Add files using upload-large-folder tool

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Subalgebra","isSimpleOrder_of_finrank_prime"],"typeFallback":"forall (F : Type.{u_1}) (A : Type.{u_2}) [inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4 : Field.{u_1} F] [inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7 : Ring.{u_2} A] [inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.10 : IsDomain.{u_2} A (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7)] [inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13 : Algebra.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7)], (Nat.Prime (Module.finrank.{u_1, u_2} F A (DivisionSemiring.toSemiring.{u_1} F (Semifield.toDivisionSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_2} A (NonAssocRing.toNonUnitalNonAssocRing.{u_2} A (Ring.toNonAssocRing.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7)))) (Algebra.toModule.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13))) -> (IsSimpleOrder.{u_2} (Subalgebra.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13) (Preorder.toLE.{u_2} (Subalgebra.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13) (PartialOrder.toPreorder.{u_2} (Subalgebra.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13) (Subalgebra.instPartialOrder.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13))) (CompleteLattice.toBoundedOrder.{u_2} (Subalgebra.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13) (Algebra.instCompleteLatticeSubalgebra.{u_1, u_2} F A (Semifield.toCommSemiring.{u_1} F (Field.toSemifield.{u_1} F inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.4)) (Ring.toSemiring.{u_2} A inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Algebra.Subalgebra.IsSimpleOrder.3087223977._hygCtx._hyg.13)))","typeFull":"∀ (F : Type u_1) (A : Type u_2) [inst : Field F] [inst_1 : Ring A] [IsDomain A] [inst_3 : Algebra F A],\n  Nat.Prime (Module.finrank F A) → IsSimpleOrder (Subalgebra F A)","typeReadable":"∀ (F : Type u_1) (A : Type u_2) [inst : Field F] [inst_1 : Ring A] [IsDomain A] [inst_3 : Algebra F A],\n  Nat.Prime (Module.finrank F A) → IsSimpleOrder (Subalgebra F A)","typeReferences":[["Semifield","toCommSemiring"],["Subalgebra"],["PartialOrder","toPreorder"],["Ring","toNonAssocRing"],["Field"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["IsSimpleOrder"],["IsDomain"],["DivisionSemiring","toSemiring"],["Ring","toSemiring"],["Algebra"],["Algebra","instCompleteLatticeSubalgebra"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Module","finrank"],["Algebra","toModule"],["Field","toSemifield"],["Semifield","toDivisionSemiring"],["Preorder","toLE"],["CompleteLattice","toBoundedOrder"],["Nat","Prime"],["Subalgebra","instPartialOrder"],["NonAssocRing","toNonUnitalNonAssocRing"],["Ring"]],"valueReferences":[["Ring","toNonAssocRing"],["Eq","trans"],["Module","finrank_mul_finrank"],["Subalgebra","bot_eq_top_iff_finrank_eq_one"],["SemigroupAction","toSMul"],["Exists","intro"],["SMulZeroClass","toSMul"],["Subalgebra","instModuleSubtypeMem","_proof_1"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["EuclideanDomain","toNontrivial"],["Eq","symm"],["IsScalarTower","right"],["Monoid","toSemigroup"],["Subalgebra"],["Exists"],["Subalgebra","isDomain"],["DivisionSemiring","toSemiring"],["Nat","not_prime_one"],["Ring","toSemiring"],["BoundedOrder","toOrderBot"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["Eq","subst"],["CompleteLattice","toBoundedOrder"],["Subalgebra","instPartialOrder"],["AddMonoid","toAddZeroClass"],["Semifield","toCommSemiring"],["EuclideanDomain","toCommRing"],["AddCommMonoid","toAddMonoid"],["Field","toEuclideanDomain"],["instOfNatNat"],["DivisionRing","toDivisionSemiring"],["Module","Free","of_divisionRing"],["Eq"],["Preorder","toLE"],["IsDomain","toNontrivial"],["Subalgebra","instSubsemiringClass"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Subalgebra","instSetLike"],["Submodule","instTop"],["Field","toDivisionRing"],["OrderEmbedding"],["DivisionSemiring","isPrincipalIdealRing"],["OfNat","ofNat"],["Subalgebra","toSemiring"],["Module","toDistribMulAction"],["Ne"],["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Membership","mem"],["Iff","mp"],["HMul","hMul"],["Algebra","id"],["Subalgebra","toRing"],["Module","finrank"],["Semiring","toNonAssocSemiring"],["Submodule","eq_top_of_finrank_eq"],["divisionRingOfFiniteDimensional"],["DistribMulAction","toMulAction"],["instFunLikeOrderEmbedding"],["IsNoetherianRing","strongRankCondition"],["Semifield","toDivisionSemiring"],["Nontrivial","mk"],["Semiring","toModule"],["Nat","Prime"],["NonAssocRing","toNonUnitalNonAssocRing"],["Algebra","toSubmodule_eq_top"],["DistribSMul","toSMulZeroClass"],["SetLike","instMembership"],["Subalgebra","isScalarTower_mid"],["CompleteLattice","toLattice"],["FiniteDimensional","of_finrank_pos"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["AddZeroClass","toAddZero"],["Bot","bot"],["Nat"],["instHMul"],["Subalgebra","eq_bot_of_finrank_one"],["AddZero","toZero"],["Subtype"],["Submodule","module"],["SemilatticeSup","toPartialOrder"],["Subalgebra","algebra"],["Subalgebra","instModuleSubtypeMem"],["DFunLike","coe"],["Submodule"],["SubsemiringClass","nontrivial"],["Algebra","instCompleteLatticeSubalgebra"],["commRing_strongRankCondition"],["Subalgebra","finrank_toSubmodule"],["Or","imp"],["Algebra","toModule"],["BoundedOrder","toOrderTop"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["instMulNat"],["PrincipalIdealRing","isNoetherianRing"],["OrderBot","toBot"],["FiniteDimensional","finiteDimensional_subalgebra"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["Submodule","instPartialOrder"],["IsSimpleOrder","mk"],["Nat","Prime","eq_one_or_self_of_dvd"],["Semiring","toMonoidWithZero"],["Subalgebra","toSubmodule"],["Nat","Prime","pos"],["Ring","toAddCommGroup"],["Submodule","setLike"],["Field","toSemifield"],["Subalgebra","moduleLeft"],["Submodule","addCommMonoid"],["MulAction","toSemigroupAction"],["OrderTop","toTop"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.BigOperators.Ring.Multiset.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Multiset","prod_eq_zero_iff","_simp_1"],"typeFallback":"forall {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.11 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.14 : Nontrivial.{u_3} M₀] {s : Multiset.{u_3} M₀}, Eq.{1} Prop (Eq.{succ u_3} M₀ (Multiset.prod.{u_3} M₀ (CommMonoidWithZero.toCommMonoid.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6) s) (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6))))))) (Membership.mem.{u_3, u_3} M₀ (Multiset.{u_3} M₀) (Multiset.instMembership.{u_3} M₀) s (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6)))))))","typeFull":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [NoZeroDivisors M₀] [Nontrivial M₀] {s : Multiset M₀},\n  (s.prod = 0) = (0 ∈ s)","typeReadable":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [NoZeroDivisors M₀] [Nontrivial M₀] {s : Multiset M₀},\n  (s.prod = 0) = (0 ∈ s)","typeReferences":[["CommMonoidWithZero"],["NoZeroDivisors"],["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["MulZeroClass","toMul"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Zero","toOfNat0"],["Eq"],["Multiset","instMembership"]],"valueReferences":[["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["Multiset","prod_eq_zero_iff"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Zero","toOfNat0"],["Eq"],["propext"],["Multiset","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Multiset","dvd_sum"],"typeFallback":"forall {R : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2140512242._hygCtx._hyg.6 : NonUnitalSemiring.{u_4} R] {s : Multiset.{u_4} R} {a : R}, (forall (x : R), (Membership.mem.{u_4, u_4} R (Multiset.{u_4} R) (Multiset.instMembership.{u_4} R) s x) -> (Dvd.dvd.{u_4} R (semigroupDvd.{u_4} R (SemigroupWithZero.toSemigroup.{u_4} R (NonUnitalSemiring.toSemigroupWithZero.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2140512242._hygCtx._hyg.6))) a x)) -> (Dvd.dvd.{u_4} R (semigroupDvd.{u_4} R (SemigroupWithZero.toSemigroup.{u_4} R (NonUnitalSemiring.toSemigroupWithZero.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2140512242._hygCtx._hyg.6))) a (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2140512242._hygCtx._hyg.6)) s))","typeFull":"∀ {R : Type u_4} [inst : NonUnitalSemiring R] {s : Multiset R} {a : R}, (∀ (x : R), x ∈ s → a ∣ x) → a ∣ s.sum","typeReadable":"∀ {R : Type u_4} [inst : NonUnitalSemiring R] {s : Multiset R} {a : R}, (∀ (x : R), x ∈ s → a ∣ x) → a ∣ s.sum","typeReferences":[["semigroupDvd"],["Dvd","dvd"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Membership","mem"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalSemiring"],["NonUnitalSemiring","toSemigroupWithZero"],["Multiset","sum"],["SemigroupWithZero","toSemigroup"],["Multiset"],["Multiset","instMembership"]],"valueReferences":[["Or","inr"],["semigroupDvd"],["Membership","mem"],["NonUnitalSemiring","toSemigroupWithZero"],["AddCommMonoid","toAddMonoid"],["congrArg"],["dvd_add"],["Multiset","sum_cons"],["Or"],["Multiset","cons"],["Multiset","sum"],["Eq"],["dvd_zero"],["NonUnitalNonAssocSemiring","toDistrib"],["Dvd","dvd"],["instHAdd"],["Multiset","mem_cons"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Multiset"],["Multiset","mem_cons_self"],["HAdd","hAdd"],["Iff","mpr"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["id"],["Eq","mpr"],["SemigroupWithZero","toSemigroup"],["Multiset","induction_on"],["Distrib","leftDistribClass"],["Multiset","instMembership"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Commute","multiset_sum_right"],"typeFallback":"forall {R : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2946427549._hygCtx._hyg.6 : NonUnitalNonAssocSemiring.{u_4} R] (s : Multiset.{u_4} R) (a : R), (forall (b : R), (Membership.mem.{u_4, u_4} R (Multiset.{u_4} R) (Multiset.instMembership.{u_4} R) s b) -> (Commute.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2946427549._hygCtx._hyg.6)) a b)) -> (Commute.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2946427549._hygCtx._hyg.6)) a (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2946427549._hygCtx._hyg.6) s))","typeFull":"∀ {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] (s : Multiset R) (a : R),\n  (∀ (b : R), b ∈ s → Commute a b) → Commute a s.sum","typeReadable":"∀ {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] (s : Multiset R) (a : R),\n  (∀ (b : R), b ∈ s → Commute a b) → Commute a s.sum","typeReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Commute"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Membership","mem"],["Multiset","sum"],["Multiset"],["NonUnitalNonAssocSemiring"],["Multiset","instMembership"]],"valueReferences":[["Membership","mem"],["Quotient","mk"],["List"],["AddCommMonoid","toAddMonoid"],["Multiset","ofList"],["congrArg"],["List","isSetoid"],["Commute"],["Multiset","sum"],["Eq"],["NonUnitalNonAssocSemiring","toDistrib"],["Quotient","inductionOn"],["Multiset","quot_mk_to_coe"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Multiset","sum_coe"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Multiset"],["List","sum"],["Commute","list_sum_right"],["id"],["Eq","mpr"],["AddZero","toZero"],["Multiset","instMembership"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Multiset","sum_map_mul_right"],"typeFallback":"forall {ι : Type.{u_1}} {R : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3331801072._hygCtx._hyg.6 : NonUnitalNonAssocSemiring.{u_4} R] {a : R} {s : Multiset.{u_1} ι} {f : ι -> R}, Eq.{succ u_4} R (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3331801072._hygCtx._hyg.6) (Multiset.map.{u_4, u_1} ι R (fun (i : ι) => HMul.hMul.{u_4, u_4, u_4} R R R (instHMul.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3331801072._hygCtx._hyg.6))) (f i) a) s)) (HMul.hMul.{u_4, u_4, u_4} R R R (instHMul.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3331801072._hygCtx._hyg.6))) (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3331801072._hygCtx._hyg.6) (Multiset.map.{u_4, u_1} ι R f s)) a)","typeFull":"∀ {ι : Type u_1} {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] {a : R} {s : Multiset ι} {f : ι → R},\n  (Multiset.map (fun i => f i * a) s).sum = (Multiset.map f s).sum * a","typeReadable":"∀ {ι : Type u_1} {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] {a : R} {s : Multiset ι} {f : ι → R},\n  (Multiset.map (fun i => f i * a) s).sum = (Multiset.map f s).sum * a","typeReferences":[["Multiset","map"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["instHMul"],["HMul","hMul"],["Multiset","sum"],["Eq"],["Multiset"],["NonUnitalNonAssocSemiring"]],"valueReferences":[["Multiset","map_cons"],["Eq","trans"],["HMul","hMul"],["Multiset","instZero"],["AddCommMonoid","toAddMonoid"],["add_mul"],["congrArg"],["Distrib","rightDistribClass"],["Multiset","sum_cons"],["congr"],["congrFun'"],["Multiset","cons"],["Zero","toOfNat0"],["Multiset","sum"],["Eq"],["Multiset","map"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["instHAdd"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["MulZeroClass","zero_mul"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Multiset"],["OfNat","ofNat"],["HAdd","hAdd"],["eq_self"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHMul"],["Multiset","induction_on"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Multiset","prod_map_sum"],"typeFallback":"forall {R : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1107748835._hygCtx._hyg.6 : CommSemiring.{u_4} R] {s : Multiset.{u_4} (Multiset.{u_4} R)}, Eq.{succ u_4} R (Multiset.prod.{u_4} R (CommSemiring.toCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1107748835._hygCtx._hyg.6) (Multiset.map.{u_4, u_4} (Multiset.{u_4} R) R (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R (CommSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1107748835._hygCtx._hyg.6))))) s)) (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R (CommSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1107748835._hygCtx._hyg.6)))) (Multiset.map.{u_4, u_4} (Multiset.{u_4} R) R (Multiset.prod.{u_4} R (CommSemiring.toCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1107748835._hygCtx._hyg.6)) (Multiset.Sections.