diff --git a/FltRegular.json b/FltRegular.json deleted file mode 100644 index 364c833b..00000000 --- a/FltRegular.json +++ /dev/null @@ -1,38571 +0,0 @@ -[{"references": - ["Ne", - "Exists", - "OfNat.ofNat", - "Int.zero_ne_one", - "Nontrivial", - "Exists.intro", - "Nontrivial.mk", - "instOfNat", - "Int"], - "name": "Int.instNontrivial", - "constType": "Nontrivial ℤ", - "constCategory": "Definition"}, - {"references": - ["Semigroup.toMul", - "Monoid.toSemigroup", - "Monoid", - "MulOneClass.toMul", - "Eq.refl", - "Dvd.dvd", - "semigroupDvd", - "Exists.intro", - "Exists.casesOn", - "Monoid.toMulOneClass", - "Eq", - "instHMul", - "Eq.mpr", - "mul_assoc", - "Eq.ndrec", - "HMul.hMul", - "Eq.symm", - "congrArg", - "id"], - "name": "mul_dvd_mul_left", - "constType": - "∀ {α : Type u_1} [inst : Monoid α] {b c : α} (a : α), b ∣ c → a * b ∣ a * c", - "constCategory": "Theorem"}, - {"references": - ["instHSub", - "AddZeroClass.toAdd", - "instHAdd", - "SubNegMonoid.toNeg", - "HAdd.hAdd", - "HSub.hSub", - "Eq.refl", - "Neg.neg", - "sub_eq_add_neg", - "AddMonoid.toAddZeroClass", - "AddGroup.toSubNegMonoid", - "Eq", - "add_neg_cancel_right", - "SubNegMonoid.toSub", - "Eq.mpr", - "AddGroup", - "SubNegMonoid.toAddMonoid", - "congrArg", - "id"], - "name": "add_sub_cancel_right", - "constType": "∀ {G : Type u_3} [inst : AddGroup G] (a b : G), a + b - b = a", - "constCategory": "Theorem"}, - {"references": - ["instHSub", - "Nat.casesAuxOn", - "OfNat.ofNat", - "HEq", - "instSubNat", - "instAddNat", - "instHAdd", - "HAdd.hAdd", - "HSub.hSub", - "Eq.refl", - "Nat.elimOffset", - "Eq", - "Iff.intro", - "Nat.add", - "Iff", - "Eq.ndrec", - "HEq.refl", - "Nat.noConfusion", - "instOfNatNat", - "Eq.casesOn", - "Nat", - "Eq.symm", - "Nat.succ"], - "name": "Nat.pred_eq_succ_iff", - "constType": "∀ {m n : ℕ}, n - 1 = m + 1 ↔ n = m + 2", - "constCategory": "Theorem"}, - {"references": - ["Zero.toOfNat0", - "AddMonoidWithOne", - "OfNat.ofNat", - "AddMonoidWithOne.toNatCast", - "Nat.cast", - "CharZero", - "propext", - "AddMonoid.toZero", - "AddMonoidWithOne.toAddMonoid", - "instOfNatNat", - "Nat", - "Nat.cast_eq_zero", - "Eq"], - "name": "Mathlib.Algebra.CharZero.Defs._auxLemma.3", - "constType": - "∀ {R : Type u_1} [inst : AddMonoidWithOne R] [inst_1 : CharZero R] {n : ℕ}, (↑n = 0) = (n = 0)", - "constCategory": "Theorem"}, - {"references": ["LinearOrderedCommMonoid", "LinearOrderedCommMonoidWithZero"], - "name": "LinearOrderedCommMonoidWithZero.toLinearOrderedCommMonoid", - "constType": - "{α : Type u_2} → [self : LinearOrderedCommMonoidWithZero α] → LinearOrderedCommMonoid α", - "constCategory": "Definition"}, - {"references": - ["NonUnitalCommRing.toNonUnitalNonAssocCommRing", - "CommMonoidWithZero.toZero", - "CommSemiring.toSemiring", - "RingHom", - "MulZeroOneClass.toMulOneClass", - "CommRing.toNonUnitalCommRing", - "IsDomain", - "Module.Finite", - "Semiring.toMonoidWithZero", - "FractionRing.field", - "Ideal.span", - "NonAssocSemiring.toMulZeroOneClass", - "CommRing", - "Eq", - "nonZeroDivisors", - 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"name": "FltRegular.NumberTheory.IdealNorm._auxLemma.11", - "constType": - "∀ (R : Type u_1) [inst : CommRing R] {S : Type u_2} [inst_1 : CommRing S] [inst_2 : Algebra R S]\n [inst_3 : IsIntegrallyClosed R] [inst_4 : IsDomain R] [inst_5 : IsDomain S] [inst_6 : NoZeroSMulDivisors R S]\n [hRS : Module.Finite R S] [inst_7 : IsIntegrallyClosed S]\n [inst_8 : Algebra.IsSeparable (FractionRing R) (FractionRing S)] (I : Ideal S) {T : Type u_3} [inst_9 : CommRing T]\n (f : R →+* T), Ideal.span (⇑f ∘ ⇑(Algebra.intNorm R S) '' ↑I) = Ideal.map f (Ideal.spanIntNorm R I)", - "constCategory": "Theorem"}, - {"references": - ["_obj", - "FltRegular.termR.«_@».FltRegular.CaseI.Statement._hyg.727._closed_5", - "FltRegular.termP.«_@».FltRegular.CaseI.Statement._hyg.15._closed_10", - "Lean.Name.str._override"], - "name": - "FltRegular.termR.«_@».FltRegular.CaseI.Statement._hyg.727._closed_6._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["CommSemiring.toSemiring", - 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"name": "Ideal.span_singleton_eq_bot", - "constType": - "∀ {α : Type u} [inst : Semiring α] {x : α}, Ideal.span {x} = ⊥ ↔ x = 0", - "constCategory": "Theorem"}, - {"references": - ["_obj", - "FltRegular._aux_FltRegular_CaseI_AuxLemmas___macroRules_FltRegular_termK_1._closed_5", - "Lean.Syntax.Preresolved.namespace"], - "name": - "FltRegular._aux_FltRegular_CaseI_AuxLemmas___macroRules_FltRegular_termK_1._closed_7._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["Iff", - "instHAdd", - "HAdd.hAdd", - "add_right_injective", - "Add", - "Function.Injective.eq_iff", - "IsLeftCancelAdd", - "Eq"], - "name": "add_right_inj", - "constType": - "∀ {G : Type u_3} [inst : Add G] [inst_1 : IsLeftCancelAdd G] (a : G) {b c : G}, a + b = a + c ↔ b = c", - "constCategory": "Theorem"}, - {"references": - ["Iff.intro", "False.elim", "False", "propext", "Not", "absurd", "Eq"], - "name": "eq_false", - "constType": "∀ {p : Prop}, ¬p → p = False", - "constCategory": "Theorem"}, - {"references": - ["Membership.mem", - "DecidablePred", - "And", - "propext", - "Finset", - "Finset.mem_filter", - "Finset.filter", - "Eq", - "Finset.instMembership"], - "name": "FltRegular.NumberTheory.Unramified._auxLemma.6", - "constType": - "∀ {α : Type u_1} {p : α → Prop} [inst : DecidablePred p] {s : Finset α} {a : α}, (a ∈ Finset.filter p s) = (a ∈ s ∧ p a)", - "constCategory": "Theorem"}, - {"references": - ["Multiset.map", - "Finset.map.proof_1", - "Finset", - "Finset.val", - "Function.Embedding", - "DFunLike.coe", - "Finset.mk", - "Function.instFunLikeEmbedding"], - "name": "Finset.map", - "constType": - "{α : Type u_1} → {β : Type u_2} → (α ↪ β) → Finset α → Finset β", - "constCategory": "Definition"}, - {"references": - ["List.cons", - "FltRegular._aux_FltRegular_CaseI_Statement___macroRules_FltRegular_termP_1._closed_39", - "_obj", - "List.nil", - "_neutral"], - "name": - "FltRegular._aux_FltRegular_CaseI_Statement___macroRules_FltRegular_termP_1._closed_40._cstage2", - "constType": "_obj", - 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"name": "zero_le", - "constType": - "∀ {α : Type u} [inst : CanonicallyOrderedAddCommMonoid α] (a : α), 0 ≤ a", - "constCategory": "Theorem"}, - {"references": - ["_obj", - "Lean.Name.num._override", - "FltRegular.termR.«_@».FltRegular.CaseI.Statement._hyg.727._closed_7"], - "name": - "FltRegular.termR.«_@».FltRegular.CaseI.Statement._hyg.727._closed_8._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": ["Part.Dom", "Part"], - "name": "Part.get", - "constType": "{α : Type u} → (self : Part α) → self.Dom → α", - "constCategory": "Definition"}, - {"references": [], - "name": "UInt8", - "constType": "Type", - "constCategory": "Other"}, - {"references": - ["CommSemiring.toSemiring", - "inferInstance", - "IdemCommSemiring", - "CommSemiring", - "Submodule.instIdemCommSemiring", - "Algebra.id", - "Ideal"], - "name": "Ideal.instIdemCommSemiring", - "constType": - "{R : Type u} → [inst : CommSemiring R] → IdemCommSemiring (Ideal R)", - "constCategory": "Definition"}, - 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"Eq"], - "name": "FltRegular.CaseI.AuxLemmas._auxLemma.2", - "constType": - "∀ {R : Type u_1} [inst : AddMonoidWithOne R] [inst_1 : CharZero R] (n : ℕ) [inst_2 : n.AtLeastTwo],\n (OfNat.ofNat n = 1) = False", - "constCategory": "Theorem"}, - {"references": ["CommSemiring", "Semifield"], - "name": "Semifield.toCommSemiring", - "constType": "{α : Type u_4} → [self : Semifield α] → CommSemiring α", - "constCategory": "Definition"}, - {"references": - ["AddCommMonoid.mk", - "SubtractionCommMonoid", - "SubtractionMonoid.toSubNegMonoid", - "SubNegMonoid.toAddMonoid", - "SubtractionCommMonoid.add_comm", - "SubtractionCommMonoid.toSubtractionMonoid", - "AddCommMonoid"], - "name": "SubtractionCommMonoid.toAddCommMonoid", - "constType": - "{G : Type u} → [self : SubtractionCommMonoid G] → AddCommMonoid G", - "constCategory": "Definition"}, - {"references": - ["CommSemiring.toSemiring", - "instHMul", - "Semiring.toNonAssocSemiring", - "Eq.ndrec", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - 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"MonoidWithZeroHom", - "Nat", - "And.casesOn", - "MonoidWithZeroHom.funLike", - "Eq.trans", - "instOfNat", - "NonUnitalNonAssocRing.toAddCommGroup", - "HPow.hPow", - "instHSub", - "Ne", - "Int.instHPowNat", - "Int.instDvd", - "Mathlib.Algebra.Divisibility.Units._auxLemma.2", - "Finset.instSingleton", - "MulZeroOneClass.toMulZeroClass", - "dvd_pow", - "SemigroupWithZero.toSemigroup", - "MonoidWithZero.toMulZeroOneClass", - "instHMul", - "And", - "eq_true", - "Prime", - "of_eq_true", - "propext", - "NormedCommRing.toNonUnitalNormedCommRing", - "Eq.mp", - "NonUnitalNormedCommRing.toNonUnitalNormedRing", - "congr", - "Int.instCommMonoid", - "Singleton.singleton", - "Not", - "Finset.gcd_singleton", - "And.intro", - "congrArg", - "Fact", - "Int", - "AddZeroClass.toAdd", - "Exists", - "NonUnitalRing.toNonUnitalSemiring", - "Mathlib.Algebra.Group.Units._auxLemma.29", - "Int.instAddGroup", - "Prime.dvd_of_dvd_pow", - "FltRegular.CaseII.Statement._auxLemma.3", - "CancelCommMonoidWithZero.toCommMonoidWithZero", - "Fact.out", - "normalize", - "Exists.intro", - "semigroupDvd", - "AddMonoid.toAddZeroClass", - "Finset.gcd", - "True", - "GCDMonoid.gcd", - "NormalizationMonoid.normUnit", - "instHPow", - "Finset", - "Monoid.toNatPow", - "instOfNatNat", - "AddCommGroup.toAddGroup", - "DFunLike.coe", - "CommMonoid.toMonoid", - "CommSemiring.toCommMonoidWithZero", - "Eq.symm", - "dvd_sub", - "id", - "CommMonoidWithZero.toMonoidWithZero", - "Monoid.toSemigroup", - "False", - "instHAdd", - "Int.instNormalizedGCDMonoid", - "HSub.hSub", - "One.toOfNat1", - "Dvd.dvd", - "Int.instMonoid", - "Monoid.toMulOneClass", - "Int.instCancelCommMonoidWithZero", - "AddGroup.toSubNegMonoid", - "Prime.not_unit", - "instNatCastInt", - "Units.val", - "Finset.gcd_insert", - "SubNegMonoid.toAddMonoid", - "letFun", - "HMul.hMul", - "NonUnitalSemiring.toSemigroupWithZero", - "IsUnit", - "Int.instCommSemiring", - "and_self", - "add_sub_cancel_right"], - "name": "FltRegular.not_exists_Int_solution'", - "constType": - "∀ {p : ℕ} [hpri : Fact (Nat.Prime p)],\n IsRegularPrime p → p ≠ 2 → ¬∃ x y z, {x, y, z}.gcd id = 1 ∧ ↑p ∣ z ∧ z ≠ 0 ∧ x ^ p + y ^ p = z ^ p", - "constCategory": "Theorem"}, - {"references": ["Zero", "Finsupp", "Finset"], - "name": "Finsupp.support", - "constType": - "{α : Type u_13} → {M : Type u_14} → [inst : Zero M] → (α →₀ M) → Finset α", - "constCategory": "Definition"}, - {"references": ["propext", "GE.ge", "ge_iff_le", "LE", "LE.le", "Eq"], - "name": "FltRegular.NumberTheory.KummersLemma.KummersLemma._auxLemma.1", - "constType": "∀ {α : Type u} [inst : LE α] {x y : α}, (x ≥ y) = (y ≤ x)", - "constCategory": "Theorem"}, - {"references": ["IdemCommSemiring", "CommSemiring"], - "name": "IdemCommSemiring.toCommSemiring", - "constType": "{α : Type u} → [self : IdemCommSemiring α] → CommSemiring