module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 43,
"column": 2
} | {
"line": 56,
"column": 48
} | {
"line": 58,
"column": 0
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : b.natDegree ≤ d\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\n⊢ ∃ i₀ i₁, i₀ ≠ i₁ ∧ A i₁ = A i₀",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants":... | [] | set f : Fin m.succ → Fin d → Fq := fun i j => (A i).coeff j
have : Fintype.card (Fin d → Fq) < Fintype.card (Fin m.succ) := by
simpa using lt_of_le_of_lt hm (Nat.lt_succ_self m)
-- Therefore, the differences have all coefficients higher than `deg b - d` equal.
obtain ⟨i₀, i₁, i_ne, i_eq⟩ := Fintype.exists_ne_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 58
} | {
"line": 274,
"column": 4
} | [
{
"pp": "n : ℕ\n⊢ ∑ m ∈ Icc 1 n, Λ m =\n ∑ m ∈ (filter Nat.Prime (Icc 1 n)).biUnion fun p ↦ image (fun x ↦ p ^ x) (Icc 1 (Nat.log p n)), Λ m",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Real",
"Nat.Prime",
"ArithmeticFunction... | [
"case refine_1\nn q : ℕ\nhq : q ∈ (filter Nat.Prime (Icc 1 n)).biUnion fun p ↦ image (fun x ↦ p ^ x) (Icc 1 (Nat.log p n))\n⊢ q ∈ Icc 1 n",
"case refine_2\nn x : ℕ\nhx : x ∈ Icc 1 n\n⊢ (x ∉ (filter Nat.Prime (Icc 1 n)).biUnion fun p ↦ image (fun x ↦ p ^ x) (Icc 1 (Nat.log p n))) → Λ x = 0"
] | refine (sum_subset (fun q hq ↦ ?_) fun x hx ↦ ?_).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Chebyshev | {
"line": 344,
"column": 6
} | {
"line": 345,
"column": 82
} | {
"line": 346,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nN : ℕ\nhN : ⌊log x / log 2⌋₊ ≤ N\n⊢ ∀ (a : ℕ × ℕ)\n (ha :\n a ∈\n {x_1 ∈ Icc 1 N ×ˢ filter Nat.Prime (Ioc 0 ⌊x⌋₊) |\n match x_1 with\n | (k, p) => p ≤ ⌊x ^ (↑k)⁻¹⌋₊}),\n (match a, ha ... | [] | simp +contextual [hx, rpow_nonneg, le_floor_iff, ← pos_iff_ne_zero, Prime.isPrimePow,
one_le_iff_ne_zero, le_rpow_inv_iff_of_pos, isPrimePow_pow_iff, prime_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Chebyshev | {
"line": 344,
"column": 6
} | {
"line": 345,
"column": 82
} | {
"line": 346,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nN : ℕ\nhN : ⌊log x / log 2⌋₊ ≤ N\n⊢ ∀ (a : ℕ × ℕ)\n (ha :\n a ∈\n {x_1 ∈ Icc 1 N ×ˢ filter Nat.Prime (Ioc 0 ⌊x⌋₊) |\n match x_1 with\n | (k, p) => p ≤ ⌊x ^ (↑k)⁻¹⌋₊}),\n (match a, ha ... | [] | simp +contextual [hx, rpow_nonneg, le_floor_iff, ← pos_iff_ne_zero, Prime.isPrimePow,
one_le_iff_ne_zero, le_rpow_inv_iff_of_pos, isPrimePow_pow_iff, prime_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Chebyshev | {
"line": 344,
"column": 6
} | {
"line": 345,
"column": 82
} | {
"line": 346,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nN : ℕ\nhN : ⌊log x / log 2⌋₊ ≤ N\n⊢ ∀ (a : ℕ × ℕ)\n (ha :\n a ∈\n {x_1 ∈ Icc 1 N ×ˢ filter Nat.Prime (Ioc 0 ⌊x⌋₊) |\n match x_1 with\n | (k, p) => p ≤ ⌊x ^ (↑k)⁻¹⌋₊}),\n (match a, ha ... | [] | simp +contextual [hx, rpow_nonneg, le_floor_iff, ← pos_iff_ne_zero, Prime.isPrimePow,
one_le_iff_ne_zero, le_rpow_inv_iff_of_pos, isPrimePow_pow_iff, prime_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bernoulli | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 79
} | {
"line": 633,
"column": 2
} | [
{
"pp": "case h\nk p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\nhcast : ↑(∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = 0\nT : ℤ\nhT_int : (∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = ↑p * T\nhT : ∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + vonStaudtIndicator (2 * k) p = ↑p * ↑T\n⊢ ber... | [
"case h\nk p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\nhcast : ↑(∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = 0\nT : ℤ\nhT_int : (∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + if p - 1 ∣ 2 * k then 1 else 0) = ↑p * T\nhT : ∑ v ∈ Ico 1 p, ↑v ^ (2 * k) + vonStaudtIndicator (2 * k) p = ↑p * ↑T\nhp_ne : ↑p ≠ 0\n⊢... | have hp_ne : (p : ℚ) ≠ 0 := Nat.cast_ne_zero.mpr (Fact.out : p.Prime).ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Chebyshev | {
"line": 460,
"column": 4
} | {
"line": 460,
"column": 44
} | {
"line": 461,
"column": 4
} | [
{
"pp": "case pos\nx : ℝ\nhx✝ : 0 ≤ x\nhx : x < 1\n⊢ ψ x ≤ (log 4 + 4) * x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne... | [
"case pos\nx : ℝ\nhx✝ : 0 ≤ x\nhx : x < 1\n⊢ 0 ≤ (log 4 + 4) * x"
] | rw [psi_eq_zero_of_lt_two (by linarith)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 218,
"column": 43
} | {
"line": 218,
"column": 59
} | {
"line": 218,
"column": 60
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nH : x.zmodRepr = 0\n⊢ ‖x - 0‖ < 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"PadicInt",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMon... | [
"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nH : x.zmodRepr = 0\n⊢ ‖x - ↑0‖ < 1"
] | ← Nat.cast_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 15
} | {
"line": 297,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : EuclideanDomain R\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝³ : Infinite R\ninst✝² : DecidableEq R\ninst✝¹ : IsDe... | [] | exact b_mem | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 298,
"column": 17
} | {
"line": 298,
"column": 19
} | {
"line": 299,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : EuclideanDomain R\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝³ : Infinite R\ninst✝² : DecidableEq R\ninst✝¹ : IsDe... | [
"R : Type u_1\nS : Type u_2\ninst✝⁹ : EuclideanDomain R\ninst✝⁸ : CommRing S\ninst✝⁷ : IsDomain S\ninst✝⁶ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝³ : Infinite R\ninst✝² : DecidableEq R\ninst✝¹ : IsDedekindDomain... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 354,
"column": 30
} | {
"line": 372,
"column": 45
} | {
"line": 374,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nK : Type u_3\nL : Type u_4\ninst✝¹⁷ : EuclideanDomain R\ninst✝¹⁶ : CommRing S\ninst✝¹⁵ : IsDomain S\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra R K\ninst✝¹¹ : IsFractionRing R K\ninst✝¹⁰ : Algebra K L\ninst✝⁹ : FiniteDimensional K L\ninst✝⁸ : Algebra.IsSeparable... | [] | by
letI := Classical.decEq L
letI := IsIntegralClosure.isFractionRing_of_finite_extension R K L S
letI := IsIntegralClosure.isDedekindDomain R K L S
choose s b hb_int using FiniteDimensional.exists_is_basis_integral R K L
have : LinearIndependent R ((Algebra.traceForm K L).dualBasis
(traceForm_nondegene... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 454,
"column": 18
} | {
"line": 454,
"column": 20
} | {
"line": 454,
"column": 21
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ : ℤ_[p]\nn : ℕ\nx : ℤ_[p]\na b : ℕ\n⊢ x - ↑a ∈ Ideal.span {↑(p ^ n)} → x - ↑b ∈ Ideal.span {↑(p ^ n)} → ↑a = ↑b",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"PadicInt",
"AddGroupWithOne.toAdd... | [