{u_4} R s)))","typeFull":"∀ {R : Type u_4} [inst : CommSemiring R] {s : Multiset (Multiset R)},\n  (Multiset.map Multiset.sum s).prod = (Multiset.map Multiset.prod s.Sections).sum","typeReadable":"∀ {R : Type u_4} [inst : CommSemiring R] {s : Multiset (Multiset R)},\n  (Multiset.map Multiset.sum s).prod = (Multiset.map Multiset.prod s.Sections).sum","typeReferences":[["Multiset","map"],["Multiset","Sections"],["Semiring","toNonAssocSemiring"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring"],["Multiset","sum"],["Multiset","prod"],["Eq"],["Multiset"],["CommSemiring","toCommMonoid"]],"valueReferences":[["Multiset","map_cons"],["Eq","trans"],["Multiset","instZero"],["HMul","hMul"],["Multiset","Sections"],["Semiring","toNonAssocSemiring"],["Multiset","sections_cons"],["funext"],["Multiset","cons"],["Multiset","sum"],["Semiring","toNonUnitalSemiring"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["Multiset","map_congr"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toCommMonoid"],["Eq","refl"],["instHMul"],["MulOneClass","toMulOne"],["CommMonoid","toMonoid"],["Multiset","map_id'"],["Multiset","sum_bind"],["Multiset","prod_cons"],["Multiset","sum_map_mul_right"],["congrArg"],["MulOne","toMul"],["Multiset","sum_singleton"],["congr"],["Monoid","toMulOneClass"],["Zero","toOfNat0"],["congrFun'"],["Eq"],["Multiset","map"],["Multiset","map_bind"],["True"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Distrib","toMul"],["Multiset","prod"],["Function","comp"],["Multiset"],["OfNat","ofNat"],["eq_self"],["One","toOfNat1"],["of_eq_true"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Multiset","induction_on"],["Multiset","sum_map_mul_left"],["Multiset","bind"],["Multiset","map_map"]]},{"isProp":true,"kind":"theorem","name":["Multiset","prod_eq_zero_iff"],"typeFallback":"forall {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.11 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.14 : Nontrivial.{u_3} M₀] {s : Multiset.{u_3} M₀}, Iff (Eq.{succ u_3} M₀ (Multiset.prod.{u_3} M₀ (CommMonoidWithZero.toCommMonoid.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6) s) (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6))))))) (Membership.mem.{u_3, u_3} M₀ (Multiset.{u_3} M₀) (Multiset.instMembership.{u_3} M₀) s (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1456635985._hygCtx._hyg.6)))))))","typeFull":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [NoZeroDivisors M₀] [Nontrivial M₀] {s : Multiset M₀},\n  s.prod = 0 ↔ 0 ∈ s","typeReadable":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [NoZeroDivisors M₀] [Nontrivial M₀] {s : Multiset M₀},\n  s.prod = 0 ↔ 0 ∈ s","typeReferences":[["CommMonoidWithZero"],["NoZeroDivisors"],["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["MulZeroClass","toMul"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["Nontrivial"],["Zero","toOfNat0"],["Eq"],["Multiset","instMembership"]],"valueReferences":[["MulOneClass","toMulOne"],["CommMonoid","toMonoid"],["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["Quotient","mk"],["MonoidWithZero","toMulZeroOneClass"],["List"],["Multiset","ofList"],["List","prod"],["congrArg"],["List","isSetoid"],["MulOne","toMul"],["Monoid","toMulOneClass"],["Zero","toOfNat0"],["Eq"],["List","prod_eq_zero_iff"],["MulOne","toOne"],["Quotient","inductionOn"],["Multiset","quot_mk_to_coe"],["Multiset","prod_coe"],["CommMonoidWithZero","toCommMonoid"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["id"],["Eq","mpr"],["Multiset","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Multiset","prod_map_neg"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6 : CommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.9 : HasDistribNeg.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6))))] (s : Multiset.{u_2} M), Eq.{succ u_2} M (Multiset.prod.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6 (Multiset.map.{u_2, u_2} M M (Neg.neg.{u_2} M (InvolutiveNeg.toNeg.{u_2} M (HasDistribNeg.toInvolutiveNeg.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6)))) inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.9))) s)) (HMul.hMul.{u_2, u_2, u_2} M M M (instHMul.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6))))) (HPow.hPow.{u_2, 0, u_2} M Nat M (instHPow.{u_2, 0} M Nat (Monoid.toPow.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6))) (Neg.neg.{u_2} M (InvolutiveNeg.toNeg.{u_2} M (HasDistribNeg.toInvolutiveNeg.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6)))) inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.9)) (OfNat.ofNat.{u_2} M 1 (One.toOfNat1.{u_2} M (MulOne.toOne.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6))))))) (Multiset.card.{u_2} M s)) (Multiset.prod.{u_2} M inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3928577373._hygCtx._hyg.6 s))","typeFull":"∀ {M : Type u_2} [inst : CommMonoid M] [inst_1 : HasDistribNeg M] (s : Multiset M),\n  (Multiset.map Neg.neg s).prod = (-1) ^ s.card * s.prod","typeReadable":"∀ {M : Type u_2} [inst : CommMonoid M] [inst_1 : HasDistribNeg M] (s : Multiset M),\n  (Multiset.map Neg.neg s).prod = (-1) ^ s.card * s.prod","typeReferences":[["instHPow"],["MulOneClass","toMulOne"],["Multiset","map"],["Multiset","card"],["MulOne","toOne"],["CommMonoid","toMonoid"],["Neg","neg"],["HMul","hMul"],["Multiset","prod"],["HPow","hPow"],["Multiset"],["OfNat","ofNat"],["Nat"],["MulOne","toMul"],["One","toOfNat1"],["Monoid","toPow"],["HasDistribNeg","toInvolutiveNeg"],["Monoid","toMulOneClass"],["HasDistribNeg"],["instHMul"],["CommMonoid"],["InvolutiveNeg","toNeg"],["Eq"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["CommMonoid","toMonoid"],["Multiset","card"],["HMul","hMul"],["Quotient","mk"],["List"],["List","prod"],["congrArg"],["List","isSetoid"],["MulOne","toMul"],["Monoid","toPow"],["forall_congr"],["Monoid","toMulOneClass"],["congrFun'"],["InvolutiveNeg","toNeg"],["Eq"],["Multiset","map"],["instHPow"],["True"],["MulOne","toOne"],["Neg","neg"],["Quotient","inductionOn"],["Multiset","prod"],["HPow","hPow"],["OfNat","ofNat"],["implies_true"],["eq_self"],["Nat"],["HasDistribNeg","toInvolutiveNeg"],["One","toOfNat1"],["of_eq_true"],["instHMul"],["List","prod_map_neg"],["List","length"]]},{"isProp":true,"kind":"theorem","name":["Multiset","sum_map_mul_left"],"typeFallback":"forall {ι : Type.{u_1}} {R : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1298557614._hygCtx._hyg.6 : NonUnitalNonAssocSemiring.{u_4} R] {a : R} {s : Multiset.{u_1} ι} {f : ι -> R}, Eq.{succ u_4} R (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1298557614._hygCtx._hyg.6) (Multiset.map.{u_4, u_1} ι R (fun (i : ι) => HMul.hMul.{u_4, u_4, u_4} R R R (instHMul.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1298557614._hygCtx._hyg.6))) a (f i)) s)) (HMul.hMul.{u_4, u_4, u_4} R R R (instHMul.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1298557614._hygCtx._hyg.6))) a (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1298557614._hygCtx._hyg.6) (Multiset.map.{u_4, u_1} ι R f s)))","typeFull":"∀ {ι : Type u_1} {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] {a : R} {s : Multiset ι} {f : ι → R},\n  (Multiset.map (fun i => a * f i) s).sum = a * (Multiset.map f s).sum","typeReadable":"∀ {ι : Type u_1} {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] {a : R} {s : Multiset ι} {f : ι → R},\n  (Multiset.map (fun i => a * f i) s).sum = a * (Multiset.map f s).sum","typeReferences":[["Multiset","map"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["instHMul"],["HMul","hMul"],["Multiset","sum"],["Eq"],["Multiset"],["NonUnitalNonAssocSemiring"]],"valueReferences":[["Multiset","map_cons"],["Eq","trans"],["HMul","hMul"],["Multiset","instZero"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Multiset","sum_cons"],["congr"],["Multiset","cons"],["Zero","toOfNat0"],["Multiset","sum"],["Eq"],["Multiset","map"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["instHAdd"],["mul_add"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["MulZeroClass","mul_zero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Multiset"],["OfNat","ofNat"],["HAdd","hAdd"],["eq_self"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHMul"],["Multiset","induction_on"],["Distrib","leftDistribClass"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Multiset","prod_map_add"],"typeFallback":"forall {ι : Type.{u_1}} {R : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3296133605._hygCtx._hyg.6 : CommSemiring.{u_4} R] {s : Multiset.{u_1} ι} {f : ι -> R} {g : ι -> R}, Eq.{succ u_4} R (Multiset.prod.{u_4} R (CommSemiring.toCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3296133605._hygCtx._hyg.6) (Multiset.map.{u_4, u_1} ι R (fun (i : ι) => HAdd.hAdd.{u_4, u_4, u_4} R R R (instHAdd.{u_4} R (Distrib.toAdd.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R (CommSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3296133605._hygCtx._hyg.6)))))) (f i) (g i)) s)) (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R (CommSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3296133605._hygCtx._hyg.6)))) (Multiset.map.{u_4, u_1} (Prod.{u_1, u_1} (Multiset.{u_1} ι) (Multiset.{u_1} ι)) R (fun (p : Prod.{u_1, u_1} (Multiset.{u_1} ι) (Multiset.{u_1} ι)) => HMul.hMul.{u_4, u_4, u_4} R R R (instHMul.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} R (Semiring.toNonAssocSemiring.{u_4} R (CommSemiring.toSemiring.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3296133605._hygCtx._hyg.6)))))) (Multiset.prod.{u_4} R (CommSemiring.toCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3296133605._hygCtx._hyg.6) (Multiset.map.{u_4, u_1} ι R f (Prod.fst.{u_1, u_1} (Multiset.{u_1} ι) (Multiset.{u_1} ι) p))) (Multiset.prod.{u_4} R (CommSemiring.toCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3296133605._hygCtx._hyg.6) (Multiset.map.{u_4, u_1} ι R g (Prod.snd.{u_1, u_1} (Multiset.{u_1} ι) (Multiset.{u_1} ι) p)))) (Multiset.antidiagonal.{u_1} ι s)))","typeFull":"∀ {ι : Type u_1} {R : Type u_4} [inst : CommSemiring R] {s : Multiset ι} {f g : ι → R},\n  (Multiset.map (fun i => f i + g i) s).prod =\n    (Multiset.map (fun p => (Multiset.map f p.1).prod * (Multiset.map g p.2).prod) s.antidiagonal).sum","typeReadable":"∀ {ι : Type u_1} {R : Type u_4} [inst : CommSemiring R] {s : Multiset ι} {f g : ι → R},\n  (Multiset.map (fun i => f i + g i) s).prod =\n    (Multiset.map (fun p => (Multiset.map f p.1).prod * (Multiset.map g p.2).prod) s.antidiagonal).sum","typeReferences":[["Multiset","map"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Multiset","antidiagonal"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["HMul","hMul"],["Prod","snd"],["Multiset","prod"],["Multiset"],["CommSemiring","toCommMonoid"],["Prod","fst"],["Prod"],["HAdd","hAdd"],["Semiring","toNonAssocSemiring"],["instHMul"],["Multiset","sum"],["Eq"]],"valueReferences":[["Multiset","instAdd"],["Multiset","map_cons"],["Eq","trans"],["Prod","mk"],["Singleton","singleton"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["Multiset","instZero"],["Multiset","sum_map_add"],["Distrib","rightDistribClass"],["Semiring","toNonAssocSemiring"],["Multiset","map_add"],["Eq","symm"],["Multiset","cons"],["Multiset","sum"],["add_comm"],["NonAssocSemiring","toMulZeroOneClass"],["NonUnitalCommSemiring","toCommSemigroup"],["NonUnitalNonAssocSemiring","toDistrib"],["MulOne","toOne"],["CommMagma","toMul"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["Multiset","map_congr"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Prod","snd"],["AddZeroClass","toAddZero"],["CommSemiring","toCommMonoid"],["Prod"],["Multiset","instSingleton"],["Eq","refl"],["id"],["instHMul"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["CommMonoid","toMonoid"],["Multiset","antidiagonal"],["Multiset","antidiagonal_cons"],["mul_assoc"],["MulZeroOneClass","toMulOneClass"],["Multiset","prod_cons"],["AddCommMonoid","toAddMonoid"],["Semigroup","toMul"],["add_mul"],["Prod","fst"],["congrArg"],["MulOne","toMul"],["Multiset","sum_singleton"],["CommSemiring","toNonUnitalCommSemiring"],["congr"],["Monoid","toMulOneClass"],["AddCommSemigroup","toAddCommMagma"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["CommSemigroup","toCommMagma"],["Multiset","sum_add"],["Multiset","map"],["Distrib","toAdd"],["True"],["instHAdd"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["mul_one"],["Function","comp"],["Multiset","prod"],["AddZero","toAdd"],["Prod","map"],["Multiset"],["mul_left_comm"],["OfNat","ofNat"],["HAdd","hAdd"],["eq_self"],["AddCommMonoid","toAddCommSemigroup"],["One","toOfNat1"],["of_eq_true"],["SemigroupWithZero","toSemigroup"],["Multiset","induction_on"],["Multiset","sum_map_mul_left"],["Multiset","map_map"]]},{"isProp":true,"kind":"theorem","name":["Multiset","prod_ne_zero"],"typeFallback":"forall {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.6 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.11 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.6)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.14 : Nontrivial.{u_3} M₀] {s : Multiset.{u_3} M₀}, (Not (Membership.mem.{u_3, u_3} M₀ (Multiset.{u_3} M₀) (Multiset.instMembership.{u_3} M₀) s (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.6)))))))) -> (Ne.{succ u_3} M₀ (Multiset.prod.{u_3} M₀ (CommMonoidWithZero.toCommMonoid.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.6) s) (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.3370944760._hygCtx._hyg.6)))))))","typeFull":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [NoZeroDivisors M₀] [Nontrivial M₀] {s : Multiset M₀},\n  ¬0 ∈ s → s.prod ≠ 0","typeReadable":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [NoZeroDivisors M₀] [Nontrivial M₀] {s : Multiset M₀},\n  ¬0 ∈ s → s.prod ≠ 0","typeReferences":[["Not"],["CommMonoidWithZero"],["NoZeroDivisors"],["CommMonoidWithZero","toMonoidWithZero"],["MulZeroClass","toMul"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Ne"],["Zero","toOfNat0"],["Multiset","instMembership"]],"valueReferences":[["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["Iff","mp"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["Multiset","prod_eq_zero_iff"],["mt"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Zero","toOfNat0"],["Eq"],["Multiset","instMembership"]]},{"isProp":true,"kind":"theorem","name":["Commute","multiset_sum_left"],"typeFallback":"forall {R : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1199817320._hygCtx._hyg.6 : NonUnitalNonAssocSemiring.{u_4} R] (s : Multiset.{u_4} R) (b : R), (forall (a : R), (Membership.mem.{u_4, u_4} R (Multiset.{u_4} R) (Multiset.instMembership.{u_4} R) s a) -> (Commute.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1199817320._hygCtx._hyg.6)) a b)) -> (Commute.{u_4} R (Distrib.toMul.{u_4} R (NonUnitalNonAssocSemiring.toDistrib.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1199817320._hygCtx._hyg.6)) (Multiset.sum.{u_4} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} R inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.1199817320._hygCtx._hyg.6) s) b)","typeFull":"∀ {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] (s : Multiset R) (b : R),\n  (∀ (a : R), a ∈ s → Commute a b) → Commute s.sum b","typeReadable":"∀ {R : Type u_4} [inst : NonUnitalNonAssocSemiring R] (s : Multiset R) (b : R),\n  (∀ (a : R), a ∈ s → Commute a b) → Commute s.sum b","typeReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Commute"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Membership","mem"],["Multiset","sum"],["Multiset"],["NonUnitalNonAssocSemiring"],["Multiset","instMembership"]],"valueReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Commute","symm"],["Multiset","sum"],["Commute","multiset_sum_right"]]},{"isProp":true,"kind":"theorem","name":["Multiset","prod_eq_zero"],"typeFallback":"forall {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2037111364._hygCtx._hyg.6 : CommMonoidWithZero.