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.Chebyshev | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 37
} | [
{
"pp": "x : ℝ\nhx : 2 ≤ x\n⊢ ψ x = θ x + ∑ n ∈ Icc 2 ⌊log x / log 2⌋₊, θ (x ^ (1 / ↑n))",
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.C... | psi_eq_sum_theta (by linarith), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 197,
"column": 6
} | {
"line": 198,
"column": 21
} | [
{
"pp": "case succ.refine_2.refine_2\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nε : ℝ\nhε : 0 < ε\nb : Fq[X]\nhb : b ≠ 0\nhbε : 0 < cardPowDegree b • ε\nn : ℕ\nih :\n ∀ (A : Fin n → Fq[X]),\n ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε\nA : ... | rw [Fin.cons_succ, Fin.cons_succ]
exact ht' i₀ i₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 197,
"column": 6
} | {
"line": 198,
"column": 21
} | [
{
"pp": "case succ.refine_2.refine_2\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nε : ℝ\nhε : 0 < ε\nb : Fq[X]\nhb : b ≠ 0\nhbε : 0 < cardPowDegree b • ε\nn : ℕ\nih :\n ∀ (A : Fin n → Fq[X]),\n ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ ↔ ↑(cardPowDegree (A i₁ % b - A i₀ % b)) < cardPowDegree b • ε\nA : ... | rw [Fin.cons_succ, Fin.cons_succ]
exact ht' i₀ i₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bernoulli | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 89
} | [
{
"pp": "case neg.hb\np d : ℕ\ninst✝ : Fact (Nat.Prime p)\nhd : d ≥ 2\nhcase : ¬(p = 2 ∧ d = 2)\nhp2 : 2 ≤ p\n⊢ d + 1 ≤ p ^ (d - 1)",
"usedConstants": [
"Nat.instMonoid",
"HSub.hSub",
"GE.ge",
"instSubNat",
"instOfNatNat",
"LE.le",
"instLENat",
"Monoid.toPow",... | suffices ∀ n : ℕ, n ≥ 2 → ¬(p = 2 ∧ n = 2) → n + 1 ≤ p ^ (n - 1) from this d hd hcase | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.Chebyshev | {
"line": 503,
"column": 4
} | {
"line": 503,
"column": 74
} | [
{
"pp": "x : ℝ\nhx : 2 ≤ x\na : ℕ → ℝ := (setOf Nat.Prime).indicator fun n ↦ log ↑n\nint_deriv :\n ∀ (f : ℝ → ℝ), ∫ (u : ℝ) in 2..x, deriv (fun x ↦ (log x)⁻¹) u * f u = ∫ (u : ℝ) in 2..x, f u * -(u * log u ^ 2)⁻¹\n⊢ (log x)⁻¹ * ∑ k ∈ Icc 0 ⌊x⌋₊, a k - ∫ (x : ℝ) in 2..x, deriv (fun n ↦ (log n)⁻¹) x * ∑ k ∈ Icc ... | simp [int_deriv, a, Set.indicator_apply, sum_filter, theta_eq_sum_Icc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 120,
"column": 2
} | {
"line": 128,
"column": 44
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : EuclideanDomain R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nabv : AbsoluteValue R ℤ\nI : ↥(Ideal S)⁰\n⊢ ∃ b ∈ ↑I, b ≠ 0 ∧ ∀ c ∈ ↑I, abv ((Algebra.norm R) c) < abv ((Algebra.norm R) b) → c = 0",
"usedConstants": [
"Submodule",
"M... | obtain ⟨_, ⟨b, b_mem, b_ne_zero, rfl⟩, min⟩ := @Int.exists_least_of_bdd
(fun a => ∃ b ∈ (I : Ideal S), b ≠ (0 : S) ∧ abv (Algebra.norm R b) = a)
(by
use 0
rintro _ ⟨b, _, _, rfl⟩
apply abv.nonneg)
(by
obtain ⟨b, b_mem, b_ne_zero⟩ := (I : Ideal S).ne_bot_iff.mp (nonZeroDivisors.coe_... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Chebyshev | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 47
} | [
{
"pp": "⊢ Real.sqrt =o[atTop] fun x ↦ x / log x ^ 2",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"NormedDivisionRing.toNorm",
"NormedDivisionRing.toDivisionRing",
"DivisionRing.toDivisionSemiring",
"DivisionRing.toDivInvMonoid",
"HDiv.hDiv",
"i... | apply isLittleO_mul_iff_isLittleO_div _ |>.mp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Chebyshev | {
"line": 601,
"column": 22
} | {
"line": 601,
"column": 50
} | [
{
"pp": "case h\nx✝ : ℝ\n| √x✝ / log 2 ^ 2",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"MulOne.toOne",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | rw [← mul_one_div, mul_comm] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 25
} | [
{
"pp": "case h.e'_2.h.e'_3\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nn : ℕ\nhn : OfNat.ofNat n < p\n⊢ OfNat.ofNat n = ↑(OfNat.ofNat n)",
"usedConstants": []
}
] | rcases n with _ | _ | n | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 400,
"column": 4
} | {
"line": 400,
"column": 29
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nm k : ℕ\nih : p ^ m ∣ x.appr (m + k) - x.appr m\nh : ¬x - ↑(x.appr (m + k)) = 0\n⊢ p ^ m ∣ p ^ (m + k) * (toZMod (↑(unitCoeff h) * ↑p ^ (↑(x - ↑(x.appr (m + k))).valuation - (↑m + ↑k)).natAbs)).val",
"usedConstants": [
"dvd_mul_of_dvd_left",
... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.NumberField.Discriminant.Defs | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 47
} | [
{
"pp": "ι : Type u_2\nι' : Type u_3\nK : Type u_1\ninst✝⁵ : Field K\ninst✝⁴ : DecidableEq ι\ninst✝³ : DecidableEq ι'\ninst✝² : Fintype ι\ninst✝¹ : Fintype ι'\ninst✝ : NumberField K\nb : Basis ι ℚ K\nb' : Basis ι' ℚ K\nh : ∀ (i : ι) (j : ι'), IsIntegral ℤ (b.toMatrix (⇑b') i j)\nh' : ∀ (i j : ι'), IsIntegral ℤ ... | exact (IsFractionRing.injective ℤ ℚ) this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 83,
"column": 8
} | {
"line": 84,
"column": 59
} | [
{
"pp": "case e_a.inl.succ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nmf : Module.Finite K L\nse : Algebra.IsSeparable K L\nhp : Fact (Nat.Prime 2)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ ((n✝ + 1).succ + 1)} K L\nhζ : IsPrimitiveRoot ζ (2 ^ ((n✝ + 1).succ + 1)... | simp_rw [_root_.pow_succ', (even_two.mul_right _).neg_one_pow,
((even_two.mul_right _).mul_right _).neg_one_pow] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 83,
"column": 8
} | {
"line": 84,
"column": 59
} | [
{
"pp": "case e_a.inl.succ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nmf : Module.Finite K L\nse : Algebra.IsSeparable K L\nhp : Fact (Nat.Prime 2)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ ((n✝ + 1).succ + 1)} K L\nhζ : IsPrimitiveRoot ζ (2 ^ ((n✝ + 1).succ + 1)... | simp_rw [_root_.pow_succ', (even_two.mul_right _).neg_one_pow,