α", - "constCategory": "Definition"}, - {"references": - ["CommSemiring.toSemiring", - "OfNat.ofNat", - "Monoid.toOne", - "MulZeroOneClass.toMulOneClass", - "dite", - "ClassGroup.mk0", - "ClassGroup", - "SetLike.instMembership", - "Eq", - "nonZeroDivisors", - "SubmonoidClass.mk_pow", - "Nat.dvd_one", - "Eq.mpr", - "Fintype", - "Nat.Prime", - "Semiring.toNonAssocSemiring", - "IsDomain.to_noZeroDivisors", - "MonoidWithZero.toMonoid", - "DivisionCommMonoid.toDivisionMonoid", - "Nat.instDvd", - "orderOf_dvd_card", - "Nat", - "Subtype", - "pow_mem", - "HPow.hPow", - "Classical.propDecidable", - "MonoidHom", - "IsDedekindDomain.toIsDomain", - "IsDedekindDomain", - "And", - "ClassGroup.mk0_eq_one_iff", - "MonoidWithZero.toMulZeroOneClass", - "mem_nonZeroDivisors_of_ne_zero", - "Submonoid.instSubmonoidClass", - "propext", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "DivInvOneMonoid.toInvOneClass", - "Eq.mp", - "CommRing.toRing", - "Nat.gcd", - "Not", - "And.intro", - "MonoidHom.instFunLike", - "Algebra.id", - "SubmonoidClass.nPow", - "Fact", - "congrArg", - "DivisionMonoid.toDivInvOneMonoid", - "Ideal", - "CommGroup.toDivisionCommMonoid", - "Nat.dvd_gcd_iff", - "Submodule.idemSemiring", - "Semiring.toMonoidWithZero", - "instCommGroupClassGroup", - "Ideal.instNoZeroDivisors", - "CommRing", - "Zero.toOfNat0", - "map_pow", - "instHPow", - "MonoidHom.instMonoidHomClass", - "Monoid.toNatPow", - "instOfNatNat", - "orderOf", - "DFunLike.coe", - "Eq.symm", - "orderOf_dvd_iff_pow_eq_one", - "Submodule.IsPrincipal", - "Submonoid.instSetLike", - "Semiring.toModule", - "Submonoid.toMonoid", - "id", - "Submonoid.toMulOneClass", - "Membership.mem", - "CommGroup.toGroup", - "bot_isPrincipal", - "Dvd.dvd", - "Fintype.card", - "One.toOfNat1", - "Submodule.pointwiseZero", - "DivInvMonoid.toMonoid", - "orderOf_eq_one_iff", - "Monoid.toMulOneClass", - "pow_ne_zero", - "CommRing.toCommSemiring", - "InvOneClass.toOne", - "Nat.Coprime", - "Submonoid", - "Ring.toAddCommGroup", - "IdemSemiring.toSemiring", - "NonUnitalNonAssocSemiring.toAddCommMonoid", - "Group.toDivInvMonoid", - "Subtype.mk"], - "name": "IsPrincipal_of_IsPrincipal_pow_of_Coprime", - "constType": - "∀ (A : Type u_1) [inst : CommRing A] [inst_1 : IsDedekindDomain A] [inst_2 : Fintype (ClassGroup A)] (p : ℕ)\n [inst_3 : Fact (Nat.Prime p)],\n p.Coprime (Fintype.card (ClassGroup A)) → ∀ (I : Ideal A), Submodule.IsPrincipal (I ^ p) → Submodule.IsPrincipal I", - "constCategory": "Theorem"}, - {"references": - ["Semigroup.toMul", - "Distrib.toAdd", - "Semiring.toNonUnitalSemiring", - "CommSemiring.toSemiring", - "OfNat.ofNat", - "IsPrimitiveRoot", - "a_eta_zero_dvd_p_pow", - "Semifield.toCommSemiring", - "HAdd.hAdd", - "Ideal.span", - "algebraRat", - "MulZeroClass.toMul", - "Eq", - "MonoidWithZero.toMonoid", - "Field", - "IsPrimitiveRoot.unit'", - "CommSemiring.toCommMonoid", - "Nat", - "HPow.hPow", - "instHSub", - "Ne", - "instOfNatPNatOfNeZeroNat", - "NonUnitalNonAssocSemiring.toMulZeroClass", - "Exists.choose_spec", - "root_div_zeta_sub_one_dvd_gcd", - "NumberField.to_charZero", - "EuclideanDomain.toCommRing", - "NonUnitalNonAssocSemiring.toDistrib", - "SemigroupWithZero.toSemigroup", - "instHMul", - "Field.toEuclideanDomain", - "NumberField.inst_ringOfIntegersAlgebra", - "NumberField", - "CommRing.toRing", - "Singleton.singleton", - "Not", - "Algebra.id", - "Fact", - "Field.toDivisionRing", - "NonUnitalSemiring.toNonUnitalNonAssocSemiring", - "Ideal", - "Submodule.idemSemiring", - "PNat.Prime", - "Set", - "Semiring.toMonoidWithZero", - "NumberField.RingOfIntegers.instCommRing", - "semigroupDvd", - "PNat.val", - "Set.instSingletonSet", - "Rat.commRing", - "instHPow", - "Ring.toSub", - "Monoid.toNatPow", - "instOfNatNat", - "zeta_sub_one_dvd_root", - "Eq.symm", - "p_pow_dvd_a_eta_zero", - "NeZero.succ", - "instAddNat", - "instHAdd", - "Rat", - "HSub.hSub", - "Field.toSemifield", - "One.toOfNat1", - "Dvd.dvd", - "Ring.toSemiring", - "IsCyclotomicExtension", - "Semiring.toOne", - "Units", - "CommRing.toCommSemiring", - "Exists.choose", - "Units.val", - "HMul.hMul", - "NonUnitalSemiring.toSemigroupWithZero", - "IdemSemiring.toSemiring", - "NumberField.RingOfIntegers", - "PNat"], - "name": "a_eta_zero_dvd_p_pow_spec", - "constType": - "∀ {K : Type u_1} {p : ℕ+} [hpri : Fact p.Prime] [inst : Field K] [inst_1 : NumberField K]\n [inst_2 : IsCyclotomicExtension {p} ℚ K] (hp : p ≠ 2) {ζ : K} (hζ : IsPrimitiveRoot ζ ↑p)\n {x y z : NumberField.RingOfIntegers K} {ε : (NumberField.RingOfIntegers K)ˣ} {m : ℕ}\n (e : x ^ ↑p + y ^ ↑p = ↑ε * ((↑hζ.unit' - 1) ^ (m + 1) * z) ^ ↑p) (hy : ¬↑hζ.unit' - 1 ∣ y),\n Ideal.span {↑hζ.unit' - 1} ^ m * a_eta_zero_dvd_p_pow hp hζ e hy =\n root_div_zeta_sub_one_dvd_gcd hp hζ e hy (zeta_sub_one_dvd_root hp hζ e hy)", - "constCategory": "Theorem"}, - {"references": ["Sup"], - "name": "Sup.sup", - "constType": "{α : Type u_1} → [self : Sup α] → α → α → α", - "constCategory": "Definition"}, - {"references": - ["Membership.mem", - "And", - "Iff", - "Preorder.toLT", - "LT.lt", - "Finset", - "LocallyFiniteOrder", - "Preorder", - "Finset.Ioo", - "Finset.instMembership", - "LocallyFiniteOrder.finset_mem_Ioo"], - "name": "Finset.mem_Ioo", - "constType": - "∀ {α : Type u_1} [inst : Preorder α] [inst_1 : LocallyFiniteOrder α] {a b x : α}, x ∈ Finset.Ioo a b ↔ a < x ∧ x < b", - "constCategory": "Theorem"}, - {"references": - ["Zero.toOfNat0", - "OfNat.ofNat", - "LinearOrderedCommMonoid.toOrderedCommMonoid", - "PartialOrder.toPreorder", - "propext", - "LinearOrderedCommMonoidWithZero.toZero", - "OrderedCommMonoid.toPartialOrder", - 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"name": "HPow", - "constType": "Type u → Type v → outParam (Type w) → Type (max (max u v) w)", - "constCategory": "Other"}, - {"references": - ["Int.ofNat", - "OfNat.ofNat", - "Decidable.casesOn", - "Decidable", - "Not", - "instOfNatNat", - "Nat", - "Nat.decEq", - "Int", - "Int.sub", - "Eq"], - "name": "FltRegular.CaseI.f0k₂._cstage1", - "constType": "ℤ → ℤ → ℕ → ℤ", - "constCategory": "Definition"}, - {"references": - ["Semigroup.toMul", - "NonUnitalCommRing.toNonUnitalNonAssocCommRing", - "Nat.succ_pred_prime", - "OfNat.ofNat", - "PartialOrder.toPreorder", - "SubNegMonoid.toNeg", - "integralClosure", - "Subalgebra.instSetLike", - "NumberField.RingOfIntegers.val", - "Finset.instMembership", - "Finset.sum_range", - "RelIso", - "lt_trans", - "IsPrimitiveRoot.integralPowerBasis'_gen", - "Function.Embedding", - "instOfNat", - "Fin.mk", - "Subtype", - "RingHomInvPair.ids", - "IsPrimitiveRoot.coe_submonoidClass_iff", - "IsPrimitiveRoot.pow_sub_one_eq", - "Nat.totient_prime", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring", - "Fin.last", - "Set.instMembership", - "SMulZeroClass.toSMul", - "Finset.sum_neg_distrib", - "False.elim", - "Nat.le_pred_of_lt", - "NumberField.RingOfIntegers.instIsDomain", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "congr", - "LinearOrder.toPartialOrder", - "MonoidWithZero.toZero", - "Field.toDivisionRing", - "Function.instFunLikeEmbedding", - "Int.instSemigroup", - "AddZeroClass.toAdd", - "AddCommGroup.toDivisionAddCommMonoid", - "NonUnitalCommRing.toNonUnitalCommSemiring", - "Int.instSemiring", - "Set", - "Nat.decLt", - "Eq.refl", - "semigroupDvd", - "Fin.cast", - "rfl", - "Basis.repr", - "AddGroup.toAddCancelMonoid", - "Monoid.toNatPow", - "Pi.Function.module", - "Basis.equivFun_symm_apply", - "Decidable.byContradiction", - "SubtractionMonoid.toSubNegZeroMonoid", - "Neg", - "Basis.coord_apply", - "OrderIso", - "Fin.coe_orderIso_apply", - "Field.toSemifield", - "SubmonoidClass.toCommMonoid", - "Neg.neg", - "Int.instMonoid", - "LE.le", - "Distrib.leftDistribClass", - "CommRing.toCommSemiring", - 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"PowerBasis.dim", - "Ring.toAddGroupWithOne", - "congr_fun", - "AddRightCancelMonoid.toZero", - "Finset.sum", - "LinearEquiv.symm", - "Nat.pred_eq_sub_one", - "SubNegZeroMonoid.toNegZeroClass", - "IsCyclotomicExtension", - "Fin.castSuccEmb", - "Int.instCommRing", - "Fact.mk", - "Finset.smul_sum", - "Ring.toAddCommGroup", - "NonUnitalNonAssocSemiring.toAddCommMonoid", - "FltRegular.NumberTheory.Cyclotomic.CyclRat._auxLemma.12", - "Module.toDistribMulAction", - "Fin.ext", - "sub_eq_add_neg", - "Exists.casesOn", - "instLTNat", - "SetLike.instMembership", - "Preorder.toLE", - "Eq", - "Finset.sum_add_distrib", - "Eq.mpr", - "Ring.toNeg", - "Nat.Prime.ne_zero", - "Nat", - "Finset.sum_congr", - "Basis.dvd_coord_smul", - "Nat.totient", - "DistribMulAction.toDistribSMul", - "AddCommMonoid", - "congr_arg", - "instModuleRingOfIntegers_fltRegular", - "Fin.val", - "EuclideanDomain.toCommRing", - "SubNegMonoid.SMulInt", - "Basis.coord_equivFun_symm", - "instHMul", - "Not", - "IsPrimitiveRoot.integralPowerBasis'", - "congrArg", - "Fact", - "PNat.pos", - "DivisionRing.toRing", - "NumberField.RingOfIntegers.instCommRing", - "NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring", - "Fact.out", - "Subalgebra", - "LinearMap.instFunLike", - "AddCommMonoid.toAddMonoid", - "IsPrimitiveRoot.toInteger.eq_1", - "Basis.instFunLike", - "Ring.toNonAssocRing", - "PowerBasis.gen", - "PNat.val", - "Nat.pred", - "instHPow", - "Preorder.toLT", - "AddCancelMonoid.toAddRightCancelMonoid", - "Equiv.instFunLike", - "Eq.symm", - "HAdd", - "Fin.castOrderIso", - "funext", - "HSMul", - "instHAdd", - "HSub.hSub", - "Rat", - "AddGroup.toSubNegMonoid", - "Fin.cast_mk", - "DivisionSemiring.toSemiring", - "Finsupp.instAddCommMonoid", - "SubtractionMonoid.toSubNegMonoid", - "SubNegMonoid.toAddMonoid", - "HMul.hMul", - "NegZeroClass.toZero"], - "name": "dvd_coeff_cycl_integer", - "constType": - "∀ {p : ℕ+} {L : Type u} [inst : Field L] [inst_1 : CharZero L] [inst_2 : IsCyclotomicExtension {p} ℚ L],\n Nat.Prime ↑p →\n ∀ {ζ : NumberField.RingOfIntegers L},\n IsPrimitiveRoot ζ ↑p →\n ∀ {f : Fin ↑p → ℤ}, (∃ i, f i = 0) → ∀ {m : ℤ}, ↑m ∣ ∑ j : Fin ↑p, f j • ζ ^ ↑j → ∀ (j : Fin ↑p), m ∣ f j", - "constCategory": "Theorem"}, - {"references": - ["OfNat.ofNat", - "Int.instDecidableEq", - "instAddNat", - "DecidableEq", - "instHAdd", - "ZMod.match_1", - "HAdd.hAdd", - "Unit", - "instDecidableEqFin", - "inferInstanceAs", - "instOfNatNat", - "Nat", - "Int", - "Fin", - "ZMod"], - "name": "ZMod.decidableEq", - "constType": "(n : ℕ) → DecidableEq (ZMod n)", - "constCategory": "Definition"}, - {"references": ["trivial", "Or", "eq_true", "Or.inl", "True", "Eq"], - "name": "true_or", - "constType": "∀ (p : Prop), (True ∨ p) = True", - "constCategory": "Theorem"}, - {"references": ["Iff"], - "name": "Iff.mp", - "constType": "∀ {a b : Prop}, (a ↔ b) → a → b", - "constCategory": "Theorem"}, - {"references": - ["Semiring.natCast_zero", - "Semiring.toOne", - "Semiring.toNonUnitalSemiring", - "Semiring.natCast_succ", - "Semiring", - "Semiring.mul_one", - "Semiring.one_mul", - "NonUnitalSemiring.toNonUnitalNonAssocSemiring", - "NonAssocSemiring", - "NonAssocSemiring.mk", - "Semiring.toNatCast"], - "name": "Semiring.toNonAssocSemiring", - "constType": "{α : Type u} → [self : Semiring α] → NonAssocSemiring α", - "constCategory": "Definition"}, - {"references": ["And", "Exists", "propext", "exists_prop", "Eq"], - "name": "FltRegular.NumberTheory.IdealNorm._auxLemma.16", - "constType": "∀ {b a : Prop}, (∃ (_ : a), b) = (a ∧ b)", - "constCategory": "Theorem"}, - {"references": ["Classical.not_not", "propext", "Not", "Eq"], - "name": "FltRegular.NumberTheory.Hilbert92._auxLemma.3", - "constType": "∀ {a : Prop}, (¬¬a) = a", - "constCategory": "Theorem"}, - {"references": - ["Units", - "Units.isUnit", - "eq_true", - "Monoid", - "Units.val", - "IsUnit", - "True", - "Eq"], - "name": "FltRegular.NumberTheory.KummersLemma.KummersLemma._auxLemma.7", - "constType": "∀ {M : Type u_1} [inst : Monoid M] (u : Mˣ), IsUnit ↑u = True", - "constCategory": "Theorem"}, - {"references": - ["FltRegular.termK.«_@».FltRegular.CaseI.AuxLemmas._hyg.250._closed_2", - "_obj", - "FltRegular.termP.«_@».FltRegular.CaseI.AuxLemmas._hyg.14._closed_5", - "Lean.Name.str._override"], - "name": - "FltRegular.termK.