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.ArithmeticFunction.Moebius | {
"line": 70,
"column": 2
} | {
"line": 73,
"column": 12
} | {
"line": 75,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ μ n ≠ 0 ↔ Squarefree n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"Arithmeti... | [] | constructor <;> intro h
· contrapose h
simp [h]
· simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 70,
"column": 2
} | {
"line": 73,
"column": 12
} | {
"line": 75,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ μ n ≠ 0 ↔ Squarefree n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NegZeroClass.toNeg",
"False",
"Mathlib.Tactic.Contrapose.contrapose₂",
"Nat.instMulZeroClass",
"IsDomain.to_noZeroDivisors",
"Arithmeti... | [] | constructor <;> intro h
· contrapose h
simp [h]
· simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 12
} | {
"line": 134,
"column": 0
} | [
{
"pp": "n m : ℕ\nhn : n ≠ 0\nhm : m ≠ 0\nhnm : n.Coprime m\n⊢ μ (n * m) = μ n * μ m",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"ite_zero_mul_ite_zero",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"MulZeroClass.toMul",
... | [] | simp only [moebius, coe_mk, squarefree_mul hnm, ite_zero_mul_ite_zero, cardFactors_mul hn hm,
pow_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 232,
"column": 6
} | {
"line": 233,
"column": 18
} | {
"line": 234,
"column": 6
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\nf' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }\ng' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }\nn : ℕ\n⊢ (μ • g') (n + 1) = f' (n + 1) ↔ n + 1 > 0 → ∑ x ... | [
"case succ\nR : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\nf' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else f x, map_zero' := ⋯ }\ng' : ArithmeticFunction R := { toFun := fun x ↦ if x = 0 then 0 else g x, map_zero' := ⋯ }\nn : ℕ\n⊢ (∑ x ∈ (n + 1).divisorsAntidiagonal, μ x.1 • if x.2 = 0 then... | simp only [forall_prop_of_true, succ_pos', smul_apply, f', g', coe_mk, succ_ne_zero,
ite_false] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 117,
"column": 6
} | {
"line": 117,
"column": 66
} | {
"line": 117,
"column": 67
} | [
{
"pp": "E : Type u_1\nV : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nf : V → E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nhf1 :\n Continuous[PseudoMetricSpace.toUniformSpace.toTopolog... | [
"E : Type u_1\nV : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nf : V → E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nhf1 :\n Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 43
} | {
"line": 309,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nP : ℕ → Prop\nn : ℕ\nh : n > 0 → n ∈ s → P n\nhn : n ∈ s\nhs₀ : n ≤ 0\n⊢ 0 ∈ s",
"ppTerm": "?m.86",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nP : ℕ → Prop\nn : ℕ\nh : n > 0 → n ∈ s → P n\nhn : n ∈ s\nhs₀ : n ≤ 0\n⊢ 0 ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 25
} | {
"line": 310,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nhs₀ : 0 ∉ s\nthis : ∀ (P : ℕ → Prop), (∀ n ∈ s, P n) ↔ ∀ n > 0, n ∈ s → P n\n⊢ (∀ n ∈ s, ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n ∈ s, ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n",
"ppTerm": "?m.6... | [
"R : Type u_1\ninst✝ : AddCommGroup R\nf g : ℕ → R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nhs₀ : 0 ∉ s\nthis : ∀ (P : ℕ → Prop), (∀ n ∈ s, P n) ↔ ∀ n > 0, n ∈ s → P n\n⊢ (∀ n > 0, n ∈ s → ∑ i ∈ n.divisors, f i = g n) ↔ ∀ n > 0, n ∈ s → ∑ x ∈ n.divisorsAntidiagonal, μ x.1 • g x.2 = f n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 26
} | {
"line": 43,
"column": 27
} | [
{
"pp": "⊢ IsPrimitiveRoot (-I) 4",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"⊢ IsPrimitiveRoot (-I) 4"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 65
} | {
"line": 117,
"column": 0
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (cyclotomic' n R).roots = (primitiveRoots n R).val",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.roots",
"congrArg",
"CommSemiring.toSemiring",
... | [] | rw [cyclotomic']; exact roots_prod_X_sub_C (primitiveRoots n R) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 65
} | {
"line": 117,
"column": 0
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (cyclotomic' n R).roots = (primitiveRoots n R).val",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.roots",
"congrArg",
"CommSemiring.toSemiring",
... | [] | rw [cyclotomic']; exact roots_prod_X_sub_C (primitiveRoots n R) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 328,
"column": 29
} | {
"line": 328,
"column": 54
} | {
"line": 328,
"column": 55
} | [
{
"pp": "M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : Nontrivial M₀\nl : ℕ\nhl : 0 ^ l = 1\n⊢ 0 ∣ l",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Nat.instSemigroupWithZero",
"SemigroupWithZero.toMulZeroClass",
"id",
"ins... | [
"M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : Nontrivial M₀\nl : ℕ\nhl : 0 ^ l = 1\n⊢ l = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 342,
"column": 27
} | {
"line": 342,
"column": 77
} | {
"line": 342,
"column": 78
} | [
{
"pp": "M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : IsCancelMulZero M₀\nn : ℕ\nζ : M₀\nhζ : IsPrimitiveRoot ζ n\nα : M₀\nhα : α ≠ 0\ni : ℕ\nhi : i ∈ ↑(range n)\nj : ℕ\nhj : j ∈ ↑(range n)\ne : (fun x ↦ ζ ^ x * α) i = (fun x ↦ ζ ^ x * α) j\n⊢ (fun x ↦ ζ ^ x) i = (fun x ↦ ζ ^ x) j",
"ppTerm": "?m.... | [
"M₀ : Type u_7\ninst✝¹ : CommMonoidWithZero M₀\ninst✝ : IsCancelMulZero M₀\nn : ℕ\nζ : M₀\nhζ : IsPrimitiveRoot ζ n\nα : M₀\nhα : α ≠ 0\ni : ℕ\nhi : i ∈ ↑(range n)\nj : ℕ\nhj : j ∈ ↑(range n)\ne : (fun x ↦ ζ ^ x * α) i = (fun x ↦ ζ ^ x * α) j\n⊢ ζ ^ i = ζ ^ j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 465,
"column": 22
} | {
"line": 465,
"column": 61
} | {
"line": 465,
"column": 62
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh✝ h : IsPrimitiveRoot ζ k\n⊢ ∀ (x y : ℤ),\n Additive.ofMul ⟨(fun x ↦ ζ ^ x) (x + y), ⋯⟩ =\n Addi... | [] | by intro i j; simp only [zpow_add]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 192,
"column": 2
} | {
"line": 198,
"column": 38
} | {
"line": 199,
"column": 2
} | [
{