((even_two.mul_right _).mul_right _).neg_one_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 83,
"column": 8
} | {
"line": 84,
"column": 59
} | [
{
"pp": "case e_a.inl.succ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nmf : Module.Finite K L\nse : Algebra.IsSeparable K L\nhp : Fact (Nat.Prime 2)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ ((n✝ + 1).succ + 1)} K L\nhζ : IsPrimitiveRoot ζ (2 ^ ((n✝ + 1).succ + 1)... | simp_rw [_root_.pow_succ', (even_two.mul_right _).neg_one_pow,
((even_two.mul_right _).mul_right _).neg_one_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 380,
"column": 17
} | {
"line": 380,
"column": 26
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\ne : m = n\n⊢ yn a1 m ≤ yn a1 n",
"usedConstants": [
"Eq.mpr",
"le_refl",
"congrArg",
"id",
"LE.le",
"instLENat",
"Nat.instPreorder",
"Pell.yn",
"Nat",
"Eq"
]
}
] | by rw [e] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 390,
"column": 17
} | {
"line": 390,
"column": 26
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : m < n + 1\ne : m = n\n⊢ xn a1 m ≤ xn a1 n",
"usedConstants": [
"Eq.mpr",
"le_refl",
"Pell.xn",
"congrArg",
"id",
"LE.le",
"instLENat",
"Nat.instPreorder",
"Nat",
"Eq"
]
}
] | by rw [e] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 321,
"column": 45
} | {
"line": 325,
"column": 28
} | [
{
"pp": "d : ℤ\na b : ℤ√d\nhcoprime : IsCoprime a.re a.im\nhdvd : b ∣ a\nz : ℤ\nhz : z ∈ nonunits ℤ\nhzdvdu : z ∣ b.re\nhzdvdv : z ∣ b.im\n⊢ z ∣ a.re ∧ z ∣ a.im",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Dvd.dvd",
"Zsqrtd.re",
"congrArg",
"semigroupDvd",
"Zsqrtd.in... | by
rw [← intCast_dvd]
apply dvd_trans _ hdvd
rw [intCast_dvd]
exact ⟨hzdvdu, hzdvdv⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 573,
"column": 54
} | {
"line": 573,
"column": 89
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn j : ℕ\n⊢ 0 ≡ xn a1 (2 * n + j) + xn a1 j [MOD xn a1 n]",
"usedConstants": [
"HMul.hMul",
"Pell.xn",
"instMulNat",
"instOfNatNat",
"instHAdd",
"Pell.xn_modEq_x2n_add",
"HAdd.hAdd",
"Nat.ModEq.symm",
"Nat",
"instAddNat",... | exact (xn_modEq_x2n_add _ _ _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Dioph | {
"line": 310,
"column": 60
} | {
"line": 310,
"column": 91
} | [
{
"pp": "α : Type u\nS : Set (α → ℕ)\nl : List (Set (α → ℕ))\nIH :\n List.Forall Dioph l →\n ∃ β pl, ∀ (v : α → ℕ), List.Forall (fun S ↦ v ∈ S) l ↔ ∃ t, List.Forall (fun p ↦ p (v ⊗ t) = 0) pl\nd : List.Forall Dioph (S :: l)\ndl : List.Forall Dioph l\nβ : Type u\np : Poly (α ⊕ β)\npe : ∀ (v : α → ℕ), v ∈ S ↔... | dsimp [Function.comp_def] at hq | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 103,
"column": 29
} | {
"line": 103,
"column": 79
} | [
{
"pp": "case pos.right\nR : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fu... | ← (equivProdNatFactoredNumbers hpp hp).hasSum_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 95,
"column": 4
} | {
"line": 111,
"column": 40
} | [
{
"pp": "case insert\nR : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fun m... | rw [filter_insert]
split_ifs with hpp
· constructor
· simp only [← (equivProdNatFactoredNumbers hpp hp).summable_iff, Function.comp_def,
equivProdNatFactoredNumbers_apply', factoredNumbers.map_prime_pow_mul hmul hpp hp]
refine Summable.of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun _ ↦ nor... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 95,
"column": 4
} | {
"line": 111,
"column": 40
} | [
{
"pp": "case insert\nR : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fun m... | rw [filter_insert]
split_ifs with hpp
· constructor
· simp only [← (equivProdNatFactoredNumbers hpp hp).summable_iff, Function.comp_def,
equivProdNatFactoredNumbers_apply', factoredNumbers.map_prime_pow_mul hmul hpp hp]
refine Summable.of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun _ ↦ nor... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 52
} | [
{
"pp": "m₁ m₂ n : ℕ\nhm1 : m₁ ∈ n.smoothNumbers\nhm2 : m₂ ∈ n.smoothNumbers\n⊢ m₁ * m₂ ∈ n.smoothNumbers",
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"Finset.range",
"Nat.smoothNumbers_eq_factoredNumbers",
"Nat.factoredNumbers",
"Nat",
"Nat.smo... | rw [smoothNumbers_eq_factoredNumbers] at hm1 hm2 ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 277,
"column": 4
} | {
"line": 278,
"column": 50
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nthis : LocallyIntegrableOn (fun x ↦ (P.ε * ↑(x ^ (-P.k))) • P.g₀) (Ioi 0) volume\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ IntegrableAtFilter ((Ioo 0 1).indicator fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) (𝓝[Ioi... | obtain ⟨s, hs, hs'⟩ := P.hf_int.sub this x hx
exact ⟨s, hs, hs'.indicator measurableSet_Ioo⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 277,
"column": 4
} | {
"line": 278,
"column": 50
} | [
{
"pp": "case refine_2\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nthis : LocallyIntegrableOn (fun x ↦ (P.ε * ↑(x ^ (-P.k))) • P.g₀) (Ioi 0) volume\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ IntegrableAtFilter ((Ioo 0 1).indicator fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) (𝓝[Ioi... | obtain ⟨s, hs, hs'⟩ := P.hf_int.sub this x hx
exact ⟨s, hs, hs'.indicator measurableSet_Ioo⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 85,
"column": 73
} | {
"line": 93,
"column": 51
} | [
{
"pp": "S T : ℝ\nhT : 0 < T\nz τ : ℂ\nhz : |z.im| ≤ S\nhτ : T ≤ τ.im\nn : ℤ\n⊢ ‖jacobiTheta₂_term n z τ‖ ≤ rexp (-π * (T * ↑n ^ 2 - 2 * S * ↑|n|))",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Distrib.leftDistribClass",
... | by
