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.FundamentalCone | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\nhx : x ∈ fundamentalCone K\nu : (𝓞 K)ˣ\n⊢ u • x ∈ fundamentalCone K ↔ u ∈ torsion K",
"usedConstants": [
"instHSMul",
"NumberField.instCommRingRingOfIntegers",
"CommSemiring.toSemiring",
"NumberField.m... | refine ⟨fun h ↦ ?_, fun h ↦ torsion_smul_mem_of_mem hx h⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 213,
"column": 2
} | {
"line": 217,
"column": 72
} | [
{
"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... | · intro _
simp_rw [of_apply, ← Real.log_pow]
rw [← Real.log_prod, Equiv.prod_comp f (fun w ↦ (w (u _) ^ (mult w))), prod_eq_abs_norm,
Units.norm, Rat.cast_one, Real.log_one]
exact fun _ _ ↦ pow_ne_zero _ <| (map_ne_zero _).mpr (coe_ne_zero _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 227,
"column": 96
} | {
"line": 228,
"column": 88
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\nw' : InfinitePlace K\ne : { w // w ≠ w' } ≃ Fin (rank K)\n⊢ regOfFamily u = |(of fun i w ↦ ↑(↑w).mult * Real.log (↑w ((algebraMap (𝓞 K) K) ↑(u (e i))))).det|",
"usedConstants": [
"NumberField.InfinitePlace.ins... | by
simp [regOfFamily_eq_det', abs_det_eq_abs_det u e (equivFinRank K).symm, logEmbedding] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 369,
"column": 83
} | {
"line": 369,
"column": 96
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx✝ : { w // w ≠ w₀ }\n⊢ 0 = 0 x✝",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddZeroClass.toAddZero",
"Pi.zero_apply",
"id",
"Sub... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 19
} | [
{
"pp": "case e_a.e_M.a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw i : InfinitePlace K\n⊢ (completeBasis K) w i =\n if h : w = w₀ then ↑i.mult\n else ↑i.mult * Real.log (i ((algebraMap (𝓞 K) K) ↑(fundSystem K (equivFinRank.symm ⟨w, h⟩))))",
"usedConstants": [
"NumberField.Infinit... | split_ifs with hw | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 379,
"column": 28
} | {
"line": 379,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : NumberField K\nh₁ : indexRealUnits K * (unitsMulComplexConjInv K).range.index = 2\nh₂ : (unitsMulComplexConjInv K).range.index = 1\n⊢ indexRealUnits K = 2",
"usedConstants": [
"MonoidHom.range",
"HMul.hMu... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 380,
"column": 28
} | {
"line": 380,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\ninst✝² : CharZero K\ninst✝¹ : IsCMField K\ninst✝ : NumberField K\nh₁ : indexRealUnits K * (unitsMulComplexConjInv K).range.index = 2\nh₂ : (unitsMulComplexConjInv K).range.index = 2\n⊢ indexRealUnits K = 1",
"usedConstants": [
"MonoidHom.range",
"HMul.hMu... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 51,
"column": 9
} | {
"line": 51,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ 0\nhj : j.Coprime 0\n⊢ ζ ^ j - 1 ∣ ζ - 1",
"usedConstants": [
"Nat.Coprime",
"Dvd.dvd",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"semigroupDv... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 51,
"column": 9
} | {
"line": 51,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ 0\nhj : j.Coprime 0\n⊢ ζ ^ j - 1 ∣ ζ - 1",
"usedConstants": [
"Nat.Coprime",
"Dvd.dvd",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"semigroupDv... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 51,
"column": 9
} | {
"line": 51,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ 0\nhj : j.Coprime 0\n⊢ ζ ^ j - 1 ∣ ζ - 1",
"usedConstants": [
"Nat.Coprime",
"Dvd.dvd",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
"semigroupDv... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 52,
"column": 9
} | {
"line": 52,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ 1\nhj : j.Coprime 1\n⊢ ζ ^ j - 1 ∣ ζ - 1",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Dvd.dvd",
"sub_self",
"Monoid.toMulOneClass",
"AddGroupWithOne.toAddGroup",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 52,
"column": 9
} | {
"line": 52,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ 1\nhj : j.Coprime 1\n⊢ ζ ^ j - 1 ∣ ζ - 1",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Dvd.dvd",
"sub_self",
"Monoid.toMulOneClass",
"AddGroupWithOne.toAddGroup",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 52,
"column": 9
} | {
"line": 52,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ 1\nhj : j.Coprime 1\n⊢ ζ ^ j - 1 ∣ ζ - 1",
"usedConstants": [
"one_pow",
"MulOne.toOne",
"Dvd.dvd",
"sub_self",
"Monoid.toMulOneClass",
"AddGroupWithOne.toAddGroup",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 98,
"column": 9
} | {
"line": 98,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhj_Unit : IsUnit ↑j\nhζ : IsPrimitiveRoot ζ 0\nhj : j.Coprime 0\n⊢ IsUnit (∑ i ∈ range j, ζ ^ i)",
"usedConstants": [