«_@».FltRegular.CaseI.AuxLemmas._hyg.250._closed_3._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["funext", "Subsingleton.intro", "Subsingleton", "Subsingleton.elim"], - "name": "instSubsingletonForall", - "constType": - "∀ {α : Sort u} {β : α → Sort v} [inst : ∀ (a : α), Subsingleton (β a)], Subsingleton ((a : α) → β a)", - "constCategory": "Definition"}, - {"references": - ["AddZeroClass.toAdd", - "MulOneClass.toMul", - "NonAssocSemiring.toAddCommMonoidWithOne", - "MulZeroOneClass.toMulOneClass", - "MulMemClass.mk", - "ZeroMemClass.mk", - "Subalgebra", - "AddSubmonoidClass.toAddMemClass", - "AddMonoid.toAddZeroClass", - "NonAssocSemiring.toMulZeroOneClass", - "Subalgebra.instSetLike", - "AddZeroClass.toZero", - "Semiring.toNonAssocSemiring", - "AddSubmonoidClass.mk", - "AddMemClass.mk", - "CommSemiring", - "Algebra", - "Subsemiring.instSetLike", - "OneMemClass.mk", - "ZeroMemClass.zero_mem", - "SubsemiringClass.addSubmonoidWithOneClass", - "SubsemiringClass.toSubmonoidClass", - "SubsemiringClass", - "Semiring", - "MulOneClass.toOne", - "Subsemiring.instSubsemiringClass", - "AddMonoidWithOne.toAddMonoid", - "Subsemiring", - "SubsemiringClass.toAddSubmonoidClass", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "SubsemiringClass.mk", - "SubmonoidClass.mk", - "AddMemClass.add_mem", - "AddSubmonoidClass.toZeroMemClass", - "OneMemClass.one_mem", - "AddMonoidWithOne.toOne", - "SubmonoidClass.toMulMemClass", - "MulMemClass.mul_mem", - "AddSubmonoidWithOneClass.toOneMemClass", - "Subalgebra.toSubsemiring"], - "name": "Subalgebra.SubsemiringClass", - "constType": - "∀ {R : Type u} {A : Type v} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Algebra R A],\n SubsemiringClass (Subalgebra R A) A", - "constCategory": "Definition"}, - {"references": ["outParam", "HAdd"], - "name": "HAdd.hAdd", - "constType": - "{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ", - "constCategory": "Definition"}, - {"references": ["CommMonoid", "Nat"], - "name": "IsPrimitiveRoot", - "constType": "{M : Type u_1} → [inst : CommMonoid M] → M → ℕ → Prop", - "constCategory": "Other"}, - {"references": - ["Zero.toOfNat0", - "CommMonoidWithZero.toZero", - "Or", - "OfNat.ofNat", - "CancelCommMonoidWithZero.toCommMonoidWithZero", - "GCDMonoid", - "propext", - "GCDMonoid.lcm", - "lcm_eq_zero_iff", - "CancelCommMonoidWithZero", - "Eq"], - "name": "FltRegular.NumberTheory.Cyclotomic.UnitLemmas._auxLemma.4", - "constType": - "∀ {α : Type u_1} [inst : CancelCommMonoidWithZero α] [inst_1 : GCDMonoid α] (a b : α), (lcm a b = 0) = (a = 0 ∨ b = 0)", - "constCategory": "Theorem"}, - {"references": - ["Lean.Omega.LinearCombo.const", - "Lean.Omega.Coeffs.dot", - "instHAdd", - "Lean.Omega.LinearCombo.coeffs", - "HAdd.hAdd", - "Lean.Omega.Coeffs", - "Int", - "Lean.Omega.LinearCombo", - "Int.instAdd"], - "name": "Lean.Omega.LinearCombo.eval", - "constType": "Lean.Omega.LinearCombo → Lean.Omega.Coeffs → ℤ", - "constCategory": "Definition"}, - {"references": ["cond.match_1", "Unit", "Bool"], - "name": "cond", - "constType": "{α : Type u} → Bool → α → α → α", - "constCategory": "Definition"}, - {"references": - ["Membership.mem", - "Finset.univ", - "eq_true", - "Fintype", - "Finset", - "Finset.mem_univ", - "True", - "Eq", - "Finset.instMembership"], - "name": "FltRegular.NumberTheory.Finrank._auxLemma.9", - "constType": - "∀ {α : Type u_1} [inst : Fintype α] (x : α), (x ∈ Finset.univ) = True", - "constCategory": "Theorem"}, - {"references": - ["eq_self", - "CommSemiring.toSemiring", - "NonAssocSemiring.toAddCommMonoidWithOne", - "OfNat.ofNat", - "MulZeroOneClass.toMulOneClass", - "Nat.rawCast", - "One.toOfNat1", - "NonUnitalNonAssocSemiring.toMul", - "NonAssocSemiring.toMulZeroOneClass", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "True", - "Eq", - "instHMul", - "Semiring.toNonAssocSemiring", - "one_mul", - "Nat.cast", - "AddMonoidWithOne.toNatCast", - "AddMonoidWithOne.toOne", - "of_eq_true", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "CommSemiring", - "HMul.hMul", - "instOfNatNat", - "Nat", - "congrArg", - "Eq.trans", - "Nat.cast_one"], - "name": "Mathlib.Tactic.Ring.one_mul", - "constType": - "∀ {R : Type u_1} [inst : CommSemiring R] (a : R), Nat.rawCast 1 * a = a", - "constCategory": "Theorem"}, - {"references": - ["Membership.mem", - "Ne", - "OfNat.ofNat", - "MulZeroOneClass.toMulOneClass", - "MulZeroOneClass.toMulZeroClass", - "MulZeroClass.toMul", - "SetLike.instMembership", - "Eq", - "nonZeroDivisors", - "NoZeroDivisors", - "Zero.toOfNat0", - "MonoidWithZero.toMulZeroOneClass", - "propext", - "mem_nonZeroDivisors_iff_ne_zero", - "Submonoid", - "Nontrivial", - "MonoidWithZero.toZero", - "Submonoid.instSetLike", - "MonoidWithZero"], - "name": "FltRegular.NumberTheory.QuotientTrace._auxLemma.8", - "constType": - "∀ {M : Type u_2} [inst : MonoidWithZero M] [inst_1 : NoZeroDivisors M] [inst_2 : Nontrivial M] {x : M},\n (x ∈ nonZeroDivisors M) = (x ≠ 0)", - "constCategory": "Theorem"}, - {"references": - ["Distrib.toAdd", - "Semiring.toNonUnitalSemiring", - "CommSemiring.toSemiring", - "OfNat.ofNat", - "IsPrimitiveRoot", - "a_eta_zero_dvd_p_pow", - "Semifield.toCommSemiring", - "HAdd.hAdd", - "Ideal.span", - "algebraRat", - "MulZeroClass.toMul", - "Eq", - "MonoidWithZero.toMonoid", - "Field", - "IsPrimitiveRoot.unit'", - "CommSemiring.toCommMonoid", - "Nat", - "HPow.hPow", - "instHSub", - "Ne", - "instOfNatPNatOfNeZeroNat", - "NonUnitalNonAssocSemiring.toMulZeroClass", - "root_div_zeta_sub_one_dvd_gcd", - "NumberField.to_charZero", - "EuclideanDomain.toCommRing", - "NonUnitalNonAssocSemiring.toDistrib", - "SemigroupWithZero.toSemigroup", - "instHMul", - "Field.toEuclideanDomain", - "NumberField.inst_ringOfIntegersAlgebra", - "NumberField", - "CommRing.toRing", - "Singleton.singleton", - "Not", - "Algebra.id", - "Fact", - "Field.toDivisionRing", - "NonUnitalSemiring.toNonUnitalNonAssocSemiring", - "Ideal", - "Submodule.idemSemiring", - "PNat.Prime", - "Set", - "Semiring.toMonoidWithZero", - "NumberField.RingOfIntegers.instCommRing", - "semigroupDvd", - "PNat.val", - "Set.instSingletonSet", - "Rat.commRing", - "instHPow", - "Ring.toSub", - "Monoid.toNatPow", - "instOfNatNat", - "zeta_sub_one_dvd_root", - "Eq.symm", - "NeZero.succ", - "instAddNat", - "instHAdd", - "Rat", - "HSub.hSub", - "Field.toSemifield", - "One.toOfNat1", - "Dvd.dvd", - "Ring.toSemiring", - "IsCyclotomicExtension", - "Semiring.toOne", - "Units", - "CommRing.toCommSemiring", - "Units.val", - "HMul.hMul", - "NonUnitalSemiring.toSemigroupWithZero", - "IdemSemiring.toSemiring", - "NumberField.RingOfIntegers", - "a_eta_zero_dvd_p_pow_spec", - "PNat"], - "name": "FltRegular.CaseII.InductionStep._auxLemma.16", - "constType": - "∀ {K : Type u_1} {p : ℕ+} [hpri : Fact p.Prime] [inst : Field K] [inst_1 : NumberField K]\n [inst_2 : IsCyclotomicExtension {p} ℚ K] (hp : p ≠ 2) {ζ : K} (hζ : IsPrimitiveRoot ζ ↑p)\n {x y z : NumberField.RingOfIntegers K} {ε : (NumberField.RingOfIntegers K)ˣ} {m : ℕ}\n (e : x ^ ↑p + y ^ ↑p = ↑ε * ((↑hζ.unit' - 1) ^ (m + 1) * z) ^ ↑p) (hy : ¬↑hζ.unit' - 1 ∣ y),\n root_div_zeta_sub_one_dvd_gcd hp hζ e hy (zeta_sub_one_dvd_root hp hζ e hy) =\n Ideal.span {↑hζ.unit' - 1} ^ m * a_eta_zero_dvd_p_pow hp hζ e hy", - "constCategory": "Theorem"}, - {"references": ["Bool.false", "Bool.true", "Bool.rec", "Bool"], - "name": "Bool.casesOn", - "constType": - "{motive : Bool → Sort u} → (t : Bool) → motive false → motive true → motive t", - "constCategory": "Definition"}, - {"references": - ["Finsupp.single", - "Finsupp.instAddCommMonoid", - "Finsupp", - "AddMonoid.toZero", - "Finsupp.sum", - "AddCommMonoid.toAddMonoid", - "AddCommMonoid"], - "name": "Finsupp.mapDomain", - "constType": - "{α : Type u_1} → {β : Type u_2} → {M : Type u_5} → [inst : AddCommMonoid M] → (α → β) → (α →₀ M) → β →₀ M", - "constCategory": "Definition"}, - {"references": ["GroupWithZero", "Div"], - "name": "GroupWithZero.toDiv", - "constType": "{G₀ : Type u} → [self : GroupWithZero G₀] → Div G₀", - "constCategory": "Definition"}, - {"references": - ["Polynomial", - "RingHom", - "Semiring.toNonAssocSemiring", - "Polynomial.X", - "Polynomial.C", - "Semiring", - "RingHom.comp", - "Polynomial.eval₂", - "Polynomial.semiring"], - "name": "Polynomial.map", - "constType": - "{R : Type u} → {S : Type v} → [inst : Semiring R] → [inst_1 : Semiring S] → (R →+* S) → Polynomial R → Polynomial S", - "constCategory": "Definition"}, - {"references": - ["Distrib.toAdd", - "CommSemiring.toSemiring", - "OfNat.ofNat", - "NonAssocSemiring.toAddCommMonoidWithOne", - "Finset.instInsert", - "PartialOrder.toPreorder", - "Mathlib.Meta.NormNum.IsNat.of_raw", - "ite_cond_eq_false", - "Nat.succ_le_succ", - "Finset.card_pos", - "Insert.insert", - "Iff.mpr", - "instDecidableEqNat", - "Nat.cast_zero", - "Mathlib.Meta.NormNum.isInt_add", - "ite", - "Semiring.toNatCast", - "Finset.instMembership", - "Mathlib.Tactic.Ring.add_pf_add_gt", - "SubNegMonoid.toSub", - "LinearOrderedCommRing.toLinearOrderedCommSemiring", - "Nat.cast", - "Mathlib.Tactic.Ring.neg_congr", - "Finset.range", - "LinearOrderedCommMonoidWithZero.toZero", - "instOfNat", - "false_or", - "StrictOrderedCommSemiring.toOrderedCommSemiring", - "instHSub", - "HPow.hPow", - "Or", - "Mathlib.Meta.NormNum.IsNat.to_raw_eq", - "Mathlib.Tactic.Ring.add_overlap_pf_zero", - "Mathlib.Tactic.Ring.add_pf_add_overlap", - "SDiff.sdiff", - "or_true", - "Int.instRing", - "LinearOrderedCommSemiring.toStrictOrderedCommSemiring", - "And", - "Mathlib.Meta.NormNum.IsNat.to_isInt", - "Mathlib.Tactic.Ring.neg_zero", - "Int.instCharZero", - "eq_true", - "Nat.zero", - "False.elim", - "Finset.card_sdiff", - "AddMonoidWithOne.toOne", - "propext", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "LT.lt", - "congr", - "instNatAtLeastTwo", - "LinearOrder.toPartialOrder", - "Mathlib.Tactic.Ring.add_pf_zero_add", - "And.intro", - "Nat.instCharZero", - "Int", - "Fin", - "AddZeroClass.toAdd", - "AddCommGroup.toDivisionAddCommMonoid", - "Exists", - "CommMonoidWithZero.toZero", - "Int.instAddGroup", - "Nat.instStrictOrderedSemiring", - "sub_nonpos_of_le", - "Int.instSemiring", - "Eq.ge", - "Init.Data.Fin.Lemmas._auxLemma.1", - "GE.ge", - "Semiring.toMonoidWithZero", - "Mathlib.Meta.NormNum.isInt_mul", - "Int.rawCast", - "Eq.refl", - "Mathlib.Tactic.Ring.sub_congr", - "IsRightCancelAdd.covariant_swap_add_lt_of_covariant_swap_add_le", - "Exists.intro", - "NeZero.one", - "Fin.is_lt", - "AddMonoid.toAddZeroClass", - "NonAssocSemiring.toMulZeroOneClass", - "Mathlib.Data.Finset.Basic._auxLemma.154", - "rfl", - "IsCancelAdd.toIsRightCancelAdd", - "True", - "Mathlib.Tactic.Ring.sub_pf", - "AddMonoidWithOne.toNatCast", - "Finset.card_insert_le", - "Int.instAddMonoid", - "Int.instCovariantClassAddLE", - "AddMonoid.toAddSemigroup", - "AddGroup.toAddCancelMonoid", - "Finset", - "Monoid.toNatPow", - "zero_lt_one", - "Nat.instAddCommMonoidWithOne", - "Nat.sub_pos_of_lt", - "SubtractionMonoid.toSubNegZeroMonoid", - "LinearOrderedCommRing.toLinearOrderedRing", - "NonUnitalNonAssocRing.toMul", - "covariant_swap_add_of_covariant_add", - "Ring.toAddGroupWithOne", - "Int.negOfNat", - "Mathlib.Tactic.Ring.neg_one_mul", - "lt_of_lt_of_le", - "Mathlib.Algebra.NeZero._auxLemma.3", - "Mathlib.Tactic.Ring.cast_pos", - "Mathlib.Tactic.Ring.mul_add", - "One.toOfNat1", - "Neg.neg", - "SubNegZeroMonoid.toNegZeroClass", - "Mathlib.Tactic.Ring.neg_add", - "LE.le", - "Finset.instSDiff", - "Finset.mem_range", - "Mathlib.Data.Finset.Basic._auxLemma.28", - "Int.instLinearOrderedCommRing", - "Mathlib.Tactic.Ring.zero_mul", - "instNatCastInt", - "Bool.false", - "Mathlib.Algebra.NeZero._auxLemma.4", - "letFun", - "not_true_eq_false", - "FltRegular.f", - "Mathlib.Tactic.Ring.add_mul", - "NegZeroClass.toNeg", - "Mathlib.Tactic.Ring.atom_pf", - "Finset.insert_subset_iff", - "Mathlib.Meta.NormNum.instAddMonoidWithOne", - "Mathlib.Tactic.Ring.mul_congr", - "Finset.instHasSubset", - "HAdd.hAdd", - "AddGroup.toSubtractionMonoid", - "CanonicallyOrderedCommSemiring.toOne", - "Exists.casesOn", - "instLTNat", - "Preorder.toLE", - "Eq", - "Eq.mpr", - "Semiring.toNonAssocSemiring", - "MonoidWithZero.toMonoid", - "StrictOrderedRing.toOrderedAddCommGroup", - "true_or", - "Mathlib.Tactic.Ring.add_pf_add_overlap_zero", - "le_trans", - "Nat", - "Eq.trans", - "Nat.cast_one", - "StrictOrderedRing.