simp_rw [norm_jacobiTheta₂_term, Real.exp_le_exp, sub_eq_add_neg, neg_mul, ← neg_add,
neg_le_neg_iff, mul_comm (2 : ℝ), mul_assoc π, ← mul_add, mul_le_mul_iff_right₀ pi_pos,
mul_comm T, mul_comm S]
refine add_le_add (mul_le_mul le_rfl hτ hT.le (sq_nonneg _)) ?_
rw [← mul_neg, mul_assoc, mul_assoc, mul_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 41,
"column": 2
} | {
"line": 52,
"column": 45
} | [
{
"pp": "case convert_3\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), a i = 0 ∨ 0 < p i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-p i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\ni : ι\n⊢ Integrable ((fun i t ↦ ↑t ^ (s - 1) * (a ... | · -- integrability of terms
rcases hp i with hai | hpi
· simp [hai]
simp_rw [← mul_assoc, mul_comm _ (a i), mul_assoc]
have := Complex.GammaIntegral_convergent hs
rw [← mul_zero (p i), ← integrableOn_Ioi_comp_mul_left_iff _ _ hpi] at this
refine (IntegrableOn.congr_fun (this.const_mul (1 / p i ^... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 77,
"column": 4
} | {
"line": 83,
"column": 11
} | [
{
"pp": "case h.e'_5.h\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\ni : ι\n⊢... | have : a i / ↑(π * q i) ^ s = π ^ (-s) * a i / q i ^ s := by
rcases hq i with h | h
· simp [h]
· rw [ofReal_mul, mul_cpow_ofReal_nonneg pi_pos.le h.le, ← div_div, cpow_neg,
← div_eq_inv_mul]
simp_rw [mul_div_assoc, this]
ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 77,
"column": 4
} | {
"line": 83,
"column": 11
} | [
{
"pp": "case h.e'_5.h\nι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nq : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhq : ∀ (i : ι), a i = 0 ∨ 0 < q i\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ a i * ↑(rexp (-π * q i * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / q i ^ s.re\nhp : ∀ (i : ι), a i = 0 ∨ 0 < π * q i\ni : ι\n⊢... | have : a i / ↑(π * q i) ^ s = π ^ (-s) * a i / q i ^ s := by
rcases hq i with h | h
· simp [h]
· rw [ofReal_mul, mul_cpow_ofReal_nonneg pi_pos.le h.le, ← div_div, cpow_neg,
← div_eq_inv_mul]
simp_rw [mul_div_assoc, this]
ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 458,
"column": 59
} | {
"line": 458,
"column": 78
} | [
{
"pp": "case e_f.h\nz τ : ℂ\nn : ℤ\n⊢ -((starRingEnd ℂ) 2 * (starRingEnd ℂ) ↑π * (starRingEnd ℂ) I * (starRingEnd ℂ) ↑n *\n (starRingEnd ℂ) (cexp (2 * ↑π * I * ↑n * z + ↑π * I * ↑n ^ 2 * τ))) =\n 2 * ↑π * I * ↑n * cexp (2 * ↑π * I * ↑n * -(starRingEnd ℂ) z + ↑π * I * ↑n ^ 2 * -(starRingEnd ℂ) τ)",
... | ← Complex.exp_conj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 334,
"column": 8
} | {
"line": 334,
"column": 26
} | [
{
"pp": "case e_a\na : UnitAddCircle\ns : ℂ\n⊢ (hurwitzEvenFEPair a).symm.Λ₀ (s / 2) / 2 - (1 / (s / 2)) • (hurwitzEvenFEPair a).symm.f₀ / 2 =\n completedCosZeta₀ a s - 1 / s",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"instHSMul",
"instHDiv",
"Complex.... | completedCosZeta₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 336,
"column": 2
} | {
"line": 338,
"column": 54
} | [
{
"pp": "case e_a\na : UnitAddCircle\ns : ℂ\n⊢ ((hurwitzEvenFEPair a).symm.ε / (↑(hurwitzEvenFEPair a).symm.k - s / 2)) • (hurwitzEvenFEPair a).symm.g₀ / 2 =\n (if a = 0 then 1 else 0) / (1 - s)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
... | · simp_rw [WeakFEPair.symm, hurwitzEvenFEPair, push_cast, inv_one, smul_eq_mul,
mul_comm _ (if _ then _ else _), mul_div_assoc, div_div, ← sub_div,
div_mul_cancel₀ _ (two_ne_zero' ℂ), mul_one_div] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 377,
"column": 33
} | {
"line": 377,
"column": 51
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\n⊢ (hurwitzEvenFEPair a).Λ₀ ((1 - s) / 2) / 2 = completedCosZeta₀ a s",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"instHDiv",
"Complex.instNormedAddCommGroup",
"WeakFEPair.Λ₀",
"congrArg",
"HSub.hSub",
"C... | completedCosZeta₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 99,
"column": 2
} | {
"line": 100,
"column": 60
} | [
{
"pp": "a : UnitAddCircle\n⊢ Tendsto (fun s ↦ hurwitzZetaEven a s - 1 / (s - 1) / s.Gammaℝ + hurwitzZetaOdd a s) (𝓝 1)\n (𝓝 (hurwitzZetaEven a 1 + hurwitzZetaOdd a 1))",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"... | refine (tendsto_hurwitzZetaEven_sub_one_div_nhds_one a).add
(differentiable_hurwitzZetaOdd a 1).continuousAt.tendsto | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 254,
"column": 31
} | {
"line": 254,
"column": 44
} | [
{
"pp": "case e_a\na t : ℝ\nht : 0 < t\nthis :\n HasSum\n (fun n ↦\n -I * ↑↑n * cexp (2 * ↑π * I * ↑a * ↑↑n) * ↑(rexp (-π * ↑↑n ^ 2 * t)) +\n -I * ↑(-↑n) * cexp (2 * ↑π * I * ↑a * ↑(-↑n)) * ↑(rexp (-π * ↑(-↑n) ^ 2 * t)))\n ↑(sinKernel (↑a) t)\nn : ℕ\n⊢ ↑n * ((cexp (-(2 * ↑π * ↑a * ↑n * I)) - ... | mul_comm _ I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 254,
"column": 58
} | {
"line": 254,
"column": 71
} | [
{
"pp": "case e_a\na t : ℝ\nht : 0 < t\nthis :\n HasSum\n (fun n ↦\n -I * ↑↑n * cexp (2 * ↑π * I * ↑a * ↑↑n) * ↑(rexp (-π * ↑↑n ^ 2 * t)) +\n -I * ↑(-↑n) * cexp (2 * ↑π * I * ↑a * ↑(-↑n)) * ↑(rexp (-π * ↑(-↑n) ^ 2 * t)))\n ↑(sinKernel (↑a) t)\nn : ℕ\n⊢ ↑n * I * (cexp (-(2 * ↑π * ↑a * ↑n * I))... | mul_comm _ I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 81,
"column": 56
} | {
"line": 81,
"column": 74
} | [
{
"pp": "s : ℂ\n⊢ completedCosZeta₀ 0 s = (hurwitzEvenFEPair 0).Λ₀ (s / 2) / 2",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHDiv",
"Complex.instNormedAddCommGroup",
"WeakFEPair.Λ₀",
"congrArg",
"AddCommGroup.toAddGroup",
... | completedCosZeta₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 395,
"column": 13
} | {
"line": 395,
"column": 15
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nc : ℤ → ℂ := fun n ↦ -I * cexp (2 * ↑π * I * ↑a * ↑n) / 2\nn : ℤ\n⊢ ‖c n‖ = 1 / 2",