"Nat.Coprime",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommR... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 98,
"column": 9
} | {
"line": 98,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhj_Unit : IsUnit ↑j\nhζ : IsPrimitiveRoot ζ 0\nhj : j.Coprime 0\n⊢ IsUnit (∑ i ∈ range j, ζ ^ i)",
"usedConstants": [
"Nat.Coprime",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommR... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 98,
"column": 9
} | {
"line": 98,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhj_Unit : IsUnit ↑j\nhζ : IsPrimitiveRoot ζ 0\nhj : j.Coprime 0\n⊢ IsUnit (∑ i ∈ range j, ζ ^ i)",
"usedConstants": [
"Nat.Coprime",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommR... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 99,
"column": 9
} | {
"line": 99,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhj_Unit : IsUnit ↑j\nhζ : IsPrimitiveRoot ζ 1\nhj : j.Coprime 1\n⊢ IsUnit (∑ i ∈ range j, ζ ^ i)",
"usedConstants": [
"one_pow",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 99,
"column": 9
} | {
"line": 99,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhj_Unit : IsUnit ↑j\nhζ : IsPrimitiveRoot ζ 1\nhj : j.Coprime 1\n⊢ IsUnit (∑ i ∈ range j, ζ ^ i)",
"usedConstants": [
"one_pow",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 99,
"column": 9
} | {
"line": 99,
"column": 17
} | [
{
"pp": "n j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhj_Unit : IsUnit ↑j\nhζ : IsPrimitiveRoot ζ 1\nhj : j.Coprime 1\n⊢ IsUnit (∑ i ∈ range j, ζ ^ i)",
"usedConstants": [
"one_pow",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"instHSMul",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 665,
"column": 6
} | {
"line": 665,
"column": 23
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\nx✝ : (∀ (w : InfinitePlace K), ¬w = w₀ → ↑expMapBasis.symm x w ∈ Set.Ico 0 1) ∧ ↑expMapBasis.symm x w₀ ≤ 0\nw : InfinitePlace K\nh₁ : ∀ (w : InfinitePlace K), ¬w = w₀ → ↑expMapBasis.... | split_ifs with hw | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.NumberTheory.NumberField.Ideal.Basic | {
"line": 117,
"column": 33
} | {
"line": 117,
"column": 47
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nP : Ideal (𝓞 K)\nhP₀ : P ≠ ⊥\nhP₁ : P.IsPrime\nhP₂ : (absNorm P).Coprime (torsionOrder K)\nthis : P.IsMaximal\nx✝ : Field (𝓞 K ⧸ P) := Quotient.field P\nhP₃ : absNorm P ≠ 1\nh : Fintype.card ↥(torsion K) ∣ Nat.card (𝓞 K ⧸ P)ˣ\n⊢ torsionOrder K ∣... | Nat.card_units | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 773,
"column": 15
} | {
"line": 773,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\n⊢ (Real.exp (y w₀) • ↑expMapBasis fun i ↦ if i = w₀ then 0 else y i) = ↑expMapBasis y",
"usedConstants": [
"Eq.mpr",
... | expMapBasis_apply'' y | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 73,
"column": 8
} | {
"line": 73,
"column": 51
} | [
{
"pp": "case neg\np k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nh : ¬p = 2\n⊢ Associated ((Algebra.norm ℤ) (hζ.toInteger - 1)) ↑p",
"usedConstants": [
"Eq.mpr",
... | hζ.norm_toInteger_sub_one_of_prime_ne_two h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 44
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\n⊢ 𝒑.ramificationIdx (span {hζ.toInteger - 1}) = p ^ k * (p - 1)",
"usedConstants": [
"LinearOrderedCommMonoidWit... | have h := isPrime_span_zeta_sub_one p k hζ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 336,
"column": 42
} | {
"line": 336,
"column": 78
} | [
{
"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✝⁴ : IsAbelianGalois ℚ K\nthis✝³ : NeZero m\nthis✝² : NeZero n\nhp' : 𝒑 ≠ ⊥\nζ : K := zeta n ℚ K\nhζ : IsPrimitiveRoot (zeta n... | Nat.eq_one_of_mul_eq_one_right this, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 54
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nhJ : ClassGroup.mk0 J = C⁻¹\nI : ↥(Ideal (𝓞 K))⁰\n⊢ ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C ↔\n IsPrincipal ↑↑((Equiv.dvd J) I) ∧ ↑(absNorm ↑↑((Equiv.dvd J) I)) ≤ s * ↑(absNorm ↑J)",
"use... | rw [← ClassGroup.mk0_eq_one_iff (SetLike.coe_mem _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 106,
"column": 12
} | {
"line": 106,
"column": 15
} | [
{
"pp": "case h.e'_5.h.e'_3\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 121,
"column": 48
} | {
"line": 121,
"column": 51
} | [
{
"pp": "case convert_4\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nh₁ :\n ∀ (s : ℝ),\n {x | x ∈ ⇑(toMixed K) ⁻¹' fundamentalCone K ∧ mixedEmbedding.norm ((toMixed K) x) ≤ s} =\n ⇑(toMixed K) ⁻¹' {x | x ∈ fundamentalCone K ∧ mixedEmbedding.norm x ≤ s}\nh₂ : {x | x ∈ f... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 52,
"column": 70
} | {
"line": 53,
"column": 47
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα β : K\n⊢ house (α * β) ≤ house α * house β",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