{u_3} M₀] {s : Multiset.{u_3} M₀}, (Membership.mem.{u_3, u_3} M₀ (Multiset.{u_3} M₀) (Multiset.instMembership.{u_3} M₀) s (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2037111364._hygCtx._hyg.6))))))) -> (Eq.{succ u_3} M₀ (Multiset.prod.{u_3} M₀ (CommMonoidWithZero.toCommMonoid.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2037111364._hygCtx._hyg.6) s) (OfNat.ofNat.{u_3} M₀ 0 (Zero.toOfNat0.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.Multiset.2037111364._hygCtx._hyg.6)))))))","typeFull":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] {s : Multiset M₀}, 0 ∈ s → s.prod = 0","typeReadable":"∀ {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] {s : Multiset M₀}, 0 ∈ s → s.prod = 0","typeReferences":[["CommMonoidWithZero"],["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Zero","toOfNat0"],["Eq"],["Multiset","instMembership"]],"valueReferences":[["MulOneClass","toMulOne"],["CommMonoidWithZero","toMonoidWithZero"],["Eq","trans"],["CommMonoid","toMonoid"],["Multiset","exists_cons_of_mem"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["Multiset","prod_cons"],["congrArg"],["MulOne","toMul"],["Monoid","toMulOneClass"],["Zero","toOfNat0"],["Multiset","cons"],["congrFun'"],["Eq"],["True"],["CommMonoidWithZero","toCommMonoid"],["MulZeroClass","zero_mul"],["Multiset","prod"],["Multiset"],["OfNat","ofNat"],["Exists","casesOn"],["eq_self"],["MulZeroOneClass","toMulZeroClass"],["of_eq_true"],["MulZeroClass","toZero"],["instHMul"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.Ring.Instances.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["CommRingCat","isLocalHom_comp"],"typeFallback":"forall {R : CommRingCat.{u_1}} {S : CommRingCat.{u_1}} {T : CommRingCat.{u_1}} (f : Quiver.Hom.{u_1, succ u_1} CommRingCat.{u_1} (CategoryTheory.CategoryStruct.toQuiver.{u_1, succ u_1} CommRingCat.{u_1} (CategoryTheory.Category.toCategoryStruct.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1})) R S) (g : Quiver.Hom.{u_1, succ u_1} CommRingCat.{u_1} (CategoryTheory.CategoryStruct.toQuiver.{u_1, succ u_1} CommRingCat.{u_1} (CategoryTheory.Category.toCategoryStruct.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1})) S T) [inst._@.Mathlib.Algebra.Category.Ring.Instances.1966972171._hygCtx._hyg.15 : IsLocalHom.{u_1, u_1, u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.carrier.{u_1} T) (RingHom.{u_1, u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.carrier.{u_1} T) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRingCat.commRing.{u_1} T))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} S) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} T) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} T) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRingCat.commRing.{u_1} T))))) (RingHom.instFunLike.{u_1, u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.carrier.{u_1} T) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRingCat.commRing.{u_1} T))))) (CommRingCat.Hom.hom.{u_1} S T g)] [inst._@.Mathlib.Algebra.Category.Ring.Instances.1966972171._hygCtx._hyg.18 : IsLocalHom.{u_1, u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (RingHom.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} S) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (RingHom.instFunLike.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (CommRingCat.Hom.hom.{u_1} R S f)], IsLocalHom.{u_1, u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} T) (RingHom.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} T) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRingCat.commRing.{u_1} T))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} T) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} T) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRingCat.commRing.{u_1} T))))) (RingHom.instFunLike.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} T) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} T) (CommRingCat.commRing.{u_1} T))))) (CommRingCat.Hom.hom.{u_1} R T (CategoryTheory.CategoryStruct.comp.{u_1, succ u_1} CommRingCat.{u_1} (CategoryTheory.Category.toCategoryStruct.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1}) R S T f g))","typeFull":"∀ {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T) [IsLocalHom (CommRingCat.Hom.hom g)]\n  [IsLocalHom (CommRingCat.Hom.hom f)], IsLocalHom (CommRingCat.Hom.hom (CategoryTheory.CategoryStruct.comp f g))","typeReadable":"∀ {R S T : CommRingCat} (f : R ⟶ S) (g : S ⟶ T) [IsLocalHom (CommRingCat.Hom.hom g)]\n  [IsLocalHom (CommRingCat.Hom.hom f)], IsLocalHom (CommRingCat.Hom.hom (CategoryTheory.CategoryStruct.comp f g))","typeReferences":[["RingHom"],["CommRing","toCommSemiring"],["CommRingCat","Hom","hom"],["IsLocalHom"],["CommSemiring","toSemiring"],["RingHom","instFunLike"],["Semiring","toMonoidWithZero"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","instCategory"],["CategoryTheory","CategoryStruct","toQuiver"],["Semiring","toNonAssocSemiring"],["CommRingCat"],["CategoryTheory","CategoryStruct","comp"],["Quiver","Hom"],["MonoidWithZero","toMonoid"],["CommRingCat","commRing"],["CommRingCat","carrier"]],"valueReferences":[["CommRingCat","Hom","hom'"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["RingHom","isLocalHom_comp"],["CommRingCat","commRing"],["CommRingCat","carrier"]]},{"isProp":true,"kind":"theorem","name":["localization_unit_isIso"],"typeFallback":"forall (R : CommRingCat.{u_1}), CategoryTheory.IsIso.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} (CommRingCat.of.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (CommRingCat.of.{u_1} (Localization.Away.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.instCommRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))))))) (CommRingCat.ofHom.{u_1} (CommRingCat.carrier.{u_1} R) (Localization.Away.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (CommRingCat.commRing.{u_1} R) (OreLocalization.instCommRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))))) (algebraMap.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (Localization.Away.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OreLocalization.instSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))))) (OreLocalization.instAlgebra.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (Ring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))))) (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Algebra.id.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))","typeFull":"∀ (R : CommRingCat), CategoryTheory.IsIso (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away 1)))","typeReadable":"∀ (R : CommRingCat), CategoryTheory.IsIso (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away 1)))","typeReferences":[["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["AddGroupWithOne","toAddMonoidWithOne"],["CategoryTheory","IsIso"],["Algebra","id"],["OreLocalization","instCommRing"],["Ring","toAddGroupWithOne"],["CommRingCat"],["OreLocalization","instAlgebra"],["CommRingCat","commRing"],["CommRingCat","carrier"],["CommRing","toCommSemiring"],["OfNat","ofNat"],["Localization","Away"],["CommRing","toCommMonoid"],["Ring","toSemiring"],["CommRingCat","instCategory"],["CommRing","toRing"],["One","toOfNat1"],["Submonoid","powers"],["AddMonoidWithOne","toOne"],["OreLocalization","instSemiring"],["CommRingCat","of"],["CommRingCat","ofHom"],["algebraMap"]],"valueReferences":[["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["CategoryTheory","Iso","isIso_hom"],["AddGroupWithOne","toAddMonoidWithOne"],["AlgEquiv","toRingEquiv"],["Algebra","id"],["Semiring","toNonAssocSemiring"],["OreLocalization","instCommRing"],["Ring","toAddGroupWithOne"],["CommRingCat"],["OreLocalization","instAlgebra"],["MonoidWithZero","toMonoid"],["Localization","isLocalization"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["CommRingCat","commRing"],["RingEquiv","toCommRingCatIso"],["NonAssocSemiring","toAddCommMonoidWithOne"],["CommRingCat","carrier"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["Localization","Away"],["Ring","toSemiring"],["CommRing","toCommMonoid"],["CommRingCat","instCategory"],["CommRing","toRing"],["One","toOfNat1"],["Submonoid","powers"],["AddMonoidWithOne","toOne"],["OreLocalization","instCommSemiring"],["CommRingCat","of"],["IsLocalization","atOne"]]},{"isProp":true,"kind":"theorem","name":["isLocalHom_of_iso"],"typeFallback":"forall {R : CommRingCat.{u_1}} {S : CommRingCat.{u_1}} (f : CategoryTheory.Iso.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} R S), IsLocalHom.{u_1, u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (RingHom.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} S) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (RingHom.instFunLike.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (CommRingCat.Hom.hom.{u_1} R S (CategoryTheory.Iso.hom.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} R S f))","typeFull":"∀ {R S : CommRingCat} (f : R ≅ S), IsLocalHom (CommRingCat.Hom.hom f.hom)","typeReadable":"∀ {R S : CommRingCat} (f : R ≅ S), IsLocalHom (CommRingCat.Hom.hom f.hom)","typeReferences":[["RingHom"],["CategoryTheory","Iso"],["CommRing","toCommSemiring"],["CommRingCat","Hom","hom"],["IsLocalHom"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["RingHom","instFunLike"],["CommRingCat","instCategory"],["Semiring","toNonAssocSemiring"],["CommRingCat"],["MonoidWithZero","toMonoid"],["CommRingCat","commRing"],["CommRingCat","carrier"],["CategoryTheory","Iso","hom"]],"valueReferences":[["RingHom"],["Eq","trans"],["RingHom","instFunLike"],["CommRingCat","instConcreteCategoryRingHomCarrier"],["DFunLike","coe"],["CategoryTheory","Iso","hom_inv_id_apply"],["congrArg"],["Semiring","toNonAssocSemiring"],["CommRingCat"],["MonoidWithZero","toMonoid"],["eq_of_heq"],["Eq","symm"],["CommRingCat","commRing"],["Eq"],["Eq","ndrec"],["CommRingCat","carrier"],["CategoryTheory","Iso","hom"],["IsLocalHom","mk"],["CommRing","toCommSemiring"],["True"],["CategoryTheory","Iso","inv"],["HEq","refl"],["CommRingCat","Hom","hom"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["IsUnit"],["Eq","casesOn"],["eq_self"],["CommRingCat","instCategory"],["of_eq_true"],["Eq","refl"],["HEq"],["Eq","mpr"],["RingHom","isUnit_map"]]},{"isProp":true,"kind":"theorem","name":["Localization","epi'"],"typeFallback":"forall {R : CommRingCat.{u_1}} (M : Submonoid.{u_1} (CommRingCat.carrier.{u_1} R) (MulZeroOneClass.toMulOneClass.{u_1} (CommRingCat.carrier.{u_1} R) (NonAssocSemiring.toMulZeroOneClass.{u_1} (CommRingCat.carrier.{u_1} R) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))), CategoryTheory.Epi.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} R (CommRingCat.of.{u_1} (Localization.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) M) (OreLocalization.instCommRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R) M (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) M))) (CommRingCat.ofHom.{u_1} (CommRingCat.carrier.{u_1} R) (Localization.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) M) (CommRingCat.commRing.{u_1} R) (OreLocalization.instCommRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R) M (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) M)) (algebraMap.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (Localization.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) M) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OreLocalization.instSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) M (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) M)) (OreLocalization.instAlgebra.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) M (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) M) (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Algebra.id.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))","typeFull":"∀ {R : CommRingCat} (M : Submonoid ↑R), CategoryTheory.Epi (CommRingCat.ofHom (algebraMap (↑R) (Localization M)))","typeReadable":"∀ {R : CommRingCat} (M : Submonoid ↑R), CategoryTheory.Epi (CommRingCat.ofHom (algebraMap (↑R) (Localization M)))","typeReferences":[["CommRing","toCommSemiring"],["Localization"],["OreLocalization","oreSetComm"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["Algebra","id"],["CommRing","toCommMonoid"],["Submonoid"],["CommRingCat","instCategory"],["OreLocalization","instCommRing"],["Semiring","toNonAssocSemiring"],["OreLocalization","instAlgebra"],["CommRingCat"],["OreLocalization","instSemiring"],["CategoryTheory","Epi"],["CommRingCat","of"],["CommRingCat","commRing"],["CommRingCat","ofHom"],["CommRingCat","carrier"],["NonAssocSemiring","toMulZeroOneClass"],["algebraMap"]],"valueReferences":[["Localization"],["CommRing","toCommSemiring"],["OreLocalization","oreSetComm"],["CommRingCat","casesOn"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["Algebra","id"],["CommRing","toCommMonoid"],["Submonoid"],["CommRingCat","instCategory"],["OreLocalization","instCommRing"],["Semiring","toNonAssocSemiring"],["OreLocalization","instAlgebra"],["CommRingCat"],["OreLocalization","instSemiring"],["CategoryTheory","Epi"],["Localization","isLocalization"],["CommRingCat","of"],["IsLocalization","epi"],["CommRingCat","commRing"],["CommRingCat","ofHom"],["CommRingCat","carrier"],["NonAssocSemiring","toMulZeroOneClass"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["IsLocalization","epi"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.3 : CommRing.{u_1} R] (M : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.3)))))) (S : Type.{u_1}) [inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.9 : CommRing.{u_1} S] [inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.12 : Algebra.{u_1, u_1} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.9))] [inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.16 : IsLocalization.{u_1, u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.3) M S (CommRing.toCommSemiring.{u_1} S inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.9) inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.12], CategoryTheory.Epi.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} (CommRingCat.of.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.3) (CommRingCat.of.{u_1} S inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.9) (CommRingCat.ofHom.{u_1} R S inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.9 (algebraMap.{u_1, u_1} R S (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.3) (CommSemiring.toSemiring.{u_1} S (CommRing.toCommSemiring.