"usedConstants": [
"Norm.norm",
"Real",
"instHDiv",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instNorm",
"id",
"HDi... | c, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 17
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n HasSum\n (fun n ↦\n (s + 1).Gammaℝ * -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑↑n) / ↑|↑n| ^ s / 2 +\n (s + 1).Gammaℝ * -I * ↑(-↑n).sign * cexp (2 * ↑π * I * ↑a * ↑(-↑n)) / ↑|↑n| ^ s / 2)\n (completedSinZeta (↑a) s)\n⊢ HasSum (fun n ↦ (s + 1).Ga... | Int.sign_neg, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.SumPrimeReciprocals | {
"line": 100,
"column": 4
} | {
"line": 104,
"column": 94
} | [
{
"pp": "case neg\nr : ℝ\nh : ¬r < -1\n⊢ (Summable fun p ↦ ↑↑p ^ r) ↔ r < -1",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"Nat.Pr... | simp only [h, iff_false]
refine fun H ↦ Nat.Primes.not_summable_one_div <| H.of_nonneg_of_le (fun _ ↦ by positivity) ?_
intro p
rw [one_div, ← Real.rpow_neg_one]
exact Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast p.prop.one_lt.le) <| not_lt.mp h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.SumPrimeReciprocals | {
"line": 100,
"column": 4
} | {
"line": 104,
"column": 94
} | [
{
"pp": "case neg\nr : ℝ\nh : ¬r < -1\n⊢ (Summable fun p ↦ ↑↑p ^ r) ↔ r < -1",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"MulOne.toOne",
"Real.instPow",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"Nat.Pr... | simp only [h, iff_false]
refine fun H ↦ Nat.Primes.not_summable_one_div <| H.of_nonneg_of_le (fun _ ↦ by positivity) ?_
intro p
rw [one_div, ← Real.rpow_neg_one]
exact Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast p.prop.one_lt.le) <| not_lt.mp h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 441,
"column": 13
} | {
"line": 441,
"column": 15
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nr : ℤ → ℝ := fun n ↦ ↑n + a\nc : ℤ → ℂ := fun n ↦ 1 / 2\nhF : ∀ (t : ℝ), 0 < t → HasSum (fun n ↦ c n * ↑(r n) * ↑(rexp (-π * r n ^ 2 * t))) (↑(oddKernel (↑a) t) / 2)\n⊢ Summable fun i ↦ ‖c i‖ / |r i| ^ s.re",
"usedConstants": [
"Norm.norm",
"Real.instPow",
... | c, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 527,
"column": 20
} | {
"line": 527,
"column": 33
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis :\n HasSum\n (fun n ↦\n -I * ↑(↑n).sign * cexp (2 * ↑π * I * ↑a * ↑↑n) / ↑|↑n| ^ s / 2 +\n -I * ↑(-↑n).sign * cexp (2 * ↑π * I * ↑a * ↑(-↑n)) / ↑|↑n| ^ s / 2)\n (sinZeta (↑a) s + -I * ↑(Int.sign 0) * cexp (2 * ↑π * I * ↑a * ↑0) / ↑|0| ^ s / 2)\n⊢ Has... | Int.sign_neg, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.FLT.Four | {
"line": 146,
"column": 2
} | {
"line": 147,
"column": 48
} | [
{
"pp": "r s : ℤ\nh : IsCoprime r s\n⊢ IsCoprime (r ^ 2 + s ^ 2) (r * s)",
"usedConstants": [
"Eq.mpr",
"isCoprime_comm",
"congrArg",
"id",
"instOfNatNat",
"Int",
"add_comm",
"Int.instMonoid",
"Monoid.toPow",
"instHAdd",
"HPow.hPow",
"H... | apply IsCoprime.mul_right (Int.isCoprime_of_sq_sum (isCoprime_comm.mp h))
rw [add_comm]; apply Int.isCoprime_of_sq_sum h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Four | {
"line": 146,
"column": 2
} | {
"line": 147,
"column": 48
} | [
{
"pp": "r s : ℤ\nh : IsCoprime r s\n⊢ IsCoprime (r ^ 2 + s ^ 2) (r * s)",
"usedConstants": [
"Eq.mpr",
"isCoprime_comm",
"congrArg",
"id",
"instOfNatNat",
"Int",
"add_comm",
"Int.instMonoid",
"Monoid.toPow",
"instHAdd",
"HPow.hPow",
"H... | apply IsCoprime.mul_right (Int.isCoprime_of_sq_sum (isCoprime_comm.mp h))
rw [add_comm]; apply Int.isCoprime_of_sq_sum h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Radical | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 31
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\np : k[X]\nhp : Prime p\ni : ℕ\n⊢ p ^ (i + 1) ∣ normalize p * (C (↑i + 1) * p ^ i) * derivative p",
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"dvd_mul_of_dvd_left",
"Polynomial.derivative",
"Polynomial.C",
... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.Radical | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 29
} | [
{
"pp": "case h₁\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b : k[X]\n⊢ divRadical a ∣ a * derivative b",
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"dvd_mul_of_dvd_left",
"Polynomial.derivative",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.Radical | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 29
} | [
{
"pp": "case h₂\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b : k[X]\n⊢ divRadical a ∣ derivative a * b",
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"dvd_mul_of_dvd_left",
"Polynomial.derivative",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
... | apply dvd_mul_of_dvd_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 48,
"column": 34
} | {
"line": 48,
"column": 56
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\na b c w : k[X]\nhw : w ≠ 0\nwab : w = a.wronskian b\nha : a ≠ 0\nhb : b ≠ 0\nhc : c ≠ 0\nab_nz : a * b ≠ 0\nabc_nz : a * b * c ≠ 0\nabc_dr : k[X] := divRadical (a * b * c)\nabc_dr_dvd_w : abc_dr ∣ w\nabc_dr_ndeg_lt : abc_dr.natDegree < a.natDegree ... | radical_mul_divRadical | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Radical.Basic | {
"line": 385,
"column": 48
} | {
"line": 385,
"column": 70
} | [
{
"pp": "case h\nE : Type u_1\ninst✝² : EuclideanDomain E\ninst✝¹ : NormalizationMonoid E\ninst✝ : UniqueFactorizationMonoid E\na b : E\nhab : IsCoprime a b\n⊢ a * (radical b * divRadical b) = a * b",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"congrArg... | radical_mul_divRadical | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 55
} | [
{
"pp": "z : ℤ\n⊢ z * z % 4 ≠ 2",
"usedConstants": [
"HMul.hMul",