"Real.instLE",
... | by
simp only [house, map_mul]; apply norm_mul_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.House | {
"line": 145,
"column": 6
} | {
"line": 146,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (K →+* ℂ)\nα : 𝓞 K\ni : K →+* ℂ\nσ : K →+* (K →+* ℂ) → ℂ := canonicalEmbedding K\n⊢ ‖↑((((integralBasis K).reindex (equivReindex K).symm).repr ↑α) i)‖ ≤\n ∑ j, ‖(basisMatrix K)ᵀ⁻¹ i j‖ * ‖σ ((algebraMap (𝓞 K) K) α) j‖",
... | rw [← inverse_basisMatrix_mulVec_eq_repr]
exact norm_sum_le_of_le _ fun _ _ ↦ (norm_mul _ _).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.House | {
"line": 145,
"column": 6
} | {
"line": 146,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (K →+* ℂ)\nα : 𝓞 K\ni : K →+* ℂ\nσ : K →+* (K →+* ℂ) → ℂ := canonicalEmbedding K\n⊢ ‖↑((((integralBasis K).reindex (equivReindex K).symm).repr ↑α) i)‖ ≤\n ∑ j, ‖(basisMatrix K)ᵀ⁻¹ i j‖ * ‖σ ((algebraMap (𝓞 K) K) α) j‖",
... | rw [← inverse_basisMatrix_mulVec_eq_repr]
exact norm_sum_le_of_le _ fun _ _ ↦ (norm_mul _ _).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Ostrowski | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 39
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nm n : ℕ\nhm : 1 < m\nhn : 1 < n\nnotbdd : ¬∀ (n : ℕ), f ↑n ≤ 1\nk : ℕ\nhk : k ≠ 0\nh_ineq1 : ∀ {m n : ℕ}, 1 < m → 1 < n → f ↑n ≤ ↑m * f ↑m / (f ↑m - 1) * f ↑m ^ logb ↑m ↑n\n⊢ f ↑n ^ k ≤ ((↑m * f ↑m / (f ↑m - 1)) ^ (↑k)⁻¹) ^ ↑k * (f ↑m ^ logb ↑m ↑n) ^ ↑k",
"usedConstants": [
... | ← rpow_mul (expr_pos hm notbdd).le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.Complex | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 22
} | [
{
"pp": "case a\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ (↑p)⁻¹ = (↑↑p)⁻¹",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"congrArg",
"Real.instInv",
"id",
"AddMonoidWithOne.toNatCast",
"NNReal",
"AddCommMonoidWithOne.toAdd... | NNReal.coe_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 469,
"column": 2
} | {
"line": 469,
"column": 45
} | [
{
"pp": "case neg\nf : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\nbdd : ¬∀ (n : ℕ), f ↑n ≤ 1\n⊢ f ≈ real ∨ ∃! p, ∃ (x : Fact (Nat.Prime p)), f ≈ padic p",
"usedConstants": [
"Real.partialOrder",
"Real",
"Nat.Prime",
"Rat",
"Rat.AbsoluteValue.equiv_real_of_unbounded",
... | · exact .inl <| equiv_real_of_unbounded bdd | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 218,
"column": 6
} | {
"line": 219,
"column": 85
} | [
{
"pp": "case succ.refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nk : ℕ\nIH : ∀ (n : ℕ... | exact div_le_div_of_nonneg_left (norm_nonneg _)
(mod_cast pow_pos hp.out.pos _) (mod_cast pow_le_pow_right₀ hp.out.one_le hk) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 218,
"column": 6
} | {
"line": 219,
"column": 85
} | [
{
"pp": "case succ.refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nk : ℕ\nIH : ∀ (n : ℕ... | exact div_le_div_of_nonneg_left (norm_nonneg _)
(mod_cast pow_pos hp.out.pos _) (mod_cast pow_le_pow_right₀ hp.out.one_le hk) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 218,
"column": 6
} | {
"line": 219,
"column": 85
} | [
{
"pp": "case succ.refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nk : ℕ\nIH : ∀ (n : ℕ... | exact div_le_div_of_nonneg_left (norm_nonneg _)
(mod_cast pow_pos hp.out.pos _) (mod_cast pow_le_pow_right₀ hp.out.one_le hk) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Pell | {
"line": 499,
"column": 74
} | {
"line": 499,
"column": 88
} | [
{
"pp": "case inr\nd : ℤ\na : Solution₁ d\nh : IsFundamental a\nH : ∀ (n : ℤ), 0 ≤ n → (a ^ n).y < (a ^ (n + 1)).y\nn : ℤ\nhn : n < 0\nm : ℤ := -n - 1\nhm : n = -m - 1\n⊢ -(a ^ (m + 1)).y < -(a ^ m).y",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMo... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 517,
"column": 2
} | {
"line": 521,
"column": 53
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nh : IsFundamental a\nn n' : ℤ\n⊢ a ^ n ≠ -a ^ n'",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"lt_neg",
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"lt_irrefl",
"AddLef... | intro hf
apply_fun Solution₁.x at hf
have H := x_zpow_pos h.x_pos n
rw [hf, x_neg, lt_neg, neg_zero] at H
exact lt_irrefl _ ((x_zpow_pos h.x_pos n').trans H) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Pell | {
"line": 517,
"column": 2
} | {
"line": 521,
"column": 53
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nh : IsFundamental a\nn n' : ℤ\n⊢ a ^ n ≠ -a ^ n'",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"lt_neg",
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"lt_irrefl",
"AddLef... | intro hf
apply_fun Solution₁.x at hf
have H := x_zpow_pos h.x_pos n
rw [hf, x_neg, lt_neg, neg_zero] at H