"pp": "case h.inr\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\n⊢ cyclotomic' k K ∈ li... | [
"case h.inr\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nBint : B ∈ lifts (Int.castRingHom ... | have Bint : B ∈ lifts (Int.castRingHom K) := by
refine Subsemiring.prod_mem (lifts (Int.castRingHom K)) ?_
intro x hx
have xsmall := (Nat.mem_properDivisors.1 hx).2
obtain ⟨d, hd⟩ := (Nat.mem_properDivisors.1 hx).1
rw [mul_comm] at hd
exact ihk x xsmall (h.pow hpos hd) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 74
} | {
"line": 206,
"column": 75
} | [
{
"pp": "case right\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ⋯\nBmo : B.Monic\nB₁ : ℤ[X]\nhB₁ : map (Int.castRingHom K) B₁ = B\nleft✝ : B₁.... | [
"case right\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nB₁ : ℤ[X]\nhB₁ : map (Int.castRing... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 553,
"column": 56
} | {
"line": 562,
"column": 24
} | {
"line": 564,
"column": 0
} | [
{
"pp": "R : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nk : ℕ\ninst✝ : NeZero k\nζ ξ : Rˣ\nh : IsPrimitiveRoot ζ k\nhξ : ξ ∈ rootsOfUnity k R\n⊢ ∃ i < k, ζ ^ i = ξ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"MulOne.toOne",
"In... | [] | by
obtain ⟨n, rfl⟩ : ∃ n : ℤ, ζ ^ n = ξ := by rwa [← h.zpowers_eq] at hξ
have hk0 : (0 : ℤ) < k := mod_cast NeZero.pos k
let i := n % k
have hi0 : 0 ≤ i := Int.emod_nonneg _ (ne_of_gt hk0)
lift i to ℕ using hi0 with i₀ hi₀
refine ⟨i₀, ?_, ?_⟩
· zify; rw [hi₀]; exact Int.emod_lt_of_pos _ hk0
· rw [← zpow... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 345,
"column": 2
} | {
"line": 346,
"column": 47
} | {
"line": 346,
"column": 48
} | [
{
"pp": "n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\ninteger : ∏ i ∈ n.divisors, cyclotomic i ℤ = X ^ n - 1\n⊢ ∏ i ∈ n.divisors, cyclotomic i R = X ^ n - 1",
"ppTerm": "?m.49",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nhpos : 0 < n\nR : Type u_1\ninst✝ : CommRing R\ninteger : ∏ i ∈ n.divisors, cyclotomic i ℤ = X ^ n - 1\n⊢ ∏ i ∈ n.divisors, cyclotomic i R = X ^ n - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 351,
"column": 4
} | {
"line": 352,
"column": 29
} | {
"line": 352,
"column": 30
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nthis : cyclotomic n ℤ ∣ X ^ n - 1\n⊢ cyclotomic n R ∣ X ^ n - 1",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nR : Type u_1\ninst✝ : Ring R\nthis : cyclotomic n ℤ ∣ X ^ n - 1\n⊢ cyclotomic n R ∣ X ^ n - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 361,
"column": 4
} | {
"line": 362,
"column": 29
} | {
"line": 362,
"column": 30
} | [
{
"pp": "n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\nthis : ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i R = ∑ i ∈ range n, X ^ i",
"ppTerm": "?m.56",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []... | [
"n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\nthis : ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i R = ∑ i ∈ range n, X ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 363,
"column": 2
} | {
"line": 364,
"column": 69
} | {
"line": 366,
"column": 0
} | [
{
"pp": "n : ℕ\nh : 0 < n\nR : Type u_1\ninst✝ : CommRing R\n⊢ ∏ i ∈ n.divisors.erase 1, cyclotomic i ℤ = ∑ i ∈ range n, X ^ i",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Int.instAddCommMonoid",
"Finset.prod_erase_mul",
"Polynomial.in... | [] | rw [← mul_left_inj' (cyclotomic_ne_zero 1 ℤ), prod_erase_mul _ _ (Nat.one_mem_divisors.2 h.ne'),
cyclotomic_one, geom_sum_mul, prod_cyclotomic_eq_X_pow_sub_one h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 99
} | {
"line": 371,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : ℕ\nhp : Fact (Nat.Prime p)\nthis : cyclotomic p ℤ = ∑ i ∈ range p, X ^ i\n⊢ cyclotomic p R = ∑ i ∈ range p, X ^ i",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Ring R\np : ℕ\nhp : Fact (Nat.Prime p)\nthis : cyclotomic p ℤ = ∑ i ∈ range p, X ^ i\n⊢ cyclotomic p R = ∑ i ∈ range p, X ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 99
} | {
"line": 389,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nthis : cyclotomic d ℤ ∣ ∑ i ∈ range n, X ^ i\n⊢ cyclotomic d R ∣ ∑ i ∈ range n, X ^ i",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : Ring R\nd n : ℕ\nhdn : d ∣ n\nhd : d ≠ 1\nthis : cyclotomic d ℤ ∣ ∑ i ∈ range n, X ^ i\n⊢ cyclotomic d R ∣ ∑ i ∈ range n, X ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 61
} | {
"line": 608,
"column": 62
} | [
{
"pp": "case neg.a\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\ne : α ^ n = a\nhn : n > 0\nhα : ¬α = 0\n⊢ (nthRoots n a).card ≤ (Multiset.map (fun x ↦ ζ ^ x * α) (Multiset.range n)).card",
"ppTerm": "?neg.a✝",
"assigned": true,
"usedConsta... | [
"case neg.a\nR : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nζ : R\nhζ : IsPrimitiveRoot ζ n\nα a : R\ne : α ^ n = a\nhn : n > 0\nhα : ¬α = 0\n⊢ (nthRoots n a).card ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 674,
"column": 20
} | {
"line": 674,
"column": 61
} | {
"line": 674,
"column": 61
} | [
{
"pp": "R : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nh : IsPrimitiveRoot ζ n\na : R\nha : a ≠ 0\nhn : n > 0\nα : R\nhα : α ^ n = a\nhα' : α = 0\n⊢ a = 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp"... | [] | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 674,
"column": 20
} | {
"line": 674,
"column": 61
} | {
"line": 674,
"column": 61
} | [
{
"pp": "R : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nh : IsPrimitiveRoot ζ n\na : R\nha : a ≠ 0\nhn : n > 0\nα : R\nhα : α ^ n = a\nhα' : α = 0\n⊢ a = 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp"... | [] | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 674,
"column": 20
} | {
"line": 674,
"column": 61
} | {
"line": 674,
"column": 61
} | [
{
"pp": "R : Type u_4\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nζ : R\nn : ℕ\nh : IsPrimitiveRoot ζ n\na : R\nha : a ≠ 0\nhn : n > 0\nα : R\nhα : α ^ n = a\nhα' : α = 0\n⊢ a = 0",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp"... | [] | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 49
} | {
"line": 168,
"column": 50