"instHMod",
"instOfNatNat",
"Int",
"Nat.cast",
"Int.instMul",
"HMod.hMod",
"instOfNat",
"Nat",
"Int.instMod",
"instNatCastInt",
"OfNat.ofNat",
"Eq",
"Not",
... | suffices ¬z * z % (4 : ℕ) = 2 % (4 : ℕ) by exact this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 19
} | [
{
"pp": "case h.e'_3\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : ⋯.IsPrimitiveClassified\n⊢ ↑(x.gcd y) ∣ z",
"usedConstants": [
"PythagoreanTriple.gcd_dvd"
]
}
] | exact h.gcd_dvd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 19
} | [
{
"pp": "case h.e'_3\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : ⋯.IsPrimitiveClassified\n⊢ ↑(x.gcd y) ∣ z",
"usedConstants": [
"PythagoreanTriple.gcd_dvd"
]
}
] | exact h.gcd_dvd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 19
} | [
{
"pp": "case h.e'_3\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : ⋯.IsPrimitiveClassified\n⊢ ↑(x.gcd y) ∣ z",
"usedConstants": [
"PythagoreanTriple.gcd_dvd"
]
}
] | exact h.gcd_dvd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 254,
"column": 4
} | {
"line": 258,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nhk : ∀ (x : K), 1 + x ^ 2 ≠ 0\nx : K\n⊢ (fun p ↦ (↑p).1 / ((↑p).2 + 1)) ((fun x ↦ ⟨(2 * x / (1 + x ^ 2), (1 - x ^ 2) / (1 + x ^ 2)), ⋯⟩) x) = x",
"usedConstants": [
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Mathlib.Tact... | have h2 : (1 + 1 : K) = 2 := by norm_num
have h3 : (2 : K) ≠ 0 := by
convert! hk 1
rw [one_pow 2, h2]
simp [field, hk x, h2, add_assoc, add_comm, add_sub_cancel, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 254,
"column": 4
} | {
"line": 258,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nhk : ∀ (x : K), 1 + x ^ 2 ≠ 0\nx : K\n⊢ (fun p ↦ (↑p).1 / ((↑p).2 + 1)) ((fun x ↦ ⟨(2 * x / (1 + x ^ 2), (1 - x ^ 2) / (1 + x ^ 2)), ⋯⟩) x) = x",
"usedConstants": [
"one_pow",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Mathlib.Tact... | have h2 : (1 + 1 : K) = 2 := by norm_num
have h3 : (2 : K) ≠ 0 := by
convert! hk 1
rw [one_pow 2, h2]
simp [field, hk x, h2, add_assoc, add_comm, add_sub_cancel, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 362,
"column": 2
} | {
"line": 364,
"column": 59
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhmn : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\n⊢ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1",
"usedConstants": [
"Int.gcd",
"_private.Mathlib.NumberTheory.PythagoreanTriples.0.coprime_sq_sub_mul_of_even_odd",
"HMul.hMul",
"HSub.hSub",
"instHMod... | rcases hmn with h1 | h2
· exact coprime_sq_sub_mul_of_even_odd h h1.left h1.right
· exact coprime_sq_sub_mul_of_odd_even h h2.left h2.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 362,
"column": 2
} | {
"line": 364,
"column": 59
} | [
{
"pp": "m n : ℤ\nh : m.gcd n = 1\nhmn : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0\n⊢ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1",
"usedConstants": [
"Int.gcd",
"_private.Mathlib.NumberTheory.PythagoreanTriples.0.coprime_sq_sub_mul_of_even_odd",
"HMul.hMul",
"HSub.hSub",
"instHMod... | rcases hmn with h1 | h2
· exact coprime_sq_sub_mul_of_even_odd h h1.left h1.right
· exact coprime_sq_sub_mul_of_odd_even h h2.left h2.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Norm | {
"line": 54,
"column": 6
} | {
"line": 54,
"column": 57
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\ninst✝⁵ : IsDedekindDomain R\ninst✝⁴ : Free ℤ R\ninst✝³ : Module.Finite ℤ R\nK : Type u_2\ninst✝² : CommRing K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nI : FractionalIdeal R⁰ K\na : ↥R⁰\nI₀ : Ideal R\nh : a • I.den • ↑I = I.den • Submodule.map (Algebra.linear... | rw [← map_mul, ← map_mul, mul_comm, ← h', mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 33
} | [
{
"pp": "case refine_1.refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nx : K\na : 𝓞 K\nd : ℤ\nhd : d ∈ ↑ℤ⁰\nh :\n x * (algebraMap (𝓞 K) K) ↑(a, ⟨(algebraMap ℤ (𝓞 K)) d, ⋯⟩).2 = (algebraMap (𝓞 K) K) (a, ⟨(algebraMap ℤ (𝓞 K)) d, ⋯⟩).1\n⊢ (algebraMap (𝓞 K) ... | rw [map_mul, map_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 33
} | [
{
"pp": "case refine_1.refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nx : K\na : 𝓞 K\nd : ℤ\nhd : d ∈ ↑ℤ⁰\nh :\n x * (algebraMap (𝓞 K) K) ↑(a, ⟨(algebraMap ℤ (𝓞 K)) d, ⋯⟩).2 = (algebraMap (𝓞 K) K) (a, ⟨(algebraMap ℤ (𝓞 K)) d, ⋯⟩).1\n⊢ (algebraMap (𝓞 K) ... | rw [map_mul, map_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 33
} | [
{
"pp": "case refine_1.refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nx : K\na : 𝓞 K\nd : ℤ\nhd : d ∈ ↑ℤ⁰\nh :\n x * (algebraMap (𝓞 K) K) ↑(a, ⟨(algebraMap ℤ (𝓞 K)) d, ⋯⟩).2 = (algebraMap (𝓞 K) K) (a, ⟨(algebraMap ℤ (𝓞 K)) d, ⋯⟩).1\n⊢ (algebraMap (𝓞 K) ... | rw [map_mul, map_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : 𝓞 K\nw : InfinitePlace K\nh₁ : x ≠ 0\nh₂ : ∀ ⦃w' : InfinitePlace K⦄, w' ≠ w → w' ↑x < 1\nh₃ : w.IsReal ∨ |(w.embedding ↑x).re| < 1\n⊢ ∀ (x : K), (Polynomial.map (algebraMap ℚ ℂ) (minpoly ℚ x)).Splits",
"usedConstants": [
"Algebra.alg... | · exact fun x ↦ IsAlgClosed.splits _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 633,
"column": 6
} | {
"line": 633,
"column": 54
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nz : (v : InfinitePlace K) → WithAbs ↑v\nr : ℝ\nhr : r > 0\na : InfinitePlace K → K\nhx : ∀ (i : InfinitePlace K), 1 < ↑i (a i) ∧ ∀ (j : InfinitePlace K), j ≠ i → ↑j (a i) < 1\ny : ℕ → K := fun n ↦ ∑ v, 1 / (1 + (a v)⁻¹ ^ n) * (WithAbs.equ... | have := u.1.tendsto_div_one_add_pow_nhds_zero hu | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 95,
"column": 5
} | {
"line": 95,
"column": 55
} | [
{
"pp": "k : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh : (w.comap f).IsReal\n⊢ ComplexEmbedding.IsReal (w.embedding.comp f)",