exact lt_irrefl _ ((x_zpow_pos h.x_pos n').trans H) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Pell | {
"line": 645,
"column": 12
} | {
"line": 645,
"column": 25
} | [
{
"pp": "case inl.inl.inr\nd : ℤ\nh₀ : 0 < d\nhd : ¬IsSquare d\na₁ : Solution₁ d\nha₁ : IsFundamental a₁\nn₁ : ℤ\nHx : 1 < (a₁ ^ (-1)).x\nHy : 0 < (a₁ ^ (-1)).y\nH : ∀ (a : Solution₁ d), ∃ n, a = (a₁ ^ (-1)) ^ n ∨ a = -(a₁ ^ (-1)) ^ n\nhn₁ : a₁ ^ (-1 * n₁ - 1) = 1\n⊢ a₁ ^ (-1) = a₁",
"usedConstants": [
... | zpow_neg_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Rayleigh | {
"line": 159,
"column": 55
} | {
"line": 159,
"column": 63
} | [
{
"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, beattySeq r k = x} ∨ j ∉ {x | ∃ k, beattySeq r... | and_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.SumFourSquares | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 67
} | [
{
"pp": "case inr\np : ℕ\nhp : Fact (Nat.Prime p)\nhodd : Odd p\na b : ℕ\nha : a ≤ p / 2\nhb : b ≤ p / 2\nhab : ↑a ^ 2 + ↑b ^ 2 ≡ -1 [ZMOD ↑p]\nk : ℕ\nhk : ↑a ^ 2 + ↑b ^ 2 + 1 = ↑k * ↑p\nhk₀ : 0 < ↑k\n⊢ k < p",
"usedConstants": [
"Int.instAddCommGroup",
"Nat.cast_mul._simp_1",
"NonUnitalNo... | replace hk : a ^ 2 + b ^ 2 + 1 ^ 2 + 0 ^ 2 = k * p := mod_cast hk | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity | {
"line": 46,
"column": 8
} | {
"line": 47,
"column": 31
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nhpi : Prime ↑p\nhp1 : p % 2 = 1\nhp3 : p % 4 ≠ 3\nk : ℕ\nk_lt_p : k < p\nhk : -1 = ↑k * ↑k\n⊢ p ∣ k ^ 2 + 1",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
"po... | rw [pow_two, ← CharP.cast_eq_zero_iff (ZMod p) p, Nat.cast_add, Nat.cast_mul, Nat.cast_one,
← hk, neg_add_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.SumFourSquares | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 61
} | [
{
"pp": "case neg.inr\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,... | exact hmin r hrm ⟨hrm.trans hmp, hr₀, _, _, _, _, this⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 215,
"column": 29
} | {
"line": 215,
"column": 37
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝² : v.IsNontrivial\ninst✝¹ : IsTrivialOn K v\nhle : v X ≤ 1\ninst✝ : v.IsRankOneDiscrete\np q : K[X]\nhq0 : q ≠ 0\nhf0 : (algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q ≠ 0\n⊢ p ≠ 0",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 223,
"column": 45
} | {
"line": 223,
"column": 53
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝² : v.IsNontrivial\ninst✝¹ : IsTrivialOn K v\nhle : v X ≤ 1\ninst✝ : v.IsRankOneDiscrete\np q : K[X]\nhq0 : q ≠ 0\nhf0 : (algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q ≠ 0\nhp0 : p ≠... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 223,
"column": 45
} | {
"line": 223,
"column": 53
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝² : v.IsNontrivial\ninst✝¹ : IsTrivialOn K v\nhle : v X ≤ 1\ninst✝ : v.IsRankOneDiscrete\np q : K[X]\nhq0 : q ≠ 0\nhf0 : (algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q ≠ 0\nhp0 : p ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.RatFunc.Ostrowski | {
"line": 223,
"column": 45
} | {
"line": 223,
"column": 53
} | [
{
"pp": "K : Type u_1\nΓ : Type u_2\ninst✝⁴ : Field K\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nv : Valuation K⟮X⟯ Γ\ninst✝² : v.IsNontrivial\ninst✝¹ : IsTrivialOn K v\nhle : v X ≤ 1\ninst✝ : v.IsRankOneDiscrete\np q : K[X]\nhq0 : q ≠ 0\nhf0 : (algebraMap K[X] K⟮X⟯) p / (algebraMap K[X] K⟮X⟯) q ≠ 0\nhp0 : p ≠... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 285,
"column": 54
} | {
"line": 285,
"column": 58
} | [
{
"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⊢ ‖↑(ih_n hnorm ⋯) - ↑(newton_seq_aux hnorm n)‖ =\n ‖(Polynomial.aeval ↑(ne... | ih_n | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 11
} | [
{
"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⊢ ‖eval (x • s) (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p))‖ ≤ c ^ p",
"usedConstants": [
... | grw [← h] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 131,
"column": 68
} | {
"line": 131,
"column": 77
} | [
{
"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⊢ ‖eval (x • s) (X ^ (p - 1)) * eval (x • s) (map (algebraMap ℤ ℂ) f ^ p)‖ ≤\n |max (x * ‖s‖) 1 ... | eval_pow, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Padics.Hensel | {
"line": 437,
"column": 50
} | {
"line": 437,
"column": 68
} | [
{
"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\nhnsol : (Polynomial.aeval a) F ≠ 0\nz : ℤ_[p]\nhev : (Polynomial.aeval z) F = 0\nhnlt... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 15
} | [
{
"pp": "case h.right.refine_2\nf : ℤ[X]\nhf : eval 0 f ≠ 0\nc' : ℂ → ℝ\nc'0 : ∀ (s : ℂ), c' s ≥ 0\nPp'_le : ∀ (s : ℂ) (p : ℕ), p ≠ 0 → ‖P (map (algebraMap ℤ ℂ) (X ^ (p - 1) * f ^ p)) s‖ ≤ c' s ^ p\np : ℕ\np_gt : p > (eval 0 f).natAbs\nprime_p : Nat.Prime p\ngp' : ℤ[X]\nh' : eval 0 (sumIDeriv (X ^ (p - 1) * f ^... | ← norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 183,