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝² : CommRing K\nμ : K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nm n✝ : ℕ\nh : IsPrimitiveRoot μ n✝\nhn : Nat.Coprime 0 n✝\n⊢ μ = μ ^ 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOne... | [
"case refine_1\nK : Type u_1\ninst✝² : CommRing K\nμ : K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\nm n✝ : ℕ\nh : IsPrimitiveRoot μ n✝\nhn : Nat.Coprime 0 n✝\n⊢ μ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 723,
"column": 6
} | {
"line": 723,
"column": 44
} | {
"line": 723,
"column": 45
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R... | [
"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R)\nhr : 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 728,
"column": 6
} | {
"line": 728,
"column": 44
} | {
"line": 728,
"column": 45
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R... | [
"M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝⁴ : CommMonoid M\ninst✝³ : CommMonoid N\ninst✝² : DivisionCommMonoid G\nk l : ℕ\ninst✝¹ : CommRing R\nζ : Rˣ\nh✝ : IsPrimitiveRoot ζ k\ninst✝ : IsDomain R\na b n : ℕ\nh : a * b ≡ 1 [MOD n]\nx : ↥(primitiveRoots n R)\nhr : 0 < ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 88,
"column": 39
} | {
"line": 88,
"column": 83
} | {
"line": 88,
"column": 84
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : X - C μ ∣ cyclotomic n K\nhμn : orderOf μ ∣ n\nhnμ : orderOf μ ≠ n\nho : 0 < orderOf μ\ni : ℕ\nhiμ : X - C μ ∣ cyclotomic i K\nhio : i ∣ orderOf μ\nkey : i < n\nkey' : i ∣ n\n⊢ {i, n} ⊆ n.divisors",
"ppTerm": "?m.1... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\ninst✝ : NeZero ↑n\nhnpos : 0 < n\nhμ : X - C μ ∣ cyclotomic n K\nhμn : orderOf μ ∣ n\nhnμ : orderOf μ ≠ n\nho : 0 < orderOf μ\ni : ℕ\nhiμ : X - C μ ∣ cyclotomic i K\nhio : i ∣ orderOf μ\nkey : i < n\nkey' : i ∣ n\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 878,
"column": 4
} | {
"line": 878,
"column": 92
} | {
"line": 880,
"column": 2
} | [
{
"pp": "G : Type u_7\nG' : Type u_8\ninst✝³ : Group G\ninst✝² : IsCyclic G\ninst✝¹ : Finite G\ninst✝ : CommGroup G'\na : G\nha : a ≠ 1\ninst : Fintype G := Fintype.ofFinite G\nζ : G'\nhζ : IsPrimitiveRoot ζ (Nat.card G)\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ orderOf ζ ∣ orderOf g",
"ppTerm": "?m... | [] | rw [← hζ.eq_orderOf, orderOf_eq_card_of_forall_mem_zpowers hg, Nat.card_eq_fintype_card] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 38
} | {
"line": 149,
"column": 38
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn m : ℕ\nhzero : n ≠ 0\nthis : NeZero n\nhnm : cyclotomic n ℂ = cyclotomic m ℂ\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n\nhroot : (cyclotomic m ℂ).IsRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n))\nhmzero... | [
"case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CharZero R\nn m : ℕ\nhzero : n ≠ 0\nthis : NeZero n\nhnm : cyclotomic n ℂ = cyclotomic m ℂ\nhprim : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) n\nhroot : IsPrimitiveRoot (Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)) m\nhmzero : NeZero m\n⊢ n =... | isRoot_cyclotomic_iff (R := ℂ) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 70
} | {
"line": 523,
"column": 4
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔\n (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n +... | [
"R : Type u_1\ninst✝ : CommRing R\np : ℕ\nhp : Nat.Prime p\nthis :\n ∀ (m : ℕ),\n cyclotomic (p ^ (m + 1)) R = ∑ i ∈ range p, (X ^ p ^ m) ^ i ↔\n (∑ i ∈ range p, (X ^ p ^ m) ^ i) * ∏ x ∈ range (m + 1), cyclotomic (p ^ x) R = X ^ p ^ (m + 1) - 1\nn_n : ℕ\nn_ih : cyclotomic (p ^ (n_n + 1)) R = ∑ i ∈ range p,... | rw [← (eq_cyclotomic_iff (pow_pos hp.pos (n_n + 1 + 1)) _).mpr ?_] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 56
} | {
"line": 157,
"column": 57
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ (aeval μ) (cyclotomic n ℤ) = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Ring.toNonAssocRing",
"congrArg",
"... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ eval μ (cyclotomic n K) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 40
} | {
"line": 175,
"column": 41
} | [
{
"pp": "n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ (cyclotomic n ℤ).natDegree ≤ (minpoly ℤ μ).natDegree",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.natDegree_cyclotomic"... | [
"n : ℕ\nK : Type u_2\ninst✝¹ : Field K\nμ : K\nh : IsPrimitiveRoot μ n\nhpos : 0 < n\ninst✝ : CharZero K\n⊢ n.totient ≤ (minpoly ℤ μ).natDegree"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 13
} | {
"line": 242,
"column": 14
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : CharZero K\np : ℕ\nζ : K\nhp : Nat.Prime p\nhζ : IsPrimitiveRoot ζ p\nα : Fin p → ℤ\n⊢ ∑ i, ↑(α i) * ζ ^ ↑i = 0 ↔ ∀ (i j : Fin p), α i = α j",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : CharZero K\np : ℕ\nζ : K\nhp : Nat.Prime p\nhζ : IsPrimitiveRoot ζ p\nα : Fin p → ℤ\n⊢ ∑ i, ↑(α i) * ζ ^ ↑i = 0 ↔ ∀ (i j : Fin p), α i = α j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 28
} | {
"line": 651,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nζ : R\nn : ℕ\nx y : R\ninst✝ : IsDomain R\nhodd : Odd n\nh : IsPrimitiveRoot ζ n\n⊢ x ^ n + y ^ n = ∏ ζ ∈ nthRootsFinset n 1, (x + ζ * y)",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\nζ : R\nn : ℕ\nx y : R\ninst✝ : IsDomain R\nhodd : Odd n\nh : IsPrimitiveRoot ζ n\n⊢ x ^ n + y ^ n = ∏ ζ ∈ nthRootsFinset n 1, (x + ζ * y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 27
} | {
"line": 166,
"column": 2
} | [
{
"pp": "case neg\nS : Type u_2\ninst✝⁶ : CommRing S\nR : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R S\nA : Type u_4\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nB : PowerBasis S A\nhB : IsIntegral R B.gen\nx : A\nhmin : minpoly S B.gen = Polynomial.map (... | [] | · exact isIntegral_zero | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 35
} | {
"line": 58,
"column": 36
} | [
{
"pp": "n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nthis : 0 < (fun x ↦ ↑x) (eval (↑(-1)) (cyclotomic n ℤ))\n⊢ 0 < eval (↑(-1)) (cyclotomic n ℤ)",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"used... | [
"n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nthis : 0 < (fun x ↦ ↑x) (eval (↑(-1)) (cyclotomic n ℤ))\n⊢ 0 < eval (↑(-1)) (cyclotomic n ℤ)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 64,