"usedConstants": [
"Eq.mpr",
"NumberField.InfinitePlace.comap_mk",
"NumberField.ComplexEmbedding.IsReal",
... | by rwa [← isReal_mk_iff, ← comap_mk, mk_embedding] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 342,
"column": 2
} | {
"line": 344,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Convex ℝ (convexBodySum K B)",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NumberField.mixedEmbedding.convexBodySumFun_add_le",
"NumberFi... | refine Convex_subadditive_le (fun _ _ => convexBodySumFun_add_le _ _) (fun c x h => ?_) B
convert! le_of_eq (convexBodySumFun_smul c x)
exact (abs_eq_self.mpr h).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 342,
"column": 2
} | {
"line": 344,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\n⊢ Convex ℝ (convexBodySum K B)",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NumberField.mixedEmbedding.convexBodySumFun_add_le",
"NumberFi... | refine Convex_subadditive_le (fun _ _ => convexBodySumFun_add_le _ _) (fun c x h => ?_) B
convert! le_of_eq (convexBodySumFun_smul c x)
exact (abs_eq_self.mpr h).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 32
} | [
{
"pp": "case pos\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : ¬(w.comap f).IsReal\nh₂ : w.IsReal\n⊢ 2 ≤ 1",
"usedConstants": [
"False.elim",
"NumberField.InfinitePlace.IsReal.comap",
"instOfNatNat",
"LE.le",
"instLENat"... | exact (h₁ (h₂.comap _)).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 32
} | [
{
"pp": "case pos\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : ¬(w.comap f).IsReal\nh₂ : w.IsReal\n⊢ 2 ≤ 1",
"usedConstants": [
"False.elim",
"NumberField.InfinitePlace.IsReal.comap",
"instOfNatNat",
"LE.le",
"instLENat"... | exact (h₁ (h₂.comap _)).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 32
} | [
{
"pp": "case pos\nk : Type u_1\ninst✝¹ : Field k\nK : Type u_2\ninst✝ : Field K\nf : k →+* K\nw : InfinitePlace K\nh₁ : ¬(w.comap f).IsReal\nh₂ : w.IsReal\n⊢ 2 ≤ 1",
"usedConstants": [
"False.elim",
"NumberField.InfinitePlace.IsReal.comap",
"instOfNatNat",
"LE.le",
"instLENat"... | exact (h₁ (h₂.comap _)).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 369,
"column": 2
} | {
"line": 379,
"column": 10
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\n⊢ Nat.card ↥(Stab w) = 1 ∨ Nat.card ↥(Stab w) = 2",
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Set.ext",
"Eq.mpr",
"Fintype.card_ofFinset",
"Fal... | rw [← SetLike.coe_sort_coe, ← mk_embedding w]
by_cases! h : ∃ σ, ComplexEmbedding.IsConj (k := k) (embedding w) σ
· obtain ⟨σ, hσ⟩ := h
rw [hσ.coe_stabilizer_mk]
simp
· left
trans Nat.card ({1} : Set Gal(K/k))
· congr with x
simp only [SetLike.mem_coe, mem_stabilizer_mk_iff, Set.mem_singleto... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 369,
"column": 2
} | {
"line": 379,
"column": 10
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra k K\nw : InfinitePlace K\n⊢ Nat.card ↥(Stab w) = 1 ∨ Nat.card ↥(Stab w) = 2",
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Set.ext",
"Eq.mpr",
"Fintype.card_ofFinset",
"Fal... | rw [← SetLike.coe_sort_coe, ← mk_embedding w]
by_cases! h : ∃ σ, ComplexEmbedding.IsConj (k := k) (embedding w) σ
· obtain ⟨σ, hσ⟩ := h
rw [hσ.coe_stabilizer_mk]
simp
· left
trans Nat.card ({1} : Set Gal(K/k))
· congr with x
simp only [SetLike.mem_coe, mem_stabilizer_mk_iff, Set.mem_singleto... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 43
} | [
{
"pp": "k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\n⊢ Set.MapsTo (fun x ↦ x.comap (algebraMap k K)) ↑{w | IsUnramified k w} ↑{w | IsUnramifiedIn K w}",
"usedConstants": [
"Finset.coe_filter... | · simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 43
} | [
{
"pp": "k : Type u_1\ninst✝⁵ : Field k\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : NumberField K\ninst✝¹ : NumberField k\ninst✝ : IsGalois k K\n⊢ Set.MapsTo (fun x ↦ x.comap (algebraMap k K)) ↑{w | IsUnramified k w}ᶜ ↑{w | IsUnramifiedIn K w}ᶜ",
"usedConstants": [
"instDecidableNo... | · simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 101,
"column": 4
} | {
"line": 109,
"column": 56
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nr : ℝ\nh✝ : 0 ≤ r\n⊢ (↑(integerLattice K) ∩ Metric.closedBall 0 r).Finite",
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Set.ext"... | have heq : ∀ x, canonicalEmbedding K x ∈ Metric.closedBall 0 r ↔
∀ φ : K →+* ℂ, ‖φ x‖ ≤ r := by
intro x; rw [← norm_le_iff, mem_closedBall_zero_iff]
convert! (Embeddings.finite_of_norm_le K ℂ r).image (canonicalEmbedding K)
ext; constructor
· rintro ⟨⟨_, ⟨x, rfl⟩, rfl⟩, hx⟩
exact ⟨x, ⟨Se... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 101,
"column": 4
} | {
"line": 109,
"column": 56
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nr : ℝ\nh✝ : 0 ≤ r\n⊢ (↑(integerLattice K) ∩ Metric.closedBall 0 r).Finite",
"usedConstants": [
"Subalgebra.instSetLike",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Set.ext"... | have heq : ∀ x, canonicalEmbedding K x ∈ Metric.closedBall 0 r ↔
∀ φ : K →+* ℂ, ‖φ x‖ ≤ r := by
intro x; rw [← norm_le_iff, mem_closedBall_zero_iff]
convert! (Embeddings.finite_of_norm_le K ℂ r).image (canonicalEmbedding K)
ext; constructor
· rintro ⟨⟨_, ⟨x, rfl⟩, rfl⟩, hx⟩
exact ⟨x, ⟨Se... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Embeddings | {
"line": 55,
"column": 45
} | {
"line": 56,
"column": 50
} | [
{
"pp": "case pos\nn : ℕ\ninst✝² : NeZero n\nK : Type u\ninst✝¹ : Field K\ninst✝ : CharZero K\nh : IsCyclotomicExtension {n} ℚ K\nthis : NumberField K\nhn : 2 < n\nk : ℕ\nhk : φ n = k + k\nkey : 2 * nrComplexPlaces K = Module.finrank ℚ K\n⊢ nrComplexPlaces K = φ n / 2",
"usedConstants": [
"Nat.instMul... | IsCyclotomicExtension.finrank (n := n) K