"column": 12
} | {
"line": 183,
"column": 41
} | [
{
"pp": "p x : ℝ\nr : ℚ\nh : LiouvilleWith p (x + ↑r)\n⊢ LiouvilleWith p x",
"usedConstants": [
"NegZeroClass.toNeg",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"add_neg_cancel_right",
"Real.instRatCast",
"Rat",
"LiouvilleWith.add_rat",
"Eq.mp",
... | by simpa using h.add_rat (-r) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.WellApproximable | {
"line": 137,
"column": 2
} | {
"line": 154,
"column": 42
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedCommGroup A\na : A\nn : ℕ\nδ : ℝ\nhn : 0 < n\nhan : orderOf a ^ 2 ∣ n\n⊢ a • approxOrderOf A n δ = approxOrderOf A n δ",
"usedConstants": [
"Metric.thickening_eq_biUnion_ball",
"Iff.mpr",
"Eq.mpr",
"Subtype.mk.congr_simp",
"DivInvMonoid.... | simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
replace han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n := by
intro b hb
rw [← hb] at han hn
rw [sq] at han
rwa [(Commute.all a b).orderOf_mul_eq_right_of_foral... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.WellApproximable | {
"line": 137,
"column": 2
} | {
"line": 154,
"column": 42
} | [
{
"pp": "A : Type u_1\ninst✝ : SeminormedCommGroup A\na : A\nn : ℕ\nδ : ℝ\nhn : 0 < n\nhan : orderOf a ^ 2 ∣ n\n⊢ a • approxOrderOf A n δ = approxOrderOf A n δ",
"usedConstants": [
"Metric.thickening_eq_biUnion_ball",
"Iff.mpr",
"Eq.mpr",
"Subtype.mk.congr_simp",
"DivInvMonoid.... | simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
replace han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n := by
intro b hb
rw [← hb] at han hn
rw [sq] at han
rwa [(Commute.all a b).orderOf_mul_eq_right_of_foral... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 12
} | [
{
"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\na : ι\nha : (∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j a → sInf s... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 59,
"column": 2
} | {
"line": 62,
"column": 12
} | [
{
"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\n⊢ e ≤ sInf s",
"usedConstants": [
"False",
... | · by_contra! hs
obtain ⟨a, ha⟩ := hs
refine ha.2.not_ge <| le_sInf_apply fun f hf hf' ↦ apply_le_of_toLex (h hf) ?_
simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.CompleteLattice.PiLex | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 12
} | [
{
"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\na : ι\nha : (∀ (j : ι), (fun x1 x2 ↦ x1 < x2) j a → e j = ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.Category.PartOrdEmb | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 82
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsFiltered J\nF : J ⥤ PartOrdEmb\nc : Cocone (F ⋙ forget PartOrdEmb)\nhc : IsColimit c\ns : Cocone F\nj : J\nx' y' : (F ⋙ forget PartOrdEmb).obj j\nhx :\n (ConcreteCategory.hom (hc.desc ((forget PartOrdEmb).mapCocone s))) ((ConcreteCategory.hom (c.ι.app j)... | exact le_of_eq_of_le (by simp [hl]) (le_of_le_of_eq h (by simp [h₁, h₂, hl'])) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Concept | {
"line": 308,
"column": 7
} | {
"line": 308,
"column": 15
} | [
{
"pp": "case h.h\nα : Type u_2\nβ : Type u_3\nr : α → β → Prop\nc : Concept α β r\ne : Set α\ni : Set β\nhe : e = c.extent\nhi : i = c.intent\nx✝ : α\n⊢ x✝ ∈ (c.copy e i he hi).extent ↔ x✝ ∈ c.extent",
"usedConstants": [
"congrArg",
"Membership.mem",
"Concept.copy.congr_simp",
"Conc... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.Filter.ListTraverse | {
"line": 52,
"column": 2
} | {
"line": 53,
"column": 51
} | [
{
"pp": "case mpr\nα β : Type u\nf : β → Filter α\nfs : List β\nt : Set (List α)\n⊢ (∃ us, Forall₂ (fun b s ↦ s ∈ f b) fs us ∧ sequence us ⊆ t) → t ∈ traverse f fs",
"usedConstants": [
"Filter.instMembership",
"Membership.mem",
"Exists",
"HasSubset.Subset",
"Filter.mem_traverse... | · rintro ⟨us, hus, hs⟩
exact mem_of_superset (mem_traverse _ _ hus) hs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Partial | {
"line": 257,
"column": 6
} | {
"line": 257,
"column": 22
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α →. β\nl₁ : Filter α\nl₂ : Filter β\nh : f.Dom ∈ l₁\nh' : ∀ s ∈ l₂, f.core s ∈ l₁\ns : Set β\nsl₂ : s ∈ l₂\n⊢ f.preimage s ∈ l₁",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"PFun.preimage_eq",
"congrArg",
"Membership.mem",
... | PFun.preimage_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Heyting.Regular | {
"line": 206,
"column": 26
} | {
"line": 206,
"column": 49
} | [
{
"pp": "α : Type u_1\ninst✝ : HeytingAlgebra α\na✝ b✝ : α\na b c : Regular α\n⊢ (↑a ⊔ ↑b)ᶜᶜ ⊓ (↑a ⊔ ↑c)ᶜᶜ ≤ ((↑a ⊔ ↑b) ⊓ (↑a ⊔ ↑c))ᶜᶜ",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"... | compl_compl_inf_distrib | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Lattice.Congruence | {