toPartialOrder", - "Ne", - "Mathlib.Meta.NormNum.IsInt.to_isNat", - "neg_neg_of_pos", - "Mathlib.Data.Finset.Basic._auxLemma.116", - "Fin.val", - "OrderedSemiring.zeroLEOneClass", - "Nat.rawCast", - "Finset.singleton_subset_iff", - "Finset.card_range", - "HasSubset.Subset", - "Finset.instSingleton", - "NonAssocRing.toNonUnitalNonAssocRing", - "lt_of_not_ge", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "Mathlib.Tactic.Ring.add_congr", - "Int.instLTInt", - "Int.instAddCommSemigroup", - "NonUnitalNonAssocSemiring.toDistrib", - "Nat.instOrderedSemiring", - "Mathlib.Tactic.Ring.neg_mul", - "instHMul", - "Mathlib.Tactic.Ring.mul_zero", - "Mathlib.Tactic.Ring.cast_zero", - "Mathlib.Tactic.Ring.of_eq", - "of_eq_true", - "AddCancelMonoid.toIsCancelAdd", - "Eq.mp", - "Not", - "Singleton.singleton", - "congrArg", - "Int.instLEInt", - "OrderedAddCommGroup.toAddCommGroup", - "Finset.card", - "StrictOrderedRing.toRing", - "Int.ofNat", - "Finset.Nonempty", - "Nat.instLinearOrderedCommMonoidWithZero", - "Linarith.lt_irrefl", - "LinearOrderedRing.toStrictOrderedRing", - "Mathlib.Meta.NormNum.IsInt.to_raw_eq", - "instSubNat", - "Mathlib.Data.Finset.Basic._auxLemma.64", - "Mathlib.Tactic.Zify._auxLemma.2", - "Mathlib.Tactic.Ring.add_pf_add_lt", - "SubtractionCommMonoid.toSubtractionMonoid", - "Ring.toNonAssocRing", - "Zero.toOfNat0", - "instLENat", - "and_false", - "Preorder.toLT", - "instHPow", - "AddMonoid.toZero", - "Mathlib.Meta.NormNum.isNat_lt_true", - "Linarith.mul_neg", - "Int.instNegInt", - "instOfNatNat", - "CommSemiring.toCommMonoidWithZero", - "add_lt_of_neg_of_le", - "Eq.symm", - "id", - "instOfNatAtLeastTwo", - "NeZero.succ", - "eq_self", - "Membership.mem", - "AddGroupWithOne.toAddMonoidWithOne", - "Mathlib.Meta.NormNum.IsInt.of_raw", - "False", - "instAddNat", - "lt_of_le_of_lt", - "AddSemigroup.toAdd", - "instHAdd", - "HSub.hSub", - "AddMonoidWithOne.toAddMonoid", - "NonUnitalNonAssocSemiring.toMul", - "Mathlib.Meta.NormNum.isNat_ofNat", - "AddGroup.toSubNegMonoid", - "GT.gt", - "Semiring.toOne", - "Mathlib.Tactic.Ring.add_pf_add_zero", - "eq_false", - "StrictOrderedSemiring.toPartialOrder", - "Finset.sdiff_subset", - "OrderedCommSemiring.toOrderedSemiring", - "Nat.instCanonicallyOrderedCommSemiring", - "Nat.instLinearOrder", - "Mathlib.Tactic.Ring.instCommSemiringNat", - "HMul.hMul", - "Bool", - "NegZeroClass.toZero", - "Int.instCommSemiring", - "Int.instNontrivial"], - "name": "FltRegular.auxf'", - "constType": - "∀ {p : ℕ}, 5 ≤ p → ∀ (a b : ℤ) (k₁ k₂ : Fin p), ∃ i ∈ Finset.range p, FltRegular.f a b (↑k₁) (↑k₂) i = 0", - "constCategory": "Theorem"}, - {"references": ["GroupWithZero", "MonoidWithZero"], - "name": "GroupWithZero.toMonoidWithZero", - "constType": "{G₀ : Type u} → [self : GroupWithZero G₀] → MonoidWithZero G₀", - "constCategory": "Definition"}, - {"references": [], - "name": "LeftCancelMonoid", - "constType": "Type u → Type u", - "constCategory": "Other"}, - {"references": - ["NormedField.toNormedCommRing.proof_2", - "NormedField.toField", - "Field.toCommRing", - "NormedCommRing", - "NormedField.toMetricSpace", - "NormedField.dist_eq", - "NormedField", - "CommRing.toRing", - "NormedRing.mk", - "NormedCommRing.mk", - "NormedField.toNormedCommRing.proof_1", - "NormedField.toNorm"], - "name": "NormedField.toNormedCommRing", - "constType": "{α : Type u_1} → [inst : NormedField α] → NormedCommRing α", - "constCategory": "Definition"}, - {"references": - ["OfNat.ofNat", - "PartialOrder.toPreorder", - "EmptyCollection.emptyCollection", - "Nat.le_of_dvd", - "Finset.Ico", - "HAdd.hAdd", - "instSubsingletonForall", - "Mathlib.Algebra.GroupWithZero.Divisibility._auxLemma.2", - "instLTNat", - "Preorder.toLE", - "Eq", - "Finset.instMembership", - "DecidablePred", - "AddZeroClass.toZero", - "Eq.mpr", - "Iff", - "Finset.range", - "Nat.instDvd", - "NonUnitalCommSemiring.toNonUnitalSemiring", - "Nat", - "Eq.trans", - "Ne", - "Or", - "eq_or_ne", - "Mathlib.Order.Basic._auxLemma.1", - "Mathlib.NumberTheory.Divisors._auxLemma.10", - "Nat.filter_dvd_eq_divisors", - "Mathlib.NumberTheory.Divisors._auxLemma.2", - "Or.casesOn", - "Mathlib.Data.Finset.Basic._auxLemma.20", - "instDecidableTrue", - "Ne.bot_lt", - "SemigroupWithZero.toSemigroup", - "Nat.instLocallyFiniteOrder", - "And", - "of_eq_true", - "LT.lt", - "congr", - "Finset.filter_empty", - "Not", - "congrArg", - "Mathlib.NumberTheory.Divisors._auxLemma.11", - "Finset.instEmptyCollection", - "instSubsingletonDecidable", - "AddZeroClass.toAdd", - "Nat.instStrictOrderedSemiring", - "Eq.refl", - "Mathlib.NumberTheory.Divisors._auxLemma.8", - "CommSemiring.toNonUnitalCommSemiring", - "semigroupDvd", - "Nat.instAddMonoid", - "SemigroupWithZero.toZero", - "AddMonoid.toAddZeroClass", - "True", - "iff_self", - "Subsingleton.elim", - "Zero.toOfNat0", - "zero_add", - "instLENat", - "and_false", - "Eq.rec", - "Finset", - "Decidable", - "Finset.filter", - "instOfNatNat", - "Eq.symm", - "Mathlib.NumberTheory.Divisors._auxLemma.9", - "Nat.succ", - "id", - "eq_self", - "Membership.mem", - "funext", - "False", - "instAddNat", - "instHAdd", - "Dvd.dvd", - "Nat.instOrderBot", - "LE.le", - "Nat.divisors", - "eq_false", - "Nat.instCommSemiring", - "Eq.ndrec", - "StrictOrderedSemiring.toPartialOrder", - "Mathlib.NumberTheory.Divisors._auxLemma.1", - "Nat.decidable_dvd", - "NonUnitalSemiring.toSemigroupWithZero", - "not_true_eq_false", - "Finset.Ico_eq_empty_of_le"], - "name": "Nat.mem_divisors", - "constType": "∀ {n m : ℕ}, n ∈ m.divisors ↔ n ∣ m ∧ m ≠ 0", - "constCategory": "Theorem"}, - {"references": - ["Algebra.toRingHom", - "CommSemiring.toSemiring", - "RingHom", - "Semiring.toNonAssocSemiring", - "Semiring", - "CommSemiring", - "Algebra"], - "name": "algebraMap", - "constType": - "(R : Type u) → (A : Type v) → [inst : CommSemiring R] → [inst_1 : Semiring A] → [inst_2 : Algebra R A] → R →+* A", - "constCategory": "Definition"}, - {"references": - ["instHSub", - "SubNegMonoid.toSub", - "instHMul", - "NonUnitalNonAssocRing.toMul", - "NonUnitalNonAssocRing", - "mul_sub_right_distrib", - "HSub.hSub", - "HMul.hMul", - "AddCommGroup.toAddGroup", - "AddGroup.toSubNegMonoid", - "NonUnitalNonAssocRing.toAddCommGroup", - "Eq"], - "name": "sub_mul", - "constType": - "∀ {α : Type u} [inst : NonUnitalNonAssocRing α] (a b c : α), (a - b) * c = a * c - b * c", - "constCategory": "Theorem"}, - {"references": - ["CommSemiring.toSemiring", - "IsDomain", - "CommSemiring", - "CommSemiring.toCommMonoidWithZero", - "IsDomain.toCancelCommMonoidWithZero.proof_1", - "CancelCommMonoidWithZero", - "CancelCommMonoidWithZero.mk"], - "name": "IsDomain.toCancelCommMonoidWithZero", - "constType": - "{α : Type u_1} → [inst : CommSemiring α] → [inst : IsDomain α] → CancelCommMonoidWithZero α", - "constCategory": "Definition"}, - {"references": - ["Units.instMul", - "instHMul", - "Units", - "MulOneClass.toMul", - "Monoid", - "Units.val", - "HMul.hMul", - "Units.val_mul", - "Eq.symm", - "Monoid.toMulOneClass", - "Eq"], - "name": "FltRegular.CaseII.InductionStep._auxLemma.21", - "constType": - "∀ {α : Type u} [inst : Monoid α] (a b : αˣ), ↑a * ↑b = ↑(a * b)", - "constCategory": "Theorem"}, - {"references": - ["AddZeroClass.toAdd", - "add_assoc", - "OfNat.ofNat", - "instHAdd", - "SubNegMonoid.toNeg", - "AddSemigroup.toAdd", - "HAdd.hAdd", - "Eq.refl", - "add_left_neg", - "Neg.neg", - "AddMonoid.toAddZeroClass", - "AddGroup.toSubNegMonoid", - "Eq", - "Zero.toOfNat0", - "AddZeroClass.toZero", - "Eq.mpr", - "AddGroup", - "AddMonoid.toAddSemigroup", - "AddMonoid.toZero", - "SubNegMonoid.toAddMonoid", - "add_zero", - "congrArg", - "id"], - "name": "neg_add_cancel_right", - "constType": "∀ {G : Type u_1} [inst : AddGroup G] (a b : G), a + -b + b = a", - "constCategory": "Theorem"}, - {"references": - ["instHMul", - "MulOneClass.toMul", - "Monoid", - "MulAction.toSMul", - "instHSMul", - "HSMul.hSMul", - "HMul.hMul", - "MulAction.mul_smul", - "Eq.symm", - "MulAction", - "Monoid.toMulOneClass", - "Eq"], - "name": "FltRegular.NumberTheory.GaloisPrime._auxLemma.18", - "constType": - "∀ {α : Type u_11} {β : Type u_12} [inst : Monoid α] [self : MulAction α β] (x y : α) (b : β), x • y • b = (x * y) • b", - "constCategory": "Theorem"}, - {"references": - ["Iff.symm", "Units", "Monoid", "Iff", "Units.val", "Units.eq_iff", "Eq"], - "name": "Units.ext_iff", - "constType": "∀ {α : Type u} [inst : Monoid α] {a b : αˣ}, a = b ↔ ↑a = ↑b", - "constCategory": "Theorem"}, - {"references": - ["Membership.mem", - "RingHom", - "CommSemiring.toSemiring", - "FractionalIdeal.coeIdeal", - "Exists", - "MulZeroOneClass.toMulOneClass", - "algebraMap", - "NonAssocSemiring.toMulZeroOneClass", - "RingHom.instFunLike", - "CommRing", - "SetLike.instMembership", - "Eq", - "CommRing.toCommSemiring", - "And", - "Semiring.toNonAssocSemiring", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "propext", - "Algebra", - "Submonoid", - "FractionalIdeal", - "FractionalIdeal.mem_coeIdeal", - "NonUnitalNonAssocSemiring.toAddCommMonoid", - "DFunLike.coe", - "Ideal", - "FractionalIdeal.instSetLike", - "Semiring.toModule", - "Submodule.setLike"], - "name": "FltRegular.NumberTheory.Different._auxLemma.3", - "constType": - "∀ {R : Type u_1} [inst : CommRing R] (S : Submonoid R) {P : Type u_2} [inst_1 : CommRing P] [inst_2 : Algebra R P]\n {x : P} {I : Ideal R}, (x ∈ ↑I) = ∃ x' ∈ I, (algebraMap R P) x' = x", - "constCategory": "Theorem"}, - {"references": - ["trivial", - "Part.ext'", - "Part.Dom", - "AddMonoidWithOne.toNatCast", - "Nat.cast", - "PartENat.instAddCommMonoidWithOne", - "Part.get", - "PartENat", - "iff_of_true", - "Nat", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "rfl", - "Eq"], - "name": "PartENat.natCast_get", - "constType": "∀ {x : PartENat} (h : x.Dom), ↑(x.get h) = x", - "constCategory": "Theorem"}, - {"references": ["_obj"], - "name": - "FltRegular.termP.«_@».FltRegular.CaseI.AuxLemmas._hyg.14._closed_10._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["Finset.card", - "Membership.mem", - "Finset.insert_eq_of_mem", - "OfNat.ofNat", - "Finset.decidableMem", - "PartialOrder.toPreorder", - "Finset.instInsert", - "Lattice.toSemilatticeInf", - "instAddNat", - "Finset.card_insert_of_not_mem", - "instHAdd", - "HAdd.hAdd", - "SemilatticeInf.toPartialOrder", - "dite", - "Insert.insert", - "LE.le", - "instDistribLatticeNat", - "Eq", - "Finset.instMembership", - "DistribLattice.toLattice", - "Eq.mpr", - "instLENat", - "DecidableEq", - "le_refl", - "Finset", - "Nat.le_succ", - "Not", - "instOfNatNat", - "Nat", - "congrArg", - "id"], - "name": "Finset.card_insert_le", - "constType": - "∀ {α : Type u_1} [inst : DecidableEq α] (a : α) (s : Finset α), (insert a s).card ≤ s.card + 1", - "constCategory": "Theorem"}, - {"references": - ["QuotientGroup.instHasQuotientSubgroup", - "Quotient.mk''", - "QuotientGroup.leftRel", - "Subgroup", - "Group", - "HasQuotient.Quotient"], - "name": "QuotientGroup.mk", - "constType": - "{α : Type u_1} → [inst : Group α] → {s : Subgroup α} → α → α ⧸ s", - "constCategory": "Definition"}, - {"references": ["LE", "Sub", "Add"], - "name": "OrderedSub", - "constType": - "(α : Type u_3) → [inst : LE α] → [inst : Add α] → [inst : Sub α] → Prop", - "constCategory": "Other"}, - {"references": - ["Units", - "UniqueFactorizationMonoid", - "CancelCommMonoidWithZero.toCommMonoidWithZero", - "MonoidWithZero.toMonoid", - "Multiset", - "UniqueFactorizationMonoid.factors", - "UniqueFactorizationMonoid.factors_eq_normalizedFactors", - "Eq.symm", - "normalizationMonoidOfUniqueUnits", - "UniqueFactorizationMonoid.normalizedFactors", - "CancelCommMonoidWithZero", - "Eq", - "CommMonoidWithZero.toMonoidWithZero", - "Unique"], - "name": "FltRegular.NumberTheory.Unramified._auxLemma.9", - "constType": - "∀ {M : Type u_2} [inst : CancelCommMonoidWithZero M] [inst_1 : UniqueFactorizationMonoid M] [inst_2 : Unique Mˣ]\n (x : M), UniqueFactorizationMonoid.normalizedFactors x = UniqueFactorizationMonoid.factors x", - "constCategory": "Theorem"}, - {"references": - ["Nat.dvd_of_mod_eq_zero", - "Or", - "OfNat.ofNat", - "Nat.mod_two_eq_zero_or_one", - "of_decide_eq_true", - "Eq.refl", - "instDecidableEqNat", - "Eq", - "Or.resolve_left", - "Nat.Prime", - "Or.imp_left", - "instHMod", - "instDecidableNot", - "Bool.true", - "HMod.hMod", - "instOfNatNat", - "Not", - "Bool", - "Nat", - "Eq.symm", - "Nat.Prime.eq_one_or_self_of_dvd", - "Nat.instMod"], - "name": "Nat.Prime.eq_two_or_odd", - "constType": "∀ {p : ℕ}, Nat.Prime p → p = 2 ∨ p % 2 = 1", - "constCategory": "Theorem"}, - {"references": - ["Distrib.toAdd", - "AddZeroClass.toAdd", - "MulOneClass.toMul", - "AlgHomClass.mk", - "NonAssocSemiring.toAddCommMonoidWithOne", - "MulZeroOneClass.toMulOneClass", - "AddMonoid.toAddZeroClass", - "NonAssocSemiring.toMulZeroOneClass", - "EquivLike", - "Semiring.toNonAssocSemiring", - "MulEquivClass.map_mul", - "AlgEquivClass.toRingEquivClass", - "RingHomClass.toMonoidWithZeroHomClass", - "CommSemiring", - "Algebra", - "MonoidHomClass.toOneHomClass", - "EquivLike.toFunLike", - "AddMonoidHomClass.mk", - "MonoidHomClass.mk", - "RingEquivClass.toRingHomClass", - "RingHomClass.mk", - "AlgHomClass", - "Semiring", - "AddMonoidWithOne.toAddMonoid", - "AlgEquivClass.commutes", - "NonUnitalNonAssocSemiring.toMul", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "MulHomClass.mk", - "NonUnitalNonAssocSemiring.toDistrib", - "AddHomClass.mk", - "RingEquivClass.map_add", - "RingHomClass.toMonoidHomClass", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "RingEquivClass.toMulEquivClass", - "AlgEquivClass", - "MonoidWithZeroHomClass.toZeroHomClass"], - "name": "AlgEquivClass.toAlgHomClass", - "constType": - "∀ (F : Type u_1) (R : Type u_2) (A : Type u_3) (B : Type u_4) [inst : CommSemiring R] [inst_1 : Semiring A]\n [inst_2 : Semiring B] [inst_3 : Algebra R A] [inst_4 : Algebra R B] [inst_5 : EquivLike F A B]\n [h : AlgEquivClass F R A B], AlgHomClass F R A B", - "constCategory": "Definition"}, - {"references": - ["instCommGroupRelativeUnits", - "CommGroup.toGroup", - "DivisionRing.toRing", - "Algebra.toModule", - "Semifield.toCommSemiring", - "LieRing.toAddCommGroup", - "Field.toSemifield", - "Eq.refl", - "RelativeUnits", - "MonoidHom", - "DivInvMonoid.toMonoid", - "AlgHom", - "Monoid.toMulOneClass", - "Eq", - "DivisionSemiring.toSemiring", - "relativeUnitsMap", - "relativeUnitsMapHom", - "AlgHom.End", - "Monoid.End.inst", - "Field", - "NumberField", - "Algebra", - "Semifield.toDivisionSemiring", - "MonoidHom.instFunLike", - "Group.toDivInvMonoid", - "DFunLike.coe", - "LieRing.ofAssociativeRing", - "Field.toDivisionRing", - "Monoid.End", - "FiniteDimensional"], - "name": "relativeUnitsMapHom_apply", - "constType": - "∀ {K : Type u_1} [inst : Field K] [inst_1 : NumberField K] {k : Type u_2} [inst_2 : Field k] [inst_3 : NumberField k]\n [inst_4 : Algebra k K] [inst_5 : FiniteDimensional k K] (σ : K →ₐ[k] K), relativeUnitsMapHom σ = relativeUnitsMap σ", - "constCategory": "Theorem"}, - {"references": ["PNat.val", "Nat.Prime", "PNat"], - "name": "PNat.Prime", - "constType": "ℕ+ → Prop", - "constCategory": "Definition"}, - {"references": - ["CommSemiring.toSemiring", - "inferInstance", - "Semiring", - "Int.instCommSemiring", - "Int"], - "name": "Int.instSemiring", - "constType": "Semiring ℤ", - "constCategory": "Definition"}, - {"references": - ["AddZeroClass.toAdd", - "OfNat.ofNat", - "Iff.rfl", - "SubNegMonoid.toNeg", - "HAdd.hAdd", - "AddGroup.toSubtractionMonoid", - "sub_eq_add_neg", - "AddMonoid.toAddZeroClass", - "AddGroup.covconv_swap", - "Iff.mpr", - "Eq", - "Zero.toOfNat0", - "SubNegMonoid.toSub", - "Function.swap", - "AddZeroClass.toZero", - "zero_add", - "Eq.mpr", - "AddGroup", - "Iff", - "LE", - "add_le_add_iff_right", - "Eq.symm", - "neg_add_cancel_right", - "SubtractionMonoid.toSubNegZeroMonoid", - "id", - "instHSub", - "instHAdd", - "HSub.hSub", - "Neg.neg", - "SubNegZeroMonoid.toNegZeroClass", - "LE.le", - "AddGroup.toSubNegMonoid", - "propext", - "SubNegMonoid.toAddMonoid", - "CovariantClass", - "NegZeroClass.toZero", - "congrArg"], - "name": "sub_nonpos_of_le", - "constType": - "∀ {α : Type u} [inst : AddGroup α] [inst_1 : LE α]\n [inst_2 : CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] {a b : α}, a ≤ b → a - b ≤ 0", - "constCategory": "Theorem"}, - {"references": - ["instHMod", - "Int.instDecidableEq", - "Int.ModEq", - "Decidable", - "HMod.hMod", - "Int.instMod", - "decEq", - "Int"], - "name": "Int.instDecidableModEq", - "constType": "{n a b : ℤ} → Decidable (a ≡ b [ZMOD n])", - "constCategory": "Definition"}, - {"references": ["_obj"], - "name": - "FltRegular._aux_FltRegular_CaseI_Statement___unexpand_CyclotomicField_1._closed_1._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["Field.zpow_succ'", - "DivisionRing", - "Field.qsmul_def", - "DivisionRing.mk", - "Field.zpow_zero'", - "Field.nnqsmul", - "Field.toNontrivial", - "Field.toInv", - "Field.zpow", - "Field.zpow_neg'", - "Field.nnqsmul_def", - "Field.toNNRatCast", - "Field.toRatCast", - "Field.ratCast_def", - "Field.toCommRing", - "Field.mul_inv_cancel", - "Field", - "CommRing.toRing", - "Field.inv_zero", - "Field.toDiv", - "Field.qsmul", - "Field.nnratCast_def", - "Field.div_eq_mul_inv"], - "name": "Field.toDivisionRing", - "constType": "{K : Type u} → [self : Field K] → DivisionRing K", - "constCategory": "Definition"}, - {"references": - ["Membership.mem", - "Exists", - "propext", - "MonoidHom.mem_range", - "Subgroup", - "Group", - "MonoidHom.instFunLike", - "Group.toDivInvMonoid", - "MonoidHom", - "DFunLike.coe", - "MonoidHom.range", - "DivInvMonoid.toMonoid", - "Subgroup.instSetLike", - "Monoid.toMulOneClass", - "SetLike.instMembership", - "Eq"], - "name": "FltRegular.NumberTheory.Hilbert92._auxLemma.8", - "constType": - "∀ {G : Type u_1} [inst : Group G] {N : Type u_5} [inst_1 : Group N] {f : G →* N} {y : N}, (y ∈ f.range) = ∃ x, f x = y", - "constCategory": "Theorem"}, - {"references": - ["Membership.mem", - "Submodule.instBot", - "Bot.bot", - "OfNat.ofNat", - 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"Semiring.toNonAssocSemiring", - "Set.image_image", - "OreLocalization.oreSetComm", - "FractionRing.liftAlgebra", - "Algebra", - "DFunLike.coe", - "Set.image", - "CommSemiring.toCommMonoidWithZero", - "IsIntegrallyClosed", - "Algebra.toSMul", - "Semiring.toModule", - "Algebra.intNorm", - "NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring", - "id", - "OreLocalization.instCommRing", - "NoZeroSMulDivisors", - "Ideal.map", - "Algebra.toModule", - "Function.comp", - "Ideal.spanIntNorm.eq_1", - "OreLocalization.instRing", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring", - "MonoidHom", - "RingHom.instFunLike", - "OreLocalization.instAlgebra", - "CommRing.toCommSemiring", - "CommRing.toCommMonoid", - "FractionRing", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "CommRing.toRing", - "FractionRing.instNoZeroSMulDivisors", - "MonoidHom.instFunLike", - "Ideal.spanIntNorm", - "NonUnitalNonAssocSemiring.toAddCommMonoid", - "Algebra.IsSeparable", - "congrArg", - "Ideal", - 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"Nat"], - "name": "instNatAtLeastTwo", - "constType": "∀ {n : ℕ}, (n + 2).AtLeastTwo", - "constCategory": "Definition"}, - {"references": - ["MonoidHomClass", - "outParam", - "FunLike", - "MulZeroOneClass.toMulOneClass", - "RingHomClass", - "NonAssocSemiring.toMulZeroOneClass", - "NonAssocSemiring"], - "name": "RingHomClass.toMonoidHomClass", - "constType": - "∀ {F : Type u_5} {α : outParam (Type u_6)} {β : outParam (Type u_7)} [inst : NonAssocSemiring α]\n [inst_1 : NonAssocSemiring β] [inst_2 : FunLike F α β] [self : RingHomClass F α β], MonoidHomClass F α β", - "constCategory": "Theorem"}, - {"references": [], - "name": "And", - "constType": "Prop → Prop → Prop", - "constCategory": "Other"}, - {"references": ["Zero", "Mul"], - "name": "NoZeroDivisors", - "constType": "(M₀ : Type u_4) → [inst : Mul M₀] → [inst : Zero M₀] → Prop", - "constCategory": "Other"}, - {"references": ["Nat.gcd", "Nat", "Int", "Int.natAbs"], - "name": "Int.gcd", - "constType": "ℤ → ℤ → ℕ", - "constCategory": "Definition"}, - {"references": ["Zero", "SMul", "SMulZeroClass"], - "name": "SMulZeroClass.toSMul", - "constType": - "{M : Type u_10} → {A : Type u_11} → [inst : Zero A] → [self : SMulZeroClass M A] → SMul M A", - "constCategory": "Definition"}, - {"references": ["_obj"], - "name": - "FltRegular._aux_FltRegular_CaseI_AuxLemmas___macroRules_FltRegular_termK_1._closed_10._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["Membership.mem", - "Ne", - "CommMonoidWithZero.toZero", - "CommSemiring.toSemiring", - "OfNat.ofNat", - "Finsupp.instFunLike", - "Finsupp", - "Nat.instLinearOrderedCommMonoidWithZero", - "MvPolynomial", - "Semiring.toMonoidWithZero", - "Mathlib.Data.Finsupp.Defs._auxLemma.1", - "MvPolynomial.coeff", - "True", - "iff_self", - "Finset.instMembership", - "Eq", - "Zero.toOfNat0", - "MvPolynomial.support", - "Iff", - "of_eq_true", - "CommSemiring", - "Finset", - "LinearOrderedCommMonoidWithZero.toZero", - "Not", - "CommSemiring.toCommMonoidWithZero", - "DFunLike.coe", - 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"constType": "∀ {α : Type u_1} {a b : α}, b ∈ {a} ↔ b = a", - "constCategory": "Theorem"}, - {"references": - ["HPow.hPow", - "PartENat.instLE", - "Monoid.toSemigroup", - "PartENat.partialOrder", - "Monoid", - "PartialOrder.toPreorder", - "PartENat", - "Dvd.dvd", - "semigroupDvd", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "LE.le", - "Eq", - "multiplicity.Finite", - "Eq.mpr", - "Nat.cast", - "AddMonoidWithOne.toNatCast", - "instHPow", - "PartENat.natCast_get", - "PartENat.instAddCommMonoidWithOne", - "Part.get", - "le_refl", - "multiplicity.pow_dvd_of_le_multiplicity", - "Monoid.toNatPow", - "multiplicity", - "Nat", - "congrArg", - "id", - "DecidableRel"], - "name": "multiplicity.pow_multiplicity_dvd", - "constType": - "∀ {α : Type u_1} [inst : Monoid α] [inst_1 : DecidableRel fun x x_1 => x ∣ x_1] {a b : α} (h : multiplicity.Finite a b),\n a ^ (multiplicity a b).get h ∣ b", - "constCategory": "Theorem"}, - {"references": [], - "name": "Subtype", - "constType": "{α : Sort u} → (α → Prop) → Sort (max 1 u)", - "constCategory": "Other"}, - {"references": ["Multiset", "Finset"], - "name": "Finset.val", - "constType": "{α : Type u_4} → Finset α → Multiset α", - "constCategory": "Definition"}, - {"references": ["InvolutiveNeg", "HasDistribNeg", "Mul"], - "name": "HasDistribNeg.toInvolutiveNeg", - "constType": - "{α : Type u_1} → [inst : Mul α] → [self : HasDistribNeg α] → InvolutiveNeg α", - "constCategory": "Definition"}, - {"references": - ["Finset.gcd_image", - "DecidableEq", - "Finset.image", - "Finset", - "NormalizedGCDMonoid", - "Eq.symm", - "Finset.gcd", - "CancelCommMonoidWithZero", - "Eq", - "id"], - "name": "Finset.gcd_eq_gcd_image", - "constType": - "∀ {α : Type u_2} {β : Type u_3} [inst : CancelCommMonoidWithZero α] [inst_1 : NormalizedGCDMonoid α] {s : Finset β}\n {f : β → α} [inst_2 : DecidableEq α], s.gcd f = (Finset.image f s).gcd id", - "constCategory": "Theorem"}, - {"references": ["And"], - "name": "And.right", - "constType": "∀ {a b : Prop}, a ∧ b → b", - "constCategory": "Theorem"}, - 