(cyclotomic.irreducible_rat (NeZero.pos _)), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 580,
"column": 79
} | {
"line": 581,
"column": 35
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ ∏ _k, 2⁻¹ * I = (2⁻¹ * I) ^ Fintype.card { w // w.IsComplex }",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.univ",
"Complex.commRing",
"congrArg",
"NumberField.InfinitePlace.IsComplex",
"Nat.ins... | by
rw [prod_const, Fintype.card] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 599,
"column": 6
} | {
"line": 602,
"column": 48
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nw : { w // w.IsReal }\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inl w)) =\n ∑ x_1,\n ... | simp_rw [stdBasis_apply_isReal, fromBlocks_apply₁₁, fromBlocks_apply₁₂,
one_apply, Matrix.zero_apply, ite_mul, one_mul, zero_mul, sum_ite_eq, mem_univ, ite_true,
add_zero, sum_const_zero, add_zero, ← conj_eq_iff_re, hx (embedding w.val),
conjugate_embedding_eq_of_isReal w.prop] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 599,
"column": 6
} | {
"line": 602,
"column": 48
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nw : { w // w.IsReal }\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inl w)) =\n ∑ x_1,\n ... | simp_rw [stdBasis_apply_isReal, fromBlocks_apply₁₁, fromBlocks_apply₁₂,
one_apply, Matrix.zero_apply, ite_mul, one_mul, zero_mul, sum_ite_eq, mem_univ, ite_true,
add_zero, sum_const_zero, add_zero, ← conj_eq_iff_re, hx (embedding w.val),
conjugate_embedding_eq_of_isReal w.prop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 599,
"column": 6
} | {
"line": 602,
"column": 48
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : (K →+* ℂ) → ℂ\nhx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)\nw : { w // w.IsReal }\n⊢ ↑(((stdBasis K).repr (fun w ↦ (x (↑w).embedding).re, fun w ↦ x (↑w).embedding)) (Sum.inl w)) =\n ∑ x_1,\n ... | simp_rw [stdBasis_apply_isReal, fromBlocks_apply₁₁, fromBlocks_apply₁₂,
one_apply, Matrix.zero_apply, ite_mul, one_mul, zero_mul, sum_ite_eq, mem_univ, ite_true,
add_zero, sum_const_zero, add_zero, ← conj_eq_iff_re, hx (embedding w.val),
conjugate_embedding_eq_of_isReal w.prop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 431,
"column": 10
} | {
"line": 432,
"column": 38
} | [
{
"pp": "case refine_2.refine_2.h₂.a\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := ⋯\nB : ℝ≥0 := ⋯\nC : ℕ := ⋯\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}\nhK₂ : |discr ↥↑... | exact (Nat.choose_le_choose _ (rank_le_rankOfDiscrBdd hK₂)).trans
(Nat.choose_le_middle _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 431,
"column": 10
} | {
"line": 432,
"column": 38
} | [
{
"pp": "case refine_2.refine_2.h₂.a\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := ⋯\nB : ℝ≥0 := ⋯\nC : ℕ := ⋯\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}\nhK₂ : |discr ↥↑... | exact (Nat.choose_le_choose _ (rank_le_rankOfDiscrBdd hK₂)).trans
(Nat.choose_le_middle _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 431,
"column": 10
} | {
"line": 432,
"column": 38
} | [
{
"pp": "case refine_2.refine_2.h₂.a\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := ⋯\nB : ℝ≥0 := ⋯\nC : ℕ := ⋯\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}\nhK₂ : |discr ↥↑... | exact (Nat.choose_le_choose _ (rank_le_rankOfDiscrBdd hK₂)).trans
(Nat.choose_le_middle _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Extended | {
"line": 119,
"column": 72
} | {
"line": 121,
"column": 78
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : CommRing A\nB : Type u_2\ninst✝⁶ : CommRing B\nf : A →+* B\nK : Type u_3\nM : Submonoid A\ninst✝⁵ : CommRing K\ninst✝⁴ : Algebra A K\ninst✝³ : IsLocalization M K\nL : Type u_4\nN : Submonoid B\ninst✝² : CommRing L\ninst✝¹ : Algebra B L\ninst✝ : IsLocalization N L\nhf : M ≤ Submon... | by
rw [one_le] at hI ⊢
exact (mem_extended_iff _ _ _ _).mpr <| subset_span ⟨1, hI, by rw [map_one]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Quotient | {
"line": 74,
"column": 2
} | {
"line": 86,
"column": 75
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\np : Ideal R\ninst✝¹¹ : p.IsMaximal\nRₚ : Type u_3\nSₚ : Type u_4\ninst✝¹⁰ : CommRing Rₚ\ninst✝⁹ : CommRing Sₚ\ninst✝⁸ : Algebra R Rₚ\ninst✝⁷ : IsLocalization.AtPrime Rₚ p\ninst✝⁶ : IsLocalRing Rₚ\ninst✝⁵ : Al... | have : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ) := by
apply IsScalarTower.of_algebraMap_eq'
rw [IsScalarTower.algebraMap_eq R Rₚ (Rₚ ⧸ _), IsScalarTower.algebraMap_eq R Rₚ (Sₚ ⧸ _),
← RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq Rₚ]
rw [Algebra.trace_eq_of_equiv_equiv (equivQuotMaximalIdea... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Quotient | {
"line": 74,
"column": 2
} | {
"line": 86,
"column": 75
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\np : Ideal R\ninst✝¹¹ : p.IsMaximal\nRₚ : Type u_3\nSₚ : Type u_4\ninst✝¹⁰ : CommRing Rₚ\ninst✝⁹ : CommRing Sₚ\ninst✝⁸ : Algebra R Rₚ\ninst✝⁷ : IsLocalization.AtPrime Rₚ p\ninst✝⁶ : IsLocalRing Rₚ\ninst✝⁵ : Al... | have : IsScalarTower R (Rₚ ⧸ maximalIdeal Rₚ) (Sₚ ⧸ pSₚ) := by
apply IsScalarTower.of_algebraMap_eq'
rw [IsScalarTower.algebraMap_eq R Rₚ (Rₚ ⧸ _), IsScalarTower.algebraMap_eq R Rₚ (Sₚ ⧸ _),
← RingHom.comp_assoc, ← IsScalarTower.algebraMap_eq Rₚ]
rw [Algebra.trace_eq_of_equiv_equiv (equivQuotMaximalIdea... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1132,
"column": 4
} | {
"line": 1132,
"column": 46
} | [
{
"pp": "case h.inr\nK : Type u_1\ninst✝ : Field K\nx✝ : realSpace K\nh : (∀ (x : { w // w.IsReal }), x✝ ↑x = 0 x) ∧ ∀ (x : { w // w.IsComplex }), ↑(x✝ ↑x) = 0 x\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ x✝ w = 0 w",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real",