"line": 95,
"column": 4
} | {
"line": 99,
"column": 40
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Lattice α\ninst✝ : Lattice β\nr : α → α → Prop\nh₁ : Std.Refl r\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)\nw x✝ y✝ ... | have compatible_left_inf {x y t : α} (hh : r x y) : r (x ⊓ t) (y ⊓ t) :=
closed_interval h₂ h₄ ((x ⊓ y) ⊓ t) _ _ ((x ⊔ y) ⊓ t)
(inf_le_inf_right _ inf_le_left) (inf_le_inf_right _ le_sup_left)
(inf_le_inf_right _ inf_le_right) (inf_le_inf_right _ le_sup_right)
(h₄ inf_le_sup (h... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Nucleus | {
"line": 215,
"column": 8
} | {
"line": 220,
"column": 74
} | [] | l = (l ⊓ m (x ⊔ k)) ⊓ (l ⊓ m (y ⊔ k)) := by
rw [← inf_inf_distrib_left, ← map_inf, ← sup_inf_right, sup_eq_right.2 hxyk,
inf_eq_left.2 hlk]
_ ≤ n (x ⊔ k) ⊓ n (y ⊔ k) := by
gcongr; exacts [hlx (x ⊔ k) le_sup_left, hly (y ⊔ k) le_sup_left]
_ = n k := by rw [← map_inf, ← sup... | Lean.Elab.Tactic.evalCalc | Lean.calcSteps |
Mathlib.Order.Types.Defs | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 42
} | [
{
"pp": "α : Type u\ninst✝² : LinearOrder α\ninst✝¹ : Nonempty α\ninst✝ : Subsingleton α\n⊢ type α = 1",
"usedConstants": [
"PUnit.instUnique",
"Unique",
"PartialOrder.toPreorder",
"OrderType.instOne",
"OrderType.type",
"SemilatticeInf.toPartialOrder",
"OrderIso.typ... | cases nonempty_unique α
exact (OrderIso.ofUnique α _).type_congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Types.Defs | {
"line": 120,
"column": 2
} | {
"line": 121,
"column": 42
} | [
{
"pp": "α : Type u\ninst✝² : LinearOrder α\ninst✝¹ : Nonempty α\ninst✝ : Subsingleton α\n⊢ type α = 1",
"usedConstants": [
"PUnit.instUnique",
"Unique",
"PartialOrder.toPreorder",
"OrderType.instOne",
"OrderType.type",
"SemilatticeInf.toPartialOrder",
"OrderIso.typ... | cases nonempty_unique α
exact (OrderIso.ofUnique α _).type_congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.Tree | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 17
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PartialOrder α\ninst✝³ : PredOrder α\ninst✝² : IsPredArchimedean α\ninst✝¹ : OrderBot α\ninst✝ : DecidableEq α\nr : α\n⊢ Order.pred (findAtom r) = ⊥",
"usedConstants": [
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Bot.bot... | unfold findAtom | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Order.Sublocale | {
"line": 150,
"column": 39
} | {
"line": 150,
"column": 79
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nι : Sort u_2\nS✝ T : Sublocale X\ns✝ : Set X\nf : ι → X\na✝ b✝ : X\nS : Sublocale X\na b : X\ns : ↥S\n⊢ b ≤ a ⇨ ↑s ↔ Sublocale.restrictAux✝ S (a ⊓ b) ≤ s",
"usedConstants": [
"congrArg",
"le_himp_iff._simp_1",
"PartialOrder.toPreorder",
"... | by simp [inf_comm, S.giAux.gc.le_iff_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Sublocale | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 47
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nι : Sort u_2\nS✝ T : Sublocale X\ns : Set X\nf : ι → X\na b : X\nS : Sublocale X\n⊢ ⊤ ≤ Sublocale.restrictAux✝ S ⊤",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"PartialOrder.toP... | rw [← Subtype.coe_le_coe, S.giAux.gc.u_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Sublocale | {
"line": 182,
"column": 17
} | {
"line": 182,
"column": 25
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\nx : X\nhx : x ∈ ↑S\n⊢ S.toNucleus x = x",
"usedConstants": [
"SetLike.mem_coe._simp_1",
"FrameHom",
"Sublocale.toNucleus",
"congrArg",
"Nucleus",
"Membership.mem",
"Sublocale.instSetLike",
"Complet... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Order.Sublocale | {
"line": 182,
"column": 17
} | {
"line": 182,
"column": 25
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\nx : X\nhx : x ∈ ↑S\n⊢ S.toNucleus x = x",
"usedConstants": [
"SetLike.mem_coe._simp_1",
"FrameHom",
"Sublocale.toNucleus",
"congrArg",
"Nucleus",
"Membership.mem",
"Sublocale.instSetLike",
"Complet... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Sublocale | {
"line": 182,
"column": 17
} | {
"line": 182,
"column": 25
} | [
{
"pp": "X : Type u_1\ninst✝ : Order.Frame X\nS : Sublocale X\nx : X\nhx : x ∈ ↑S\n⊢ S.toNucleus x = x",
"usedConstants": [
"SetLike.mem_coe._simp_1",
"FrameHom",
"Sublocale.toNucleus",
"congrArg",
"Nucleus",
"Membership.mem",
"Sublocale.instSetLike",
"Complet... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Predictable | {
"line": 114,
"column": 12
} | {
"line": 114,
"column": 17
} | [
{
"pp": "case h\nΩ : Type u_1\nm : MeasurableSpace Ω\n𝓕 : Filtration ℕ m\ns : Set (ℕ × Ω)\nhs : MeasurableSet s\nn i : ℕ\nA : Set Ω\nhA : MeasurableSet A\nhni : n < i\np : ↑(Set.singleton (n + 1)) × Ω\n⊢ i < ↑p.1 → p.2 ∉ A",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