{u_1} S inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411403._hygCtx._hyg.12))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (M : Submonoid R) (S : Type u_1) [inst_1 : CommRing S] [inst_2 : Algebra R S]\n  [IsLocalization M S], CategoryTheory.Epi (CommRingCat.ofHom (algebraMap R S))","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (M : Submonoid R) (S : Type u_1) [inst_1 : CommRing S] [inst_2 : Algebra R S]\n  [IsLocalization M S], CategoryTheory.Epi (CommRingCat.ofHom (algebraMap R S))","typeReferences":[["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["CommRing"],["Algebra"],["IsLocalization"],["Submonoid"],["CommRingCat","instCategory"],["Semiring","toNonAssocSemiring"],["CommRingCat"],["CategoryTheory","Epi"],["CommRingCat","of"],["CommRingCat","ofHom"],["NonAssocSemiring","toMulZeroOneClass"],["algebraMap"]],"valueReferences":[["CommRingCat","Hom"],["RingHom"],["CommRing","toCommSemiring"],["CategoryTheory","Epi","mk"],["CommSemiring","toSemiring"],["CommRingCat","Hom","hom"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","hom_ext"],["congrArg"],["CommRingCat","instCategory"],["Semiring","toNonAssocSemiring"],["CommRingCat"],["CategoryTheory","CategoryStruct","comp"],["IsLocalization","ringHom_ext"],["id"],["CommRingCat","of"],["RingHom","comp"],["Eq"],["CommRingCat","commRing"],["CommRingCat","ofHom"],["CommRingCat","carrier"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["isLocalHom_of_isIso"],"typeFallback":"forall {R : CommRingCat.{u_1}} {S : CommRingCat.{u_1}} (f : Quiver.Hom.{u_1, succ u_1} CommRingCat.{u_1} (CategoryTheory.CategoryStruct.toQuiver.{u_1, succ u_1} CommRingCat.{u_1} (CategoryTheory.Category.toCategoryStruct.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1})) R S) [inst._@.Mathlib.Algebra.Category.Ring.Instances.786933521._hygCtx._hyg.9 : CategoryTheory.IsIso.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} R S f], IsLocalHom.{u_1, u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (RingHom.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))) (MonoidWithZero.toMonoid.{u_1} (CommRingCat.carrier.{u_1} S) (Semiring.toMonoidWithZero.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (RingHom.instFunLike.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.carrier.{u_1} S) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))) (Semiring.toNonAssocSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommSemiring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} S) (CommRingCat.commRing.{u_1} S))))) (CommRingCat.Hom.hom.{u_1} R S f)","typeFull":"∀ {R S : CommRingCat} (f : R ⟶ S) [CategoryTheory.IsIso f], IsLocalHom (CommRingCat.Hom.hom f)","typeReadable":"∀ {R S : CommRingCat} (f : R ⟶ S) [CategoryTheory.IsIso f], IsLocalHom (CommRingCat.Hom.hom f)","typeReferences":[["RingHom"],["CommRing","toCommSemiring"],["CommRingCat","Hom","hom"],["IsLocalHom"],["CommSemiring","toSemiring"],["RingHom","instFunLike"],["Semiring","toMonoidWithZero"],["CategoryTheory","IsIso"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","instCategory"],["CategoryTheory","CategoryStruct","toQuiver"],["Semiring","toNonAssocSemiring"],["CommRingCat"],["Quiver","Hom"],["MonoidWithZero","toMonoid"],["CommRingCat","commRing"],["CommRingCat","carrier"]],"valueReferences":[["CommRingCat","instCategory"],["CommRingCat"],["isLocalHom_of_iso"],["CategoryTheory","asIso"]]},{"isProp":true,"kind":"theorem","name":["localization_unit_isIso'"],"typeFallback":"forall (R : CommRingCat.{u_1}), CategoryTheory.IsIso.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} R (CommRingCat.of.{u_1} (Localization.Away.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.instCommRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))))))) (CommRingCat.ofHom.{u_1} (CommRingCat.carrier.{u_1} R) (Localization.Away.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (CommRingCat.commRing.{u_1} R) (OreLocalization.instCommRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))))) (algebraMap.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (Localization.Away.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (OreLocalization.instSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))))) (OreLocalization.instAlgebra.{u_1, u_1} (CommRingCat.carrier.{u_1} R) (Ring.toSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)))))))) (OreLocalization.oreSetComm.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Submonoid.powers.{u_1} (CommRingCat.carrier.{u_1} R) (CommMonoid.toMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommMonoid.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))) (OfNat.ofNat.{u_1} (CommRingCat.carrier.{u_1} R) 1 (One.toOfNat1.{u_1} (CommRingCat.carrier.{u_1} R) (AddMonoidWithOne.toOne.{u_1} (CommRingCat.carrier.{u_1} R) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (Ring.toAddGroupWithOne.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toRing.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))))) (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R)) (Algebra.id.{u_1} (CommRingCat.carrier.{u_1} R) (CommRing.toCommSemiring.{u_1} (CommRingCat.carrier.{u_1} R) (CommRingCat.commRing.{u_1} R))))))","typeFull":"∀ (R : CommRingCat), CategoryTheory.IsIso (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away 1)))","typeReadable":"∀ (R : CommRingCat), CategoryTheory.IsIso (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away 1)))","typeReferences":[["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["AddGroupWithOne","toAddMonoidWithOne"],["CategoryTheory","IsIso"],["Algebra","id"],["OreLocalization","instCommRing"],["Ring","toAddGroupWithOne"],["CommRingCat"],["OreLocalization","instAlgebra"],["CommRingCat","commRing"],["CommRingCat","carrier"],["CommRing","toCommSemiring"],["OfNat","ofNat"],["Localization","Away"],["CommRing","toCommMonoid"],["Ring","toSemiring"],["CommRingCat","instCategory"],["CommRing","toRing"],["One","toOfNat1"],["Submonoid","powers"],["AddMonoidWithOne","toOne"],["OreLocalization","instSemiring"],["CommRingCat","of"],["CommRingCat","ofHom"],["algebraMap"]],"valueReferences":[["CommMonoid","toMonoid"],["OreLocalization","oreSetComm"],["localization_unit_isIso"],["CategoryTheory","IsIso"],["AddGroupWithOne","toAddMonoidWithOne"],["Algebra","id"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["CommRingCat"],["Ring","toAddGroupWithOne"],["Eq","symm"],["Eq","ndrec"],["CommRingCat","commRing"],["Eq"],["CommRingCat","carrier"],["CommRing","toCommSemiring"],["CommRingCat","casesOn"],["OfNat","ofNat"],["Localization","Away"],["CommRing","toCommMonoid"],["Ring","toSemiring"],["CommRingCat","instCategory"],["CommRing","toRing"],["One","toOfNat1"],["Eq","refl"],["Submonoid","powers"],["AddMonoidWithOne","toOne"],["OreLocalization","instSemiring"],["CommRingCat","of"],["CommRingCat","ofHom"],["algebraMap"]]},{"isProp":true,"kind":"theorem","name":["Localization","epi"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3 : CommRing.{u_1} R] (M : Submonoid.{u_1} R (MulZeroOneClass.toMulOneClass.{u_1} R (NonAssocSemiring.toMulZeroOneClass.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3)))))), CategoryTheory.Epi.{u_1, succ u_1} CommRingCat.{u_1} CommRingCat.instCategory.{u_1} (CommRingCat.of.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) (CommRingCat.of.{u_1} (Localization.{u_1} R (CommRing.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) M) (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3 M (OreLocalization.oreSetComm.{u_1} R (CommRing.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) M))) (CommRingCat.ofHom.{u_1} R (Localization.{u_1} R (CommRing.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) M) inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3 (OreLocalization.instCommRing.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3 M (OreLocalization.oreSetComm.{u_1} R (CommRing.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) M)) (algebraMap.{u_1, u_1} R (Localization.{u_1} R (CommRing.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) M) (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) (OreLocalization.instSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3)) M (OreLocalization.oreSetComm.{u_1} R (CommRing.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) M)) (OreLocalization.instAlgebra.{u_1, u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3)) M (OreLocalization.oreSetComm.{u_1} R (CommRing.toCommMonoid.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) M) R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3) (Algebra.id.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.Category.Ring.Instances.3351411404._hygCtx._hyg.3)))))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (M : Submonoid R),\n  CategoryTheory.Epi (CommRingCat.ofHom (algebraMap R (Localization M)))","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (M : Submonoid R),\n  CategoryTheory.Epi (CommRingCat.ofHom (algebraMap R (Localization M)))","typeReferences":[["CommRing","toCommSemiring"],["Localization"],["OreLocalization","oreSetComm"],["CommSemiring","toSemiring"],["MulZeroOneClass","toMulOneClass"],["CommRing"],["Algebra","id"],["CommRing","toCommMonoid"],["Submonoid"],["CommRingCat","instCategory"],["OreLocalization","instCommRing"],["Semiring","toNonAssocSemiring"],["OreLocalization","instAlgebra"],["CommRingCat"],["OreLocalization","instSemiring"],["CategoryTheory","Epi"],["CommRingCat","of"],["CommRingCat","ofHom"],["NonAssocSemiring","toMulZeroOneClass"],["algebraMap"]],"valueReferences":[["CommRing","toCommSemiring"],["Localization"],["OreLocalization","instCommRing"],["OreLocalization","instAlgebra"],["OreLocalization","oreSetComm"],["CommSemiring","toSemiring"],["Localization","isLocalization"],["IsLocalization","epi"],["Algebra","id"],["CommRing","toCommMonoid"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Field.Rat.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["NNRat","coe_zpow"],"typeFallback":"forall (p : NNRat) (n : Int), Eq.{1} Rat (NNRat.cast.{0} Rat Rat.instNNRatCast (HPow.hPow.{0, 0, 0} NNRat Int NNRat (instHPow.{0, 0} NNRat Int NNRat.instZPow) p n)) (HPow.hPow.{0, 0, 0} Rat Int Rat (instHPow.{0, 0} Rat Int Rat.instPowInt) (NNRat.cast.{0} Rat Rat.instNNRatCast p) n)","typeFull":"∀ (p : ℚ≥0) (n : ℤ), ↑(p ^ n) = ↑p ^ n","typeReadable":"∀ (p : ℚ≥0) (n : ℤ), ↑(p ^ n) = ↑p ^ n","typeReferences":[["instHPow"],["Rat","instNNRatCast"],["NNRat"],["NNRat","instZPow"],["Rat"],["Rat","instPowInt"],["NNRat","cast"],["HPow","hPow"],["Eq"],["Int"]],"valueReferences":[["instHPow"],["rfl"],["Rat","instNNRatCast"],["NNRat"],["NNRat","instZPow"],["Rat"],["NNRat","cast"],["HPow","hPow"],["Int"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_5"],"typeFallback":"Eq.{1} NNRat (Inv.inv.{0} NNRat NNRat.instInv (OfNat.ofNat.{0} NNRat 0 (Zero.toOfNat0.{0} NNRat (AddMonoid.toZero.{0} NNRat (AddCommMonoid.toAddMonoid.{0} NNRat (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} NNRat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonUnitalSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))))))) (OfNat.ofNat.{0} NNRat 0 (Zero.toOfNat0.{0} NNRat (AddMonoid.toZero.{0} NNRat (AddCommMonoid.toAddMonoid.{0} NNRat (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} NNRat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonUnitalSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))))))","typeFull":"0⁻¹ = 0","typeReadable":"0⁻¹ = 0","typeReferences":[["AddMonoid","toZero"],["Inv","inv"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["AddCommMonoid","toAddMonoid"],["OfNat","ofNat"],["NNRat"],["instCommSemiringNNRat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Zero","toOfNat0"],["NNRat","instInv"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["NNRat","ext"],["inv_zero"],["AddMonoid","toZero"],["Rat","instNNRatCast"],["Eq","trans"],["MonoidWithZero","toMulZeroOneClass"],["AddCommMonoid","toAddMonoid"],["Rat","instOfNat"],["congrArg"],["NNRat"],["GroupWithZero","toMonoidWithZero"],["NNRat","cast"],["congrFun'"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toNonUnitalSemiring"],["Rat","instField"],["Inv","inv"],["True"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DivisionSemiring","toGroupWithZero"],["OfNat","ofNat"],["eq_self"],["instCommSemiringNNRat"],["of_eq_true"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Field","toSemifield"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Rat"],["NNRat","instInv"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_7"],"typeFallback":"forall (q : NNRat), Eq.{1} NNRat q (HDiv.hDiv.{0, 0, 0} NNRat NNRat NNRat (instHDiv.{0} NNRat NNRat.instDiv) (Nat.cast.{0} NNRat (AddMonoidWithOne.toNatCast.{0} NNRat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNRat (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))) (NNRat.num q)) (Nat.cast.{0} NNRat (AddMonoidWithOne.toNatCast.{0} NNRat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNRat (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))) (NNRat.den q)))","typeFull":"∀ (q : ℚ≥0), q = ↑q.num / ↑q.den","typeReadable":"∀ (q : ℚ≥0), q = ↑q.num / ↑q.den","typeReferences":[["NNRat","den"],["Nat","cast"],["CommSemiring","toSemiring"],["instHDiv"],["HDiv","hDiv"],["NNRat"],["AddMonoidWithOne","toNatCast"],["Semiring","toNonAssocSemiring"],["instCommSemiringNNRat"],["NNRat","instDiv"],["NNRat","num"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"]],"valueReferences":[["NNRat","den"],["Nat","cast"],["CommSemiring","toSemiring"],["instHDiv"],["HDiv","hDiv"],["NNRat"],["AddMonoidWithOne","toNatCast"],["Semiring","toNonAssocSemiring"],["instCommSemiringNNRat"],["NNRat","instDiv"],["NNRat","num_div_den"],["NNRat","num"],["Eq","symm"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instInv","_proof_1"],"typeFallback":"forall (x : NNRat), LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) (Inv.inv.{0} Rat Rat.instInv (NNRat.cast.{0} Rat Rat.instNNRatCast x))","typeFull":"∀ (x : ℚ≥0), 0 ≤ (↑x)⁻¹","typeReadable":"∀ (x : ℚ≥0), 0 ≤ (↑x)⁻¹","typeReferences":[["Rat","instLE"],["Rat","instNNRatCast"],["Inv","inv"],["NNRat"],["LE","le"],["Rat"],["NNRat","cast"],["Rat","instInv"],["Rat","instOfNat"],["OfNat","ofNat"]],"valueReferences":[["Rat","instLE"],["Rat","instNNRatCast"],["Rat","inv_nonneg"],["LE","le"],["Subtype","property"],["Rat"],["NNRat","cast"],["Rat","instOfNat"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["NNRat","coe_inv","_simp_1"],"typeFallback":"forall (q : NNRat), Eq.{1} Rat (Inv.inv.{0} Rat Rat.instInv (NNRat.cast.{0} Rat Rat.instNNRatCast q)) (NNRat.cast.{0} Rat Rat.instNNRatCast (Inv.inv.{0} NNRat NNRat.instInv q))","typeFull":"∀ (q : ℚ≥0), (↑q)⁻¹ = ↑q⁻¹","typeReadable":"∀ (q : ℚ≥0), (↑q)⁻¹ = ↑q⁻¹","typeReferences":[["Rat","instNNRatCast"],["Inv","inv"],["NNRat"],["Rat"],["NNRat","cast"],["Rat","instInv"],["NNRat","instInv"],["Eq"]],"valueReferences":[["Rat","instNNRatCast"],["Inv","inv"],["NNRat"],["Rat"],["Eq","symm"],["Rat","instInv"],["NNRat","cast"],["NNRat","instInv"],["NNRat","coe_inv"]]},{"isProp":false,"kind":"definition","name":["Rat","instField"],"typeFallback":"Field.