"column": 69
} | {
"line": 64,
"column": 85
} | {
"line": 64,
"column": 86
} | [
{
"pp": "n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nh0 : eval 0 (cyclotomic n ℝ) = 1\nhx : eval (-1) (cyclotomic n ℝ) ≤ 0\nthis : Set.Icc (eval (-1) (cyclotomic n ℝ)) (eval 0 (cyclotomic n ℝ)) ⊆ Set.range fun x ↦ eval x (cyclot... | [
"n : ℕ\nhn : 2 < n\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nthis✝ : NeZero n\nh0 : eval 0 (cyclotomic n ℝ) = 1\nhx : eval (-1) (cyclotomic n ℝ) ≤ 0\nthis : Set.Icc (eval (-1) (cyclotomic n ℝ)) (eval 0 (cyclotomic n ℝ)) ⊆ Set.range fun x ↦ eval x (cyclotomic n ℝ)\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 91,
"column": 8
} | {
"line": 91,
"column": 69
} | {
"line": 91,
"column": 70
} | [
{
"pp": "case h.inl.hb.inr.inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x ... | [
"case h.inl.hb.inr.inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 15
} | {
"line": 112,
"column": 16
} | [
{
"pp": "case inl\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 ≤ n\n⊢ Irreducible (cyclotomic (p ^ 0) R)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Po... | [
"case inl\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nn : ℕ\nh : Irreducible (cyclotomic (p ^ n) R)\nhmn : 0 ≤ n\n⊢ Irreducible (X - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 115,
"column": 12
} | {
"line": 115,
"column": 23
} | {
"line": 115,
"column": 24
} | [
{
"pp": "case inr.zero\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nm : ℕ\nhm : m > 0\nhmn : m ≤ m + 0\nh : Irreducible (cyclotomic (p ^ (m + 0)) R)\n⊢ Irreducible (cyclotomic (p ^ m) R)",
"ppTerm": "?inr.zero",
"assigned": false,
"usedConstants": [],
"usedFVa... | [
"case inr.zero\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nm : ℕ\nhm : m > 0\nhmn : m ≤ m + 0\nh : Irreducible (cyclotomic (p ^ (m + 0)) R)\n⊢ Irreducible (cyclotomic (p ^ m) R)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 67
} | {
"line": 111,
"column": 68
} | [
{
"pp": "case hb\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ ran... | [
"case hb\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ range n, x ^ i\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 108,
"column": 35
} | {
"line": 108,
"column": 94
} | {
"line": 108,
"column": 95
} | [
{
"pp": "n : ℕ\ninst✝⁶ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : IsCyclotomicExtension ∅ A B\n⊢ ⊥ = ⊤",
"ppTerm": "?m.18",
"assigned": true,
"us... | [
"n : ℕ\ninst✝⁶ : NeZero n\nS T : Set ℕ\nA : Type u\nB : Type v\nK : Type w\nL : Type z\ninst✝⁵ : CommRing A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra A B\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh : IsCyclotomicExtension ∅ A B\n⊢ ∀ (x : B), x ∈ ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 38
} | {
"line": 117,
"column": 39
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension {1} A B\nx : B\n⊢ x ∈ ⊥",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nh : IsCyclotomicExtension {1} A B\nx : B\n⊢ x ∈ ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 66
} | {
"line": 144,
"column": 67
} | [
{
"pp": "case inr.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nh✝ : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n\nhn' : n > 0\nhn : 1 < n\nh : eval 1 (cyclotomic n ℤ) = 1\nthis : eval (↑1) (map (Int.castRingHom R) (cyclotomic n ℤ)) = (Int.castRingHom R) (eval (↑1) (cyclotomic n ℤ))\n⊢ eval 1 (cyclotomic n R) = 1"... | [
"case inr.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\nh✝ : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n\nhn' : n > 0\nhn : 1 < n\nh : eval 1 (cyclotomic n ℤ) = 1\nthis : eval (↑1) (map (Int.castRingHom R) (cyclotomic n ℤ)) = (Int.castRingHom R) (eval (↑1) (cyclotomic n ℤ))\n⊢ eval 1 (cyclotomic n R) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 240,
"column": 4
} | {
"line": 240,
"column": 22
} | {
"line": 240,
"column": 23
} | [
{
"pp": "case inr\nn : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\nr : ℕ\nhr : GCDMonoid.lcm l n = 2 * n\nineq : φ n * φ r ≤ φ (GCDMo... | [
"case inr\nn : ℕ\ninst✝³ : NeZero n\nK : Type u\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nζ : K\nl : ℕ\nhζ : IsPrimitiveRoot ζ l\nhl✝ : l ≠ 0\nhl : NeZero l\nhroot : IsPrimitiveRoot (zeta n ℚ K) n\nr : ℕ\nhr : GCDMonoid.lcm l n = 2 * n\nineq : φ n * φ r ≤ φ (GCDMonoid.lcm l n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 195,
"column": 47
} | {
"line": 195,
"column": 58
} | {
"line": 195,
"column": 59
} | [
{
"pp": "n : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\n⊢ ¬eval (↑q) (cyclotomic n ℂ) = 0",
"ppTerm": "?m.... | [
"n : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\n⊢ ¬eval q (cyclotomic n ℝ) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 50
} | {
"line": 212,
"column": 51
} | [
{
"pp": "case convert_5\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0... | [
"case convert_5\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\nx : ℂ\nhx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 69
} | {
"line": 215,
"column": 70
} | [
{
"pp": "case convert_6\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0... | [
"case convert_6\nn : ℕ\nq : ℝ\nhn' : 2 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, q - 1 ≤ ‖↑q - ζ'‖\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, q - 1 < ‖↑q - ζ'‖\nthis : ¬eval (↑q) (cyclotomic n ℂ) = 0\n⊢ ∃ a ∈ pr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 40
} | {
"line": 269,
"column": 41
} | [
{
"pp": "case neg.refine_1\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nhS : ¬∃ s ∈ S, s ≠ 0\nH : IsCyclotomicExtension ∅ A B\n⊢ adjoin A {b | ∃ n ∈ {1}, n ≠ 0 ∧ b ^ n = 1} = ⊤",
"ppTerm": "?neg.refine_1✝",
"assigned": true,
"usedConstants": [
... | [
"case neg.refine_1\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nhS : ¬∃ s ∈ S, s ≠ 0\nH : IsCyclotomicExtension ∅ A B\n⊢ ⊥ = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 49
} | {
"line": 111,
"column": 4
} | [
{
"pp": "case succ.refine_1.inl\nn : ℕ\nh : n ≠ 0\nn✝ : ℕ\nh_prime : Fact (Nat.Prime (n✝ + 1))\nthis : Fintype (GaloisField (n✝ + 1) n)\ng_poly : (ZMod (n✝ + 1))[X] := X ^ (n✝ + 1) ^ n - X\nhp : 1 < n✝ + 1\naux : X ^ (n✝ + 1) ^ n - X ≠ 0\nkey : Fintype.card ↑(g_poly.rootSet (GaloisField (n✝ + 1) n)) = (n✝ + 1) ... | [] | rw [← map_pow, ZMod.pow_card_pow, sub_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 32