... | p.1.2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Predictable | {
"line": 123,
"column": 12
} | {
"line": 123,
"column": 17
} | [
{
"pp": "case h\nΩ : Type u_1\nm : MeasurableSpace Ω\n𝓕 : Filtration ℕ m\ns : Set (ℕ × Ω)\nhs : MeasurableSet s\nn i : ℕ\nA : Set Ω\nhA : MeasurableSet A\nhin : i ≤ n\np : ↑(Set.singleton (n + 1)) × Ω\nhp2 : p.2 ∈ A\n⊢ i < ↑p.1 ↔ p.1 = ⟨n + 1, ⋯⟩",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
... | p.1.2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Basic | {
"line": 260,
"column": 33
} | {
"line": 270,
"column": 53
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁸ : Preorder ι\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nℱ : Filtration ι m0\ninst✝⁵ : CompleteSpace E\ninst✝⁴ : Lattice E\ninst✝³ : ContinuousSup E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoi... | by
refine ⟨fun i =>
@StronglyMeasurable.sup _ _ _ _ (ℱ i) _ _ _ (hf.stronglyAdapted i) (hg.stronglyAdapted i),
fun i j hij => ?_, fun i => Integrable.sup (hf.integrable _) (hg.integrable _)⟩
refine EventuallyLE.sup_le ?_ ?_
· exact EventuallyLE.trans (hf.2.1 i j hij)
(condExp_mono (hf.integrable _) ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.Basic | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 18
} | [
{
"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 → ... | ← neg_neg c, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Basic | {
"line": 373,
"column": 6
} | {
"line": 373,
"column": 18
} | [
{
"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 → ... | ← neg_neg c, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Centering | {
"line": 205,
"column": 6
} | {
"line": 207,
"column": 32
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝¹ : CompleteSpace E\nhf : StronglyAdapted ℱ f\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\ninst✝ : SigmaFiniteFiltration μ ℱ\ni... | rw [condExp_of_stronglyMeasurable]
· exact stronglyMeasurable_condExp.mono (ℱ.mono hk.le)
· exact integrable_condExp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Centering | {
"line": 205,
"column": 6
} | {
"line": 207,
"column": 32
} | [
{
"pp": "case refine_2\nΩ : Type u_1\nE : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : ℕ → Ω → E\nℱ : Filtration ℕ m0\ninst✝¹ : CompleteSpace E\nhf : StronglyAdapted ℱ f\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\ninst✝ : SigmaFiniteFiltration μ ℱ\ni... | rw [condExp_of_stronglyMeasurable]
· exact stronglyMeasurable_condExp.mono (ℱ.mono hk.le)
· exact integrable_condExp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.HittingTime | {
"line": 125,
"column": 2
} | {
"line": 129,
"column": 91
} | [
{
"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 k : ι\nhk₁ : k < hittingBtwn u s n m ω\nhk₂ : n ≤ k\n⊢ u k ω ∉ s",
"usedConstants": [
"not_le",
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"congrA... | intro h
have hexists : ∃ j ∈ Set.Icc n m, u j ω ∈ s := ⟨k, ⟨hk₂, le_trans hk₁.le <| hittingBtwn_le _⟩, h⟩
refine not_le.2 hk₁ ?_
simp_rw [hittingBtwn, if_pos hexists]
exact csInf_le bddBelow_Icc.inter_of_left ⟨⟨hk₂, le_trans hk₁.le <| hittingBtwn_le _⟩, h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.HittingTime | {
"line": 125,
"column": 2
} | {
"line": 129,
"column": 91
} | [
{
"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 k : ι\nhk₁ : k < hittingBtwn u s n m ω\nhk₂ : n ≤ k\n⊢ u k ω ∉ s",
"usedConstants": [
"not_le",
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"congrA... | intro h
have hexists : ∃ j ∈ Set.Icc n m, u j ω ∈ s := ⟨k, ⟨hk₂, le_trans hk₁.le <| hittingBtwn_le _⟩, h⟩
refine not_le.2 hk₁ ?_
simp_rw [hittingBtwn, if_pos hexists]
exact csInf_le bddBelow_Icc.inter_of_left ⟨⟨hk₂, le_trans hk₁.le <| hittingBtwn_le _⟩, h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 419,
"column": 6
} | {
"line": 419,
"column": 9
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nh₁ : 0 ≤ ∫ (x : Ω), (∑ k ∈ Finset.range n, (1 - upcrossingStrat a b f N k) * (f (k + 1) - f k)) x ∂μ\nh₂ :\n ∫ (x : Ω), (∑ k ∈ Finset.range n,... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.HittingTime | {
"line": 354,
"column": 2
} | {
"line": 360,
"column": 14
} | [
{
"pp": "case neg\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : ConditionallyCompleteLinearOrder ι\nω : Ω\nu : ι → Ω → β\ns : Set β\nn m₁ m₂ : ι\nhm : m₁ ≤ m₂\nh : ¬∃ j ∈ Set.Icc n m₁, u j ω ∈ s\n⊢ (fun x ↦ hittingBtwn u s n x ω) m₁ ≤ (fun x ↦ hittingBtwn u s n x ω) m₂",
"usedConstants": [
"Mathl... | · simp_rw [hittingBtwn, if_neg h]
split_ifs with h'
· obtain ⟨j, hj₁, hj₂⟩ := h'
refine le_csInf ⟨j, hj₁, hj₂⟩ ?_
by_contra! ⟨i, hi₁, hi₂⟩
exact h ⟨i, ⟨hi₁.1.1, hi₂.le⟩, hi₁.2⟩
· exact hm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 24
} | [
{