{0} Rat","typeFull":"Field ℚ","typeReadable":"Field ℚ","typeReferences":[["Field"],["Rat"]],"valueReferences":[["Rat","instNNRatCast"],["NonUnitalNonAssocSemiring","toMul"],["Field","mk"],["CommGroupWithZero","inv_zero"],["CommGroupWithZero","toNontrivial"],["CommGroupWithZero","mul_inv_cancel"],["HMul","hMul"],["CommGroupWithZero","zpow_neg'"],["Rat","instField","_proof_3"],["CommGroupWithZero","toInv"],["Rat","commGroupWithZero"],["Rat","cast"],["NNRat","cast"],["instRatCastRat"],["Rat","instField","_proof_4"],["CommGroupWithZero","zpow_zero'"],["Rat","instField","_proof_1"],["Semiring","toNonUnitalSemiring"],["Rat","commRing"],["CommGroupWithZero","zpow_succ'"],["Rat","instField","_proof_2"],["CommGroupWithZero","toDiv"],["CommGroupWithZero","zpow"],["Ring","toSemiring"],["CommRing","toRing"],["Rat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["CommGroupWithZero","div_eq_mul_inv"]]},{"isProp":true,"kind":"theorem","name":["Rat","instField","_proof_1"],"typeFallback":"forall (q : NNRat), Eq.{1} Rat (NNRat.cast.{0} Rat Rat.instNNRatCast q) (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat (CommGroupWithZero.toDiv.{0} Rat Rat.commGroupWithZero)) (Nat.cast.{0} Rat (Semiring.toNatCast.{0} Rat (Ring.toSemiring.{0} Rat (CommRing.toRing.{0} Rat Rat.commRing))) (NNRat.num q)) (Nat.cast.{0} Rat (Semiring.toNatCast.{0} Rat (Ring.toSemiring.{0} Rat (CommRing.toRing.{0} Rat Rat.commRing))) (NNRat.den q)))","typeFull":"∀ (q : ℚ≥0), ↑q = ↑q.num / ↑q.den","typeReadable":"∀ (q : ℚ≥0), ↑q = ↑q.num / ↑q.den","typeReferences":[["Rat","instNNRatCast"],["NNRat","den"],["Nat","cast"],["CommGroupWithZero","toDiv"],["instHDiv"],["Ring","toSemiring"],["HDiv","hDiv"],["CommRing","toRing"],["NNRat"],["Rat","commGroupWithZero"],["Semiring","toNatCast"],["NNRat","num"],["Rat"],["NNRat","cast"],["Eq"],["Rat","commRing"]],"valueReferences":[["Rat","instNNRatCast"],["Nat","cast"],["NNRat","den"],["AddGroupWithOne","toAddMonoidWithOne"],["instHDiv"],["NNRat","den_coe"],["Int","cast"],["Rat","num"],["congrArg"],["Rat","instIntCast"],["HDiv","hDiv"],["Rat","instNatCast"],["Semiring","toNatCast"],["Rat","commGroupWithZero"],["Ring","toAddGroupWithOne"],["NNRat","num"],["Eq","symm"],["NNRat","cast"],["Eq"],["Rat","instDiv"],["NNRat","num_coe"],["instNatCastInt"],["Rat","commRing"],["Rat","den"],["CommGroupWithZero","toDiv"],["Int"],["Ring","toSemiring"],["CommRing","toRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["Rat","num_div_den"],["Rat"],["id"],["AddGroupWithOne","toIntCast"],["Eq","mpr"],["Int","cast_natCast"]]},{"isProp":false,"kind":"definition","name":["NNRat","instSemifield"],"typeFallback":"Semifield.{0} NNRat","typeFull":"Semifield ℚ≥0","typeReadable":"Semifield ℚ≥0","typeReferences":[["NNRat"],["Semifield"]],"valueReferences":[["HMul","hMul"],["NNRat","instNontrivial"],["NNRat","instNNRatCast"],["NNRat","instSemifield","_proof_4"],["NNRat"],["Semiring","toNonAssocSemiring"],["NNRat","instSemifield","_proof_7"],["Semifield","mk"],["NNRat","instSemifield","_proof_8"],["instHPow"],["NNRat","instSemifield","_proof_5"],["NonUnitalNonAssocSemiring","toDistrib"],["NNRat","instSemifield","_proof_1"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["NNRat","instSemifield","_proof_3"],["HPow","hPow"],["Int"],["instCommSemiringNNRat"],["NNRat","instDiv"],["NNRat","instZPow"],["NNRat","instSemifield","_proof_2"],["instHMul"],["NNRat","instSemifield","_proof_6"],["NNRat","instInv"]]},{"isProp":true,"kind":"theorem","name":["Rat","instField","_proof_4"],"typeFallback":"forall (x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.44 : Rat) (x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.46 : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (NonUnitalNonAssocSemiring.toMul.{0} Rat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} Rat (Semiring.toNonUnitalSemiring.{0} Rat (Ring.toSemiring.{0} Rat (CommRing.toRing.{0} Rat Rat.commRing)))))) (Rat.cast.{0} Rat instRatCastRat x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.44) x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.46) (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (NonUnitalNonAssocSemiring.toMul.{0} Rat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} Rat (Semiring.toNonUnitalSemiring.{0} Rat (Ring.toSemiring.{0} Rat (CommRing.toRing.{0} Rat Rat.commRing)))))) (Rat.cast.{0} Rat instRatCastRat x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.44) x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.46)","typeFull":"∀ (x x_1 : ℚ), ↑x * x_1 = ↑x * x_1","typeReadable":"∀ (x x_1 : ℚ), ↑x * x_1 = ↑x * x_1","typeReferences":[["CommRing","toRing"],["NonUnitalNonAssocSemiring","toMul"],["Rat","cast"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Rat"],["instHMul"],["HMul","hMul"],["instRatCastRat"],["Eq"],["Semiring","toNonUnitalSemiring"],["Rat","commRing"],["Ring","toSemiring"]],"valueReferences":[["rfl"],["CommRing","toRing"],["NonUnitalNonAssocSemiring","toMul"],["Rat","cast"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Rat"],["instHMul"],["HMul","hMul"],["instRatCastRat"],["Semiring","toNonUnitalSemiring"],["Rat","commRing"],["Ring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["NNRat","coe_div","_simp_1"],"typeFallback":"forall (p : NNRat) (q : NNRat), Eq.{1} Rat (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) (NNRat.cast.{0} Rat Rat.instNNRatCast p) (NNRat.cast.{0} Rat Rat.instNNRatCast q)) (NNRat.cast.{0} Rat Rat.instNNRatCast (HDiv.hDiv.{0, 0, 0} NNRat NNRat NNRat (instHDiv.{0} NNRat NNRat.instDiv) p q))","typeFull":"∀ (p q : ℚ≥0), ↑p / ↑q = ↑(p / q)","typeReadable":"∀ (p q : ℚ≥0), ↑p / ↑q = ↑(p / q)","typeReferences":[["HDiv","hDiv"],["Rat","instNNRatCast"],["NNRat"],["NNRat","instDiv"],["Rat"],["NNRat","cast"],["Eq"],["instHDiv"],["Rat","instDiv"]],"valueReferences":[["HDiv","hDiv"],["Rat","instNNRatCast"],["NNRat"],["NNRat","instDiv"],["Rat"],["Eq","symm"],["NNRat","cast"],["instHDiv"],["NNRat","coe_div"],["Rat","instDiv"]]},{"isProp":true,"kind":"theorem","name":["NNRat","div_def"],"typeFallback":"forall (p : NNRat) (q : NNRat), Eq.{1} NNRat (HDiv.hDiv.{0, 0, 0} NNRat NNRat NNRat (instHDiv.{0} NNRat NNRat.instDiv) p q) (NNRat.divNat (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) (NNRat.num p) (NNRat.den q)) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) (NNRat.den p) (NNRat.num q)))","typeFull":"∀ (p q : ℚ≥0), p / q = NNRat.divNat (p.num * q.den) (p.den * q.num)","typeReadable":"∀ (p q : ℚ≥0), p / q = NNRat.divNat (p.num * q.den) (p.den * q.num)","typeReferences":[["HDiv","hDiv"],["Nat"],["NNRat","den"],["NNRat"],["NNRat","divNat"],["NNRat","instDiv"],["NNRat","num"],["instMulNat"],["instHMul"],["HMul","hMul"],["Eq"],["instHDiv"]],"valueReferences":[["NNRat","ext"],["Rat","instNNRatCast"],["NNRat","den"],["Nat","cast"],["Eq","trans"],["Rat","divInt"],["HMul","hMul"],["instHDiv"],["Rat","num"],["congrArg"],["Int","instMul"],["HDiv","hDiv"],["Semiring","toNonAssocSemiring"],["NNRat"],["congr"],["NNRat","num"],["Rat","div_def'"],["instMulNat"],["NNRat","cast"],["Nat","cast_mul"],["congrFun'"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NNRat","num_coe"],["Rat","den"],["Rat","instDiv"],["instNatCastInt"],["True"],["NonUnitalNonAssocSemiring","toDistrib"],["NNRat","divNat"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["Int"],["eq_self"],["Nat"],["AddMonoidWithOne","toNatCast"],["of_eq_true"],["Int","instSemiring"],["NNRat","instDiv"],["Rat"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["NNRat","coe_zpow","_simp_1"],"typeFallback":"forall (p : NNRat) (n : Int), Eq.{1} Rat (HPow.hPow.{0, 0, 0} Rat Int Rat (instHPow.{0, 0} Rat Int Rat.instPowInt) (NNRat.cast.{0} Rat Rat.instNNRatCast p) n) (NNRat.cast.{0} Rat Rat.instNNRatCast (HPow.hPow.{0, 0, 0} NNRat Int NNRat (instHPow.{0, 0} NNRat Int NNRat.instZPow) p n))","typeFull":"∀ (p : ℚ≥0) (n : ℤ), ↑p ^ n = ↑(p ^ n)","typeReadable":"∀ (p : ℚ≥0) (n : ℤ), ↑p ^ n = ↑(p ^ n)","typeReferences":[["instHPow"],["Rat","instNNRatCast"],["NNRat"],["NNRat","instZPow"],["Rat"],["NNRat","cast"],["Rat","instPowInt"],["HPow","hPow"],["Eq"],["Int"]],"valueReferences":[["instHPow"],["Rat","instNNRatCast"],["NNRat","coe_zpow"],["NNRat"],["NNRat","instZPow"],["Rat"],["Eq","symm"],["Rat","instPowInt"],["NNRat","cast"],["HPow","hPow"],["Int"]]},{"isProp":true,"kind":"theorem","name":["NNRat","coe_div"],"typeFallback":"forall (p : NNRat) (q : NNRat), Eq.{1} Rat (NNRat.cast.{0} Rat Rat.instNNRatCast (HDiv.hDiv.{0, 0, 0} NNRat NNRat NNRat (instHDiv.{0} NNRat NNRat.instDiv) p q)) (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) (NNRat.cast.{0} Rat Rat.instNNRatCast p) (NNRat.cast.{0} Rat Rat.instNNRatCast q))","typeFull":"∀ (p q : ℚ≥0), ↑(p / q) = ↑p / ↑q","typeReadable":"∀ (p q : ℚ≥0), ↑(p / q) = ↑p / ↑q","typeReferences":[["HDiv","hDiv"],["Rat","instNNRatCast"],["NNRat"],["NNRat","instDiv"],["Rat"],["NNRat","cast"],["Eq"],["instHDiv"],["Rat","instDiv"]],"valueReferences":[["HDiv","hDiv"],["rfl"],["Rat","instNNRatCast"],["NNRat"],["NNRat","instDiv"],["Rat"],["NNRat","cast"],["instHDiv"]]},{"isProp":true,"kind":"theorem","name":["NNRat","num_inv_of_ne_zero"],"typeFallback":"forall {q : NNRat}, (Ne.{1} NNRat q (OfNat.ofNat.{0} NNRat 0 (Zero.toOfNat0.{0} NNRat (MulZeroClass.toZero.{0} NNRat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNRat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat)))))))) -> (Eq.{1} Nat (NNRat.num (Inv.inv.{0} NNRat NNRat.instInv q)) (NNRat.den q))","typeFull":"∀ {q : ℚ≥0}, q ≠ 0 → q⁻¹.num = q.den","typeReadable":"∀ {q : ℚ≥0}, q ≠ 0 → q⁻¹.num = q.den","typeReferences":[["Inv","inv"],["NNRat","den"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["OfNat","ofNat"],["Nat"],["NNRat"],["Semiring","toNonAssocSemiring"],["instCommSemiringNNRat"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NNRat","num"],["Ne"],["Zero","toOfNat0"],["NNRat","instInv"],["Eq"]],"valueReferences":[["NNRat","den"],["Rat","divInt"],["Rat","instLE"],["Semiring","toNonAssocSemiring"],["NNRat","num"],["Eq","symm"],["NNRat","cast"],["NNRat","coprime_num_den"],["NNRat","divNat"],["NNRat","num","eq_1"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Rat","divInt_ofNat"],["Nat"],["NNRat","inv_def"],["NNRat","coe_mk"],["Eq","refl"],["Iff","mpr"],["Rat"],["id"],["Rat","mk_eq_mkRat"],["Eq","mpr"],["Rat","mk'"],["Rat","instNNRatCast"],["Nat","cast"],["NNRat","divNat","_proof_1"],["Rat","instOfNat"],["Rat","num"],["congrArg"],["mkRat"],["Nat","Coprime"],["NNRat"],["NNRat","divNat","eq_1"],["instOfNatNat"],["Int","natAbs_natCast"],["Int","natAbs"],["Zero","toOfNat0"],["Eq"],["Nat","Coprime","symm"],["instNatCastInt"],["Inv","inv"],["CommSemiring","toSemiring"],["NNRat","num_ne_zero"],["OfNat","ofNat"],["Int"],["instCommSemiringNNRat"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["NNRat","instInv"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_1"],"typeFallback":"forall (a : NNRat) (b : NNRat), Eq.{1} NNRat (HDiv.hDiv.{0, 0, 0} NNRat NNRat NNRat (instHDiv.{0} NNRat NNRat.instDiv) a b) (HMul.hMul.{0, 0, 0} NNRat NNRat NNRat (instHMul.{0} NNRat (NonUnitalNonAssocSemiring.toMul.{0} NNRat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonUnitalSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))) a (Inv.inv.{0} NNRat NNRat.instInv b))","typeFull":"∀ (a b : ℚ≥0), a / b = a * b⁻¹","typeReadable":"∀ (a b : ℚ≥0), a / b = a * b⁻¹","typeReferences":[["Inv","inv"],["NonUnitalNonAssocSemiring","toMul"],["CommSemiring","toSemiring"],["HMul","hMul"],["instHDiv"],["HDiv","hDiv"],["NNRat"],["instCommSemiringNNRat"],["NNRat","instDiv"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["NNRat","instInv"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["HDiv","hDiv"],["NNRat"],["NNRat","instDiv"],["Eq","refl"],["instHDiv"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_6"],"typeFallback":"forall (q : NNRat), (Ne.{1} NNRat q (OfNat.ofNat.{0} NNRat 0 (Zero.toOfNat0.{0} NNRat (AddMonoid.toZero.{0} NNRat (AddCommMonoid.toAddMonoid.{0} NNRat (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} NNRat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonUnitalSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))))))) -> (Eq.{1} NNRat (HMul.hMul.{0, 0, 0} NNRat NNRat NNRat (instHMul.{0} NNRat (NonUnitalNonAssocSemiring.toMul.{0} NNRat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonUnitalSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))) q (Inv.inv.{0} NNRat NNRat.instInv q)) (OfNat.ofNat.{0} NNRat 1 (One.toOfNat1.{0} NNRat (Semiring.toOne.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat)))))","typeFull":"∀ (q : ℚ≥0), q ≠ 0 → q * q⁻¹ = 1","typeReadable":"∀ (q : ℚ≥0), q ≠ 0 → q * q⁻¹ = 1","typeReferences":[["AddMonoid","toZero"],["Inv","inv"],["Semiring","toOne"],["NonUnitalNonAssocSemiring","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["HMul","hMul"],["AddCommMonoid","toAddMonoid"],["OfNat","ofNat"],["NNRat"],["One","toOfNat1"],["instCommSemiringNNRat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["Ne"],["Zero","toOfNat0"],["NNRat","instInv"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["mul_inv_cancel₀"],["AddMonoid","toZero"],["Eq","trans"],["NonUnitalNonAssocSemiring","toMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["Semiring","toNonAssocSemiring"],["not_false_eq_true"],["NNRat","cast"],["Semifield","toDivisionSemiring"],["Semiring","toNonUnitalSemiring"],["NNRat","coe_eq_zero","_simp_1"],["MulOne","toOne"],["Semiring","toOne"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DivisionSemiring","toGroupWithZero"],["MulZeroOneClass","toMulZeroClass"],["eq_false"],["Rat"],["instHMul"],["MulOneClass","toMulOne"],["NNRat","ext"],["Rat","instNNRatCast"],["MulZeroOneClass","toMulOneClass"],["AddCommMonoid","toAddMonoid"],["Rat","instOfNat"],["congrArg"],["NNRat"],["GroupWithZero","toMonoidWithZero"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["Rat","instField"],["Not"],["Inv","inv"],["True"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["OfNat","ofNat"],["eq_self"],["of_eq_true"],["instCommSemiringNNRat"],["One","toOfNat1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Field","toSemifield"],["False"],["NNRat","instInv"]]},{"isProp":true,"kind":"theorem","name":["NNRat","num_div_den"],"typeFallback":"forall (q : NNRat), Eq.{1} NNRat (HDiv.hDiv.{0, 0, 0} NNRat NNRat NNRat (instHDiv.{0} NNRat NNRat.instDiv) (Nat.cast.{0} NNRat (AddMonoidWithOne.toNatCast.{0} NNRat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNRat (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))) (NNRat.num q)) (Nat.cast.{0} NNRat (AddMonoidWithOne.toNatCast.{0} NNRat (AddCommMonoidWithOne.toAddMonoidWithOne.{0} NNRat (NonAssocSemiring.toAddCommMonoidWithOne.