} | {
"line": 346,
"column": 4
} | [
{
"pp": "case mem\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx y : B\nhy : y ∈ {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n = 1}\n⊢ f y = ... | [
"case mem\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx y : B\nn : ℕ\nhn : n ∈ S\nh1 : n ≠ 0\nh2 : y ^ n = 1\n⊢ f y = g y"
] | obtain ⟨n, hn, h1, h2⟩ := hy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 341,
"column": 2
} | {
"line": 352,
"column": 44
} | {
"line": 354,
"column": 0
} | [
{
"pp": "S : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\n⊢ f = g",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | ext x
have hx := ‹IsCyclotomicExtension S A B›.adjoin_roots x
induction hx using Algebra.adjoin_induction with
| mem y hy =>
obtain ⟨n, hn, h1, h2⟩ := hy
obtain ⟨r, hr1, hr2⟩ := H n hn h1
have := NeZero.mk h1
obtain ⟨m, -, rfl⟩ := hr1.eq_pow_of_pow_eq_one h2
simp [hr2]
| algebraMap y => simp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 341,
"column": 2
} | {
"line": 352,
"column": 44
} | {
"line": 354,
"column": 0
} | [
{
"pp": "S : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\n⊢ f = g",
"ppTerm": "?m.35",
"assigned": true,
"usedConsta... | [] | ext x
have hx := ‹IsCyclotomicExtension S A B›.adjoin_roots x
induction hx using Algebra.adjoin_induction with
| mem y hy =>
obtain ⟨n, hn, h1, h2⟩ := hy
obtain ⟨r, hr1, hr2⟩ := H n hn h1
have := NeZero.mk h1
obtain ⟨m, -, rfl⟩ := hr1.eq_pow_of_pow_eq_one h2
simp [hr2]
| algebraMap y => simp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 259,
"column": 47
} | {
"line": 259,
"column": 58
} | {
"line": 259,
"column": 59
} | [
{
"pp": "n : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ < q + 1\n⊢ ¬eval (↑q) (cyclotomic n ℂ) = 0",
"ppTerm": "?m.... | [
"n : ℕ\nq : ℝ\nhn' : 3 ≤ n\nhq' : 1 < q\nhn : 0 < n\nhq : 0 < q\nhfor : ∀ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ ≤ q + 1\nζ : ℂ := Complex.exp (2 * ↑Real.pi * Complex.I / ↑n)\nhζ : IsPrimitiveRoot ζ n\nhex : ∃ ζ' ∈ primitiveRoots n ℂ, ‖↑q - ζ'‖ < q + 1\n⊢ ¬eval q (cyclotomic n ℝ) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 377,
"column": 34
} | {
"line": 377,
"column": 63
} | {
"line": 377,
"column": 64
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ {b | n ≠ 0 ∧ b ^ n = 1}\n⊢ x ∈ ↑(nthRoots n 1).toFinset",
"ppTerm": "?m.94",
"assigned": true,
"usedConsta... | [
"n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ {b | n ≠ 0 ∧ b ^ n = 1}\n⊢ x ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 378,
"column": 19
} | {
"line": 378,
"column": 61
} | {
"line": 378,
"column": 62
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ ↑(nthRoots n 1).toFinset\n⊢ x ∈ {b | n ≠ 0 ∧ b ^ n = 1}",
"ppTerm": "?m.97",
"assigned": true,
"usedConsta... | [
"n : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh✝ : IsCyclotomicExtension {n} A B\nx : B\nh : x ∈ ↑(nthRoots n 1).toFinset\n⊢ x ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 383,
"column": 52
} | {
"line": 383,
"column": 77
} | {
"line": 383,
"column": 78
} | [
{
"pp": "n✝ : ℕ\ninst✝⁴ : NeZero n✝\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh : IsCyclotomicExtension {n✝} A B\nb : B\nx✝ : b ∈ {b | ∃ n ∈ {n✝}, n ≠ 0 ∧ b ^ n = 1}\nn : ℕ\nhb : n = n✝ ∧ n ≠ 0 ∧ b ^ n = 1\n⊢ eval₂ (algebraMap A B) b (X ^ n - 1)... | [
"n✝ : ℕ\ninst✝⁴ : NeZero n✝\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nh : IsCyclotomicExtension {n✝} A B\nb : B\nx✝ : b ∈ {b | ∃ n ∈ {n✝}, n ≠ 0 ∧ b ^ n = 1}\nn : ℕ\nhb : n = n✝ ∧ n ≠ 0 ∧ b ^ n = 1\n⊢ b ^ n = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 416,
"column": 63
} | {
"line": 416,
"column": 74
} | {
"line": 416,
"column": 75
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ... | [
"p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhtwo : p ^ (k - s + 1) ≠ 2\nhirr₁ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 86
} | {
"line": 458,
"column": 6
} | [
{
"pp": "case refine_2\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : IsDomain B\nζ : B\nhζ : IsPrimitiveRoot ζ n\nx : B\nhx : x = ζ\n⊢ x ∈ (cyclotomic n A).rootSet B",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants... | [
"case refine_2\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : IsDomain B\nζ : B\nhζ : IsPrimitiveRoot ζ n\nx : B\nhx : x = ζ\n⊢ cyclotomic n B ≠ 0 ∧ eval ζ (cyclotomic n B) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 13
} | {
"line": 461,
"column": 14
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nh : p ≠ 2\n⊢ (Algebra.norm K) (ζ - 1)... | [
"p : ℕ\nK : Type u\nL : Type v\ninst✝³ : Field L\nζ : L\ninst✝² : Field K\ninst✝¹ : Algebra K L\nk : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\ninst✝ : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nh : p ≠ 2\n⊢ (Algebra.norm K) (ζ - 1) = ↑p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 13
} | {
"line": 471,
"column": 14
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nhpri : Fact (Nat.Prime p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p ≠ 2\nhirr : Irreducible (cyclotomic (p ^ (0 + 1)) K)\nhζ : IsPrimitiveRoot ζ (p ^ (0 + 1))\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K ... | [
"p : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nhpri : Fact (Nat.Prime p)\nhcyc : IsCyclotomicExtension {p} K L\nh : p ≠ 2\nhirr : Irreducible (cyclotomic (p ^ (0 + 1)) K)\nhζ : IsPrimitiveRoot ζ (p ^ (0 + 1))\nthis : IsCyclotomicExtension {p ^ (0 + 1)} K L\n⊢ (Algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 557,
"column": 61
} | {
"line": 557,
"column": 72
} | {
"line": 557,
"column": 73
} | [
{
"pp": "S : Set ℕ\nK : Type w\nL : Type z\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension S K L\nM : Type u_1\ninst✝² : Field M\ninst✝¹ : Algebra K M\ninst✝ : IsSepClosed M\nthis : Algebra.IsSeparable K L\ni : L →ₐ[K] M := IsSepClosed.lift\nhtop : IntermediateField.adj... | [