"p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx✝ : ℤ_[p]\nn : ℕ\nx : ℤ_[p]\na b : ℕ\nha : x - ↑a ∈ Ideal.span {↑(p ^ n)}\n⊢ x - ↑b ∈ Ideal.span {↑(p ^ n)} → ↑a = ↑b"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 134,
"column": 49
} | {
"line": 134,
"column": 64
} | {
"line": 134,
"column": 65
} | [
{
"pp": "L : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ng : L ≃+* L\nn : ℕ\ninst✝ : NeZero n\nt : ↥(rootsOfUnity n L)\n⊢ ↑(↑t ^ aux g n) = ↑(↑t ^ (χ₀ n g).val)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Units.val",
"Eq.mpr",
"modularCycl... | [
"L : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\ng : L ≃+* L\nn : ℕ\ninst✝ : NeZero n\nt : ↥(rootsOfUnity n L)\n⊢ ↑(↑t ^ aux g n) = ↑(↑t ^ ↑(χ₀ n g).val)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 819,
"column": 56
} | {
"line": 821,
"column": 12
} | {
"line": 823,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 1 < x\n⊢ ((x - 1) * log 2 - log (x + 2)) / log x ≤ ↑(π ⌊x⌋₊)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_le_iff₀",
"le_refl",
"Real.partialOrder",
"... | [] | by
grw [div_le_iff₀ (log_pos hx), ← psi_le_primeCounting_mul_log', psi_ge']
positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 365,
"column": 6
} | {
"line": 365,
"column": 57
} | {
"line": 366,
"column": 0
} | [
{
"pp": "case pos.refine_2\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : ∀ (i : ℕ), HasEnoughRootsOfUnity L (p ^ i)\nζ : ℕ → L\nhζ : ∀ (i : ℕ), IsPrimitiveRoot (ζ i) (p ^ i)\nε : ℝ\nhε : 0 < ε\nk : ℕ\nhk' : k ≠ 0\nhk : ↑p ^ (-↑k) <... | [] | exact (one_lt_pow₀ ‹Fact p.Prime›.1.one_lt hk').ne' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 365,
"column": 6
} | {
"line": 365,
"column": 57
} | {
"line": 366,
"column": 0
} | [
{
"pp": "case pos.refine_2\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : ∀ (i : ℕ), HasEnoughRootsOfUnity L (p ^ i)\nζ : ℕ → L\nhζ : ∀ (i : ℕ), IsPrimitiveRoot (ζ i) (p ^ i)\nε : ℝ\nhε : 0 < ε\nk : ℕ\nhk' : k ≠ 0\nhk : ↑p ^ (-↑k) <... | [] | exact (one_lt_pow₀ ‹Fact p.Prime›.1.one_lt hk').ne' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 365,
"column": 6
} | {
"line": 365,
"column": 57
} | {
"line": 366,
"column": 0
} | [
{
"pp": "case pos.refine_2\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx✝ : ∀ (i : ℕ), HasEnoughRootsOfUnity L (p ^ i)\nζ : ℕ → L\nhζ : ∀ (i : ℕ), IsPrimitiveRoot (ζ i) (p ^ i)\nε : ℝ\nhε : 0 < ε\nk : ℕ\nhk' : k ≠ 0\nhk : ↑p ^ (-↑k) <... | [] | exact (one_lt_pow₀ ‹Fact p.Prime›.1.one_lt hk').ne' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 215,
"column": 13
} | {
"line": 217,
"column": 51
} | {
"line": 219,
"column": 0
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\n⊢ IsPell (pellZd a1 (n + 1))",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Zsqrtd.instMul",
"Eq.mpr",
"HMul.hMul",
"Pell.isPell_mul",
"congrArg",
"Pell.pellZd_succ",
"id",
"_private.Mathlib.NumberTheory.P... | [] | by
let o := isPell_one a1
simpa using Pell.isPell_mul (isPell_pellZd n) o | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 299,
"column": 8
} | {
"line": 307,
"column": 48
} | {
"line": 309,
"column": 0
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\na1p : 0 ≤ { re := ↑a, im := 1 }\nam1p : 0 ≤ { re := ↑a, im := -1 }\na1m : { re := ↑a, im := 1 } * { re := ↑a, im := -1 } = 1\nx y : ℤ\nh1 : 1 ≤ { re := x, im := y }\nhp : IsPell { re := x, im := y }\nh : { re := x, im := y } ≤ pellZd a1 (n + 1)\nha : ¬{ re := ↑a, im := 1 } ≤ {... | [] | exact
match y, y0l, (yl2 : (⟨_, _⟩ : ℤ√_) < ⟨_, _⟩) with
| 0, y0l, _ => y0l (le_refl 0)
| (y + 1 : ℕ), _, yl2 =>
yl2
(Zsqrtd.le_of_le_le (by simp)
(let t := Int.ofNat_le_ofNat_of_le (Nat.succ_pos y)
add_le_add t t))
| Int.... | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 571,
"column": 2
} | {
"line": 571,
"column": 58
} | {
"line": 573,
"column": 0
} | [
{
"pp": "d x y z w : ℕ\nxy : { re := ↑x, im := -↑y }.Nonneg\nzw : { re := -↑z, im := ↑w }.Nonneg\nthis : { re := subNatNat x z, im := subNatNat w y }.Nonneg\n⊢ { re := ↑x + -↑z, im := ↑w - ↑y }.Nonneg",
"ppTerm": "?m.187",
"assigned": true,
"usedConstants": [
"congrArg",
"HSub.hSub",
... | [] | rwa [Int.subNatNat_eq_coe, Int.subNatNat_eq_coe] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 577,
"column": 4
} | {
"line": 577,
"column": 81
} | {
"line": 578,
"column": 4
} | [
{
"pp": "case inl.inr.inl\nd x y : ℕ\nha : { re := ↑x, im := ↑y }.Nonneg\nz w : ℕ\nhb : { re := ↑z, im := -↑w }.Nonneg\n⊢ ({ re := ↑x, im := ↑y } + { re := ↑z, im := -↑w }).Nonneg",
"ppTerm": "?inl.inr.inl",
"assigned": true,
"usedConstants": [
"Zsqrtd.sqLe_of_le",
"Zsqrtd.nonnegg_cases_... | [
"case inl.inr.inl.refine_1\nd x y : ℕ\nha : { re := ↑x, im := ↑y }.Nonneg\nz w : ℕ\nhb : { re := ↑z, im := -↑w }.Nonneg\ni : ℕ\nh : { re := ↑x, im := ↑y }.im + { re := ↑z, im := -↑w }.im = -↑i\n⊢ i ≤ w",
"case inl.inr.inl.refine_2\nd x y : ℕ\nha : { re := ↑x, im := ↑y }.Nonneg\nz w : ℕ\nhb : { re := ↑z, im := -↑w... | refine nonnegg_cases_right fun i h => sqLe_of_le ?_ ?_ (nonnegg_pos_neg.1 hb) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 520,
"column": 12
} | {
"line": 520,
"column": 25
} | {
"line": 521,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ny : ℕ\n⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 0 * (↑a - ↑y) + ↑(y ^ 0) - xz a1 0",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"sub_self",
"Monoid.toMulOne... | [] | simp [xz, yz] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 520,
"column": 12
} | {
"line": 520,
"column": 25
} | {
"line": 521,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ny : ℕ\n⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 0 * (↑a - ↑y) + ↑(y ^ 0) - xz a1 0",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"sub_self",
"Monoid.toMulOne... | [] | simp [xz, yz] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 520,
"column": 12
} | {
"line": 520,
"column": 25
} | {
"line": 521,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ny : ℕ\n⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 0 * (↑a - ↑y) + ↑(y ^ 0) - xz a1 0",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"sub_self",
"Monoid.toMulOne... | [] | simp [xz, yz] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 521,
"column": 12
} | {
"line": 521,
"column": 25
} | {
"line": 522,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ny : ℕ\n⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 1 * (↑a - ↑y) + ↑(y ^ 1) - xz a1 1",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"sub_self",
"Pell.xn... | [] | simp [xz, yz] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 521,
"column": 12
} | {
"line": 521,
"column": 25
} | {
"line": 522,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ny : ℕ\n⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 1 * (↑a - ↑y) + ↑(y ^ 1) - xz a1 1",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"sub_self",