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))) (NNRat.den q))) q","typeFull":"∀ (q : ℚ≥0), ↑q.num / ↑q.den = q","typeReadable":"∀ (q : ℚ≥0), ↑q.num / ↑q.den = q","typeReferences":[["NNRat","den"],["Nat","cast"],["CommSemiring","toSemiring"],["instHDiv"],["HDiv","hDiv"],["NNRat"],["AddMonoidWithOne","toNatCast"],["Semiring","toNonAssocSemiring"],["instCommSemiringNNRat"],["NNRat","instDiv"],["NNRat","num"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"]],"valueReferences":[["NNRat","ext"],["Rat","instNNRatCast"],["DivisionSemiring","toNNRatCast"],["Nat","cast"],["NNRat","den"],["GroupWithZero","toDivInvMonoid"],["AddGroupWithOne","toAddMonoidWithOne"],["Int","cast"],["instHDiv"],["Rat","num"],["congrArg"],["HDiv","hDiv"],["Semiring","toNonAssocSemiring"],["NNRat"],["Rat","instNatCast"],["Ring","toAddGroupWithOne"],["NNRat","num"],["NNRat","coe_natCast"],["Eq","symm"],["NNRat","cast"],["Int","natAbs"],["Semifield","toDivisionSemiring"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["instNatCastInt"],["Rat","instDiv"],["Rat","instField"],["NNRat","num","eq_1"],["DivisionRing","toRing"],["CommSemiring","toSemiring"],["NNRat","cast_def"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["DivInvMonoid","toDiv"],["Int"],["Nat"],["AddMonoidWithOne","toNatCast"],["instCommSemiringNNRat"],["NNRat","instDiv"],["Field","toSemifield"],["AddGroupWithOne","toIntCast"],["Rat"],["id"],["Eq","mpr"],["Int","cast_natCast"],["NNRat","coe_div"],["Rat","instDivisionRing"]]},{"isProp":true,"kind":"theorem","name":["Rat","div_nonneg"],"typeFallback":"forall {a : Rat} {b : Rat}, (LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) a) -> (LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) b) -> (LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) a b))","typeFull":"∀ {a b : ℚ}, 0 ≤ a → 0 ≤ b → 0 ≤ a / b","typeReadable":"∀ {a b : ℚ}, 0 ≤ a → 0 ≤ b → 0 ≤ a / b","typeReferences":[["HDiv","hDiv"],["Rat","instLE"],["LE","le"],["Rat"],["instHDiv"],["Rat","instOfNat"],["OfNat","ofNat"],["Rat","instDiv"]],"valueReferences":[["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Rat","instPreorder"],["Rat","inv_nonneg"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["mul_nonneg"],["Rat"],["Rat","instPosMulMono"],["CommRing","toNonUnitalCommRing"],["Rat","inv"],["Rat","commRing"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_8"],"typeFallback":"forall (q : NNRat) (a : NNRat), Eq.{1} NNRat (HMul.hMul.{0, 0, 0} NNRat NNRat NNRat (instHMul.{0} NNRat (Distrib.toMul.{0} NNRat (NonUnitalNonAssocSemiring.toDistrib.{0} NNRat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat)))))) q a) (HMul.hMul.{0, 0, 0} NNRat NNRat NNRat (instHMul.{0} NNRat (Distrib.toMul.{0} NNRat (NonUnitalNonAssocSemiring.toDistrib.{0} NNRat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat)))))) q a)","typeFull":"∀ (q a : ℚ≥0), q * a = q * a","typeReadable":"∀ (q a : ℚ≥0), q * a = q * a","typeReferences":[["Semiring","toNonAssocSemiring"],["NNRat"],["instCommSemiringNNRat"],["NonUnitalNonAssocSemiring","toDistrib"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["instHMul"],["HMul","hMul"],["Eq"]],"valueReferences":[["rfl"],["Semiring","toNonAssocSemiring"],["NNRat"],["instCommSemiringNNRat"],["NonUnitalNonAssocSemiring","toDistrib"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["instHMul"],["HMul","hMul"]]},{"isProp":false,"kind":"definition","name":["NNRat","instZPow"],"typeFallback":"Pow.{0, 0} NNRat Int","typeFull":"Pow ℚ≥0 ℤ","typeReadable":"Pow ℚ≥0 ℤ","typeReferences":[["NNRat"],["Pow"],["Int"]],"valueReferences":[["instHPow"],["Rat","instNNRatCast"],["HPow","hPow"],["OfNat","ofNat"],["Rat","instOfNat"],["Int"],["Rat","instLE"],["NNRat"],["NNRat","instZPow","_proof_1"],["LE","le"],["Rat"],["Rat","instPowInt"],["NNRat","cast"],["Subtype","mk"],["Pow","mk"]]},{"isProp":true,"kind":"theorem","name":["Rat","instField","_proof_3"],"typeFallback":"forall (x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.34 : Rat), Eq.{1} Rat (Rat.cast.{0} Rat instRatCastRat x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.34) (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) (Int.cast.{0} Rat Rat.instIntCast (Rat.num (Rat.cast.{0} Rat instRatCastRat x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.34))) (Nat.cast.{0} Rat Rat.instNatCast (Rat.den (Rat.cast.{0} Rat instRatCastRat x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.34))))","typeFull":"∀ (x : ℚ), ↑x = ↑(↑x).num / ↑(↑x).den","typeReadable":"∀ (x : ℚ), ↑x = ↑(↑x).num / ↑(↑x).den","typeReferences":[["Nat","cast"],["instHDiv"],["Int","cast"],["Rat","num"],["HDiv","hDiv"],["Rat","instIntCast"],["Rat","instNatCast"],["Rat","cast"],["Rat"],["instRatCastRat"],["Eq"],["Rat","instDiv"],["Rat","den"]],"valueReferences":[["Nat","cast"],["instHDiv"],["Int","cast"],["Rat","num"],["HDiv","hDiv"],["Rat","instIntCast"],["Rat","instNatCast"],["Rat","num_div_den"],["Rat","cast"],["Eq","symm"],["Rat"],["instRatCastRat"],["Rat","instDiv"],["Rat","den"]]},{"isProp":true,"kind":"theorem","name":["NNRat","inv_def"],"typeFallback":"forall (q : NNRat), Eq.{1} NNRat (Inv.inv.{0} NNRat NNRat.instInv q) (NNRat.divNat (NNRat.den q) (NNRat.num q))","typeFull":"∀ (q : ℚ≥0), q⁻¹ = NNRat.divNat q.den q.num","typeReadable":"∀ (q : ℚ≥0), q⁻¹ = NNRat.divNat q.den q.num","typeReferences":[["Inv","inv"],["NNRat","den"],["NNRat"],["NNRat","divNat"],["NNRat","num"],["NNRat","instInv"],["Eq"]],"valueReferences":[["NNRat","ext"],["Rat","instNNRatCast"],["NNRat","den"],["Nat","cast"],["Eq","trans"],["Rat","divInt"],["Rat","num"],["congrArg"],["mkRat"],["NNRat"],["congr"],["NNRat","num"],["NNRat","cast"],["Rat","instInv"],["Eq"],["NNRat","num_coe"],["instNatCastInt"],["Rat","den"],["Inv","inv"],["True"],["NNRat","divNat"],["Rat","divInt_ofNat"],["Int"],["eq_self"],["Rat","inv_def"],["of_eq_true"],["Rat"],["NNRat","instInv"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_4"],"typeFallback":"forall (n : Nat) (a : NNRat), Eq.{1} NNRat (HPow.hPow.{0, 0, 0} NNRat Int NNRat (instHPow.{0, 0} NNRat Int NNRat.instZPow) a (Int.negSucc n)) (Inv.inv.{0} NNRat NNRat.instInv (HPow.hPow.{0, 0, 0} NNRat Int NNRat (instHPow.{0, 0} NNRat Int NNRat.instZPow) a (Nat.cast.{0} Int instNatCastInt (Nat.succ n))))","typeFull":"∀ (n : ℕ) (a : ℚ≥0), a ^ Int.negSucc n = (a ^ ↑n.succ)⁻¹","typeReadable":"∀ (n : ℕ) (a : ℚ≥0), a ^ Int.negSucc n = (a ^ ↑n.succ)⁻¹","typeReferences":[["instHPow"],["Inv","inv"],["Nat","cast"],["Int","negSucc"],["HPow","hPow"],["Int"],["Nat"],["NNRat"],["Nat","succ"],["NNRat","instZPow"],["NNRat","instInv"],["Eq"],["instNatCastInt"]],"valueReferences":[["Rat","instField"],["NNRat","ext"],["instHPow"],["Rat","instNNRatCast"],["Inv","inv"],["Nat","cast"],["Int","negSucc"],["HPow","hPow"],["Field","zpow_neg'"],["Int"],["NNRat"],["Nat","succ"],["Rat"],["NNRat","instZPow"],["NNRat","cast"],["NNRat","instInv"],["instNatCastInt"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_3"],"typeFallback":"forall (n : Nat) (a : NNRat), Eq.{1} NNRat (HPow.hPow.{0, 0, 0} NNRat Int NNRat (instHPow.{0, 0} NNRat Int NNRat.instZPow) a (Nat.cast.{0} Int instNatCastInt (Nat.succ n))) (HMul.hMul.{0, 0, 0} NNRat NNRat NNRat (instHMul.{0} NNRat (NonUnitalNonAssocSemiring.toMul.{0} NNRat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonUnitalSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))) (HPow.hPow.{0, 0, 0} NNRat Int NNRat (instHPow.{0, 0} NNRat Int NNRat.instZPow) a (Nat.cast.{0} Int instNatCastInt n)) a)","typeFull":"∀ (n : ℕ) (a : ℚ≥0), a ^ ↑n.succ = a ^ ↑n * a","typeReadable":"∀ (n : ℕ) (a : ℚ≥0), a ^ ↑n.succ = a ^ ↑n * a","typeReferences":[["instHPow"],["Nat","cast"],["NonUnitalNonAssocSemiring","toMul"],["CommSemiring","toSemiring"],["HMul","hMul"],["HPow","hPow"],["Int"],["Nat"],["NNRat"],["instCommSemiringNNRat"],["Nat","succ"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NNRat","instZPow"],["instHMul"],["Eq"],["Semiring","toNonUnitalSemiring"],["instNatCastInt"]],"valueReferences":[["Rat","instField"],["NNRat","ext"],["instHPow"],["Rat","instNNRatCast"],["Nat","cast"],["NonUnitalNonAssocSemiring","toMul"],["CommSemiring","toSemiring"],["Field","zpow_succ'"],["HMul","hMul"],["HPow","hPow"],["Int"],["NNRat"],["instCommSemiringNNRat"],["Nat","succ"],["Rat"],["NNRat","instZPow"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NNRat","cast"],["instHMul"],["Semiring","toNonUnitalSemiring"],["instNatCastInt"]]},{"isProp":true,"kind":"theorem","name":["NNRat","coe_inv"],"typeFallback":"forall (q : NNRat), Eq.{1} Rat (NNRat.cast.{0} Rat Rat.instNNRatCast (Inv.inv.{0} NNRat NNRat.instInv q)) (Inv.inv.{0} Rat Rat.instInv (NNRat.cast.{0} Rat Rat.instNNRatCast q))","typeFull":"∀ (q : ℚ≥0), ↑q⁻¹ = (↑q)⁻¹","typeReadable":"∀ (q : ℚ≥0), ↑q⁻¹ = (↑q)⁻¹","typeReferences":[["Rat","instNNRatCast"],["Inv","inv"],["NNRat"],["Rat"],["Rat","instInv"],["NNRat","cast"],["NNRat","instInv"],["Eq"]],"valueReferences":[["rfl"],["Rat","instNNRatCast"],["Inv","inv"],["NNRat"],["Rat"],["NNRat","cast"],["NNRat","instInv"]]},{"isProp":false,"kind":"definition","name":["NNRat","instInv"],"typeFallback":"Inv.{0} NNRat","typeFull":"Inv ℚ≥0","typeReadable":"Inv ℚ≥0","typeReferences":[["NNRat"],["Inv"]],"valueReferences":[["Rat","instNNRatCast"],["Inv","inv"],["Inv","mk"],["OfNat","ofNat"],["Rat","instOfNat"],["NNRat","instInv","_proof_1"],["Rat","instLE"],["NNRat"],["LE","le"],["Rat"],["Rat","instInv"],["NNRat","cast"],["Subtype","mk"]]},{"isProp":false,"kind":"definition","name":["Rat","instDivisionRing"],"typeFallback":"DivisionRing.{0} Rat","typeFull":"DivisionRing ℚ","typeReadable":"DivisionRing ℚ","typeReferences":[["Rat"],["DivisionRing"]],"valueReferences":[["Rat","instField"],["Rat"],["Field","toDivisionRing"],["inferInstance"],["DivisionRing"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instSemifield","_proof_2"],"typeFallback":"forall (a : NNRat), Eq.{1} NNRat (HPow.hPow.{0, 0, 0} NNRat Int NNRat (instHPow.{0, 0} NNRat Int NNRat.instZPow) a (OfNat.ofNat.{0} Int 0 (instOfNat 0))) (OfNat.ofNat.{0} NNRat 1 (One.toOfNat1.{0} NNRat (Semiring.toOne.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat))))","typeFull":"∀ (a : ℚ≥0), a ^ 0 = 1","typeReadable":"∀ (a : ℚ≥0), a ^ 0 = 1","typeReferences":[["instHPow"],["NNRat"],["instCommSemiringNNRat"],["Semiring","toOne"],["One","toOfNat1"],["instOfNat"],["CommSemiring","toSemiring"],["NNRat","instZPow"],["HPow","hPow"],["Eq"],["OfNat","ofNat"],["Int"]],"valueReferences":[["Rat","instField"],["NNRat","ext"],["instHPow"],["Rat","instNNRatCast"],["Semiring","toOne"],["CommSemiring","toSemiring"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["NNRat"],["instOfNat"],["One","toOfNat1"],["instCommSemiringNNRat"],["Field","zpow_zero'"],["Rat"],["NNRat","instZPow"],["NNRat","cast"]]},{"isProp":true,"kind":"theorem","name":["Rat","inv_nonneg"],"typeFallback":"forall {a : Rat}, (LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) a) -> (LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) (Inv.inv.{0} Rat Rat.instInv a))","typeFull":"∀ {a : ℚ}, 0 ≤ a → 0 ≤ a⁻¹","typeReadable":"∀ {a : ℚ}, 0 ≤ a → 0 ≤ a⁻¹","typeReferences":[["Rat","instLE"],["Inv","inv"],["LE","le"],["Rat"],["Rat","instInv"],["Rat","instOfNat"],["OfNat","ofNat"]],"valueReferences":[["Inv","inv"],["Nat","cast"],["Int","natCast_nonneg"],["Rat","num_nonneg"],["Rat","divInt"],["Rat","num"],["OfNat","ofNat"],["Rat","instOfNat"],["congrArg"],["Int"],["Rat","instLE"],["Rat","inv_def"],["instOfNat"],["Iff","mpr"],["LE","le"],["Int","instLEInt"],["id"],["Rat"],["Rat","instInv"],["Eq","mpr"],["Eq"],["Rat","divInt_nonneg"],["instNatCastInt"],["Rat","den"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instDiv","_proof_1"],"typeFallback":"forall (x : NNRat) (y : NNRat), LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) (NNRat.cast.{0} Rat Rat.instNNRatCast x) (NNRat.cast.{0} Rat Rat.instNNRatCast y))","typeFull":"∀ (x y : ℚ≥0), 0 ≤ ↑x / ↑y","typeReadable":"∀ (x y : ℚ≥0), 0 ≤ ↑x / ↑y","typeReferences":[["HDiv","hDiv"],["Rat","instLE"],["Rat","instNNRatCast"],["NNRat"],["LE","le"],["Rat"],["NNRat","cast"],["instHDiv"],["Rat","instOfNat"],["OfNat","ofNat"],["Rat","instDiv"]],"valueReferences":[["Rat","instLE"],["Rat","instNNRatCast"],["Rat","div_nonneg"],["LE","le"],["Subtype","property"],["Rat"],["NNRat","cast"],["Rat","instOfNat"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["NNRat","instDiv"],"typeFallback":"Div.{0} NNRat","typeFull":"Div ℚ≥0","typeReadable":"Div ℚ≥0","typeReferences":[["NNRat"],["Div"]],"valueReferences":[["Rat","instNNRatCast"],["NNRat","instDiv","_proof_1"],["Div","mk"],["instHDiv"],["OfNat","ofNat"],["Rat","instOfNat"],["Rat","instLE"],["HDiv","hDiv"],["NNRat"],["LE","le"],["Rat"],["NNRat","cast"],["Subtype","mk"],["Rat","instDiv"]]},{"isProp":true,"kind":"theorem","name":["NNRat","den_inv_of_ne_zero"],"typeFallback":"forall {q : NNRat}, (Ne.{1} NNRat q (OfNat.ofNat.{0} NNRat 0 (Zero.toOfNat0.{0} NNRat (MulZeroClass.toZero.{0} NNRat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} NNRat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} NNRat (Semiring.toNonAssocSemiring.{0} NNRat (CommSemiring.toSemiring.{0} NNRat instCommSemiringNNRat)))))))) -> (Eq.{1} Nat (NNRat.den (Inv.inv.{0} NNRat NNRat.instInv q)) (NNRat.num q))","typeFull":"∀ {q : ℚ≥0}, q ≠ 0 → q⁻¹.den = q.num","typeReadable":"∀ {q : ℚ≥0}, q ≠ 0 → q⁻¹.den = q.num","typeReferences":[["Inv","inv"],["NNRat","den"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["OfNat","ofNat"],["Nat"],["NNRat"],["Semiring","toNonAssocSemiring"],["instCommSemiringNNRat"],["MulZeroClass","toZero"],["NNRat","num"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["Zero","toOfNat0"],["NNRat","instInv"],["Eq"]],"valueReferences":[["NNRat","den"],["Rat","divInt"],["Rat","instLE"],["Semiring","toNonAssocSemiring"],["NNRat","num"],["Eq","symm"],["NNRat","cast"],["NNRat","coprime_num_den"],["NNRat","divNat"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Rat","divInt_ofNat"],["Nat"],["NNRat","inv_def"],["NNRat","coe_mk"],["Eq","refl"],["Iff","mpr"],["Rat"],["id"],["Rat","mk_eq_mkRat"],["Eq","mpr"],["Rat","mk'"],["Rat","instNNRatCast"],["Nat","cast"],["NNRat","divNat","_proof_1"],["Rat","instOfNat"],["congrArg"],["mkRat"],["Nat","Coprime"],["NNRat"],["NNRat","divNat","eq_1"],["instOfNatNat"],["NNRat","den","eq_1"],["Zero","toOfNat0"],["Int","natAbs"],["Eq"],["Nat","Coprime","symm"],["Rat","den"],["instNatCastInt"],["Inv","inv"],["CommSemiring","toSemiring"],["NNRat","num_ne_zero"],["OfNat","ofNat"],["Int"],["instCommSemiringNNRat"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Ne"],["NNRat","instInv"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["NNRat","instZPow","_proof_1"],"typeFallback":"forall (x : NNRat) (n : Int), LE.le.{0} Rat Rat.instLE (OfNat.ofNat.{0} Rat 0 (Rat.instOfNat 0)) (HPow.hPow.{0, 0, 0} Rat Int Rat (instHPow.{0, 0} Rat Int Rat.instPowInt) (NNRat.cast.{0} Rat Rat.instNNRatCast x) n)","typeFull":"∀ (x : ℚ≥0) (n : ℤ), 0 ≤ ↑x ^ n","typeReadable":"∀ (x : ℚ≥0) (n : ℤ), 0 ≤ ↑x ^ n","typeReferences":[["instHPow"],["Rat","instLE"],["Rat","instNNRatCast"],["NNRat"],["LE","le"],["Rat"],["NNRat","cast"],["Rat","instPowInt"],["HPow","hPow"],["Rat","instOfNat"],["OfNat","ofNat"],["Int"]],"valueReferences":[["Rat","instLE"],["Rat","instNNRatCast"],["LE","le"],["Subtype","property"],["Rat"],["NNRat","cast"],["Rat","zpow_nonneg"],["Rat","instOfNat"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Rat","instField","_proof_2"],"typeFallback":"forall (x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.27 : NNRat) (x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.29 : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (NonUnitalNonAssocSemiring.toMul.