"S : Set ℕ\nK : Type w\nL : Type z\ninst✝⁶ : Field K\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\ninst✝³ : IsCyclotomicExtension S K L\nM : Type u_1\ninst✝² : Field M\ninst✝¹ : Algebra K M\ninst✝ : IsSepClosed M\nthis : Algebra.IsSeparable K L\ni : L →ₐ[K] M := IsSepClosed.lift\nhtop : IntermediateField.adjoin K {x | ∃... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 32
} | {
"line": 68,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommMonoid R'\nR'' : Type u_1\ninst✝ : CommMonoid R''\nφ : AddChar R R'\nf : R' →* R''\nhφ : φ.IsPrimitive\nhf : Function.Injective ⇑f\na : R\nha : a ≠ 0\n⊢ (f.compAddChar φ).mulShift a ≠ 1",
"ppTerm": "?m.24",
"assigned": true,
"usedCo... | [
"R : Type u\ninst✝² : CommRing R\nR' : Type v\ninst✝¹ : CommMonoid R'\nR'' : Type u_1\ninst✝ : CommMonoid R''\nφ : AddChar R R'\nf : R' →* R''\nhφ : φ.IsPrimitive\nhf : Function.Injective ⇑f\na : R\nha : a ≠ 0\n⊢ ∃ x, ¬f (φ (a * x)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 45
} | {
"line": 77,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nR' : Type v\ninst✝ : CommMonoid R'\nψ : AddChar R R'\nhψ : ψ.IsPrimitive\na b : R\nh : ψ.mulShift (a + -b) = 1\n⊢ a = b",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : CommRing R\nR' : Type v\ninst✝ : CommMonoid R'\nψ : AddChar R R'\nhψ : ψ.IsPrimitive\na b : R\nh : ψ.mulShift (a + -b) = 1\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 19
} | {
"line": 502,
"column": 20
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk : ℕ\nhk : 2 ≤ k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (2 ^ k) K)\nhζ : IsPrimitiveRoot ζ (2 ^ k)\nk₁ : ℕ\nhk₁ : k = k₁.succ\n⊢ (Algebra.norm K) (ζ - 1) = 2",
... | [
"K : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk : ℕ\nhk : 2 ≤ k\nH : IsCyclotomicExtension {2 ^ k} K L\nthis : 2 < 2 ^ k\nhirr : Irreducible (cyclotomic (2 ^ k) K)\nhζ : IsPrimitiveRoot ζ (2 ^ k)\nk₁ : ℕ\nhk₁ : k = k₁.succ\n⊢ (Algebra.norm K) (ζ - 1) = 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 66
} | {
"line": 192,
"column": 67
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝ : Field K\nval✝ : Fintype K\nthis : (Polynomial.map (algebraMap K K) (X ^ Fintype.card K - X)).Splits\n⊢ (X ^ Nat.card K - X).Splits",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"Field.toDiv... | [
"case inl\nK : Type u_1\ninst✝ : Field K\nval✝ : Fintype K\nthis : (Polynomial.map (algebraMap K K) (X ^ Fintype.card K - X)).Splits\n⊢ (X ^ Fintype.card K - X).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 65
} | {
"line": 194,
"column": 66
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝ : Field K\nval✝ : Infinite K\n⊢ (-(X ^ Nat.card K - X)).Splits",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"neg_sub",
"Polynomial.instNeg",
"Monoid.toMulOneClass",
"congrArg",
"Nat... | [
"case inr\nK : Type u_1\ninst✝ : Field K\nval✝ : Infinite K\n⊢ (X - 1).Splits"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 509,
"column": 48
} | {
"line": 519,
"column": 62
} | {
"line": 521,
"column": 0
} | [
{
"pp": "p : ℕ\nK : Type u\nL : Type v\ninst✝² : Field L\nζ : L\ninst✝¹ : Field K\ninst✝ : Algebra K L\nk s : ℕ\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nhpri : Fact (Nat.Prime p)\nhcycl : IsCyclotomicExtension {p ^ (k + 1)} K L\nhirr : Irreducible (cyclotomic (p ^ (k + 1)) K)\nhs : s ≤ k\nhk : k ≠ 0\n⊢ (Algebra.n... | [] | by
by_cases htwo : p ^ (k - s + 1) = 2
· obtain ⟨hp, hks⟩ := (Nat.prime_two.pow_eq_iff).1 htwo
simp only [add_eq_right] at hks
replace hs : s = k := le_antisymm hs (Nat.sub_eq_zero_iff_le.mp hks)
simp only [hp, hs] at hζ hirr hcycl ⊢
obtain ⟨k₁, hk₁⟩ := Nat.exists_eq_succ_of_ne_zero hk
rw [hζ.no... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 64
} | {
"line": 63,
"column": 65
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : HasEnoughRootsOfUnity M n\nthis : IsCyclic ↥(rootsOfUnity n M)\ng : ↥(rootsOfUnity n M)\nhg : ∀ (x : ↥(rootsOfUnity n M)), x ∈ Subgroup.zpowers g\nhg' : g ^ n = 1\nf : ZMod n → ↥(rootsOfUnity n M) := fun j ↦ g ^ ↑j.val\nx : ↥(rootsO... | [
"M : Type u_1\ninst✝² : CommMonoid M\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : HasEnoughRootsOfUnity M n\nthis : IsCyclic ↥(rootsOfUnity n M)\ng : ↥(rootsOfUnity n M)\nhg : ∀ (x : ↥(rootsOfUnity n M)), x ∈ Subgroup.zpowers g\nhg' : g ^ n = 1\nf : ZMod n → ↥(rootsOfUnity n M) := fun j ↦ g ^ ↑j.val\nx : ↥(rootsOfUnity n M)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 73
} | {
"line": 272,
"column": 2
} | [
{
"pp": "p✝ : ℕ\ninst✝⁶ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : Field K'\ninst✝³ : Fintype K\ninst✝² : Fintype K'\np : ℕ\nh_prime : Fact (Nat.Prime p)\ninst✝¹ : Algebra (ZMod p) K\ninst✝ : Algebra (ZMod p) K'\nthis✝ : CharP K p\nthis : CharP K' p\nn : ℕ+\na : Nat.P... | [
"p✝ : ℕ\ninst✝⁶ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : Field K'\ninst✝³ : Fintype K\ninst✝² : Fintype K'\np : ℕ\nh_prime : Fact (Nat.Prime p)\ninst✝¹ : Algebra (ZMod p) K\ninst✝ : Algebra (ZMod p) K'\nthis✝ : CharP K p\nthis : CharP K' p\nn : ℕ+\na : Nat.Prime p\nhK :... | have hK'Gal := (GaloisField.algEquivGaloisFieldOfFintype p n' hK').symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 45
} | {
"line": 284,
"column": 46
} | [
{
"pp": "p✝ : ℕ\ninst✝⁴ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Fintype K\ninst✝ : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : ℕ\n_char_p_K : CharP K p\np' : ℕ\n_char_p'_K' : CharP K' p'\nn : ℕ+\nhp : Nat.Prime p\nhK : Fintype.card K... | [
"p✝ : ℕ\ninst✝⁴ : Fact (Nat.Prime p✝)\nn✝ : ℕ\nK : Type u_1\nK' : Type u_2\ninst✝³ : Field K\ninst✝² : Field K'\ninst✝¹ : Fintype K\ninst✝ : Fintype K'\nhKK' : Fintype.card K = Fintype.card K'\np : ℕ\n_char_p_K : CharP K p\np' : ℕ\n_char_p'_K' : CharP K' p'\nn : ℕ+\nhp : Nat.Prime p\nhK : Fintype.card K = p ^ ↑n\nn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 26
} | {
"line": 322,
"column": 2
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nh : Module.finrank F K ∣ Module.finrank F L\nf : K →ₐ[F] L\n⊢ Nat.card (K →ₐ[F] L) = Module.finrank F K",
"ppTerm": "?m.40",
"assigned": ... | [
"F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nh : Module.finrank F K ∣ Module.finrank F L\nf : K →ₐ[F] L\nalgInst✝ : Algebra K L := f.toAlgebra\n⊢ Nat.card (K →ₐ[F] L) = Module.finrank F K"
] | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 26
} | {
"line": 334,