"Pell.xn... | [] | simp [xz, yz] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 521,
"column": 12
} | {
"line": 521,
"column": 25
} | {
"line": 522,
"column": 2
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ny : ℕ\n⊢ 2 * ↑a * ↑y - ↑y * ↑y - 1 ∣ yz a1 1 * (↑a - ↑y) + ↑(y ^ 1) - xz a1 1",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Dvd.dvd",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"sub_self",
"Pell.xn... | [] | simp [xz, yz] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 631,
"column": 26
} | {
"line": 635,
"column": 53
} | {
"line": 637,
"column": 0
} | [
{
"pp": "d : ℕ\na b : ℤ√↑d\n⊢ a < b ↔ a ≤ b ∧ ¬b ≤ a",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"neg_sub",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"Eq.mp",
"Or.resolve_left",
"Int",
"SubtractionMonoid.toSubNegMonoid",
... | [] | by
have ht : b ≤ a ∨ a ≤ b := by
have t := (a - b).nonneg_total
rwa [neg_sub] at t
exact (and_iff_right_of_imp ht.resolve_left).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Dioph | {
"line": 284,
"column": 2
} | {
"line": 314,
"column": 83
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u\nl : List (Set (α → ℕ))\nd : List.Forall Dioph l\n⊢ Dioph {v | List.Forall (fun S ↦ v ∈ S) l}",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.NumberTheory.Dioph.0.Dioph.DiophList.forall.match_1_3",
"Dioph",
"Poly",
... | [] | suffices ∃ (β : _) (pl : List (Poly (α ⊕ β))), ∀ v, List.Forall (fun S : Set _ => v ∈ S) l ↔
∃ t, List.Forall (fun p : Poly (α ⊕ β) => p (v ⊗ t) = 0) pl
from
let ⟨β, pl, h⟩ := this
⟨β, Poly.sumsq pl, fun v => (h v).trans <| exists_congr fun t => (Poly.sumsq_eq_zero _ _).symm⟩
induction l with | ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Dioph | {
"line": 284,
"column": 2
} | {
"line": 314,
"column": 83
} | {
"line": 316,
"column": 0
} | [
{
"pp": "α : Type u\nl : List (Set (α → ℕ))\nd : List.Forall Dioph l\n⊢ Dioph {v | List.Forall (fun S ↦ v ∈ S) l}",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.NumberTheory.Dioph.0.Dioph.DiophList.forall.match_1_3",
"Dioph",
"Poly",
... | [] | suffices ∃ (β : _) (pl : List (Poly (α ⊕ β))), ∀ v, List.Forall (fun S : Set _ => v ∈ S) l ↔
∃ t, List.Forall (fun p : Poly (α ⊕ β) => p (v ⊗ t) = 0) pl
from
let ⟨β, pl, h⟩ := this
⟨β, Poly.sumsq pl, fun v => (h v).trans <| exists_congr fun t => (Poly.sumsq_eq_zero _ _).symm⟩
induction l with | ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Dioph | {
"line": 400,
"column": 6
} | {
"line": 401,
"column": 69
} | {
"line": 401,
"column": 69
} | [
{
"pp": "α : Type u\nS : Set (Option α → ℕ)\nd : Dioph S\nf : (α → ℕ) →. ℕ\ndf : DiophPFun f\nv : α → ℕ\nx✝ : v ∈ {v | ∃ x, x ::ₒ v ∈ S ∩ {v | (v ∘ some, v none) ∈ f.graph}}\nx : ℕ\nhS : x ::ₒ v ∈ S\nh :\n ∃ (h : (f ((x ::ₒ v) ∘ some, (x ::ₒ v) none).1).Dom),\n (f ((x ::ₒ v) ∘ some, (x ::ₒ v) none).1).get h... | [] | rw [show (x ::ₒ v) ∘ some = v from funext fun s => rfl] at h
obtain ⟨hf, h⟩ := h; refine ⟨hf, ?_⟩; rw [PFun.fn, h]; exact hS | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Dioph | {
"line": 400,
"column": 6
} | {
"line": 401,
"column": 69
} | {
"line": 401,
"column": 69
} | [
{
"pp": "α : Type u\nS : Set (Option α → ℕ)\nd : Dioph S\nf : (α → ℕ) →. ℕ\ndf : DiophPFun f\nv : α → ℕ\nx✝ : v ∈ {v | ∃ x, x ::ₒ v ∈ S ∩ {v | (v ∘ some, v none) ∈ f.graph}}\nx : ℕ\nhS : x ::ₒ v ∈ S\nh :\n ∃ (h : (f ((x ::ₒ v) ∘ some, (x ::ₒ v) none).1).Dom),\n (f ((x ::ₒ v) ∘ some, (x ::ₒ v) none).1).get h... | [] | rw [show (x ::ₒ v) ∘ some = v from funext fun s => rfl] at h
obtain ⟨hf, h⟩ := h; refine ⟨hf, ?_⟩; rw [PFun.fn, h]; exact hS | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 114,
"column": 2
} | {
"line": 116,
"column": 36
} | {
"line": 118,
"column": 0
} | [
{
"pp": "m : ℕ\n⊢ m ∈ factoredNumbers ∅ ↔ m ∈ {1}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.SmoothNumbers.0.Nat.factoredNumbers_empty._simp_1_2",
"False",
"Nat.instMulZeroClass",
"One",
"Nat.instOne",
"congrArg",
... | [] | simp only [mem_factoredNumbers, Finset.notMem_empty, ← List.eq_nil_iff_forall_not_mem,
primeFactorsList_eq_nil, and_or_left, not_and_self_iff, ne_and_eq_iff_right zero_ne_one,
false_or, Set.mem_singleton_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 404,
"column": 2
} | {
"line": 406,
"column": 30
} | {
"line": 407,
"column": 2
} | [
{
"pp": "N k : ℕ\n⊢ #(N.smoothNumbersUpTo k) + #(N.roughNumbersUpTo k) = N",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instDecidableNot",
"Nat.roughNumbersUpTo",
"Finset.instUnion",
"congrArg",
"Finset",
"Nat.smoothNu... | [
"N k : ℕ\n⊢ #({x ∈ Finset.range (N + 1) | x ∈ k.smoothNumbers ∨ x ≠ 0 ∧ x ∉ k.smoothNumbers}) = N"
] | rw [smoothNumbersUpTo, roughNumbersUpTo,
← Finset.card_union_of_disjoint <| Finset.disjoint_filter.mpr fun n _ hn₂ h ↦ h.2 hn₂,
Finset.filter_union_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Dioph | {
"line": 657,
"column": 2
} | {
"line": 657,
"column": 48
} | {
"line": 659,
"column": 0
} | [
{
"pp": "this :\n Dioph\n {v |\n 1 < v &0 ∧\n v &1 ≤ v &3 ∧\n (v &2 = 1 ∧ v &3 = 0 ∨\n ∃ u w s t b,\n v &2 * v &2 - (v &0 * v &0 - 1) * v &3 * v &3 = 1 ∧\n u * u - (v &0 * v &0 - 1) * w * w = 1 ∧\n s * s - (b * b - 1) * t * t = 1 ∧... | [] | exact Dioph.ext this fun v => matiyasevic.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 371,
"column": 2
} | {
"line": 374,
"column": 76
} | {
"line": 375,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (𝓝 (∑' (n : ℕ), f n... | [
"F : Type u_1\ninst✝¹ : NormedField F\ninst✝ : CompleteSpace F\nf : ℕ →*₀ F\nhsum : Summable fun x ↦ ‖f x‖\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nthis :\n Tendsto (fun n ↦ ∏ i ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ ∑' (e : ℕ), f (p ^ e)) i) atTop\n (𝓝 (∑' (n : ℕ), f n))\nH : ∀ (n... | have H (n : ℕ) : ∏ p ∈ range n, {p | Nat.Prime p}.mulIndicator (fun p ↦ (1 - f p)⁻¹) p =
∏ p ∈ primesBelow n, (1 - f p)⁻¹ :=
prod_mulIndicator_eq_prod_filter
(range n) (fun _ ↦ fun p ↦ (1 - f p)⁻¹) (fun _ ↦ {p | Nat.Prime p}) id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 201,
"column": 16
} | {
"line": 201,
"column": 41
} | {
"line": 201,
"column": 41
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nstep1 : mellin (fun t ↦ P.g (1 / t)) (-s) = mellin P.g s\nstep2 : mellin (fun t ↦ ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = mellin P.g s\nstep3 : mellin (fun t ↦ P.ε • ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\ns : ℂ\nstep1 : mellin (fun t ↦ P.g (1 / t)) (-s) = mellin P.g s\nstep2 : mellin (fun t ↦ ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = mellin P.g s\nstep3 : mellin (fun t ↦ P.ε • ↑t ^ (-↑P.k) • P.g (1 / t)) (↑P.k - s) = P.ε • mellin... | ofReal_cpow (le_of_lt ht) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 272,
