^ 2 * mgf X μ v =\n ∫ (ω : Ω),\n ... | [
"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_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * rexp (v * X ω)) μ\n⊢ ∫ (ω : Ω), X ω ^ 2 * rexp (v * X ω) ∂μ - (2 * ∫ (ω : Ω), X ω * rexp (v * X ω) ∂μ) * deriv (cgf X μ) v ... | have h_int : Integrable (fun ω ↦ 2 * X ω * deriv (cgf X μ) v * exp (v * X ω)) μ := by
simp_rw [mul_assoc, mul_comm (deriv (cgf X μ) v)]
refine Integrable.const_mul ?_ _
simp_rw [← mul_assoc]
refine Integrable.mul_const ?_ _
convert! integrable_pow_mul_exp_of_mem_interior_integrableExpSet h... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 61
} | {
"line": 491,
"column": 2
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nt : ℝ\n⊢ ↑t * I * ↑μ + ↑↑v * (↑t * I) ^ 2 / 2 = ↑t * ↑μ * I - ↑↑v * ↑t ^ 2 / 2",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Complex.commRing",
... | [
"μ : ℝ\nv : ℝ≥0\nt : ℝ\n⊢ ↑t * I * ↑μ + -(↑↑v * ↑t ^ 2) / 2 = ↑t * ↑μ * I + -(↑↑v * ↑t ^ 2 / 2)"
] | simp only [mul_pow, I_sq, mul_neg, mul_one, sub_eq_add_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 75
} | {
"line": 176,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nh : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2)\nL : StrongDual ℝ E\nu : ℝ\n⊢ charFun (Me... | [
"E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nh : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2)\nL : StrongDual ℝ E\nu : ℝ\n⊢ cexp ((∫ (x : E), ↑((u ... | rw [charFun_map_eq_charFunDual_smul L u, h (u • L), charFun_gaussianReal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Distributions.Fernique | {
"line": 182,
"column": 47
} | {
"line": 184,
"column": 30
} | {
"line": 186,
"column": 0
} | [
{
"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\ninst✝ : SFinite μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\na : ℝ\nn : ℕ\n⊢ μ {... | [] | by
convert! measure_le_mul_measure_gt_le_of_map_rotation_eq_self h_rot _ _
simp [normThreshold_add_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 278,
"column": 42
} | {
"line": 278,
"column": 72
} | {
"line": 278,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : NormedSpace ℝ E\nc : E := ∫ (x : E), x ∂μ\nthis✝ : id = fun x ↦ x - c + c\nhx : ¬c = 0\nI : Integrable (fun x ↦ ‖x‖) μ\ny : E\np : ℝ≥0\nh_Lp : MemLp (fun x ↦ x - c) (↑p) μ\nhp0 : ↑p ≠ 0\nthis : Integrable (fun x... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : NormedSpace ℝ E\nc : E := ∫ (x : E), x ∂μ\nthis✝ : id = fun x ↦ x - c + c\nhx : ¬c = 0\nI : Integrable (fun x ↦ ‖x‖) μ\ny : E\np : ℝ≥0\nh_Lp : MemLp (fun x ↦ x - c) (↑p) μ\nhp0 : ↑p ≠ 0\nthis : Integrable (fun x ↦ 2 ^ (↑p).... | Real.rpow_add (by positivity), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 88,
"column": 11
} | {
"line": 88,
"column": 30
} | {
"line": 88,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ\nhf : f.toBilinForm.Is... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ\nhf : f.toBilinForm.IsPosSemidef\n... | h'.charFunDual_eq', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Distributions.Gaussian.Multivariate | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 44
} | {
"line": 151,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝⁵ : Fintype ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nb : OrthonormalBasis ι ℝ E\n⊢ (fun x ↦ ∑ i, x i • b i) = ⇑((EuclideanSpace.basisFun ι ℝ).equiv b (Equiv.refl ι... | [
"ι : Type u_1\ninst✝⁵ : Fintype ι\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nb : OrthonormalBasis ι ℝ E\n⊢ (fun x ↦ ((EuclideanSpace.basisFun ι ℝ).equiv b (Equiv.refl ι)) (toLp 2 x)) =\n ⇑((Euclid... | simp_rw [← b.equiv_apply_euclideanSpace] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Distributions.Gaussian.Fernique | {
"line": 233,
"column": 8
} | {
"line": 233,
"column": 56
} | {
"line": 234,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nμ : Measure E\ninst✝² : IsGaussian μ\ninst✝¹ : CompleteSpace E\ninst✝ : SecondCountableTopology E\nh : ∀ (x : E), μ ≠ Measure.dirac x\nx : E\nL : StrongDual ℝ E\nhL : Var[⇑L; μ] ≠ 0... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nμ : Measure E\ninst✝² : IsGaussian μ\ninst✝¹ : CompleteSpace E\ninst✝ : SecondCountableTopology E\nh : ∀ (x : E), μ ≠ Measure.dirac x\nx : E\nL : StrongDual ℝ E\nhL : Var[⇑L; μ] ≠ 0\n⊢ 0 < Var[... | simp only [ne_eq, Real.toNNReal_eq_zero, not_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.BrownianMotion.GaussianProjectiveFamily | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 32
} | {
"line": 132,
"column": 33
} | [
{
"pp": "I : Finset ℝ≥0\n⊢ ∫ (x : ↥I → ℝ), x ∂projectiveFamily I = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"LinearOrder.toDecidableEq",
"Real.instZero",... | [
"I : Finset ℝ≥0\n⊢ ∫ (x : EuclideanSpace ℝ ↥I), x.ofLp ∂multivariateGaussian 0 (covMatrix I) = 0"