"column": 2
} | [
{
"pp": "F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nx✝ : Nonempty (K →ₐ[F] L)\nf : K →ₐ[F] L\n⊢ Module.finrank F K ∣ Module.finrank F L",
"ppTerm": "?m.24",
"assigned": true,
"usedConst... | [
"F : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝⁵ : Field F\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Field L\ninst✝¹ : Algebra F L\ninst✝ : Finite L\nx✝ : Nonempty (K →ₐ[F] L)\nf : K →ₐ[F] L\nalgInst✝ : Algebra K L := f.toAlgebra\n⊢ Module.finrank F K ∣ Module.finrank F L"
] | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 13
} | {
"line": 62,
"column": 14
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod... | [] | simpa using toCircle_intCast (N := N) j | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod... | [] | simpa using toCircle_intCast (N := N) j | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 41
} | {
"line": 64,
"column": 0
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod... | [] | simpa using toCircle_intCast (N := N) j | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 54
} | {
"line": 147,
"column": 55
} | [
{
"pp": "n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈ ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 7 then 1 else -1\n⊢ χ₈ ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"_private.Mathlib.NumberTheory.LegendreSymbol... | [
"n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈ ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 7 then 1 else -1\n⊢ χ₈ ↑n = if n % 8 % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 7 then 1 else -1"
] | ← Int.emod_emod_of_dvd n (by lia : (2 : ℤ) ∣ 8), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 54
} | {
"line": 178,
"column": 55
} | [
{
"pp": "n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈' ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 3 then 1 else -1\n⊢ χ₈' ↑n = if n % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 3 then 1 else -1",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.com... | [
"n : ℤ\nhelp : ∀ (m : ℤ), 0 ≤ m → m < 8 → χ₈' ↑m = if m % 2 = 0 then 0 else if m = 1 ∨ m = 3 then 1 else -1\n⊢ χ₈' ↑n = if n % 8 % 2 = 0 then 0 else if n % 8 = 1 ∨ n % 8 = 3 then 1 else -1"
] | ← Int.emod_emod_of_dvd n (by lia : (2 : ℤ) ∣ 8), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 57
} | {
"line": 167,
"column": 8
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\nx y : R\nhx : IsUnit x\n⊢ (if hx : IsUnit (x * y) then ↑(f hx.unit) else 0) =\n (if hx : IsUnit x then ↑(f hx.unit) else 0) * if hx : IsUnit y then ↑(f hx.unit) else 0",
"ppTerm": "?pos✝",... | [
"case pos\nR : Type u_1\ninst✝¹ : CommMonoid R\nR' : Type u_2\ninst✝ : CommMonoidWithZero R'\nf : Rˣ →* R'ˣ\nx y : R\nhx : IsUnit x\n⊢ (if h : IsUnit y then ↑(f ⋯.unit) else 0) = ↑(f ⋯.unit) * if hx : IsUnit y then ↑(f hx.unit) else 0"
] | simp only [hx, IsUnit.mul_iff, true_and, dif_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 31
} | {
"line": 86,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn m : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\na₁✝ a₂✝ : DirichletCharacter R n\nh : ∀ (a : (ZMod m)ˣ), ((changeLevel hm) a₁✝) ↑a = ((changeLevel hm) a₂✝) ↑a\nz : (ZMod m)ˣ\n⊢ a₁✝ ↑((ZMod.unitsMap hm) z) = a₂✝ ↑((ZMod.unitsMap hm) z)",
"ppTerm": "?m.59",
"as... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn m : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\na₁✝ a₂✝ : DirichletCharacter R n\nh : ∀ (a : (ZMod m)ˣ), ((changeLevel hm) a₁✝) ↑a = ((changeLevel hm) a₂✝) ↑a\nz : (ZMod m)ˣ\n⊢ a₁✝ ↑((ZMod.unitsMap hm) z) = a₂✝ ↑((ZMod.unitsMap hm) z)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 17
} | {
"line": 215,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ χ a ≠ 0 → IsUnit a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"IsUnit",
"id",
"MulChar",
"MulChar.instFunLike",
... | [
"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ ¬χ a = 0 → IsUnit a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 13
} | {
"line": 219,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ χ a = 0 ↔ ¬IsUnit a",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝² : CommMonoid R\nR' : Type u_2\ninst✝¹ : CommMonoidWithZero R'\ninst✝ : Nontrivial R'\nχ : MulChar R R'\na : R\n⊢ χ a = 0 ↔ ¬IsUnit a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 183,
"column": 8
} | {
"line": 183,
"column": 51
} | {
"line": 183,
"column": 52
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero n\nψ : DirichletCharacter R m\nh : (changeLevel ⋯) χ = (changeLevel ⋯) ψ\nx : (ZMod n)ˣ\nhx : x ∈ (ZMod.unitsMap ⋯).ker\nthis : (↑x).val ≡ 1 [MOD n.gcd m]\nz : ℕ\nhz₁ : z ≡ (↑x).val [MOD... | [
"case refine_1\nR : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero n\nψ : DirichletCharacter R m\nh : (changeLevel ⋯) χ = (changeLevel ⋯) ψ\nx : (ZMod n)ˣ\nhx : x ∈ (ZMod.unitsMap ⋯).ker\nthis : (↑x).val ≡ 1 [MOD n.gcd m]\nz : ℕ\nhz₁ : z ≡ (↑x).val [MOD n]\nhz₂ : z... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 900,
"column": 4
} | {
"line": 901,
"column": 25
} | {
"line": 901,
"column": 26
} | [
{
"pp": "case hS\nn : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\n⊢ ∀ n_1 ∈ {n}, n_1 ≠ 0 → ∃ r, IsPrimitiveRoot r n_1",
"ppTerm": "?hS",
"assigned": true,
"usedC... | [
"case hS\nn : ℕ\ninst✝⁴ : NeZero n\nA : Type u\nB : Type v\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra A B\ninst✝ : IsDomain B\nC : Subalgebra A B\nζ : B\nhζ : IsPrimitiveRoot ζ n\n⊢ ∃ r, IsPrimitiveRoot r n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 610,
"column": 2
} | {
"line": 610,
"column": 46
} | {
"line": 610,
"column": 47
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommMonoid R\ninst✝² : Fintype R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\nb : Rˣ\nhb : χ ↑b ≠ 1\n⊢ χ ↑b * ∑ a, χ a = ∑ a, χ a",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
... | [
"R : Type u_1\ninst✝³ : CommMonoid R\ninst✝² : Fintype R\nR' : Type u_2\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\nb : Rˣ\nhb : χ ↑b ≠ 1\n⊢ ∑ x, χ (↑b * x) = ∑ a, χ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 367,
"column": 2
} | {
"line": 370,
"column": 82
} | {