"column": 2
} | {
"line": 277,
"column": 80
} | {
"line": 278,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : 1 < x\n⊢ P.f_modif (1 / x) = (P.ε * ↑(x ^ P.k)) • P.g_modif x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.notMem_Ioi",
"E... | [
"case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhx : 0 < 1\n⊢ P.f_modif (1 / 1) = (P.ε * ↑(1 ^ P.k)) • P.g_modif 1",
"case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : x < 1\n⊢ P.f_m... | · have : 1 / x < 1 := by rwa [one_div_lt hx one_pos, div_one]
rw [f_modif, Pi.add_apply, indicator_of_notMem (notMem_Ioi.mpr this.le),
zero_add, indicator_of_mem (mem_Ioo.mpr ⟨div_pos one_pos hx, this⟩), g_modif, Pi.add_apply,
indicator_of_mem (mem_Ioi.mpr hx'), indicator_of_notMem
(notMem_Ioo_of_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 305,
"column": 2
} | {
"line": 305,
"column": 84
} | {
"line": 306,
"column": 2
} | [
{
"pp": "z τ : ℂ\nhτ : 0 < τ.im\nT : ℝ\nhT : 0 < T\nhτ' : T < τ.im\nS : ℝ\nhz : |z.im| < S\nV : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}\nhVo : IsOpen V\nhVmem : (z, τ) ∈ V\nhVp : IsPreconnected V\nf : ℤ → ℂ × ℂ → ℂ := fun n p ↦ jacobiTheta₂_term n p.1 p.2\n⊢ HasFDerivAt (fun p ↦ jacobiTheta₂ p.1 p.2) ... | [
"z τ : ℂ\nhτ : 0 < τ.im\nT : ℝ\nhT : 0 < T\nhτ' : T < τ.im\nS : ℝ\nhz : |z.im| < S\nV : Set (ℂ × ℂ) := {u | |u.im| < S} ×ˢ {v | T < v.im}\nhVo : IsOpen V\nhVmem : (z, τ) ∈ V\nhVp : IsPreconnected V\nf : ℤ → ℂ × ℂ → ℂ := fun n p ↦ jacobiTheta₂_term n p.1 p.2\nf' : ℤ → ℂ × ℂ → ℂ × ℂ →L[ℂ] ℂ := fun n p ↦ jacobiTheta₂_... | let f' : ℤ → ℂ × ℂ → ℂ × ℂ →L[ℂ] ℂ := fun n p ↦ jacobiTheta₂_term_fderiv n p.1 p.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 63
} | {
"line": 123,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nr : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re\n⊢ HasSum (fun i ↦ s.Gammaℝ * a i / ↑|r i| ^ s) (mellin F (s / 2))",
... | [
"ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nr : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re\nhs' : 0 < (s / 2).re\n⊢ HasSum (fun i ↦ s.Gammaℝ * a i / ↑|r i| ^ s) (mellin F (... | have hs' : 0 < (s / 2).re := by rw [div_ofNat_re]; positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 34
} | {
"line": 213,
"column": 2
} | [
{
"pp": "a t : ℝ\nht : 0 < t\nthis :\n HasSum\n (fun n ↦\n (cexp (2 * ↑π * I * ↑a * ↑(n + 1)) + cexp (-(2 * ↑π * I * ↑a * ↑(n + 1)))) * ↑(rexp (-π * ↑(n + 1) ^ 2 * t)))\n (↑(cosKernel (↑a) t) - 1)\n⊢ HasSum (fun x ↦ ↑(2 * Real.cos (2 * π * a * (↑x + 1)) * rexp (-π * (↑x + 1) ^ 2 * t))) (↑(cosKernel ... | [
"a t : ℝ\nht : 0 < t\nthis :\n HasSum\n (fun n ↦\n (cexp (2 * ↑π * I * ↑a * ↑(n + 1)) + cexp (-(2 * ↑π * I * ↑a * ↑(n + 1)))) * ↑(rexp (-π * ↑(n + 1) ^ 2 * t)))\n (↑(cosKernel (↑a) t) - 1)\nn : ℕ\n⊢ ↑(2 * Real.cos (2 * π * a * (↑n + 1)) * rexp (-π * (↑n + 1) ^ 2 * t)) =\n (cexp (2 * ↑π * I * ↑a * ↑(n... | refine this.congr_fun fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 126,
"column": 50
} | {
"line": 126,
"column": 58
} | {
"line": 126,
"column": 59
} | [
{
"pp": "⊢ Function.update (fun s ↦ completedCosZeta 0 s / s.Gammaℝ) 0 (-1 / 2) =\n Function.update (fun s ↦ completedHurwitzZetaEven 0 s / s.Gammaℝ) 0 (if True then -1 / 2 else 0)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Func... | [
"⊢ Function.update (fun s ↦ completedCosZeta 0 s / s.Gammaℝ) 0 (-1 / 2) =\n Function.update (fun s ↦ completedHurwitzZetaEven 0 s / s.Gammaℝ) 0 (-1 / 2)"
] | if_true, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 34
} | {
"line": 249,
"column": 2
} | [
{
"pp": "a 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)\n⊢ HasSum (fun x ↦ ↑(2 * ↑x * Real.sin (2 * π * a * ↑x) * re... | [
"a 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⊢ ↑(2 * ↑n * Real.sin (2 * π * a * ↑n) * rexp (-π * ↑n ^ 2 * t))... | refine this.congr_fun fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 466,
"column": 2
} | {
"line": 471,
"column": 68
} | {
"line": 473,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\n⊢ Tendsto (fun s ↦ s * completedHurwitzZetaEven a s) (𝓝[≠] 0) (𝓝 (if a = 0 then -1 else 0))",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.t... | [] | have h1 : Tendsto (fun s : ℂ ↦ s * _) (𝓝[≠] 0)
(𝓝 (-(if a = 0 then 1 else 0))) := (hurwitzEvenFEPair a).Λ_residue_zero
have : -(if a = 0 then (1 : ℂ) else 0) = (if a = 0 then -1 else 0) := by { split_ifs <;> simp }
simp only [this, push_cast] at h1
refine (h1.comp <| zero_div (2 : ℂ) ▸ (tendsto_div_two_punc... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 466,
"column": 2
} | {
"line": 471,
"column": 68
} | {
"line": 473,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\n⊢ Tendsto (fun s ↦ s * completedHurwitzZetaEven a s) (𝓝[≠] 0) (𝓝 (if a = 0 then -1 else 0))",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.t... | [] | have h1 : Tendsto (fun s : ℂ ↦ s * _) (𝓝[≠] 0)
(𝓝 (-(if a = 0 then 1 else 0))) := (hurwitzEvenFEPair a).Λ_residue_zero
have : -(if a = 0 then (1 : ℂ) else 0) = (if a = 0 then -1 else 0) := by { split_ifs <;> simp }
simp only [this, push_cast] at h1
refine (h1.comp <| zero_div (2 : ℂ) ▸ (tendsto_div_two_punc... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SumPrimeReciprocals | {
"line": 99,
"column": 70
} | {
"line": 115,
"column": 52
} | {
"line": 117,
"column": 0
} | [
{
"pp": "⊢ ¬Summable ({p | Nat.Prime p}.indicator fun n ↦ 1 / ↑n)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.inter_eq_left",
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"Finset.mem_range._simp_1",
"False",
"Set.mem_Iio",
"Real.pa... | [] | by
intro h
obtain ⟨k, hk⟩ := h.nat_tsum_vanishing (Iio_mem_nhds one_half_pos : Iio (1 / 2 : ℝ) ∈ 𝓝 0)
specialize hk ({p | Nat.Prime p} ∩ {p | k ≤ p}) inter_subset_right
rw [tsum_subtype, indicator_indicator, inter_eq_left.mpr fun n hn ↦ hn.1, mem_Iio] at hk
have h' : Summable (indicator ({p | Nat.Prime p} ∩ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 373,
"column": 2
} | {
"line": 375,
"column": 56
} | {
"line": 377,
"column": 0
} | [
{
"pp": "⊢ ((fun n ↦ ↑(Λ n)) ⍟ fun n ↦ ↑(ζ n)) = fun n ↦ Complex.log ↑n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"CharP.cast_eq_zero",
"Eq.mpr",
"Complex.log",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Re... | [] | ext n
simpa [apply_ite, LSeries.convolution_def, -vonMangoldt_mul_zeta]
using congr_arg (ofReal <| · n) vonMangoldt_mul_zeta | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 373,
"column": 2
} | {
"line": 375,
"column": 56
} | {
"line": 377,
"column": 0
} | [
{
"pp": "⊢ ((fun n ↦ ↑(Λ n)) ⍟ fun n ↦ ↑(ζ n)) = fun n ↦ Complex.log ↑n",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"CharP.cast_eq_zero",
"Eq.mpr",
"Complex.log",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Re... | [] | ext n
simpa [apply_ite, LSeries.convolution_def, -vonMangoldt_mul_zeta]
