module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 330,
"column": 27
} | {
"line": 330,
"column": 77
} | [
{
"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",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 65
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (cyclotomic' n R).roots = (primitiveRoots n R).val",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.roots",
"congrArg",
"CommSemiring.toSemiring",
"HSub.hSub",
"RingHom",
"Multi... | 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
} | [
{
"pp": "n : ℕ\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ (cyclotomic' n R).roots = (primitiveRoots n R).val",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.roots",
"congrArg",
"CommSemiring.toSemiring",
"HSub.hSub",
"RingHom",
"Multi... | rw [cyclotomic']; exact roots_prod_X_sub_C (primitiveRoots n R) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 192,
"column": 2
} | {
"line": 198,
"column": 38
} | [
{
"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... | 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
} | [
{
"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.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 453,
"column": 22
} | {
"line": 453,
"column": 61
} | [
{
"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": 345,
"column": 2
} | {
"line": 346,
"column": 47
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 351,
"column": 4
} | {
"line": 352,
"column": 29
} | [
{
"pp": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\nthis : cyclotomic n ℤ ∣ X ^ n - 1\n⊢ cyclotomic n R ∣ X ^ n - 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 361,
"column": 4
} | {
"line": 362,
"column": 29
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 363,
"column": 2
} | {
"line": 364,
"column": 69
} | [
{
"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",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Int.instAddCommMonoid",
"Finset.prod_erase_mul",
"Polynomial.instOne",
"Int.instIsStrictOrderedRing",
... | 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.RootsOfUnity.PrimitiveRoots | {
"line": 545,
"column": 56
} | {
"line": 554,
"column": 24
} | [
{
"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",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"MulOne.toOne",
"Int.instDiv",
"Int.ofNat_lt._simp_2",
... | 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": 370,
"column": 4
} | {
"line": 370,
"column": 99
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 99
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 600,
"column": 4
} | {
"line": 600,
"column": 61
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 672,
"column": 20
} | {
"line": 672,
"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",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"instOfNatNat",
"eq_comm",
... | 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": 672,
"column": 20
} | {
"line": 672,
"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",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"instOfNatNat",
"eq_comm",
... | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 672,
"column": 20
} | {
"line": 672,
"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",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"Eq.mp",
"instOfNatNat",
"eq_comm",
... | rwa [hα', zero_pow hn.ne', eq_comm] at hα | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 70
} | [
{
"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 +... | 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.RootsOfUnity.PrimitiveRoots | {
"line": 721,
"column": 6
} | {
"line": 721,
"column": 44
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 726,
"column": 6
} | {
"line": 726,
"column": 44
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 49
} | [
{
"pp": "case refine_1.e_x\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",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 792,
"column": 41
} | {
"line": 792,
"column": 80
} | [
{
"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✝⁷ : CommMonoid M\ninst✝⁶ : CommMonoid N\ninst✝⁵ : DivisionCommMonoid G\nk l : ℕ\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\nμ : S\nn : ℕ\nhμ : IsPrimitiveRoot μ n\ninst✝² : CommRing R\ninst✝¹ : Algebra R S\ninst✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 807,
"column": 41
} | {
"line": 807,
"column": 80
} | [
{
"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✝⁷ : CommMonoid M\ninst✝⁶ : CommMonoid N\ninst✝⁵ : DivisionCommMonoid G\nk l : ℕ\ninst✝⁴ : CommRing S\ninst✝³ : IsDomain S\nμ : S\nn : ℕ\nhμ : IsPrimitiveRoot μ n\ninst✝² : CommRing R\ninst✝¹ : Algebra R S\ninst✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 88,
"column": 39
} | {
"line": 88,
"column": 83
} | [
{
"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",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 28
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 845,
"column": 4
} | {
"line": 845,
"column": 92
} | [
{
"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",
"usedConstant... | 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
} | [
{
"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... | isRoot_cyclotomic_iff (R := ℂ) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 56
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Ring.toNonAssocRing",
"congrArg",
"CommSemiring.toSemiring",
"AlgHom",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 40
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.natDegree_cyclotomic",
"Int.instNontrivial",
"Polynomi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Roots | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.GaloisClosure | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 29
} | [
{
"pp": "k : Type u_1\nK : Type u_2\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\ntoIntermediateField✝¹ : IntermediateField k K\nfiniteDimensional✝¹ : FiniteDimensional k ↥toIntermediateField✝¹\nisGalois✝¹ : IsGalois k ↥toIntermediateField✝¹\ntoIntermediateField✝ : IntermediateField k K\nfiniteDimen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.GaloisClosure | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 40
} | [
{
"pp": "k : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nf : K →ₐ[k] K\nx : K\n⊢ adjoin k {f x} = adjoin k {x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KrullTopology | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 13
} | [
{