"pp": "Ω : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nω : Ω\nhf : a ≤ f N ω\nhab : a < b\nhN : ¬N = 0\nk : ℕ\n⊢ f (upperCrossingTime a b f N (k + 1) ω) ω - f 0 ω + (f 0 ω - f (lowerCrossingTime a b f N k ω) ω) =\n stoppedValue f (fun ω ↦ ↑(upperCrossingTime a b f N (k + 1) ω)) ω -\n stoppedValue f (fun ... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Convergence | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 87
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\ng : Ω → ℝ\nμ : Measure Ω\nhf : Martingale f ℱ μ\nhg : Integrable g μ\nhgtends : Tendsto (fun n ↦ eLpNorm (f n - g) 1 μ) atTop (𝓝 0)\nn : ℕ\nht : Tendsto (fun m ↦ eLpNorm μ[f m - g | ↑ℱ n] 1 μ) atTop (𝓝 0)\n⊢ ∀ m ≥ n, eLpNorm μ[... | refine fun m hm => eLpNorm_congr_ae ((condExp_sub (hf.integrable m) hg _).trans ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 648,
"column": 4
} | {
"line": 648,
"column": 12
} | [
{
"pp": "case zero\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\n⊢ (fun x ↦ if ∃ j ∈ Set.Icc ⊥ N, (f j x - a)⁺ ∈ Set.Iic 0 then sInf (Set.Icc ⊥ N ∩ {i | (f i x - a)⁺ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 133,
"column": 6
} | {
"line": 134,
"column": 11
} | [
{
"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\nht : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n ↦ stoppedAbove f (↑i) n ω... | simp only [le_top, inf_of_le_left]
congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 133,
"column": 6
} | {
"line": 134,
"column": 11
} | [
{
"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\nht : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), ∃ c, Tendsto (fun n ↦ stoppedAbove f (↑i) n ω... | simp only [le_top, inf_of_le_left]
congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Stopping | {
"line": 570,
"column": 66
} | {
"line": 570,
"column": 80
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : FirstCountableTopology ι\nhτ : IsStoppingTime f τ\ni : ι\n⊢ MeasurableSet (Set.univ ∩ {ω | τ ω = ↑i})",
"usedConstants": [... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 601,
"column": 66
} | {
"line": 601,
"column": 80
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : LinearOrder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhτ : IsStoppingTime f τ\nh_countable : (Set.range τ).Countable\ni : ι\n⊢ MeasurableSet (Set.univ ∩ {ω | τ ω = ↑i})",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"Meas... | Set.univ_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 159,
"column": 42
} | {
"line": 159,
"column": 61
} | [
{
"pp": "case h.e'_1.a.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\nhf : Submartingale f ℱ μ\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\ng : ℕ → Ω → ℝ := fun n ω ↦ f n ω - f 0 ω\nhg : Submartingale g ℱ μ\... | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Probability.Martingale.BorelCantelli | {
"line": 159,
"column": 42
} | {
"line": 159,
"column": 61
} | [
{
"pp": "case h.e'_1.a.refine_2\nΩ : Type u_2\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nℱ : Filtration ℕ m0\nf : ℕ → Ω → ℝ\nR : ℝ≥0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R\ng : ℕ → Ω → ℝ := fun n ω ↦ f n ω - f 0 ω\nhg : Submartingale g ℱ μ\... | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 816,
"column": 17
} | {
"line": 816,
"column": 35
} | [
{
"pp": "case pos\nΩ : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\nf : ℕ → Ω → ℝ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\na b : ℝ\nhf : Submartingale f ℱ μ\nhab : a < b\nthis : ∀ (N : ℕ), ∫⁻ (ω : Ω), ENNReal.ofReal (f N ω - a)⁺ ∂μ = ENNReal.ofReal (∫ (ω : Ω), (f N ω - a)⁺ ∂μ)\nN : ℕ\n⊢ ENNReal.ofRe... | NNReal.coe_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Process.Stopping | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\ns : Set Ω\nhs : MeasurableSet s ∧ ∀ (i : ι), Me... | exact ((hτ.min hπ).min_const i).measurable_of_le fun _ => min_le_right _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Stopping | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\ns : Set Ω\nhs : MeasurableSet s ∧ ∀ (i : ι), Me... | exact ((hτ.min hπ).min_const i).measurable_of_le fun _ => min_le_right _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.Stopping | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 78
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\ns : Set Ω\nhs : MeasurableSet s ∧ ∀ (i : ι), Me... | exact ((hτ.min hπ).min_const i).measurable_of_le fun _ => min_le_right _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Stopping | {
"line": 718,
"column": 8
} | {
"line": 718,
"column": 35
} | [
{
"pp": "case mpr\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\ns : Set Ω\nh : MeasurableSet (s ∩ {ω ... | measurableSet_min_iff hτ hπ | Lean.Elab.Tactic.evalRewriteSeq | null |
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