using congr_arg (ofReal <| · n) vonMangoldt_mul_zeta | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 589,
"column": 2
} | {
"line": 592,
"column": 26
} | {
"line": 594,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nh : a ≠ 0 ∨ s ≠ 0\n⊢ hurwitzZetaEven a s = completedHurwitzZetaEven a s / s.Gammaℝ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"False",
"ne_or_eq",
"... | [] | rw [hurwitzZetaEven]
rcases ne_or_eq s 0 with h' | rfl
· rw [Function.update_of_ne h']
· simpa [Gammaℝ] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 589,
"column": 2
} | {
"line": 592,
"column": 26
} | {
"line": 594,
"column": 0
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nh : a ≠ 0 ∨ s ≠ 0\n⊢ hurwitzZetaEven a s = completedHurwitzZetaEven a s / s.Gammaℝ",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"False",
"ne_or_eq",
"... | [] | rw [hurwitzZetaEven]
rcases ne_or_eq s 0 with h' | rfl
· rw [Function.update_of_ne h']
· simpa [Gammaℝ] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.FLT.Basic | {
"line": 184,
"column": 14
} | {
"line": 184,
"column": 16
} | {
"line": 184,
"column": 17
} | [
{
"pp": "R : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nn : ℕ\nh : FermatLastTheoremWith' R n\nhn : ∀ (a b c : R), IsUnit a → IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n\na b c : R\n⊢ a ≠ 0 → b ≠ 0 → c ≠ 0 → a ^ n + b ^ n ≠ c ^ n",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : IsDomain R\nn : ℕ\nh : FermatLastTheoremWith' R n\nhn : ∀ (a b c : R), IsUnit a → IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n\na b c : R\nha : a ≠ 0\n⊢ b ≠ 0 → c ≠ 0 → a ^ n + b ^ n ≠ c ^ n"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.FLT.Basic | {
"line": 199,
"column": 16
} | {
"line": 199,
"column": 18
} | {
"line": 199,
"column": 19
} | [
{
"pp": "n a✝ b✝ c✝ : ℕ\n⊢ a✝ = 1 → b✝ = 1 → c✝ = 1 → a✝ ^ n + b✝ ^ n ≠ c✝ ^ n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"OfNat.ofNat",
"Eq"
],
"usedFVars": [
"a✝"
],
"usedGoals": [
{
"new": true,
... | [
"n a✝ b✝ c✝ : ℕ\nha : a✝ = 1\n⊢ b✝ = 1 → c✝ = 1 → a✝ ^ n + b✝ ^ n ≠ c✝ ^ n"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.FLT.Basic | {
"line": 205,
"column": 16
} | {
"line": 205,
"column": 18
} | {
"line": 205,
"column": 19
} | [
{
"pp": "n : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\n⊢ IsUnit a → IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"CommSemiring.toSemiring",
"IsUnit",
"Int",
"Semiring.toMonoid",
... | [
"n : ℕ\ntfae_2_iff_1 : FermatLastTheoremWith' ℕ n ↔ FermatLastTheoremFor n\na b c : ℤ\nha : IsUnit a\n⊢ IsUnit b → IsUnit c → a ^ n + b ^ n ≠ c ^ n"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 394,
"column": 77
} | {
"line": 412,
"column": 6
} | {
"line": 414,
"column": 0
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\n⊢ HasSum (fun n ↦ (s + 1).Gammaℝ * -I * ↑n.sign * cexp (2 * ↑π * I * ↑a * ↑n) / ↑|n| ^ s / 2) (completedSinZeta (↑a) s)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"sign_intCast",
"hasSum_mellin_pi_mul_sq'"... | [] | by
let c (n : ℤ) : ℂ := -I * cexp (2 * π * I * a * n) / 2
have hc (n : ℤ) : ‖c n‖ = 1 / 2 := by
simp_rw [c, (by { push_cast; ring } : 2 * π * I * a * n = ↑(2 * π * a * n) * I), norm_div,
RCLike.norm_ofNat, norm_mul, norm_neg, norm_I, one_mul, norm_exp_ofReal_mul_I]
have hF t (ht : 0 < t) :
HasSum ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 422,
"column": 2
} | {
"line": 422,
"column": 34
} | {
"line": 423,
"column": 2
} | [
{
"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 + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /\n ↑n ^ s /\n 2)\n (completedSinZeta (↑a) s)\n⊢ HasSum (fun n ↦ (s + 1)... | [
"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 + 1).Gammaℝ * -I * -↑(↑n).sign * cexp (2 * ↑π * I * ↑a * -↑n)) /\n ↑n ^ s /\n 2)\n (completedSinZeta (↑a) s)\nn : ℕ\n⊢ (s + 1).Gammaℝ * ↑(Real.sin ... | refine this.congr_fun fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 747,
"column": 2
} | {
"line": 747,
"column": 89
} | {
"line": 749,
"column": 0
} | [
{
"pp": "a : ℝ\ns : ℂ\nhs : 1 < s.re\nthis : HasSum (fun n ↦ (cexp (2 * ↑π * I * ↑a * ↑n) + cexp (-(2 * ↑π * I * ↑a * ↑n))) / 2 / ↑n ^ s) (cosZeta (↑a) s)\n⊢ HasSum (fun n ↦ (cexp (2 * ↑π * ↑a * ↑n * I) + cexp (-(2 * ↑π * ↑a * ↑n * I))) / 2 / ↑n ^ s) (cosZeta (↑a) s)",
"ppTerm": "?m.96",
"assigned": tru... | [] | exact this.congr_fun fun n ↦ by rw [show 2 * π * a * n * I = 2 * π * I * a * n by ring] | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.FLT.Basic | {
"line": 223,
"column": 14
} | {
"line": 223,
"column": 16
} | {
"line": 223,
"column": 17
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\na b c : R\n⊢ a ≠ 0 → b ≠ 0 → c ≠ 0 → a ^ n + b ^ n ≠ c ^ n",
"ppTerm": "?m.54",
"as... | [
"n : ℕ\nR : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : IsDomain R\ninst✝¹ : DecidableEq R\ninst✝ : NormalizedGCDMonoid R\nhn : ∀ (a b c : R), a ≠ 0 → b ≠ 0 → c ≠ 0 → {a, b, c}.gcd id = 1 → a ^ n + b ^ n ≠ c ^ n\na b c : R\nha : a ≠ 0\n⊢ b ≠ 0 → c ≠ 0 → a ^ n + b ^ n ≠ c ^ n"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.FLT.Four | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 40
} | {
"line": 49,
"column": 4
} | [
{
"pp": "case mpr.right.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ b ≠ 0",
"ppTerm": "?mpr.right.left",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"right_ne_zero_of_mul",
"Ne",
"instOfNatNat",
"Int... | [] | exact right_ne_zero_of_mul f42.2.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.FLT.Four | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 40
} | {
"line": 49,
"column": 4
} | [
{
"pp": "case mpr.right.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ b ≠ 0",
"ppTerm": "?mpr.right.left",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"right_ne_zero_of_mul",
"Ne",
"instOfNatNat",
"Int... | [] | exact right_ne_zero_of_mul f42.2.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FLT.Four | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 40
} | {
"line": 49,
"column": 4
} | [
{
"pp": "case mpr.right.left\na b c k : ℤ\nhk0 : k ≠ 0\nf42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2\n⊢ b ≠ 0",
"ppTerm": "?mpr.right.left",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"right_ne_zero_of_mul",
"Ne",
"instOfNatNat",
"Int... | [] | exact right_ne_zero_of_mul f42.2.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 34
} | {
"line": 450,
"column": 2
} | [
{
"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)\nh_sum : Summable fun i ↦ ‖c i‖ / |r i| ^ s.re\nthis :\n HasSum (fun i ↦ (s + 1).Gammaℝ * c i * ↑(SignType.... | [
"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)\nh_sum : Summable fun i ↦ ‖c i‖ / |r i| ^ s.re\nthis :\n HasSum (fun i ↦ (s + 1).Gammaℝ * c i * ↑(SignType.sign (r i)) ... | refine this.congr_fun fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 236,