"pp": "case h.e'_3.h.e'_4.h\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsIntegral K L\nx : L\nE : IntermediateField K L := K⟮x⟯\nhL : FiniteDimensional K ↥E\ng : Gal(L/K)\n⊢ g ∈ MulAction.stabilizer Gal(L/K) x ↔ g ∈ E.fixingSubgroup",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KrullTopology | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 47
} | [
{
"pp": "case h.mpr\nk : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁷ : Field k\ninst✝⁶ : Field E\ninst✝⁵ : Field K\ninst✝⁴ : Algebra k E\ninst✝³ : Algebra k K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower k E K\nL : IntermediateField k E\ninst✝ : Normal k E\nf : Gal(K/k)\nx : E\nhx : x ∈ ↑L.toSubsemiring\nh : ... | rwa [AlgEquiv.restrictNormal_commutes] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.KrullTopology | {
"line": 326,
"column": 25
} | {
"line": 326,
"column": 58
} | [
{
"pp": "k : Type u_1\nK : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nL : IntermediateField k K\nhnfd : FiniteDimensional k ↥L\nE : IntermediateField k K := normalClosure k (↥L) K\n⊢ L ≤ E",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 55
} | [
{
"pp": "case h\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nσ : Gal(K/k)\nh : σ ∈ (↑L.fixingSubgroup)ᶜ\ny : K\nyL : y ∈ ↑L\nne : y ∉ fun a ↦ σ • a = a\nf : Gal(K/k)\nhf : f ∈ σ • ↑(adjoin k {y}).fixingSubgroup\ng : Gal(K... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 27
} | [
{
"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": 57,
"column": 4
} | {
"line": 57,
"column": 35
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 63,
"column": 69
} | {
"line": 63,
"column": 85
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 15
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Polynomial.instOne",
"Monoid.toMulOneCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 116,
"column": 12
} | {
"line": 116,
"column": 23
} | [
{
"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)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 90,
"column": 8
} | {
"line": 90,
"column": 69
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 67
} | [
{
"pp": "case h.inr.inr.hb.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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 191,
"column": 2
} | {
"line": 192,
"column": 54
} | [
{
"pp": "case h\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nH : ClosedSubgroup Gal(K/k)\ninst✝ : IsGalois k K\nσ : Gal(K/k)\nhσ : σ ∈ (fixedField ↑H).fixingSubgroup\nh✝ : σ ∉ ↑H\nb : Set Gal(K/k)\nsub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ\ngp : Subgroup Gal(K/k)\nL :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 66
} | [
{
"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"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 275,
"column": 8
} | {
"line": 276,
"column": 44
} | [
{
"pp": "case a.h\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) :=\n fun x ↦ Subgroup.map (restrictNormalHom ↥(adjoin k {x}).t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 279,
"column": 8
} | {
"line": 281,
"column": 15
} | [
{
"pp": "case a.a\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) := ⋯\nf : ↥L → IntermediateField k K := ⋯\nthis✝ : ∀ (x : K), (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 284,
"column": 4
} | {
"line": 285,
"column": 38
} | [
{
"pp": "case refine_2\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : IsGalois k ↥L\n⊢ L.fixingSubgroup.Normal",
"usedConstants": [
"Eq.mpr",
"IsGalois.to_normal",
"IntermediateField.isScalarTower_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 194,
"column": 47
} | {
"line": 194,
"column": 58
} | [
{
"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",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 211,
"column": 4
} | {
"line": 211,
"column": 50
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 69
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 49
} | [
{
"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.PrimitiveRoots | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 13
} | [
{
"pp": "case h.e'_3\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : CommRing L\ninst✝² : IsDomain L\ninst✝¹ : Algebra K L\ninst✝ : FiniteDimensional K L\np q : ℕ\nx y : L\nhx : IsPrimitiveRoot x p\nhy : IsPrimitiveRoot y q\nhppos : p > 0\nhqpos : q > 0\nz : L := x ^ (p / p.factorizationLCMLeft q) * y ^ (q ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 258,
"column": 47
} | {
"line": 258,
"column": 58
} | [
{
"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",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 22
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 66
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"Field.toDivisionRing",
"Fintype.card",
"id"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 65
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝ : Field K\nval✝ : Infinite K\n⊢ (-(X ^ Nat.card K - X)).Splits",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"neg_sub",
"Polynomial.instNeg",
"Monoid.toMulOneClass",
"congrArg",
"Nat.card_eq_zero_of_infinite",
"HSub.hSub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 32
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MulOne.toO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 73
} | [
{
"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... | 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.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 45
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.AddCharacter | {
"line": 86,
"column": 24
} | {
"line": 86,
"column": 59
} | [
{
"pp": "R' : Type v\ninst✝¹ : CommMonoid R'\nF : Type u\ninst✝ : Field F\nψ : AddChar F R'\nhψ : ψ ≠ 1\na : F\nha : a ≠ 0\nh : ψ.mulShift a = 1\n⊢ ψ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 45
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 26
} | [
{
"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",
"usedConstants": [
"Algebra",... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 415,
"column": 63
} | {
"line": 415,
"column": 74
} | [
{
"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) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Finite.GaloisField | {
"line": 334,
"column": 2
} | {
"line": 334,
"column": 26
} | [
{
"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",
"usedConstants": [
"Algebra",
"Field.toSemi... | algebraize [f.toRingHom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.RootsOfUnity.EnoughRootsOfUnity | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 64