{0} Rat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} Rat (Semiring.toNonUnitalSemiring.{0} Rat (Ring.toSemiring.{0} Rat (CommRing.toRing.{0} Rat Rat.commRing)))))) (NNRat.cast.{0} Rat Rat.instNNRatCast x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.27) x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.29) (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (NonUnitalNonAssocSemiring.toMul.{0} Rat (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} Rat (Semiring.toNonUnitalSemiring.{0} Rat (Ring.toSemiring.{0} Rat (CommRing.toRing.{0} Rat Rat.commRing)))))) (NNRat.cast.{0} Rat Rat.instNNRatCast x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.27) x._@.Mathlib.Algebra.Field.Rat.310067971._hygCtx._hyg.29)","typeFull":"∀ (x : ℚ≥0) (x_1 : ℚ), ↑x * x_1 = ↑x * x_1","typeReadable":"∀ (x : ℚ≥0) (x_1 : ℚ), ↑x * x_1 = ↑x * x_1","typeReferences":[["Rat","instNNRatCast"],["NonUnitalNonAssocSemiring","toMul"],["HMul","hMul"],["Ring","toSemiring"],["CommRing","toRing"],["NNRat"],["Rat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["NNRat","cast"],["Eq"],["Semiring","toNonUnitalSemiring"],["Rat","commRing"]],"valueReferences":[["rfl"],["Rat","instNNRatCast"],["CommRing","toRing"],["NonUnitalNonAssocSemiring","toMul"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Rat"],["NNRat","cast"],["instHMul"],["HMul","hMul"],["Semiring","toNonUnitalSemiring"],["Rat","commRing"],["Ring","toSemiring"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Invertible.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["inv_mul_cancel_of_invertible"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2034676686._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2034676686._hygCtx._hyg.7 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2034676686._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2034676686._hygCtx._hyg.3)))) a], Eq.{succ u} α (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2034676686._hygCtx._hyg.3))))) (Inv.inv.{u} α (InvOneClass.toInv.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2034676686._hygCtx._hyg.3)))) a) a) (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2034676686._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [Invertible a], a⁻¹ * a = 1","typeReadable":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [Invertible a], a⁻¹ * a = 1","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Invertible","ne_zero"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"],["inv_mul_cancel₀"],["GroupWithZero","toNontrivial"]]},{"isProp":true,"kind":"theorem","name":["div_mul_cancel_of_invertible"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1313076185._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) (b : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1313076185._hygCtx._hyg.8 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1313076185._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1313076185._hygCtx._hyg.3)))) b], Eq.{succ u} α (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1313076185._hygCtx._hyg.3))))) (HDiv.hDiv.{u, u, u} α α α (instHDiv.{u} α (DivInvMonoid.toDiv.{u} α (GroupWithZero.toDivInvMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1313076185._hygCtx._hyg.3))) a b) b) a","typeFull":"∀ {α : Type u} [inst : GroupWithZero α] (a b : α) [Invertible b], a / b * b = a","typeReadable":"∀ {α : Type u} [inst : GroupWithZero α] (a b : α) [Invertible b], a / b * b = a","typeReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Invertible","ne_zero"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"],["div_mul_cancel₀"],["GroupWithZero","toNontrivial"]]},{"isProp":true,"kind":"theorem","name":["invertibleDiv","_proof_3"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3 : GroupWithZero.{u_1} α] (a : α) (b : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.8 : Invertible.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) a] [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.11 : Invertible.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) b], Eq.{succ u_1} α (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))))) (HDiv.hDiv.{u_1, u_1, u_1} α α α (instHDiv.{u_1} α (DivInvMonoid.toDiv.{u_1} α (GroupWithZero.toDivInvMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))) a b) (HDiv.hDiv.{u_1, u_1, u_1} α α α (instHDiv.{u_1} α (DivInvMonoid.toDiv.{u_1} α (GroupWithZero.toDivInvMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))) b a)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u_1} [inst : GroupWithZero α] (a b : α) [Invertible a] [Invertible b], a / b * (b / a) = 1","typeReadable":"∀ {α : Type u_1} [inst : GroupWithZero α] (a b : α) [Invertible a] [Invertible b], a / b * (b / a) = 1","typeReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["div_self_of_invertible"],["instHDiv"],["congrArg"],["HDiv","hDiv"],["MulOne","toMul"],["GroupWithZero","toMonoidWithZero"],["Monoid","toMulOneClass"],["congrFun'"],["Eq"],["True"],["InvOneClass","toOne"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["eq_self"],["DivInvOneMonoid","toInvOneClass"],["DivInvMonoid","toMonoid"],["MulZeroOneClass","toMulZeroClass"],["of_eq_true"],["One","toOfNat1"],["_private","Mathlib","Algebra","GroupWithZero","Invertible",0,"invertibleDiv","_simp_1"],["div_mul_cancel_of_invertible"],["instHMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["invertibleInv","_proof_2"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3 : GroupWithZero.{u_1} α] {a : α} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.7 : Invertible.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) a], Eq.{succ u_1} α (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3))))) (Inv.inv.{u_1} α (InvOneClass.toInv.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) a) a) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u_1} [inst : GroupWithZero α] {a : α} [Invertible a], a⁻¹ * a = 1","typeReadable":"∀ {α : Type u_1} [inst : GroupWithZero α] {a : α} [Invertible a], a⁻¹ * a = 1","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Inv","inv"],["InvOneClass","toInv"],["True"],["Eq","trans"],["InvOneClass","toOne"],["MulZeroClass","toMul"],["HMul","hMul"],["inv_mul_cancel_of_invertible"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["congrArg"],["eq_self"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["of_eq_true"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["congrFun'"],["Eq"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":false,"kind":"definition","name":["invertibleInv"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3 : GroupWithZero.{u} α] {a : α} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.7 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) a], Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) (Inv.inv.{u} α (InvOneClass.toInv.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) a)","typeFull":"{α : Type u} → [inst : GroupWithZero α] → {a : α} → [Invertible a] → Invertible a⁻¹","typeReadable":"{α : Type u} → [inst : GroupWithZero α] → {a : α} → [Invertible a] → Invertible a⁻¹","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["InvOneClass","toOne"],["GroupWithZero","toMonoidWithZero"],["MulZeroClass","toMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Inv","inv"],["InvOneClass","toInv"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["invertibleInv","_proof_1"],["invertibleInv","_proof_2"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["Invertible","mk"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["Invertible","ne_zero"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2223990205._hygCtx._hyg.3 : MulZeroOneClass.{u} α] (a : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2223990205._hygCtx._hyg.7 : Nontrivial.{u} α] [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2223990205._hygCtx._hyg.10 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2223990205._hygCtx._hyg.3)) (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (MulZeroOneClass.toMulOneClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2223990205._hygCtx._hyg.3))) a], Ne.{succ u} α a (OfNat.ofNat.{u} α 0 (Zero.toOfNat0.{u} α (MulZeroClass.toZero.{u} α (MulZeroOneClass.toMulZeroClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2223990205._hygCtx._hyg.3))))","typeFull":"∀ {α : Type u} [inst : MulZeroOneClass α] (a : α) [Nontrivial α] [Invertible a], a ≠ 0","typeReadable":"∀ {α : Type u} [inst : MulZeroOneClass α] (a : α) [Nontrivial α] [Invertible a], a ≠ 0","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toOne"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MulZeroClass","toMul"],["Nontrivial"],["MulZeroOneClass"],["Zero","toOfNat0"],["Ne"],["MulZeroOneClass","toMulOneClass"],["Invertible"],["OfNat","ofNat"]],"valueReferences":[["MulOneClass","toMulOne"],["instTransEq"],["Trans","trans"],["True"],["Eq","trans"],["Invertible","invOf"],["MulOne","toOne"],["MulZeroClass","toMul"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["NeZero","one"],["MulZeroClass","mul_zero"],["zero_ne_one"],["OfNat","ofNat"],["congrArg"],["eq_self"],["of_eq_true"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Invertible","invOf_mul_self"],["instHMul"],["Zero","toOfNat0"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["invertibleDiv","_proof_2"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3 : GroupWithZero.{u_1} α] (a : α) (b : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.8 : Invertible.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) a] [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.11 : Invertible.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) b], Eq.{succ u_1} α (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))))) (HDiv.hDiv.{u_1, u_1, u_1} α α α (instHDiv.{u_1} α (DivInvMonoid.toDiv.{u_1} α (GroupWithZero.toDivInvMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))) b a) (HDiv.hDiv.{u_1, u_1, u_1} α α α (instHDiv.{u_1} α (DivInvMonoid.toDiv.{u_1} α (GroupWithZero.toDivInvMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))) a b)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u_1} [inst : GroupWithZero α] (a b : α) [Invertible a] [Invertible b], b / a * (a / b) = 1","typeReadable":"∀ {α : Type u_1} [inst : GroupWithZero α] (a b : α) [Invertible a] [Invertible b], b / a * (a / b) = 1","typeReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["div_self_of_invertible"],["instHDiv"],["congrArg"],["HDiv","hDiv"],["MulOne","toMul"],["GroupWithZero","toMonoidWithZero"],["Monoid","toMulOneClass"],["congrFun'"],["Eq"],["True"],["InvOneClass","toOne"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["eq_self"],["DivInvOneMonoid","toInvOneClass"],["DivInvMonoid","toMonoid"],["MulZeroOneClass","toMulZeroClass"],["of_eq_true"],["One","toOfNat1"],["_private","Mathlib","Algebra","GroupWithZero","Invertible",0,"invertibleDiv","_simp_1"],["div_mul_cancel_of_invertible"],["instHMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["invOf_div"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) (b : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.8 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) a] [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.11 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) b] [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.14 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) (HDiv.hDiv.{u, u, u} α α α (instHDiv.{u} α (DivInvMonoid.toDiv.{u} α (GroupWithZero.toDivInvMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3))) a b)], Eq.{succ u} α (Invertible.invOf.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3)))) (HDiv.hDiv.{u, u, u} α α α (instHDiv.{u} α (DivInvMonoid.toDiv.{u} α (GroupWithZero.toDivInvMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3))) a b) inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.14) (HDiv.hDiv.{u, u, u} α α α (instHDiv.{u} α (DivInvMonoid.toDiv.{u} α (GroupWithZero.toDivInvMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3666226329._hygCtx._hyg.3))) b a)","typeFull":"∀ {α : Type u} [inst : GroupWithZero α] (a b : α) [Invertible a] [Invertible b] [inst_3 : Invertible (a / b)],\n  ⅟(a / b) = b / a","typeReadable":"∀ {α : Type u} [inst : GroupWithZero α] (a b : α) [Invertible a] [Invertible b] [inst_3 : Invertible (a / b)],\n  ⅟(a / b) = b / a","typeReferences":[["Invertible","invOf"],["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["GroupWithZero","toDivInvMonoid"],["HMul","hMul"],["div_self_of_invertible"],["instHDiv"],["congrArg"],["HDiv","hDiv"],["MulOne","toMul"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["congrFun'"],["Eq"],["MulOne","toOne"],["True"],["InvOneClass","toOne"],["DivInvMonoid","toDiv"],["OfNat","ofNat"],["invOf_eq_right_inv"],["eq_self"],["DivInvOneMonoid","toInvOneClass"],["DivInvMonoid","toMonoid"],["of_eq_true"],["One","toOfNat1"],["_private","Mathlib","Algebra","GroupWithZero","Invertible",0,"invertibleDiv","_simp_1"],["div_mul_cancel_of_invertible"],["instHMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","GroupWithZero","Invertible",0,"invertibleDiv","_simp_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.2458317242._hygCtx._hyg.3 : DivInvMonoid.{u_1} G] (a : G) (b : G) (c : G), Eq.{succ u_1} G (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2458317242._hygCtx._hyg.3))))) a (HDiv.hDiv.{u_1, u_1, u_1} G G G (instHDiv.{u_1} G (DivInvMonoid.toDiv.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2458317242._hygCtx._hyg.3)) b c)) (HDiv.hDiv.{u_1, u_1, u_1} G G G (instHDiv.{u_1} G (DivInvMonoid.toDiv.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2458317242._hygCtx._hyg.3)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (MulOne.toMul.{u_1} G (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2458317242._hygCtx._hyg.3))))) a b) c)","typeFull":"∀ {G : Type u_1} [inst : DivInvMonoid G] (a b c : G), a * (b / c) = a * b / c","typeReadable":"∀ {G : Type u_1} [inst : DivInvMonoid G] (a b c : G), a * (b / c) = a * b / c","typeReferences":[["HDiv","hDiv"],["MulOneClass","toMulOne"],["DivInvMonoid"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["Monoid","toMulOneClass"],["instHMul"],["HMul","hMul"],["Eq"],["instHDiv"],["DivInvMonoid","toDiv"]],"valueReferences":[["MulOneClass","toMulOne"],["HDiv","hDiv"],["MulOne","toMul"],["DivInvMonoid","toMonoid"],["Monoid","toMulOneClass"],["Eq","symm"],["instHMul"],["HMul","hMul"],["instHDiv"],["mul_div_assoc"],["DivInvMonoid","toDiv"]]},{"isProp":true,"kind":"theorem","name":["invOf_eq_inv"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.7 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.3)))) a], Eq.{succ u} α (Invertible.invOf.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.3)))) a inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.7) (Inv.inv.{u} α (InvOneClass.toInv.