"column": 56
} | {
"line": 236,
"column": 64
} | {
"line": 236,
"column": 64
} | [
{
"pp": "N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nhpow_le : ∀ (p : Primes), ‖χ ↑↑p * ↑↑p ^ (-s)‖ < 1\nf : ℕ → ℂ := fun n ↦ χ ↑n * ↑(Λ n) / ↑(Real.log ↑n) * ↑n ^ (-s)\np : Primes\nk : ℕ\n⊢ χ (↑↑p ^ (k + 1)) * ↑(↑p ^ (k + 1)) ^ (-s) * ↑(Λ (↑p ^ (k + 1))) / ↑(Real.log (↑↑p ^ (k + 1))) = f (↑p ^ (k ... | [
"N : ℕ\nχ : DirichletCharacter ℂ N\ns : ℂ\nhs : 1 < s.re\nhpow_le : ∀ (p : Primes), ‖χ ↑↑p * ↑↑p ^ (-s)‖ < 1\nf : ℕ → ℂ := fun n ↦ χ ↑n * ↑(Λ n) / ↑(Real.log ↑n) * ↑n ^ (-s)\np : Primes\nk : ℕ\n⊢ χ (↑↑p ^ (k + 1)) * ↑(↑p ^ (k + 1)) ^ (-s) * ↑(Λ (↑p ^ (k + 1))) / ↑(Real.log (↑↑p ^ (k + 1))) =\n χ ↑(↑p ^ (k + 1)) ... | unfold f | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.NumberTheory.LSeries.HurwitzZetaOdd | {
"line": 534,
"column": 2
} | {
"line": 534,
"column": 34
} | {
"line": 535,
"column": 2
} | [
{
"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)\n⊢ HasSum (fun n ↦ (cexp (-(2 * ↑π * ↑a * ↑n) * I) - cexp (2 * ↑π * ↑a * ↑n * ... | [
"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)\nn : ℕ\n⊢ (cexp (-(2 * ↑π * ↑a * ↑n) * I) - cexp (2 * ↑π * ↑a * ↑n * I)) * I / 2 / ↑n ^ s ... | refine this.congr_fun fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Polynomial.Radical | {
"line": 37,
"column": 2
} | {
"line": 39,
"column": 18
} | {
"line": 40,
"column": 2
} | [
{
"pp": "case h0\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\n⊢ divRadical 0 ∣ derivative 0",
"ppTerm": "?h0",
"assigned": true,
"usedConstants": [
"Polynomial.instNormalizationMonoid",
"Polynomial.derivative",
"Eq.mpr",
"dvd_zero",
"IsDomain.to_noZeroDivisor... | [
"case h1\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\nx✝ : k[X]\na✝ : IsUnit x✝\n⊢ divRadical x✝ ∣ derivative x✝",
"case hpr\nk : Type u_1\ninst✝¹ : Field k\ninst✝ : DecidableEq k\np✝ : k[X]\ni✝ : ℕ\na✝ : Prime p✝\n⊢ divRadical (p✝ ^ i✝) ∣ derivative (p✝ ^ i✝)",
"case hcp\nk : Type u_1\ninst✝¹ : Fiel... | · case h0 =>
rw [derivative_zero]
apply dvd_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Radical.Basic | {
"line": 185,
"column": 8
} | {
"line": 185,
"column": 10
} | {
"line": 185,
"column": 10
} | [
{
"pp": "case pos\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nha : a = 0\n⊢ radical a ∣ a",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"semigroupDvd",
... | [
"case pos\nM : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nha : a = 0\n⊢ radical 0 ∣ 0"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Radical.Basic | {
"line": 198,
"column": 37
} | {
"line": 200,
"column": 27
} | {
"line": 202,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nha : Prime a\nn : ℕ\nhn : n ≠ 0\n⊢ radical (a ^ n) = normalize a",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
... | [] | by
rw [radical_pow a hn]
exact radical_of_prime ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.FLT.MasonStothers | {
"line": 37,
"column": 2
} | {
"line": 40,
"column": 95
} | {
"line": 42,
"column": 2
} | [
{
"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\n⊢ c.natDegree + 1 ≤ (radical (a * b * c)).natDeg... | [
"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 + b.natDegre... | have abc_dr_ndeg_lt : abc_dr.natDegree < a.natDegree + b.natDegree := by
calc
abc_dr.natDegree ≤ w.natDegree := Polynomial.natDegree_le_of_dvd abc_dr_dvd_w hw
_ < a.natDegree + b.natDegree := by rw [wab] at hw ⊢; exact natDegree_wronskian_lt_add hw | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Radical.Basic | {
"line": 223,
"column": 12
} | {
"line": 223,
"column": 14
} | {
"line": 224,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nn : ℕ\np : M\n⊢ radical a ∣ p ^ n → radical a ∣ p",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"semigroupDvd",
"SemigroupWithZer... | [
"M : Type u_1\ninst✝² : CommMonoidWithZero M\ninst✝¹ : NormalizationMonoid M\ninst✝ : UniqueFactorizationMonoid M\na : M\nn : ℕ\np : M\nha : radical a ∣ p ^ n\n⊢ radical a ∣ p"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.FLT.Four | {
"line": 269,
"column": 14
} | {
"line": 269,
"column": 16
} | {
"line": 269,
"column": 17
} | [
{
"pp": "a b c : ℤ\n⊢ a ≠ 0 → b ≠ 0 → c ≠ 0 → a ^ 4 + b ^ 4 ≠ c ^ 4",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Ne",
"Int",
"Zero.toOfNat0",
"OfNat.ofNat",
"MulZeroClass.toZero",
"Int.instSemiring",
"instMulZeroClassOfSemiring"
],
"use... | [
"a b c : ℤ\nha : a ≠ 0\n⊢ b ≠ 0 → c ≠ 0 → a ^ 4 + b ^ 4 ≠ c ^ 4"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 436,
"column": 2
} | {
"line": 438,
"column": 88
} | {
"line": 439,
"column": 2
} | [
{
"pp": "case neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEq... | [
"case neg\nx y z : ℤ\nh : PythagoreanTriple x y z\nhc : x.gcd y = 1\nhyo : y % 2 = 1\nhzpos : 0 < z\nh0 : ¬x = 0\nv : ℚ := ↑x / ↑z\nw : ℚ := ↑y / ↑z\nhq : v ^ 2 + w ^ 2 = 1\nhvz : v ≠ 0\nhw1 : w ≠ -1\nhQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0\nhp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}\nq : ℚ := (circleEquivGen hQ).s... | have ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2) := by
apply Prod.mk.inj
exact congr_arg Subtype.val ((circleEquivGen hQ).apply_symm_apply ⟨⟨v, w⟩, hp⟩).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.FractionalIdeal | {
"line": 95,
"column": 6
} | {
"line": 95,
"column": 39
} | {
"line": 95,
"column": 40
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ finrank ℤ ↥↑↑I = finrank ℤ (𝓞 K)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Submodule",
"NumberField.instFreeIntSubtypeMemSubmoduleRingO... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\n⊢ Fintype.card (Free.ChooseBasisIndex ℤ ↥↑↑I) = finrank ℤ (𝓞 K)"
] | finrank_eq_card_chooseBasisIndex, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Norm | {
"line": 69,
"column": 30
} | {
"line": 69,
"column": 63
} | {
"line": 70,
"column": 4
} | [
{
"pp": "L : Type u_1\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : 𝓞 K\n⊢ ↑((norm K) ((algebraMap (𝓞 K) (𝓞 L)) x)) = ↑(x ^ finrank K L)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"Algebra.algebraMap... | [
"L : Type u_1\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : 𝓞 K\n⊢ (algebraMap (𝓞 K) K) ((norm K) ((algebraMap (𝓞 K) (𝓞 L)) x)) = ↑(x ^ finrank K L)"
] | RingOfIntegers.coe_eq_algebraMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Norm | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 37
} | {
"line": 71,
"column": 38
} | [
{
"pp": "L : Type u_1\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : 𝓞 K\n⊢ (algebraMap (𝓞 K) K) x ^ finrank K L = ↑(x ^ finrank K L)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"NumberField.instCommRingRi... | [