"NumberField.m... | exact Complex.ofReal_inj.mp <| h.2 ⟨w, hw⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1132,
"column": 4
} | {
"line": 1132,
"column": 46
} | [
{
"pp": "case h.inr\nK : Type u_1\ninst✝ : Field K\nx✝ : realSpace K\nh : (∀ (x : { w // w.IsReal }), x✝ ↑x = 0 x) ∧ ∀ (x : { w // w.IsComplex }), ↑(x✝ ↑x) = 0 x\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ x✝ w = 0 w",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real",
"NumberField.m... | exact Complex.ofReal_inj.mp <| h.2 ⟨w, hw⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1132,
"column": 4
} | {
"line": 1132,
"column": 46
} | [
{
"pp": "case h.inr\nK : Type u_1\ninst✝ : Field K\nx✝ : realSpace K\nh : (∀ (x : { w // w.IsReal }), x✝ ↑x = 0 x) ∧ ∀ (x : { w // w.IsComplex }), ↑(x✝ ↑x) = 0 x\nw : InfinitePlace K\nhw : w.IsComplex\n⊢ x✝ w = 0 w",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real",
"NumberField.m... | exact Complex.ofReal_inj.mp <| h.2 ⟨w, hw⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1138,
"column": 4
} | {
"line": 1138,
"column": 63
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\nx : realSpace K\nw : InfinitePlace K\nhx : 0 ≤ x w\nhw : w.IsReal\n⊢ (normAtPlace w) (fun w ↦ x ↑w, fun w ↦ ↑(x ↑w)) = x w",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"NumberField.mixedEmbedding.normAtPlace_a... | rw [normAtPlace_apply_of_isReal hw, Real.norm_of_nonneg hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1138,
"column": 4
} | {
"line": 1138,
"column": 63
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\nx : realSpace K\nw : InfinitePlace K\nhx : 0 ≤ x w\nhw : w.IsReal\n⊢ (normAtPlace w) (fun w ↦ x ↑w, fun w ↦ ↑(x ↑w)) = x w",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"NumberField.mixedEmbedding.normAtPlace_a... | rw [normAtPlace_apply_of_isReal hw, Real.norm_of_nonneg hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1138,
"column": 4
} | {
"line": 1138,
"column": 63
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\nx : realSpace K\nw : InfinitePlace K\nhx : 0 ≤ x w\nhw : w.IsReal\n⊢ (normAtPlace w) (fun w ↦ x ↑w, fun w ↦ ↑(x ↑w)) = x w",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"NumberField.mixedEmbedding.normAtPlace_a... | rw [normAtPlace_apply_of_isReal hw, Real.norm_of_nonneg hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 1212,
"column": 45
} | {
"line": 1218,
"column": 70
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx y : mixedSpace K\nh : normAtComplexPlaces x = normAtComplexPlaces y\n⊢ normAtAllPlaces x = normAtAllPlaces y",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NumberField.InfinitePlace.isReal_or_isComplex",
"Real",
"Real.lattice",
"abs... | by
ext w
obtain hw | hw := isReal_or_isComplex w
· simpa [normAtAllPlaces_apply, normAtPlace_apply_of_isReal hw,
normAtComplexPlaces_apply_isReal ⟨w, hw⟩] using congr_arg (|·|) (congr_fun h w)
· simpa [normAtAllPlaces_apply, normAtPlace_apply_of_isComplex hw,
normAtComplexPlaces_apply_isComplex ⟨w, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.PID | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 13
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\nP : Ideal R\nhP : P.IsPrime\ninst✝ : IsDedekindDomain R\nx : R\nx_mem : x ∈ P\nhxP2 : x ∉ P ^ 2\nhxQ : ∀ (Q : Ideal R), Q.IsPrime → Q ≠ P → x ∉ Q\nhP0 : P = ⊥\n⊢ P = span {x}",
"usedConstants": []
}
] | subst hP0 | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 481,
"column": 10
} | {
"line": 482,
"column": 38
} | [
{
"pp": "case refine_2.refine_2.h₂\nA : Type u_2\ninst✝¹ : Field A\ninst✝ : CharZero A\nN : ℕ\nD : ℕ := rankOfDiscrBdd N\nB : ℝ≥0 := boundOfDiscBdd N\nC : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx... | exact (Nat.choose_le_choose _ (rank_le_rankOfDiscrBdd hK₂)).trans
(Nat.choose_le_middle _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 96,
"column": 40
} | {
"line": 96,
"column": 59
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁴⁶ : CommRing A\ninst✝⁴⁵ : Field K\ninst✝⁴⁴ : Algebra A K\ninst✝⁴³ : IsFractionRing A K\ninst✝⁴² : CommRing B\ninst✝⁴¹ : Field L\ninst✝⁴⁰ : Algebra B L\ninst✝³⁹ : Algebra A L\ninst✝³⁸ : Algebra K L\ninst✝³⁷ : FiniteDimensional K L\ninst✝³⁶ : ... | span_span_of_tower, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 96,
"column": 60
} | {
"line": 96,
"column": 79
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁴⁶ : CommRing A\ninst✝⁴⁵ : Field K\ninst✝⁴⁴ : Algebra A K\ninst✝⁴³ : IsFractionRing A K\ninst✝⁴² : CommRing B\ninst✝⁴¹ : Field L\ninst✝⁴⁰ : Algebra B L\ninst✝³⁹ : Algebra A L\ninst✝³⁸ : Algebra K L\ninst✝³⁷ : FiniteDimensional K L\ninst✝³⁶ : ... | span_span_of_tower, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Norm.RelNorm | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 13
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : IsDomain R\nS : Type u_3\ninst✝⁹ : CommRing S\ninst✝⁸ : IsDomain S\ninst✝⁷ : IsIntegrallyClosed R\ninst✝⁶ : IsIntegrallyClosed S\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module.Finite R S\ninst✝³ : IsTorsionFree R S\ninst✝² : IsDedekindDomain R\ninst✝¹ : I... | subst hP0 | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.RingTheory.DedekindDomain.LinearDisjoint | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 23
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝⁵⁵ : CommRing A\ninst✝⁵⁴ : Field K\ninst✝⁵³ : Algebra A K\ninst✝⁵² : IsFractionRing A K\ninst✝⁵¹ : CommRing B\ninst✝⁵⁰ : Field L\ninst✝⁴⁹ : Algebra B L\ninst✝⁴⁸ : Algebra A L\ninst✝⁴⁷ : Algebra K L\ninst✝⁴⁶ : FiniteDimensional K L\ninst✝⁴⁵ : ... | span_span_of_tower, | Lean.Elab.Tactic.evalRewriteSeq | null |
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