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"NonUnitalNonAssocRing.toHasDistribNeg", - "AddCommSemigroup.toAddCommMagma", - "NonAssocSemiring.toAddCommMonoidWithOne", - "mul_neg", - "Mathlib.Meta.NormNum.IsNat.of_raw", - "Int.instDecidableEq", - "Finset.instInsert", - "neg_pow", - "pow_one", - "Mathlib.Meta.NormNum.isInt_add", - "MulZeroClass.toMul", - "Mathlib.Tactic.Ring.add_pf_add_gt", - "SubNegMonoid.toSub", - "Nat.Prime", - "LinearOrderedCommRing.toLinearOrderedCommSemiring", - "Nat.cast", - "Int.natAbs_neg", - "Mathlib.Tactic.Ring.neg_congr", - "Linarith.lt_of_lt_of_eq", - "MonoidWithZeroHom.funLike", - "FltRegular.MayAssume.Lemmas._auxLemma.7", - "instHSub", - "HPow.hPow", - "Or", - "eq_of_heq", - "FltRegular.MayAssume.Lemmas._auxLemma.4", - "Int.instRing", - "LinearOrderedCommSemiring.toStrictOrderedCommSemiring", - "FltRegular.MayAssume.Lemmas._auxLemma.5", - "Mathlib.Tactic.Ring.neg_zero", - "MonoidWithZero.toMulZeroOneClass", - "eq_true", - "LT.lt", - "Mathlib.Tactic.Ring.mul_pf_right", - "Mathlib.Tactic.Ring.add_pf_zero_add", - 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"Mathlib.Meta.NormNum.instAddMonoidWithOne", - "Monoid.toOne", - "MulZeroOneClass.toMulOneClass", - "Int.instNormedCommRing", - "Exists.casesOn", - "FltRegular.MayAssume.Lemmas._auxLemma.8", - "Int.instAdd", - "Preorder.toLE", - "Mathlib.Tactic.RingNF.mul_assoc_rev", - "Eq", - "Eq.mpr", - "Mathlib.Tactic.Zify._auxLemma.1", - "Nat.Prime.eq_two_or_odd'", - "Nat", - "Odd", - "Ne", - "Mathlib.Meta.NormNum.IsInt.to_isNat", - "Nat.rawCast", - "Int.ModEq", - "OrderedSemiring.zeroLEOneClass", - "Or.casesOn", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "Mathlib.Tactic.Ring.add_congr", - "Int.instAddCommSemigroup", - "Iff.intro", - "instHMul", - "AddCancelMonoid.toIsCancelAdd", - "Not", - "add_zero", - "congrArg", - "Int.instLEInt", - "OrderedAddCommGroup.toAddCommGroup", - "Mathlib.Meta.NormNum.IsInt.to_raw_eq", - "Mathlib.Data.Finset.Basic._auxLemma.64", - "Mathlib.Tactic.Ring.add_pf_add_lt", - "Ring.toNonAssocRing", - "Zero.toOfNat0", - "instLENat", - "one_mul", - "NormalizationMonoid.normUnit", - "instHPow", - "Preorder.toLT", - "Linarith.mul_neg", - "Int.instNegInt", - "add_lt_of_neg_of_le", - "Eq.symm", - "instHAdd", - "AddSemigroup.toAdd", - "HSub.hSub", - "Mathlib.Meta.NormNum.isNat_ofNat", - "AddGroup.toSubNegMonoid", - "Int.instCancelCommMonoidWithZero", - "GT.gt", - "Semiring.toOne", - "Mathlib.Tactic.Ring.add_pf_add_zero", - "Nat.instCommSemiring", - "Mathlib.Tactic.Ring.instCommSemiringNat", - "OrderedSemiring.toSemiring", - "SubNegMonoid.toAddMonoid", - "HMul.hMul", - "Int.instCommSemiring", - "NegZeroClass.toZero", - "ZMod.commRing", - "Int.instNontrivial"], - "name": "FltRegular.a_not_cong_b", - "constType": - "∀ {p : ℕ} {a b c : ℤ},\n Nat.Prime p →\n 5 ≤ p →\n a * b * c ≠ 0 →\n a ^ p + b ^ p = c ^ p →\n {a, b, c}.gcd id = 1 →\n ¬↑p ∣ a * b * c →\n ∃ x y z, x ^ p + y ^ p = z ^ p ∧ {x, y, z}.gcd id = 1 ∧ ¬x ≡ y [ZMOD ↑p] ∧ x * y * z ≠ 0 ∧ ¬↑p ∣ x * y * z", - "constCategory": "Theorem"}, - {"references": [], - "name": "Finset", - "constType": "Type u_4 → Type u_4", - "constCategory": "Other"}, - {"references": [], - "name": "Decidable", - "constType": "Prop → Type", - "constCategory": "Other"}, - {"references": - ["Lean.Name.anonymous._impl", - "Lean.ParserDescr.node", - "Lean.Name.num._override", - "_obj", - "Lean.Name.str._override", - "Lean.ParserDescr.symbol"], - "name": "FltRegular.termK.«_@».FltRegular.CaseI.AuxLemmas._hyg.250._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": ["propext", "forall_apply_eq_imp_iff₂", "Eq"], - "name": "FltRegular.NumberTheory.Different._auxLemma.6", - "constType": - "∀ {α : Sort u_2} {β : Sort u_1} {f : α → β} {p : α → Prop} {q : β → Prop},\n (∀ (b : β) (a : α), p a → f a = b → q b) = ∀ (a : α), p a → q (f a)", - "constCategory": "Theorem"}, - {"references": - ["_obj", - "FltRegular.termR.«_@».FltRegular.CaseI.AuxLemmas._hyg.717._closed_9", - "Lean.ParserDescr.symbol"], - "name": - "FltRegular.termR.«_@».FltRegular.CaseI.AuxLemmas._hyg.717._closed_10._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": [], - "name": "LT", - "constType": "Type u → Type u", - "constCategory": "Other"}, - {"references": [], - "name": "MulOneClass", - "constType": "Type u → Type u", - "constCategory": "Other"}, - {"references": ["Semiring", "Module", "AddCommMonoid"], - "name": "Submodule", - "constType": - "(R : Type u) → (M : Type v) → [inst : Semiring R] → [inst_1 : AddCommMonoid M] → [inst : Module R M] → Type v", - "constCategory": "Other"}, - {"references": - ["SemigroupWithZero.toSemigroup", - "dvd_gcd_iff", - "And", - "GCDMonoid.gcd", - "CancelCommMonoidWithZero.toCommMonoidWithZero", - "GCDMonoid", - "MonoidWithZero.toSemigroupWithZero", - "propext", - "Dvd.dvd", - "semigroupDvd", - "CancelCommMonoidWithZero", - "Eq", - "CommMonoidWithZero.toMonoidWithZero"], - "name": "FltRegular.CaseII.Statement._auxLemma.3", - "constType": - "∀ {α : Type u_1} [inst : CancelCommMonoidWithZero α] [inst_1 : GCDMonoid α] (a b c : α), (a ∣ gcd b c) = (a ∣ b ∧ a ∣ c)", - "constCategory": "Theorem"}, - {"references": ["outParam", "SetLike", "Set"], - "name": "SetLike.coe", - "constType": - "{A : Type u_1} → {B : outParam (Type u_2)} → [self : SetLike A B] → A → Set B", - "constCategory": "Definition"}, - {"references": - ["HPow.hPow", - "instHMul", - "pow_mul", - "Monoid", - "instHPow", - "HMul.hMul", - "Monoid.toNatPow", - "instMulNat", - "Nat", - "Eq.symm", - "Eq"], - "name": "FltRegular.NumberTheory.Different._auxLemma.1", - "constType": - "∀ {M : Type u_2} [inst : Monoid M] (a : M) (m n : ℕ), (a ^ m) ^ n = a ^ (m * n)", - "constCategory": "Theorem"}, - {"references": ["Nat"], - "name": "Fin", - "constType": "ℕ → Type", - "constCategory": "Other"}, - {"references": - ["CommSemiring.toSemiring", - "NonUnitalCommRing.toNonUnitalNonAssocCommRing", - "Submodule", - "MulZeroOneClass.toMulOneClass", - "Algebra.toModule", - "CommRing.toNonUnitalCommRing", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring", - "NonAssocSemiring.toMulZeroOneClass", - "CommRing", - "CommRing.toCommSemiring", - "Semiring.toNonAssocSemiring", - "Algebra", - "IsFractional", - "Submonoid", - "NonUnitalNonAssocSemiring.toAddCommMonoid", - "Subtype", - "NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring"], - "name": "FractionalIdeal", - "constType": - "{R : Type u_1} →\n [inst : CommRing R] → Submonoid R → (P : Type u_2) → [inst_1 : CommRing P] → [inst : Algebra R P] → Type u_2", - "constCategory": "Definition"}, - {"references": ["Field"], - "name": "NumberField", - "constType": "(K : Type u_1) → [inst : Field K] → Prop", - "constCategory": "Other"}, - {"references": ["LT"], - "name": "LT.lt", - "constType": "{α : Type u} → [self : LT α] → α → α → Prop", - "constCategory": "Definition"}, - {"references": - ["eq_self", - "Semigroup.toMul", - "instHAdd", - "HAdd.hAdd", - "LeftDistribClass", - "Dvd.dvd", - "semigroupDvd", - "Dvd.intro", - "True", - "Eq", - "instHMul", - "of_eq_true", - "Dvd.elim", - "HMul.hMul", - "congr", - "Add", - "congrArg", - "left_distrib", - "Eq.trans", - "Semigroup"], - "name": "dvd_add", - "constType": - "∀ {α : Type u_1} [inst : Add α] [inst_1 : Semigroup α] [inst_2 : LeftDistribClass α] {a b c : α},\n a ∣ b → a ∣ c → a ∣ b + c", - "constCategory": "Theorem"}, - {"references": - ["_obj", - "String.toSubstring'", - "FltRegular._aux_FltRegular_CaseI_Statement___macroRules_FltRegular_termP_1._closed_24"], - "name": - "FltRegular._aux_FltRegular_CaseI_Statement___macroRules_FltRegular_termP_1._closed_25._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["Zero.toOfNat0", - "Zero", - "ZeroLEOneClass", - "eq_true", - "OfNat.ofNat", - "NeZero", - "PartialOrder.toPreorder", - "One", - "Preorder.toLT", - "LT.lt", - "one_pos", - "One.toOfNat1", - "True", - "Preorder.toLE", - "Eq", - "PartialOrder"], - "name": "FltRegular.CaseII.InductionStep._auxLemma.11", - "constType": - "∀ {α : Type u_1} [inst : Zero α] [inst_1 : One α] [inst_2 : PartialOrder α] [inst_3 : ZeroLEOneClass α]\n [inst_4 : NeZero 1], (0 < 1) = True", - "constCategory": "Theorem"}, - {"references": ["AddMonoidWithOne", "One"], - "name": "AddMonoidWithOne.toOne", - "constType": "{R : Type u_2} → [self : AddMonoidWithOne R] → One R", - "constCategory": "Definition"}, - {"references": - ["HPow.hPow", - "FiniteField.pow_card", - "DivisionSemiring.toGroupWithZero", - "Semiring.toMonoidWithZero", - "Field.toSemifield", - "NeZero.of_gt'", - "Fintype.card", - "CanonicallyOrderedCommSemiring.toOne", - "ZMod.card", - "GroupWithZero.toMonoidWithZero", - "ZMod", - "Eq", - "ZMod.instField", - "DivisionSemiring.toSemiring", - "CanonicallyOrderedCommSemiring.toCanonicallyOrderedAddCommMonoid", - "Nat.Prime", - "Nat.Prime.one_lt'", - "MonoidWithZero.toMonoid", - "instHPow", - "Eq.mp", - "Nat.instCanonicallyOrderedCommSemiring", - "letFun", - "Monoid.toNatPow", - "Semifield.toDivisionSemiring", - "ZMod.fintype", - "Nat", - "congrArg", - "Fact"], - "name": "ZMod.pow_card", - "constType": "∀ {p : ℕ} [inst : Fact (Nat.Prime p)] (x : ZMod p), x ^ p = x", - "constCategory": "Theorem"}, - {"references": - ["Semigroup.toMul", - "Monoid.toSemigroup", - "Monoid.toOne", - "MulZeroOneClass.mk", - "MonoidWithZero.toMonoid", - "Monoid.one_mul", - "MonoidWithZero.mul_zero", - "MonoidWithZero.zero_mul", - "MonoidWithZero.toZero", - "Monoid.mul_one", - "MulOneClass.mk", - "MulZeroOneClass", - "MonoidWithZero"], - "name": "MonoidWithZero.toMulZeroOneClass", - "constType": - "{M₀ : Type u} → [self : MonoidWithZero M₀] → MulZeroOneClass M₀", - "constCategory": "Definition"}, - {"references": [], - "name": "Unique", - "constType": "Sort u → Sort (max 1 u)", - "constCategory": "Other"}, - {"references": - ["AddZeroClass.toAdd", - "instHSub", - "SubNegMonoid.toSub", - "AddGroup", - "instHAdd", - "propext", - "HAdd.hAdd", - "SubNegMonoid.toAddMonoid", - "HSub.hSub", - "sub_eq_iff_eq_add", - "AddMonoid.toAddZeroClass", - "AddGroup.toSubNegMonoid", - "Eq"], - "name": "FltRegular.MayAssume.Lemmas._auxLemma.5", - "constType": - "∀ {G : Type u_3} [inst : AddGroup G] {a b c : G}, (a - b = c) = (a = c + b)", - "constCategory": "Theorem"}, - {"references": - ["OfNat.ofNat", - "Nat.add_succ", - "HAdd.hAdd", - "Eq.refl", - "AddMonoid.toAddZeroClass", - "True", - "Nat.cast_zero", - "Eq", - "Zero.toOfNat0", - "Eq.mpr", - "AddMonoidWithOne.toNatCast", - "Nat.cast", - "Nat.recAux", - "AddMonoid.toAddSemigroup", - "AddMonoid.toZero", - "Nat.cast_succ", - "instOfNatNat", - "Nat", - "Eq.trans", - "Nat.succ", - "id", - "eq_self", - "add_assoc", - "instAddNat", - "AddSemigroup.toAdd", - "instHAdd", - "AddMonoidWithOne.toAddMonoid", - "One.toOfNat1", - "AddMonoidWithOne", - "of_eq_true", - "AddMonoidWithOne.toOne", - "congrArg", - "add_zero"], - "name": "Nat.cast_add", - "constType": - "∀ {R : Type u_1} [inst : AddMonoidWithOne R] (m n : ℕ), ↑(m + n) = ↑m + ↑n", - "constCategory": "Theorem"}, - {"references": - ["instHSub", - "Int.instDvd", - "Iff.rfl", - "Int.ModEq", - "HSub.hSub", - "Dvd.dvd", - "ZMod", - "Eq", - "Eq.mpr", - "Iff", - "Nat.cast", - "instNatCastInt", - "propext", - "Int.cast", - "Ring.toIntCast", - "CommRing.toRing", - "Int.modEq_iff_dvd", - "Nat", - "congrArg", - "Int.instSub", - "Int", - "ZMod.intCast_eq_intCast_iff", - "id", - "ZMod.commRing"], - "name": "ZMod.intCast_eq_intCast_iff_dvd_sub", - "constType": "∀ (a b : ℤ) (c : ℕ), ↑a = ↑b ↔ ↑c ∣ b - a", - "constCategory": "Theorem"}, - {"references": ["Nat.gcd_comm", "Nat", "Int", "Int.gcd", "Eq", "Int.natAbs"], - "name": "Int.gcd_comm", - "constType": "∀ (i j : ℤ), i.gcd j = j.gcd i", - "constCategory": "Theorem"}, - {"references": - ["CommSemiring.toSemiring", - "Semiring.toNonAssocSemiring", - "MulZeroOneClass.toMulOneClass", - "CommSemiring", - "Algebra", - "Submonoid", - "NonAssocSemiring.toMulZeroOneClass"], - "name": "IsLocalization", - "constType": - "{R : Type u_1} →\n [inst : CommSemiring R] → Submonoid R → (S : Type u_2) → [inst_1 : CommSemiring S] → [inst : Algebra R