"L : Type u_1\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : 𝓞 K\n⊢ (algebraMap (𝓞 K) K) x ^ finrank K L = (algebraMap (𝓞 K) K) (x ^ finrank K L)"
] | RingOfIntegers.coe_eq_algebraMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Norm | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 61
} | {
"line": 88,
"column": 0
} | [
{
"pp": "L : Type u_1\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : FiniteDimensional K L\ninst✝ : IsGalois K L\nx : 𝓞 L\nhint : IsIntegral ℤ (∏ σ ∈ univ.erase AlgEquiv.refl, σ ↑x)\n⊢ ∏ σ, σ ↑x = ↑(x * ⟨∏ σ ∈ univ.erase AlgEquiv.refl, σ ↑x, hint⟩)",
"ppTerm": "?m.106",
... | [] | simp [← Finset.mul_prod_erase _ _ (mem_univ AlgEquiv.refl)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Instances.Complex | {
"line": 83,
"column": 14
} | {
"line": 83,
"column": 49
} | {
"line": 83,
"column": 50
} | [
{
"pp": "case e'_3\nK : Subfield ℂ\nψ : ↥K →+* ℂ\nhc : UniformContinuous ⇑ψ\nthis✝¹ : IsTopologicalDivisionRing ℂ :=\n { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,\n toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }\nthis✝... | [
"case e'_3\nK : Subfield ℂ\nψ : ↥K →+* ℂ\nhc : UniformContinuous ⇑ψ\nthis✝¹ : IsTopologicalDivisionRing ℂ :=\n { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,\n toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }\nthis✝ : IsTopolog... | ← ofRealHom.coe_rangeRestrictField, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 478,
"column": 27
} | {
"line": 478,
"column": 44
} | {
"line": 478,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ card { φ // ComplexEmbedding.IsReal φ } + (card (K →+* ℂ) - card { x // ComplexEmbedding.IsReal x }) = card (K →+* ℂ)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberField.ComplexEmbedding.IsReal... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ card (K →+* ℂ) = card (K →+* ℂ)",
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ card { φ // ComplexEmbedding.IsReal φ } ≤ card (K →+* ℂ)"
] | Nat.add_sub_of_le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Basic | {
"line": 520,
"column": 2
} | {
"line": 520,
"column": 22
} | {
"line": 521,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\nk : ℕ\nhk : 2 < k\nhζ : IsPrimitiveRoot ζ k\nx✝ : { w // w.IsReal }\nw : InfinitePlace K\nhwreal : ComplexEmbedding.IsReal w.embedding\n⊢ False",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"RingHom",
"Num... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nζ : K\nk : ℕ\nhk : 2 < k\nhζ : IsPrimitiveRoot ζ k\nx✝ : { w // w.IsReal }\nw : InfinitePlace K\nhwreal : ComplexEmbedding.IsReal w.embedding\nf : K →+* ℂ := w.embedding\n⊢ False"
] | let f := w.embedding | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.NumberTheory.FLT.Polynomial | {
"line": 243,
"column": 14
} | {
"line": 243,
"column": 16
} | {
"line": 243,
"column": 17
} | [
{
"pp": "k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\na b c : k[X]\n⊢ a ≠ 0 →\n b ≠ 0 →\n c ≠ 0 →\n a ^ n + b ^ n = c ^ n → ∃ d a' b' c', (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants... | [
"k : Type u_1\ninst✝ : Field k\nn : ℕ\nhn : 3 ≤ n\nchn : ↑n ≠ 0\na b c : k[X]\nha : a ≠ 0\n⊢ b ≠ 0 →\n c ≠ 0 →\n a ^ n + b ^ n = c ^ n → ∃ d a' b' c', (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.NumberTheory.NumberField.InfinitePlace.TotallyRealComplex | {
"line": 159,
"column": 34
} | {
"line": 159,
"column": 61
} | {
"line": 159,
"column": 61
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nw : InfinitePlace ↥(maximalRealSubfield K)\n⊢ ComplexEmbedding.IsReal w.embedding",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NumberFiel... | [
"F : Type u_1\ninst✝³ : Field F\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : CharZero K\ninst✝ : Algebra.IsAlgebraic ℚ K\nw : InfinitePlace ↥(maximalRealSubfield K)\n⊢ conjugate w.embedding = w.embedding"
] | ComplexEmbedding.isReal_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 243,
"column": 6
} | {
"line": 246,
"column": 10
} | {
"line": 247,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\n⊢ (↑2 ^ nrRealPlaces K * ∏ x, ENNReal.ofReal ↑(f ↑x)) *\n ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x)... | [] | rw [show (4 : ℝ≥0∞) = (2 : ℝ≥0) ^ 2 by norm_num, convexBodyLT'Factor, pow_add,
← Finset.prod_erase_mul _ _ (Finset.mem_univ w₀), ofReal_coe_nnreal]
simp_rw [coe_mul, ENNReal.coe_pow]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 243,
"column": 6
} | {
"line": 246,
"column": 10
} | {
"line": 247,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nf : InfinitePlace K → ℝ≥0\nw₀ : { w // w.IsComplex }\ninst✝ : NumberField K\nvol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2\n⊢ (↑2 ^ nrRealPlaces K * ∏ x, ENNReal.ofReal ↑(f ↑x)) *\n ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x)... | [] | rw [show (4 : ℝ≥0∞) = (2 : ℝ≥0) ^ 2 by norm_num, convexBodyLT'Factor, pow_add,
← Finset.prod_erase_mul _ _ (Finset.mem_univ w₀), ofReal_coe_nnreal]
simp_rw [coe_mul, ENNReal.coe_pow]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 301,
"column": 82
} | {
"line": 308,
"column": 31
} | {
"line": 310,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : mixedSpace K\n⊢ ‖x‖ ≤ convexBodySumFun x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Finset.mem_univ",
"Real.instIsOrderedRing",
"Norm.norm",
... | [] | by
rw [norm_eq_sup'_normAtPlace]
refine (Finset.sup'_le_iff _ _).mpr fun w _ ↦ ?_
rw [convexBodySumFun_apply, ← Finset.univ.add_sum_erase _ (Finset.mem_univ w)]
refine le_add_of_le_of_nonneg ?_ ?_
· exact le_mul_of_one_le_left (normAtPlace_nonneg w x) one_le_mult
· exact Finset.sum_nonneg (fun _ _ => mul_no... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody | {
"line": 424,
"column": 73
} | {
"line": 424,
"column": 83
} | {
"line": 424,
"column": 84
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nhB : 0 < B\n⊢ (2 * Gamma 2) ^ nrRealPlaces K * (π * 2 ^ (-2) * (2 * Gamma 2)) ^ nrComplexPlaces K =\n 2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K",
"ppTerm": "?m.900",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nB : ℝ\nhB : 0 < B\n⊢ (2 * 1) ^ nrRealPlaces K * (π * 2 ^ (-2) * (2 * 1)) ^ nrComplexPlaces K =\n 2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K"
] | Gamma_two, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 494,
"column": 4
} | {
"line": 494,
"column": 43
} | {
"line": 496,
"column": 0
} | [
{
"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}",
"ppTerm": "?m.83",
"assigned": true,
... | [] | simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 494,
"column": 4
} | {
"line": 494,
"column": 43
} | {
"line": 496,
"column": 0
} | [
{
"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}",
"ppTerm": "?m.83",
"assigned": true,
... | [] | simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 494,
"column": 4
} | {
"line": 494,
"column": 43