"line": 371,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nthis : NeZero n\n⊢ ((changeLevel hm) χ).conductor = χ.conductor",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"DirichletCharacter.conductor",
"Eq.mpr"... | [
"R : Type u_1\ninst✝¹ : CommMonoidWithZero R\nn : ℕ\nχ : DirichletCharacter R n\nm : ℕ\ninst✝ : NeZero m\nhm : n ∣ m\nthis : NeZero n\nh : ((changeLevel hm) χ).conductor ∣ χ.conductor\n⊢ ((changeLevel hm) χ).conductor = χ.conductor"
] | have h : (changeLevel hm χ).conductor ∣ χ.conductor := by
refine conductor_dvd_of_mem_conductorSet _
⟨χ.conductor_dvd_level.trans hm, χ.primitiveCharacter, ?_⟩
rw [changeLevel_trans _ χ.conductor_dvd_level, changeLevel_primitiveCharacter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 923,
"column": 2
} | {
"line": 924,
"column": 74
} | {
"line": 925,
"column": 4
} | [
{
"pp": "A : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain B\nn₁ n₂ : ℕ\nC₁ C₂ : Subalgebra A B\nh₁ : IsCyclotomicExtension {n₁} A ↥C₁\nh₂ : IsCyclotomicExtension {n₂} A ↥C₂\ninst✝ : NeZero n₂\nthis : NeZero n₁\nζ₂ : ↥C₂\nhζ₂ : IsPrimitiveRoot ((algebraMap... | [
"A : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsDomain B\nn₁ n₂ : ℕ\nC₁ C₂ : Subalgebra A B\nh₁ : IsCyclotomicExtension {n₁} A ↥C₁\nh₂ : IsCyclotomicExtension {n₂} A ↥C₂\ninst✝ : NeZero n₂\nthis : NeZero n₁\nζ₂ : ↥C₂\nhζ₂ : IsPrimitiveRoot ((algebraMap (↥C₂) B) ζ₂... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 475,
"column": 17
} | {
"line": 475,
"column": 71
} | {
"line": 475,
"column": 72
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nH : Set (ZMod n)ˣ\nχ : DirichletCharacter R n\nhχ : ∀ (x : (↥(Submonoid.map (Units.coeHom (ZMod n)) (Submonoid.closure H)))ˣ), χ ↑↑x = 1\nx : (ZMod n)ˣ\nhx : x ∈ Submonoid.closure H\n⊢ x⁻¹ ∈ ↑(Submonoid.closure H)",
"ppTerm": "?m.54",
"assigned... | [
"R : Type u_1\ninst✝ : CommMonoidWithZero R\nn : ℕ\nH : Set (ZMod n)ˣ\nχ : DirichletCharacter R n\nhχ : ∀ (x : (↥(Submonoid.map (Units.coeHom (ZMod n)) (Submonoid.closure H)))ˣ), χ ↑↑x = 1\nx : (ZMod n)ˣ\nhx : x ∈ Submonoid.closure H\n⊢ x ∈ Subgroup.closure H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 30,
"column": 4
} | {
"line": 31,
"column": 11
} | {
"line": 31,
"column": 12
} | [
{
"pp": "N : ℕ\ninst✝¹ : NeZero N\nR : Type u_1\ninst✝ : CommRing R\ne : AddChar (ZMod N) R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nu : (ZMod N)ˣ\nhu : (ZMod.unitsMap hd) u = 1\na : ℤ\nha : ↑(↑u).val - 1 = ↑d * a\nthis : ↑u - 1 = ↑(↑(↑u).val - 1)\ny : ZMod N\n⊢ (e.mulShift ↑(↑d *... | [
"N : ℕ\ninst✝¹ : NeZero N\nR : Type u_1\ninst✝ : CommRing R\ne : AddChar (ZMod N) R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nu : (ZMod N)ˣ\nhu : (ZMod.unitsMap hd) u = 1\na : ℤ\nha : ↑(↑u).val - 1 = ↑d * a\nthis : ↑u - 1 = ↑(↑(↑u).val - 1)\ny : ZMod N\n⊢ e (↑d * (↑a * y)) = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 24
} | {
"line": 116,
"column": 25
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\ninst✝³ : CommRing R\ninst✝² : Fintype R\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\n⊢ gaussSum χ 1 = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Finset.... | [
"R : Type u_1\nR' : Type u_2\ninst✝³ : CommRing R\ninst✝² : Fintype R\ninst✝¹ : CommRing R'\ninst✝ : IsDomain R'\nχ : MulChar R R'\nhχ : χ ≠ 1\n⊢ ∑ a, χ a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 38
} | [
{
"pp": "N : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nh_ne : gaussSum χ e ≠ 0\nx✝ : (ZMod N)ˣ\nhu : x✝ ∈ (ZMod.unitsMap hd).ker\n⊢ x✝ ∈ (MulChar.toUnitHom χ).ker",
"ppTerm": "?m... | [
"N : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nd : ℕ\nhd : d ∣ N\nhe : e.mulShift ↑d = 1\nh_ne : gaussSum χ e ≠ 0\nx✝ : (ZMod N)ˣ\nhu : x✝ ∈ (ZMod.unitsMap hd).ker\n⊢ χ ↑x✝ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 30
} | {
"line": 61,
"column": 31
} | [
{
"pp": "case pos\nN : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nhχ : χ.IsPrimitive\na : ZMod N\nha : IsUnit a\n⊢ gaussSum χ (e.mulShift a) = χ⁻¹ a * gaussSum χ e",
"ppTerm": "?pos✝",
"assigned": false,
"usedConst... | [
"case pos\nN : ℕ\ninst✝² : NeZero N\nR : Type u_1\ninst✝¹ : CommRing R\ne : AddChar (ZMod N) R\ninst✝ : IsDomain R\nχ : DirichletCharacter R N\nhχ : χ.IsPrimitive\na : ZMod N\nha : IsUnit a\n⊢ gaussSum χ (e.mulShift a) = χ⁻¹ a * gaussSum χ e"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 246,
"column": 51
} | {
"line": 249,
"column": 5
} | {
"line": 251,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nχ : MulChar R R'\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = gaussSum (χ ^ p) (ψ ^ p)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr"... | [] | by
rw [← frobenius_def, gaussSum, gaussSum, map_sum]
simp_rw [pow_apply' χ fp.1.ne_zero, map_mul, frobenius_def]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.NormPow | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | {
"line": 65,
"column": 14
} | [
{
"pp": "x p : ℝ\nhp : 1 < p\n⊢ HasDerivAt (fun x ↦ |x| ^ p) (p * |x| ^ (p - 2) * x) x",
"ppTerm": "?m.44",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x p : ℝ\nhp : 1 < p\n⊢ HasDerivAt (fun x ↦ |x| ^ p) (p * |x| ^ (p - 2) * x) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.GaussSum | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ",
"ppTerm": "?m.32",
"assigned": true,
"u... | [] | rw [_root_.gaussSum_frob, pow_mulShift, hχ.pow_char p, ← gaussSum_mulShift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ two_ne_zero, hχ.sq_eq_one, ← hp.unit_spec,
one_apply_coe, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.GaussSum | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 27
} | {
"line": 261,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\ninst✝¹ : Fintype R\nR' : Type v\ninst✝ : CommRing R'\np : ℕ\nfp : Fact (Nat.Prime p)\nhch : CharP R' p\nhp : IsUnit ↑p\nχ : MulChar R R'\nhχ : χ.IsQuadratic\nψ : AddChar R R'\n⊢ gaussSum χ ψ ^ p = χ ↑p * gaussSum χ ψ",
"ppTerm": "?m.32",
"assigned": true,
"u... | [] | rw [_root_.gaussSum_frob, pow_mulShift, hχ.pow_char p, ← gaussSum_mulShift χ ψ hp.unit,
← mul_assoc, hp.unit_spec, ← pow_two, ← pow_apply' _ two_ne_zero, hχ.sq_eq_one, ← hp.unit_spec,
one_apply_coe, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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