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 108,
"column": 35
} | {
"line": 108,
"column": 94
} | [
{
"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⊢ ⊥ = ⊤",
"usedConstants": [
"Subalgebra.instSetLike... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 38
} | [
{
"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 ∈ ⊥",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 460,
"column": 2
} | {
"line": 460,
"column": 13
} | [
{
"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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 13
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 501,
"column": 2
} | {
"line": 501,
"column": 19
} | [
{
"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",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 508,
"column": 48
} | {
"line": 518,
"column": 62
} | [
{
"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.NumberTheory.Cyclotomic.Basic | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 40
} | [
{
"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} = ⊤",
"usedConstants": [
"Eq.mpr",
"False",
"Nat.instMulZeroCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 13
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Complex.CircleAddChar | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 41
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"Nat.inst... | 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
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"Nat.inst... | 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
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nj : ℕ\n⊢ ↑(toCircle ↑j) = cexp (2 * ↑π * I * ↑j / ↑N)",
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"instHDiv",
"Real.pi",
"HMul.hMul",
"ZMod.commRing",
"congrArg",
"Nat.inst... | simpa using toCircle_intCast (N := N) j | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 32
} | [
{
"pp": "case h.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 ... | 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
} | [
{
"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",
"usedConstants": [
"Subalgebra.instSetLike",
... | 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
} | [
{
"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",
"usedConstants": [
"Subalgebra.instSetLike",
... | 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.NumberTheory.Cyclotomic.Basic | {
"line": 377,
"column": 34
} | {
"line": 377,
"column": 63
} | [
{
"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",
"usedConstants": [
"Multiset.toFinset",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 378,
"column": 19
} | {
"line": 378,
"column": 61
} | [
{
"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}",
"usedConstants": [
"Eq.mpr",
"False",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 383,
"column": 52
} | {
"line": 383,
"column": 77
} | [
{
"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)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 31
} | [
{
"pp": "case h\nR : 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)",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.Basic | {
"line": 170,
"column": 8
} | {
"line": 170,
"column": 51
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 86
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 54
} | [
{
"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",
"usedConstants": [
"_private.Mathlib.NumberTheory.LegendreSymbol.ZModChar.0.ZMod.χ₈_int_eq_if_mod_eight._proo... | ← Int.emod_emod_of_dvd n (by lia : (2 : ℤ) ∣ 8), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 165,
"column": 8
} | {
"line": 165,
"column": 57
} | [
{
"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",
"usedConstants": [... | simp only [hx, IsUnit.mul_iff, true_and, dif_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LegendreSymbol.ZModChar | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 54
} | [
{
"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",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"CommSemiring... | ← Int.emod_emod_of_dvd n (by lia : (2 : ℤ) ∣ 8), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.MulChar.Basic | {
"line": 562,
"column": 2
} | {
"line": 562,
"column": 46
} | [
{
"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",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Finset.mul_sum",
"NonUnitalCommRi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 558,
"column": 61
} | {
"line": 558,
"column": 72
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 30,
"column": 4
} | {
"line": 31,
"column": 11
} | [
{
"pp": "case h\nN : 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.mulShif... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 37
} | [
{
"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",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.DirichletCharacter.GaussSum | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 30
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 55,
"column": 10
} | {
"line": 55,
"column": 35
} | [
{
"pp": "case h\nN : ℕ\ninst✝² : NeZero N\nE : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℂ E\nΦ : ZMod N → E\nk : ZMod N\n⊢ auxDFT (fun j ↦ Φ (-j)) k = auxDFT Φ (-k)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"HMul.hMul",
"Finset.univ",
"ZMo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 73
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Even f → Function.Even (𝓕 f)\nhΦ : Function.Even (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = Φ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 85
} | [
{
"pp": "N : ℕ\ninst✝ : NeZero N\nΦ : ZMod N → ℂ\nh : ∀ {f : ZMod N → ℂ}, Function.Odd f → Function.Odd (𝓕 f)\nhΦ : Function.Odd (𝓕 Φ)\nx : ZMod N\n⊢ Φ (-x) = -Φ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 900,
"column": 4
} | {
"line": 901,
"column": 25
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"False",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Fourier.ZMod | {
"line": 216,
"column": 43
} | {
"line": 216,
"column": 60
} | [
{
"pp": "case e_f.h\nN : ℕ\ninst✝ : NeZero N\nχ : DirichletCharacter ℂ N\nk j : ZMod N\n⊢ stdAddChar (-(k * j)) * χ j = χ j * stdAddChar (-(k * j))",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSemi... | stdAddChar_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 923,
"column": 2
} | {
"line": 924,
"column": 74
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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