S] → Prop", - "constCategory": "Other"}, - {"references": - ["Distrib.toAdd", - "NonUnitalCommRing.toNonUnitalNonAssocCommRing", - "CommSemiring.toSemiring", - "Exists", - "MulOneClass.toMul", - "OfNat.ofNat", - "Monoid.toOne", - "MulZeroOneClass.toMulOneClass", - "CommRing.toNonUnitalCommRing", - "IsDomain", - "HAdd.hAdd", - "Semiring.toMonoidWithZero", - "NonAssocSemiring.toMulZeroOneClass", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing", - "Exists.casesOn", - "AddGroupWithOne.toAddGroup", - "True", - "CommRing", - "Eq", - "Semiring.toNonAssocSemiring", - "MonoidWithZero.toMonoid", - "Ring.toSub", - "LocalRing.of_isUnit_or_isUnit_one_sub_self", - "IsDomain.toNontrivial", - "CommMonoid.toMonoid", - "add_sub_cancel", - "Eq.trans", - "NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring", - "eq_self", - "instHSub", - "NonUnitalNonAssocRing.toMul", - "Or", - "Ring.toAddGroupWithOne", - "mul_add", - "Or.inr", - "isUnit_of_mul_eq_one", - "MulOneClass.toOne", - "instHAdd", - "HSub.hSub", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring", - "One.toOfNat1", - "LocalRing", - "Or.casesOn", - "ValuationRing.cond", - "Monoid.toMulOneClass", - "NonUnitalNonAssocSemiring.toDistrib", - "Distrib.leftDistribClass", - "Semiring.toOne", - "sub_add_cancel", - "instHMul", - "CommRing.toCommSemiring", - "CommRing.toCommMonoid", - "Or.inl", - "of_eq_true", - "congr", - "HMul.hMul", - "CommRing.toRing", - "Ring.toAddCommGroup", - "mul_one", - "IsUnit", - "congrArg", - "ValuationRing"], - "name": "ValuationRing.localRing", - "constType": - "∀ (A : Type u) [inst : CommRing A] [inst_1 : IsDomain A] [inst_2 : ValuationRing A], LocalRing A", - "constCategory": "Definition"}, - {"references": - ["AddZeroClass.toAdd", - "CommSemiring.toSemiring", - "OfNat.ofNat", - "DivisionRing.toRing", - "IsPrimitiveRoot", - "HAdd.hAdd", - "Set", - "NumberField.RingOfIntegers.instCommRing", - "Eq.refl", - "Nat.instAddMonoid", - "AddMonoid.toAddZeroClass", - "algebraRat", - "pow_one", - "Eq", - "Zero.toOfNat0", - "PNat.val", - "zero_add", - "AddZeroClass.toZero", - "Rat.commRing", - "Set.instSingletonSet", - "Nat.Prime", - "Eq.mpr", - 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↑((hζ.unit' ^ k) ^ 2) * (↑x + ↑y * ↑(hζ.unit' ^ (-i))) ∈ Ideal.span {↑↑p}", - "constCategory": "Theorem"}, - {"references": - ["SModEq.def", - "Submodule", - "propext", - "Submodule.Quotient.mk", - "Submodule.hasQuotient", - "Ring", - "AddCommGroup.toAddCommMonoid", - "HasQuotient.Quotient", - "Module", - "AddCommGroup", - "Eq.symm", - "Ring.toSemiring", - "SModEq", - "Eq"], - "name": "FltRegular.NumberTheory.Cyclotomic.MoreLemmas._auxLemma.10", - "constType": - "∀ {R : Type u_1} [inst : Ring R] {M : Type u_3} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {U : Submodule R M}\n {x y : M}, (Submodule.Quotient.mk x = Submodule.Quotient.mk y) = (x ≡ y [SMOD U])", - "constCategory": "Theorem"}, - {"references": ["_obj"], - "name": - "FltRegular._aux_FltRegular_CaseI_AuxLemmas___macroRules_FltRegular_termP_1._closed_24._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["Field.zpow_succ'", - "Field.zpow_zero'", - "Field.nnqsmul", - "CommSemiring.mk", - "Field.toSemifield.proof_1", - 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"Mathlib.Meta.NormNum.isNat_ofNat", - "AddGroup.toSubNegMonoid", - "Rat.instCharZero", - "Semiring.toOne", - "Mathlib.Tactic.Ring.add_pf_add_zero", - "Nat.instCommSemiring", - "Mathlib.Tactic.Ring.instCommSemiringNat", - "SubNegMonoid.toAddMonoid", - "HMul.hMul", - "NegZeroClass.toZero", - "Int.instCommSemiring", - "ZMod.commRing"], - "name": "FltRegular.CaseI.aux1k₂", - "constType": - "∀ {p : ℕ} (hpri : Nat.Prime p) {a b c : ℤ} {ζ : NumberField.RingOfIntegers (CyclotomicField ⟨p, ⋯⟩ ℚ)},\n 5 ≤ p →\n IsPrimitiveRoot ζ p →\n ¬↑p ∣ a * b * c →\n ∀ {k₁ k₂ : Fin p}, ↑↑k₂ ≡ ↑↑k₁ - 1 [ZMOD ↑p] → ↑p ∣ ↑a + ↑b * ζ - ↑a * ζ ^ ↑k₁ - ↑b * ζ ^ ↑k₂ → 1 ≠ ↑k₂", - "constCategory": "Theorem"}, - {"references": - ["SubNegMonoid", "SMul", "SMul.mk", "SubNegMonoid.zsmul", "Int"], - "name": "SubNegMonoid.SMulInt", - "constType": "{M : Type u_2} → [inst : SubNegMonoid M] → SMul ℤ M", - "constCategory": "Definition"}, - {"references": - ["Zero.toOfNat0", - "Membership.mem", - "Finset.sum_eq_zero_iff_of_nonneg", - 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"name": "Int.ModEq", - "constType": "ℤ → ℤ → ℤ → Prop", - "constCategory": "Definition"}, - {"references": [], - "name": "Monoid", - "constType": "Type u → Type u", - "constCategory": "Other"}, - {"references": [], - "name": "SubNegMonoid", - "constType": "Type u → Type u", - "constCategory": "Other"}, - {"references": - ["eq_self", - "Int.ofNat", - "Int.cast_natCast", - "AddGroupWithOne.toAddMonoidWithOne", - "Ring.toAddGroupWithOne", - "AddGroupWithOne.toIntCast", - "True", - "Eq", - "AddMonoidWithOne.toNatCast", - "Nat.cast", - "Mathlib.Meta.NormNum.IsNat", - "of_eq_true", - "instNatCastInt", - "Int.cast", - "Mathlib.Meta.NormNum.IsInt.to_isNat.match_1", - "Ring.toIntCast", - "Ring", - "Mathlib.Meta.NormNum.IsInt", - "Nat", - "congrArg", - "Mathlib.Meta.NormNum.IsNat.mk", - "Eq.trans", - "Int"], - "name": "Mathlib.Meta.NormNum.IsInt.to_isNat", - "constType": - "∀ {α : Type u_1} [inst : Ring α] {a : α} {n : ℕ},\n Mathlib.Meta.NormNum.IsInt a (Int.ofNat n) → Mathlib.Meta.NormNum.IsNat a n", - 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"PNat.val", - "GCDMonoid.gcd", - "Set.instSingletonSet", - "Rat.commRing", - "instHPow", - "Finset", - "Ring.toSub", - "Monoid.toNatPow", - "instOfNatNat", - "div_zeta_sub_one", - "NeZero.succ", - "Membership.mem", - "Ideal.cancelCommMonoidWithZero", - "instAddNat", - "instHAdd", - "Ideal.instNormalizedGCDMonoid", - "Rat", - "HSub.hSub", - "Field.toSemifield", - "One.toOfNat1", - "Dvd.dvd", - "Ring.toSemiring", - "Polynomial.nthRootsFinset", - "IsCyclotomicExtension", - "Semiring.toOne", - "Units", - "CommRing.toCommSemiring", - "Exists.choose", - "Units.val", - "HMul.hMul", - "NonUnitalSemiring.toSemigroupWithZero", - "IdemSemiring.toSemiring", - "NumberField.RingOfIntegers", - "PNat"], - "name": "div_zeta_sub_one_dvd_gcd", - "constType": - "{K : Type u_1} →\n {p : ℕ+} →\n [hpri : Fact p.Prime] →\n [inst : Field K] →\n [inst_1 : NumberField K] →\n [inst_2 : IsCyclotomicExtension {p} ℚ K] →\n p ≠ 2 →\n {ζ : K} →\n (hζ : IsPrimitiveRoot ζ ↑p) →\n {x y z : NumberField.RingOfIntegers K} →\n {ε : (NumberField.RingOfIntegers K)ˣ} →\n {m : ℕ} →\n x ^ ↑p + y ^ ↑p = ↑ε * ((↑hζ.unit' - 1) ^ (m + 1) * z) ^ ↑p →\n ¬↑hζ.unit' - 1 ∣ y →\n { x // x ∈ Polynomial.nthRootsFinset (↑p) (NumberField.RingOfIntegers K) } →\n Ideal (NumberField.RingOfIntegers K)", - "constCategory": "Definition"}, - {"references": ["_obj"], - "name": - "FltRegular._aux_FltRegular_CaseI_Statement___macroRules_FltRegular_termP_1._closed_7._cstage2", - "constType": "_obj", - "constCategory": "Definition"}, - {"references": - ["HPow.hPow", - "Monoid", - "Units.instInv", - "Units.instMonoid", - "DivInvMonoid", - "DivInvMonoid.mk", - "Units.instDivInvMonoid.proof_1", - "Units", - "Units.instDivInvMonoid.match_1", - "instHPow", - "Units.instDiv", - "Monoid.toNatPow", - "Units.instDivInvMonoid.proof_2", - "Units.instDivInvMonoid.proof_4", - "Nat", - "Int", - "Nat.succ", - "Units.instDivInvMonoid.proof_3", - "Inv.inv"], - "name": "Units.instDivInvMonoid", - "constType": "{α : Type u} → [inst : Monoid α] → DivInvMonoid αˣ", - "constCategory": "Definition"}, - {"references": - ["Finset.empty", "EmptyCollection.mk", "Finset", "EmptyCollection"], - "name": "Finset.instEmptyCollection", - "constType": "{α : Type u_1} → EmptyCollection (Finset α)", - "constCategory": "Definition"}, - {"references": [], - "name": "Semigroup", - "constType": "Type u → Type u", - "constCategory": "Other"}, - {"references": - ["LinearOrderedCommMonoid.toOrderedCommMonoid", - "OfNat.ofNat", - "PartialOrder.toPreorder", - "zero_le'", - "LinearOrderedCommMonoidWithZero", - "le_rfl", - "LE.le", - "Preorder.toLE", - "Eq", - "Iff.intro", - "Zero.toOfNat0", - "Iff", - "Eq.rec", - "OrderedCommMonoid.toPartialOrder", - "LinearOrderedCommMonoidWithZero.toZero", - "LinearOrderedCommMonoidWithZero.toLinearOrderedCommMonoid", - "le_antisymm"], - "name": "le_zero_iff", - "constType": - "∀ {α : Type u_1} [inst : LinearOrderedCommMonoidWithZero α] {a : α}, a ≤ 0 ↔ a = 0", - "constCategory": "Theorem"}, - {"references": ["outParam", "Singleton"], - "name": "Singleton.singleton", - "constType": - "{α : outParam (Type u)} → {β : Type v} → [self : Singleton α β] → α → β", - "constCategory": "Definition"}, - {"references": - ["RingHom.instRingHomClass", - "NonUnitalCommRing.toNonUnitalNonAssocCommRing", - "CommSemiring.toSemiring", - "NonUnitalNonAssocRing.toHasDistribNeg", - "Units.instInv", - "OfNat.ofNat", - "Nat.Prime.odd_of_ne_two", - "SubNegMonoid.toNeg", - "HasDistribNeg", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing", - "algebraRat", - "AddGroupWithOne.toAddGroup", - "Nat.Prime", - "NumberField.Embeddings.pow_eq_one_of_norm_eq_one", - "Iff", - "normedAlgebraRat", - "IsPrimitiveRoot.unit'", - "Field", - "HasDistribNeg.toInvolutiveNeg", - "map_mul", - "NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring", - "HPow.hPow", - "Or", - "instOfNatPNatOfNeZeroNat", - "MulOneClass.toOne", - "Units.instMonoid", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring", - "MonoidHom", - "NumberField.to_charZero", - "CommMagma.toMul", - "CommRing.toCommMonoid", - "False.elim", - "RingHomClass.toMonoidHomClass", - "unit_inv_conj_not_neg_zeta_runity", - "LT.lt", - "NumberField", - "Field.toDivisionRing", - "Int", - "NumberField.RingOfIntegers.instAlgebra_1", - "Inv.inv", - "RingHom.IsIntegralElem.mul", - "AddCommGroup.toDivisionAddCommMonoid", - "Exists", - "Set", - "Semiring.toMonoidWithZero", - "Eq.refl", - "NonAssocSemiring.toMulZeroOneClass", - "algebraInt", - "True", - "RingHomClass.toNonUnitalRingHomClass", - "RingHomClass.toAddMonoidHomClass", - "Units.instNeg", - "zeta_runity_pow_even", - "NumberField.RingOfIntegers.isIntegral_coe", - "map_pow", - "InvolutiveNeg.toNeg", - "Monoid.toNatPow", - "neg_mul", - "RCLike.toNormedAlgebra", - "NonUnitalNonAssocRing.toMul", - "Ring.toAddGroupWithOne", - "Or.inr", - "Odd.neg_one_pow", - "algebraMap", - "Field.toSemifield", - "One.toOfNat1", - "Neg.neg", - "Even", - "Monoid.toMulOneClass", - "IsCyclotomicExtension", - "Units", - "CommRing.toCommSemiring", - "Int.instCommRing", - "Or.inl", - "letFun", - "not_true_eq_false", - "Ring.toAddCommGroup", - "Semifield.toDivisionSemiring", - "Nat.even_or_odd", - "PNat", - "MulOneClass.toMul", - "Monoid.toOne", - "MulZeroOneClass.toMulOneClass", - "IsPrimitiveRoot", - "Semifield.toCommSemiring", - "Units.val_mul", - "NonUnitalRingHomClass.toMulHomClass", - "Exists.casesOn", - "instLTNat", - "unit_lemma_val_one", - "Classical.byContradiction", - "Eq", - "Semiring.toNonAssocSemiring", - "Eq.mpr", - "Mathlib.Algebra.Group.Units._auxLemma.1", - "Iff.mp", - "Ring.toNeg", - "MonoidWithZero.toMonoid", - "FltRegular.NumberTheory.Cyclotomic.UnitLemmas._auxLemma.15", - "Nat", - "Odd", - "Eq.trans", - "Even.neg_one_pow", - "Units.instMul", - "Ne", - "Monoid", - "Complex.instNormedField", - "Nat.instSemiring", - "Or.casesOn", - "EuclideanDomain.toCommRing", - "RingHom.instFunLike", - "unitGalConj", - "instHMul", - "Mathlib.RingTheory.Localization.FractionRing._auxLemma.3", - "norm_cast_ne_two", - "Field.toEuclideanDomain", - "Complex.instRCLike", - "Eq.mp", - "CommRing.toRing", - "Nat.instCommSemigroup", - "Not", - "Singleton.singleton", - "MonoidHom.instFunLike", - "Algebra.id", - "congrArg", - "NormedField.toNormedDivisionRing", - "NumberField.RingOfIntegers.instIsFractionRing", - "RingHom", - "DivisionRing.toRing", - "Iff.rfl", - "CommRing.toNonUnitalCommRing", - "roots_of_unity_in_cyclo", - "Complex.instCharZero", - "NumberField.RingOfIntegers.instCommRing", - "SubtractionCommMonoid.toSubtractionMonoid", - "Complex.isAlgClosed", - "mul_comm", - "PNat.val", - "Rat.commRing", - "Set.instSingletonSet", - "one_mul", - "instHPow", - "instOfNatNat", - "Units.instMulOneClass", - "Units.val_pow_eq_pow_val", - "DFunLike.coe", - "Eq.symm", - "id", - "eq_self", - "NeZero.succ", - "funext", - "False", - "instAddNat", - "Rat", - "Algebra.cast", - "exists_congr", - "Ring.toSemiring", - "AddGroup.toSubNegMonoid", - "Semiring.toOne", - "DivisionSemiring.toSemiring", - "SubtractionMonoid.toSubNegMonoid", - "CommSemigroup.toCommMagma", - "map_neg", - "Units.val", - "HMul.hMul", - "Complex", - "instMulNat", - "NumberField.RingOfIntegers"], - "name": "unit_inv_conj_is_root_of_unity", - "constType": - "∀ {p : ℕ+} {K : Type u_1} [inst : Field K] {ζ : K} (hζ : IsPrimitiveRoot ζ ↑p) [inst_1 : NumberField K]\n [inst_2 : IsCyclotomicExtension {p} ℚ K],\n p ≠ 2 → Nat.Prime ↑p → ∀ (u : (NumberField.RingOfIntegers K)ˣ), ∃ m, u * ((unitGalConj K p) u)⁻¹ = (hζ.unit' ^ m) ^ 2", - "constCategory": "Theorem"}, - {"references": - ["CancelCommMonoidWithZero.toCommMonoidWithZero", - "HEq", - "Eq.refl", - "Multiset.map_id", - "normalize", - "normalize_eq", - "CancelCommMonoidWithZero", - "Eq", - "Multiset.map_congr", - "Multiset.instMembership", - "Eq.mpr", - "MonoidWithZero.toMonoid", - "Multiset", - "UniqueFactorizationMonoid.factors", - "MonoidWithZeroHom", - "DFunLike.coe", - "Eq.symm", - "MonoidWithZeroHom.funLike", - "id", - "CommMonoidWithZero.toMonoidWithZero", - "Membership.mem", - "funext", - "eq_of_heq", - "UniqueFactorizationMonoid.normalizedFactors", - "Unique", - "Multiset.map", - "Units", - "UniqueFactorizationMonoid", - "MonoidWithZero.toMulZeroOneClass", - "Eq.ndrec", - "HEq.refl", - "Eq.casesOn", - "normalizationMonoidOfUniqueUnits"], - "name": "UniqueFactorizationMonoid.factors_eq_normalizedFactors", - "constType": - "∀ {M : Type u_2} [inst : CancelCommMonoidWithZero M] [inst_1 : UniqueFactorizationMonoid M] [inst_2 : Unique Mˣ]\n (x : M), UniqueFactorizationMonoid.factors x = UniqueFactorizationMonoid.normalizedFactors x", - "constCategory": "Theorem"}, - {"references": - ["InvOneClass.mk", - "DivInvOneMonoid", - "Monoid.toOne", - "DivInvOneMonoid.inv_one", - "InvOneClass", - "DivInvMonoid.toMonoid", - "DivInvOneMonoid.toDivInvMonoid", - "DivInvMonoid.toInv"], - "name": "DivInvOneMonoid.toInvOneClass", - "constType": "{G : Type u_2} → [self : DivInvOneMonoid G] → InvOneClass G", - "constCategory": "Definition"}, - {"references": - ["inferInstance", - "Monoid", - "Nat.instCommMonoid", - "CommMonoid.toMonoid", - "Nat"], - "name": "Nat.instMonoid", - "constType": "Monoid ℕ", - "constCategory": "Definition"}, - {"references": - ["SMulZeroClass.mk", - "Zero", - "MulAction.toSMul", - "SMulWithZero", - "MonoidWithZero.toMonoid", - "MulActionWithZero.smul_zero", - "MulActionWithZero", - "MulActionWithZero.zero_smul", - "MonoidWithZero.toZero", - "SMulWithZero.mk", - "MulActionWithZero.toMulAction", - "MonoidWithZero"], - "name": "MulActionWithZero.toSMulWithZero", - "constType": - "(R : Type u_1) →\n (M : Type u_3) → [inst : MonoidWithZero R] → [inst_1 : Zero M] → [m : MulActionWithZero R M] → SMulWithZero R M", - "constCategory": "Definition"}, - {"references": ["Ne", "Ne.eq_def", "Not", "Eq.symm", "Eq"], - "name": "FltRegular.CaseI.Statement._auxLemma.9", - "constType": "∀ {α : Sort u} (a b : α), (¬a = b) = (a ≠ b)", - "constCategory": "Theorem"}, - {"references": - ["Membership.mem", - "Finset.mem_cons", - "Or", - "propext", - "Finset", - "Finset.cons", - "Not", - "Eq", - "Finset.instMembership"], - "name": "FltRegular.ReadyForMathlib.Homogenization._auxLemma.31", - "constType": - "∀ {α : Type u_1} {s : Finset α} {a b : α} {h : a ∉ s}, (b ∈ Finset.cons a s h) = (b = a ∨ b ∈ s)", - "constCategory": "Theorem"}, - {"references": - ["RingHom.instRingHomClass", - "CommSemiring.toSemiring", - "RingHom", - "DivisionRing.toRing", - "IsPrimitiveRoot", - "Semifield.toCommSemiring", - "Semiring.toMonoidWithZero", - "NumberField.RingOfIntegers.instCommRing", - "IsPrimitiveRoot.of_map_of_injective", - "rfl", - "Eq", - "PNat.val", - "Semiring.toNonAssocSemiring", - "IsFractionRing.injective", - "MonoidWithZero.toMonoid", - "IsPrimitiveRoot.unit'", - "Field", - "DFunLike.coe", - "Eq.symm", - "algebraMap", - "Field.toSemifield", - "EuclideanDomain.toCommRing", - "RingHom.instFunLike", - "CommRing.toCommSemiring", - "CommRing.toCommMonoid", - "DivisionSemiring.toSemiring", - "RingHomClass.toMonoidHomClass", - "Field.toEuclideanDomain", - "Units.val", - "Eq.mp", - "letFun", - "NumberField", - "Semifield.toDivisionSemiring", - "Algebra.id", - "congrArg", - "NumberField.RingOfIntegers", - "Field.toDivisionRing", - "NumberField.RingOfIntegers.instIsFractionRing", - "NumberField.RingOfIntegers.instAlgebra_1", - "PNat"], - "name": "IsPrimitiveRoot.unit'_coe", - "constType": - "∀ {p : ℕ+} {K : Type u_1} [inst : Field K] {ζ : K} (hζ : IsPrimitiveRoot ζ ↑p) [inst_1 : NumberField K],\n IsPrimitiveRoot ↑hζ.unit' ↑p", - "constCategory": "Theorem"}, - {"references": - ["Set.range", - "Subgroup.copy", - "MonoidHom.range.proof_1", - "Top.top", - "Subgroup.instTop", - "Subgroup", - "Group", - "MonoidHom.instFunLike", - "Group.toDivInvMonoid", - "MonoidHom", - "DFunLike.coe", - "Subgroup.map", - "DivInvMonoid.toMonoid", - "Monoid.toMulOneClass"], - "name": "MonoidHom.range", - "constType": - "{G : Type u_1} → [inst : Group G] → {N : Type u_5} → [inst_1 : Group N] → (G →* N) → Subgroup N", - "constCategory": "Definition"}, - {"references": - ["Monoid.toSemigroup", - "OfNat.ofNat", - "Monoid", - "Monoid.toOne", - "Dvd.dvd", - "mul_one", - "One.toOfNat1", - "semigroupDvd", - "Dvd.intro", - "Monoid.toMulOneClass"], - "name": "dvd_refl", - "constType": "∀ {α : Type u_1} [inst : Monoid α] (a : α), a ∣ a", - "constCategory": "Theorem"}, - {"references": - ["eq_self", - "NonUnitalNonAssocRing.toHasDistribNeg", - "NonUnitalNonAssocRing.toMul", - "mul_neg", - "Neg.neg", - "NonAssocRing.toNonUnitalNonAssocRing", - "Ring.toNonAssocRing", - "True", - "Eq", - "instHMul", - "Ring.toNeg", - "of_eq_true", - "InvolutiveNeg.toNeg", - "Ring", - "HasDistribNeg.toInvolutiveNeg", - "HMul.hMul", - "congrArg", - "Eq.trans"], - "name": "Mathlib.Tactic.RingNF.mul_neg", - "constType": "∀ {R : Type u_2} [inst : Ring R] (a b : R), a * -b = -(a * b)", - "constCategory": "Theorem"}, - {"references": - ["Mathlib.Tactic.Ring.one_pow", - "Int.cast_one", - "RingHom.instRingHomClass", - "Distrib.toAdd", - "sub_add_eq_sub_sub", - "CommSemiring.toSemiring", - "NonUnitalCommRing.toNonUnitalNonAssocCommRing", - "AddCommSemigroup.toAddCommMagma", - "NonAssocSemiring.toAddCommMonoidWithOne", - "OfNat.ofNat", - "Mathlib.Meta.NormNum.IsNat.of_raw", - "sub_self", - "SubNegMonoid.toNeg", - "Mathlib.Tactic.Ring.pow_zero", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing", - "Subalgebra.instSetLike", - "AddGroupWithOne.toAddGroup", - "Semiring.toNatCast", - "Mathlib.Tactic.Ring.add_pf_add_gt", - "mul_pow", - "SubNegMonoid.toSub", - "Nat.Prime", - "Nat.cast", - "Mathlib.Tactic.Ring.neg_congr", - "Int.castRingHom", - "map_mul", - "Mathlib.Tactic.Ring.pow_add", - "NonUnitalCommSemiring.toNonUnitalSemiring", - "Ideal.instHasQuotient", - "instOfNat", - "NonUnitalNonAssocRing.toAddCommGroup", - "NonUnitalNonAssocCommSemiring.toNonUnitalNonAssocSemiring", - "instHSub", - "HPow.hPow", - "add_assoc", - "add_comm", - "eq_of_heq", - "NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring", - "Set.instMembership", - "AddCommMagma.toAdd", - "CommRing.toCommMonoid", - "Mathlib.Tactic.Ring.neg_zero", - "Mathlib.Meta.NormNum.IsNat.to_isInt", - "And", - "AddMonoidWithOne.toOne", - "propext", - "NonAssocSemiring.toNonUnitalNonAssocSemiring", - "Mathlib.Tactic.Ring.mul_pf_right", - "congr", - "Mathlib.Tactic.Ring.add_pf_zero_add", - "Int", - "Submodule.setLike", - "AddZeroClass.toAdd", - "CommMonoidWithZero.toZero", - "Exists", - "AddCommGroup.toDivisionAddCommMonoid", - "HEq", - "NonUnitalCommRing.toNonUnitalCommSemiring", - "Set", - "Mathlib.Meta.NormNum.isInt_mul", - "Semiring.toMonoidWithZero", - "Int.rawCast", - "Eq.refl", - "Mathlib.Tactic.Ring.sub_congr", - "semigroupDvd", - "Exists.intro", - "AddMonoid.toAddZeroClass", - "algebraInt", - "CommRing", - "True", - "Mathlib.Tactic.Ring.sub_pf", - "RingHomClass.toNonUnitalRingHomClass", - "RingHomClass.toAddMonoidHomClass", - "IsCyclotomicExtension.adjoin_roots", - "AddMonoid.toAddSemigroup", - "Eq.rec", - "Monoid.toNatPow", - "HasQuotient.Quotient", - "Semiring.toModule", - "SubtractionMonoid.toSubNegZeroMonoid", - "NonUnitalNonAssocRing.toMul", - "Mathlib.Tactic.Ring.mul_pf_left", - "Ring.toAddGroupWithOne", - "Mathlib.Tactic.Ring.neg_one_mul", - "Int.negOfNat", - "Int.instMul", - "Mathlib.Tactic.Ring.mul_pow", - "algebraMap", - "Mathlib.Tactic.Ring.mul_add", - "Mathlib.Algebra.Module.Submodule.Basic._auxLemma.6", - "One.toOfNat1", - "Neg.neg", - "SubNegZeroMonoid.toNegZeroClass", - "Mathlib.Tactic.Ring.neg_add", - "Monoid.toMulOneClass", - "IsCyclotomicExtension", - "CommRing.toCommSemiring", - "Int.instCommRing", - "Mathlib.Tactic.Ring.zero_mul", - "Eq.ndrec", - "Mathlib.Tactic.Ring.one_mul", - "letFun", - "HSub", - "Ring.toAddCommGroup", - "NonUnitalNonAssocSemiring.toAddCommMonoid", - "Mathlib.Tactic.Ring.add_mul", - "Mathlib.Tactic.Ring.atom_pf", - "Ideal.mem_span_singleton", - "PNat", - "FltRegular.NumberTheory.Cyclotomic.CyclRat._auxLemma.1", - "MulOneClass.toMul", - "Int.cast_pow", - "HAdd.hAdd", - "dvd_mul_right", - "AddGroup.toSubtractionMonoid", - "sub_eq_add_neg", - "Ideal.span", - "NonUnitalRingHomClass.toMulHomClass", - "Int.cast_add", - "Exists.casesOn", - "SetLike.instMembership", - "Int.instAdd", - "Eq", - "Eq.mpr", - "Semiring.toNonAssocSemiring", - "Iff.mp", - "MonoidWithZero.toMonoid", - "Ring.toIntCast", - "Int.cast_mul", - "Nat", - "Ring.toSubtractionMonoid", - "And.casesOn", - "Eq.trans", - "exists_add_pow_prime_eq", - "setOf", - "Int.instHPowNat", - "Nat.rawCast", - "Ideal.add_mem", - "NonAssocRing.toNonUnitalNonAssocRing", - "Ideal.Quotient.eq_zero_iff_mem", - "AddCommMonoidWithOne.toAddMonoidWithOne", - "Mathlib.Tactic.Ring.add_congr", - "RingHom.instFunLike", - "NonUnitalNonAssocSemiring.toDistrib", - "Mathlib.Tactic.Ring.neg_mul", - "SemigroupWithZero.toSemigroup", - "instHMul", - "Mathlib.Tactic.Ring.mul_zero", - "NonAssocRing.toIntCast", - "Mathlib.Tactic.Ring.of_eq", - "HEq.refl", - "of_eq_true", - "Int.cast", - "Eq.mp", - "CommRing.toRing", - "Ideal.Quotient.mk", - "Singleton.singleton", - "sub_eq_zero", - "congrArg", - "Fact", - "Ideal", - "Int.ofNat", - "RingHom", - "Mathlib.Meta.NormNum.IsInt.to_raw_eq", - "AddGroupWithOne.toIntCast", - "CommRing.toNonUnitalCommRing", - "Subalgebra", - "Fact.out", - "Mathlib.Tactic.Ring.add_pf_add_lt", - "CommSemiring.toNonUnitalCommSemiring", - "SubtractionCommMonoid.toSubtractionMonoid", - "Membership", - "Submodule.Quotient.instZeroQuotient", - "Ring.toNonAssocRing", - "PNat.val", - "Zero.toOfNat0", - "outParam", - "Set.instSingletonSet", - "instHPow", - "Ring.toSub", - "AddCommGroup.toAddGroup", - "Algebra.adjoin_induction", - "CommMonoid.toMonoid", - "Mathlib.Tactic.Ring.pow_congr", - "CommSemiring.toCommMonoidWithZero", - "DFunLike.coe", - "Eq.symm", - "AddCommMonoid.toAddCommSemigroup", - "id", - "Ideal.Quotient.commRing", - "Membership.mem", - "Mathlib.Meta.NormNum.IsInt.of_raw", - "funext", - "AddGroupWithOne.toAddMonoidWithOne", - "instHAdd", - "AddSemigroup.toAdd", - "HSub.hSub", - "AddMonoidWithOne.toAddMonoid", - "eq_intCast", - "Mathlib.Tactic.Ring.single_pow", - "Dvd.dvd", - "NonUnitalNonAssocSemiring.toMul", - "Ring.toSemiring", - "AddGroup.toSubNegMonoid", - "Ideal.mul_mem_left", - "Semiring.toOne", - "Mathlib.Tactic.Ring.add_pf_add_zero", - "mul_sub", - "SubtractionMonoid.toSubNegMonoid", - "Mathlib.Tactic.Ring.instCommSemiringNat", - "SubNegMonoid.toAddMonoid", - "NonUnitalSemiring.toSemigroupWithZero", - "HMul.hMul", - 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