] | integral_projectiveFamily, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Multivariate | {
"line": 262,
"column": 4
} | {
"line": 266,
"column": 10
} | {
"line": 267,
"column": 4
} | [
{
"pp": "case hv\nι : Type u_2\ninst✝ : DecidableEq ι\nI J : Finset ι\nμ : EuclideanSpace ℝ ↥I\nS : Matrix ↥I ↥I ℝ\nhS : S.PosSemidef\nhJI : J ⊆ I\ni j : ↥J\n⊢ cov[(fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis i, u⟫) ∘ ⇑(EuclideanSpace.restrict₂ hJI),\n (fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis j... | [
"case hv.hX\nι : Type u_2\ninst✝ : DecidableEq ι\nI J : Finset ι\nμ : EuclideanSpace ℝ ↥I\nS : Matrix ↥I ↥I ℝ\nhS : S.PosSemidef\nhJI : J ⊆ I\ni j : ↥J\n⊢ AEStronglyMeasurable (fun u ↦ ⟪(EuclideanSpace.basisFun ↥J ℝ).toBasis i, u⟫)\n (Measure.map (⇑(EuclideanSpace.restrict₂ hJI)) (multivariateGaussian μ S))",
... | · have (i : J) : (fun u ↦ ⟪(EuclideanSpace.basisFun J ℝ).toBasis i, u⟫) ∘
EuclideanSpace.restrict₂ hJI = fun u ↦ u ⟨i.1, hJI i.2⟩ := by ext; simp [PiLp.inner_apply]
simp_rw [this, covariance_eval_multivariateGaussian hS,
covarianceBilin_multivariateGaussian (hS.submatrix _)]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Fernique | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 67
} | {
"line": 431,
"column": 2
} | [
{
"pp": "case neg\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 neg\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... | have ha_lt : μ {x | ‖x‖ ≤ a} < 1 := lt_of_le_of_ne prob_le_one ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 50
} | {
"line": 68,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝¹⁰ : IsFiniteMeasure P\nE : Type u_2\nF : Type u_3\nmE : MeasurableSpace E\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\nmF : MeasurableSpace F\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : CompleteSpace F... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝¹⁰ : IsFiniteMeasure P\nE : Type u_2\nF : Type u_3\nmE : MeasurableSpace E\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\nmF : MeasurableSpace F\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : CompleteSpace F\ninst✝⁴ : B... | indepFun_iff_map_prod_eq_prod_map_map hX hY, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 50
} | {
"line": 105,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝¹⁰ : IsFiniteMeasure P\nE : Type u_2\nF : Type u_3\nmE : MeasurableSpace E\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\nmF : MeasurableSpace F\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : CompleteSpace F... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝¹⁰ : IsFiniteMeasure P\nE : Type u_2\nF : Type u_3\nmE : MeasurableSpace E\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\nmF : MeasurableSpace F\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : CompleteSpace F\ninst✝⁴ : B... | indepFun_iff_map_prod_eq_prod_map_map hX hY, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 50
} | {
"line": 118,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\np : ℝ≥0∞\ninst✝¹¹ : Fact (1 ≤ p)\ninst✝¹⁰ : IsFiniteMeasure P\nE : Type u_2\nF : Type u_3\nmE : MeasurableSpace E\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\nmF : MeasurableSpace F\ninst✝⁶ : NormedAddCom... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\np : ℝ≥0∞\ninst✝¹¹ : Fact (1 ≤ p)\ninst✝¹⁰ : IsFiniteMeasure P\nE : Type u_2\nF : Type u_3\nmE : MeasurableSpace E\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\nmF : MeasurableSpace F\ninst✝⁶ : NormedAddCommGroup F\nin... | indepFun_iff_map_prod_eq_prod_map_map hX hY, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 169,
"column": 89
} | {
"line": 171,
"column": 6
} | {
"line": 173,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ns : Finset ι\nE : Type u_3\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nX : ι → Ω → E\ninst✝ : NormedSpace ℝ E\nmX : ∀ i ∈ s, AEMeasurable (X i) P\nhX : iIndepFun... | [] | by
convert! hX.charFunDual_map_finsetSum_eq_prod mX
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Process.Basic | {
"line": 46,
"column": 41
} | {
"line": 64,
"column": 38
} | {
"line": 66,
"column": 0
} | [
{
"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\n𝓧 : S → Type u_5\n𝓨 : Type u_6\ninst✝¹ : (i : S) → MeasurableSpace (𝓧 i)\ninst✝ : MeasurableSpace 𝓨\nX X' : (i : S) → Ω → 𝓧 i\nY : Ω → 𝓨\nh1 : IndepFun (fun ω i ↦ X i ω) Y κ P\... | [] | by
rintro - - ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
have : ∀ᵐ a ∂P, κ a (((fun ω i ↦ X i ω) ⁻¹' s) ∩ (Y ⁻¹' t)) =
κ a ((fun ω i ↦ X i ω) ⁻¹' s) * κ a (Y ⁻¹' t) :=
h1 ((fun ω i ↦ X i ω) ⁻¹' s) (Y ⁻¹' t) ⟨s, hs, rfl⟩ ⟨t, ht, rfl⟩
obtain ⟨I, u, hI, rfl⟩ : ∃ (I : Set S) (u : Set (Π i : I, 𝓧 i)),
I.Countable ∧ s ... | [anonymous] | Lean.Parser.Term.byTactic |
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