} | {
"line": 496,
"column": 0
} | [
{
"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}",
"ppTerm": "?m.83",
"assigned": true,
... | [] | simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 115,
"column": 8
} | {
"line": 116,
"column": 61
} | {
"line": 117,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=\n (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))\ne : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm\nM : Matrix (index K) (index K) ℝ := (mixedEm... | [] | rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
← coe_discr, map_intCast, ← Complex.nnnorm_intCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 115,
"column": 8
} | {
"line": 116,
"column": 61
} | {
"line": 117,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=\n (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))\ne : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm\nM : Matrix (index K) (index K) ℝ := (mixedEm... | [] | rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
← coe_discr, map_intCast, ← Complex.nnnorm_intCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 115,
"column": 8
} | {
"line": 116,
"column": 61
} | {
"line": 117,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nf : Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=\n (canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ (K →+* ℂ))\ne : index K ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm\nM : Matrix (index K) (index K) ℝ := (mixedEm... | [] | rw [← Algebra.discr_eq_det_embeddingsMatrixReindex_pow_two, Algebra.discr_reindex,
← coe_discr, map_intCast, ← Complex.nnnorm_intCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 43
} | {
"line": 519,
"column": 0
} | [
{
"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",
"ppTerm": "?m.98",
"assigned": true,
... | [] | simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 43
} | {
"line": 519,
"column": 0
} | [
{
"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",
"ppTerm": "?m.98",
"assigned": true,
... | [] | simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 43
} | {
"line": 519,
"column": 0
} | [
{
"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",
"ppTerm": "?m.98",
"assigned": true,
... | [] | simp [Set.MapsTo, isUnramifiedIn_comap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 180,
"column": 65
} | {
"line": 180,
"column": 80
} | {
"line": 180,
"column": 81
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(FractionalIdeal.absNorm ↑I) * (2 ^ (-1)) ^ nrComplexPlac... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nI : (FractionalIdeal (𝓞 K)⁰ K)ˣ\nB : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))\nh_le : minkowskiBound K I ≤ volume (convexBodySum K B)\nx✝ : K\n⊢ ↑(FractionalIdeal.absNorm ↑I) * (2 ^ (-1)) ^ ↑(nrComplexPlaces K) * ↑(... | ← zpow_natCast, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic | {
"line": 933,
"column": 4
} | {
"line": 933,
"column": 32
} | {
"line": 935,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ns : Set { w // w.IsReal }\ninst✝ : NumberField K\nw : { w // w.IsReal }\nhw : w ∉ s\n⊢ MeasurePreserving (↑↑(ContinuousLinearEquiv.refl ℝ ℝ)).toAddHom.1 volume volume",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Real",
"Mea... | [] | exact MeasurePreserving.id _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Invariant.Galois | {
"line": 94,
"column": 11
} | {
"line": 94,
"column": 35
} | {
"line": 94,
"column": 35
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_3\ninst✝¹⁶ : Finite G\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.LiesOver P\ninst✝¹¹ : P.IsPrime\ninst✝¹⁰ : Q.Is... | [
"A : Type u_1\nB : Type u_2\ninst✝¹⁹ : CommRing A\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nG : Type u_3\ninst✝¹⁶ : Finite G\ninst✝¹⁵ : Group G\ninst✝¹⁴ : MulSemiringAction G B\ninst✝¹³ : Algebra.IsInvariant A B G\nP : Ideal A\nQ : Ideal B\ninst✝¹² : Q.LiesOver P\ninst✝¹¹ : P.IsPrime\ninst✝¹⁰ : Q.IsPrime\nK : T... | ← Quotient.algebraMap_eq | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 428,
"column": 26
} | {
"line": 428,
"column": 41
} | {
"line": 428,
"column": 42
} | [
{
"pp": "case refine_2.refine_2\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 B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | ... | [
"case refine_2.refine_2\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 B 1 ^ D * ↑(D.choose (D / 2))⌉₊\nx✝¹ : { F // FiniteDimensional ℚ ↥F }\nK : IntermediateField ℚ A\nhK₀ : FiniteDimensional ℚ ↥K\nx✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsRea... | NNReal.coe_max, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.Index | {
"line": 80,
"column": 2
} | {
"line": 81,
"column": 57
} | {
"line": 82,
"column": 2
} | [
{
"pp": "case h₁\nR : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Ideal R\nN : Submodule R M\ninst✝ : Finite (R ⧸ I)\ns : Finset M\nhs : span R ↑s = N\nval✝ : Fintype (R ⧸ I)\ne : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=\n (comap N.subtype (I • ... | [
"case h₁\nR : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nI : Ideal R\nN : Submodule R M\ninst✝ : Finite (R ⧸ I)\ns : Finset M\nhs : span R ↑s = N\nval✝ : Fintype (R ⧸ I)\ne : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=\n (comap N.subtype (I • N)).quotEqui... | have hf : Function.Surjective f := fun x ↦ by
obtain ⟨y, hy⟩ := H.ge x.2; exact ⟨y, Subtype.ext hy⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 476,
"column": 56
} | {
"line": 476,
"column": 71
} | {
"line": 476,
"column": 72
} | [
{
"pp": "case refine_2.refine_2\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✝ :... | [
"case refine_2.refine_2\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✝ : ⟨K, hK₀⟩ ∈ ... | NNReal.coe_max, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LocalRing.Quotient | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 15
} | {
"line": 126,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsLocalRing R\nI : Ideal R\ninst✝ : IsArtinianRing (R ⧸ I)\nhI : ¬I = ⊤\nthis✝² : Nontrivial (R ⧸ I)\nthis✝¹ : IsLocalRing (R ⧸ I)\nthis✝ : IsLocalHom (Ideal.Quotient.mk I)\nn : ℕ\nhn : p ^ n ≤ I\nthis : Ideal.map (Ideal.Quotient.mk I) p = maximalId... | [] | exact ⟨n, hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Discriminant.Basic | {
"line": 481,
"column": 8
} | {
"line": 483,
"column": 38
} | {
"line": 484,
"column": 4
} | [
{
"pp": "case h₂\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.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}\nhK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N\nt... | [] | · rw [NNReal.coe_natCast, Nat.cast_le]
exact (Nat.choose_le_choose _ (rank_le_rankOfDiscrBdd hK₂)).trans
(Nat.choose_le_middle _ _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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