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1921227998._hygCtx._hyg.3)))) a)","typeFull":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [inst_1 : Invertible a], ⅟a = a⁻¹","typeReadable":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [inst_1 : Invertible a], ⅟a = a⁻¹","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["Invertible","invOf"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["mul_inv_cancel₀"],["invOf_eq_right_inv"],["Inv","inv"],["DivInvOneMonoid","toInvOneClass"],["InvOneClass","toInv"],["Invertible","ne_zero"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["MonoidWithZero","toMulZeroOneClass"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toNontrivial"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["mul_inv_cancel_of_invertible"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3307290002._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3307290002._hygCtx._hyg.7 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3307290002._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3307290002._hygCtx._hyg.3)))) a], Eq.{succ u} α (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3307290002._hygCtx._hyg.3))))) a (Inv.inv.{u} α (InvOneClass.toInv.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3307290002._hygCtx._hyg.3)))) a)) (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.3307290002._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [Invertible a], a * a⁻¹ = 1","typeReadable":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [Invertible a], a * a⁻¹ = 1","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["mul_inv_cancel₀"],["Invertible","ne_zero"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero","toNontrivial"]]},{"isProp":true,"kind":"theorem","name":["div_self_of_invertible"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1829993385._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1829993385._hygCtx._hyg.7 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1829993385._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1829993385._hygCtx._hyg.3)))) a], Eq.{succ u} α (HDiv.hDiv.{u, u, u} α α α (instHDiv.{u} α (DivInvMonoid.toDiv.{u} α (GroupWithZero.toDivInvMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1829993385._hygCtx._hyg.3))) a a) (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1829993385._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [Invertible a], a / a = 1","typeReadable":"∀ {α : Type u} [inst : GroupWithZero α] (a : α) [Invertible a], a / a = 1","typeReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["OfNat","ofNat"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["div_self"],["Invertible","ne_zero"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero","toNontrivial"]]},{"isProp":true,"kind":"theorem","name":["mul_div_cancel_of_invertible"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2688134727._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) (b : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2688134727._hygCtx._hyg.8 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2688134727._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2688134727._hygCtx._hyg.3)))) b], Eq.{succ u} α (HDiv.hDiv.{u, u, u} α α α (instHDiv.{u} α (DivInvMonoid.toDiv.{u} α (GroupWithZero.toDivInvMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2688134727._hygCtx._hyg.3))) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2688134727._hygCtx._hyg.3))))) a b) b) a","typeFull":"∀ {α : Type u} [inst : GroupWithZero α] (a b : α) [Invertible b], a * b / b = a","typeReadable":"∀ {α : Type u} [inst : GroupWithZero α] (a b : α) [Invertible b], a * b / b = a","typeReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["GroupWithZero","toMulDivCancelClass"],["mul_div_cancel_right₀"],["Invertible","ne_zero"],["GroupWithZero","toDivInvMonoid"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMulZeroOneClass"],["DivInvMonoid","toDiv"],["GroupWithZero","toNontrivial"]]},{"isProp":false,"kind":"definition","name":["invertibleOfNonzero"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.261082214._hygCtx._hyg.3 : GroupWithZero.{u} α] {a : α}, (Ne.{succ u} α a (OfNat.ofNat.{u} α 0 (Zero.toOfNat0.{u} α (MulZeroClass.toZero.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.261082214._hygCtx._hyg.3))))))) -> (Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.261082214._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.261082214._hygCtx._hyg.3)))) a)","typeFull":"{α : Type u} → [inst : GroupWithZero α] → {a : α} → a ≠ 0 → Invertible a","typeReadable":"{α : Type u} → [inst : GroupWithZero α] → {a : α} → a ≠ 0 → Invertible a","typeReferences":[["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["GroupWithZero","toMonoidWithZero"],["Ne"],["Zero","toOfNat0"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["mul_inv_cancel₀"],["Inv","inv"],["InvOneClass","toInv"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["Invertible","mk"],["inv_mul_cancel₀"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["Ring","inverse_invertible"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.3 : MonoidWithZero.{u} α] (x : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.7 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.3))) (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (MulZeroOneClass.toMulOneClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.3)))) x], Eq.{succ u} α (Ring.inverse.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.3 x) (Invertible.invOf.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.3))) (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (MulZeroOneClass.toMulOneClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.3)))) x inst._@.Mathlib.Algebra.GroupWithZero.Invertible.2241952683._hygCtx._hyg.7)","typeFull":"∀ {α : Type u} [inst : MonoidWithZero α] (x : α) [inst_1 : Invertible x], Ring.inverse x = ⅟x","typeReadable":"∀ {α : Type u} [inst : MonoidWithZero α] (x : α) [inst_1 : Invertible x], Ring.inverse x = ⅟x","typeReferences":[["MulOneClass","toMulOne"],["Invertible","invOf"],["MulOne","toOne"],["MulZeroOneClass","toMulZeroClass"],["Ring","inverse"],["MulZeroClass","toMul"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Eq"],["Invertible"],["MonoidWithZero"]],"valueReferences":[["unitOfInvertible"],["MonoidWithZero","toMonoid"],["Ring","inverse_unit"]]},{"isProp":true,"kind":"theorem","name":["Invertible","toNeZero"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.626397855._hygCtx._hyg.3 : MulZeroOneClass.{u} α] [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.626397855._hygCtx._hyg.6 : Nontrivial.{u} α] (a : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.626397855._hygCtx._hyg.10 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.626397855._hygCtx._hyg.3)) (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (MulZeroOneClass.toMulOneClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.626397855._hygCtx._hyg.3))) a], NeZero.{u} α (MulZeroClass.toZero.{u} α (MulZeroOneClass.toMulZeroClass.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.626397855._hygCtx._hyg.3)) a","typeFull":"∀ {α : Type u} [inst : MulZeroOneClass α] [Nontrivial α] (a : α) [Invertible a], NeZero a","typeReadable":"∀ {α : Type u} [inst : MulZeroOneClass α] [Nontrivial α] (a : α) [Invertible a], NeZero a","typeReferences":[["NeZero"],["MulOneClass","toMulOne"],["MulOne","toOne"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MulZeroClass","toMul"],["Nontrivial"],["MulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Invertible"]],"valueReferences":[["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Invertible","ne_zero"],["NeZero","mk"]]},{"isProp":true,"kind":"theorem","name":["invertibleInv","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3 : GroupWithZero.{u_1} α] {a : α} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.7 : Invertible.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) a], Eq.{succ u_1} α (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α (MonoidWithZero.toMulZeroOneClass.{u_1} α (GroupWithZero.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3))))) a (Inv.inv.{u_1} α (InvOneClass.toInv.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3)))) a)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (InvOneClass.toOne.{u_1} α (DivInvOneMonoid.toInvOneClass.{u_1} α (DivisionMonoid.toDivInvOneMonoid.{u_1} α (GroupWithZero.toDivisionMonoid.{u_1} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1092684251._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u_1} [inst : GroupWithZero α] {a : α} [Invertible a], a * a⁻¹ = 1","typeReadable":"∀ {α : Type u_1} [inst : GroupWithZero α] {a : α} [Invertible a], a * a⁻¹ = 1","typeReferences":[["Inv","inv"],["InvOneClass","toInv"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["OfNat","ofNat"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Eq"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Inv","inv"],["InvOneClass","toInv"],["True"],["Eq","trans"],["InvOneClass","toOne"],["MulZeroClass","toMul"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["mul_inv_cancel_of_invertible"],["OfNat","ofNat"],["congrArg"],["eq_self"],["DivInvOneMonoid","toInvOneClass"],["One","toOfNat1"],["of_eq_true"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["congrFun'"],["Eq"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":false,"kind":"definition","name":["invertibleDiv"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3 : GroupWithZero.{u} α] (a : α) (b : α) [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.8 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) a] [inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.11 : Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) b], Invertible.{u} α (MulZeroClass.toMul.{u} α (MulZeroOneClass.toMulZeroClass.{u} α (MonoidWithZero.toMulZeroOneClass.{u} α (GroupWithZero.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (InvOneClass.toOne.{u} α (DivInvOneMonoid.toInvOneClass.{u} α (DivisionMonoid.toDivInvOneMonoid.{u} α (GroupWithZero.toDivisionMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3)))) (HDiv.hDiv.{u, u, u} α α α (instHDiv.{u} α (DivInvMonoid.toDiv.{u} α (GroupWithZero.toDivInvMonoid.{u} α inst._@.Mathlib.Algebra.GroupWithZero.Invertible.1458439108._hygCtx._hyg.3))) a b)","typeFull":"{α : Type u} → [inst : GroupWithZero α] → (a b : α) → [Invertible a] → [Invertible b] → Invertible (a / b)","typeReadable":"{α : Type u} → [inst : GroupWithZero α] → (a b : α) → [Invertible a] → [Invertible b] → Invertible (a / b)","typeReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["GroupWithZero"],["instHDiv"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["GroupWithZero","toMonoidWithZero"],["Invertible"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["GroupWithZero","toDivInvMonoid"],["MulZeroClass","toMul"],["InvOneClass","toOne"],["MonoidWithZero","toMulZeroOneClass"],["instHDiv"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["invertibleDiv","_proof_2"],["DivInvOneMonoid","toInvOneClass"],["MulZeroOneClass","toMulZeroClass"],["invertibleDiv","_proof_3"],["GroupWithZero","toMonoidWithZero"],["Invertible","mk"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.DivisionRing.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","surjective_iff_ne_zero","_simp_1_1"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18)} {f : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41} {g : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41}, Eq.{1} Prop (Eq.{max (succ u_11) (succ u_8)} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41) f g) (forall (x : M), Eq.{succ u_11} M₃ (DFunLike.coe.{max (succ u_11) (succ u_8), succ u_8, succ u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41 σ) f x) (DFunLike.coe.{max (succ u_11) (succ u_8), succ u_8, succ u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41 σ) g x))","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  {f g : M →ₛₗ[σ] M₃}, (f = g) = ∀ (x : M), f x = g x","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  {f g : M →ₛₗ[σ] M₃}, (f = g) = ∀ (x : M), f x = g x","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["Module"],["Eq"],["LinearMap"],["DFunLike","coe"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["LinearMap","ext_iff"],["LinearMap"],["Eq"],["DFunLike","coe"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","_simp_1_2"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_4}} [inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9 : Semiring.{u_1} R] [inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12 : AddCommMonoid.{u_4} M] [inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.15 : Module.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12] {x : M} {y : M}, Eq.{1} Prop (Membership.mem.{u_4, u_4} M (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.15) (SetLike.instMembership.{u_4, u_4} (Submodule.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.15) M (Submodule.setLike.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.15)) (Submodule.span.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.15 (Singleton.singleton.{u_4, u_4} M (Set.{u_4} M) (Set.instSingletonSet.{u_4} M) y)) x) (Exists.{succ u_1} R (fun (a : R) => Eq.{succ u_4} M (HSMul.hSMul.{u_1, u_4, u_4} R M M (instHSMul.{u_1, u_4} R M (SMulZeroClass.toSMul.{u_1, u_4} R M (AddZero.toZero.{u_4} M (AddZeroClass.toAddZero.{u_4} M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12)))) (DistribSMul.toSMulZeroClass.{u_1, u_4} R M (AddMonoid.toAddZeroClass.{u_4} M (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12)) (DistribMulAction.toDistribSMul.{u_1, u_4} R M (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9)) (AddCommMonoid.toAddMonoid.{u_4} M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12) (Module.toDistribMulAction.{u_1, u_4} R M inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.9 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.12 inst._@.Mathlib.LinearAlgebra.Span.Defs.2364764113._hygCtx._hyg.15))))) a y) x))","typeFull":"∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {x y : M},\n  (x ∈ R ∙ y) = ∃ a, a • y = x","typeReadable":"∀ {R : Type u_1} {M : Type u_4} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M] {x y : M},\n  (x ∈ R ∙ y) = ∃ a, a • y = x","typeReferences":[["Module"],["Singleton","singleton"],["Membership","mem"],["Submodule","span"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["DistribSMul","toSMulZeroClass"],["Exists"],["SetLike","instMembership"],["Set"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Set","instSingletonSet"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["Submodule","setLike"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SetLike","instMembership"],["Exists"],["Set"],["Singleton","singleton"],["Membership","mem"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","instSingletonSet"],["Submodule"],["Module","toDistribMulAction"],["Submodule","setLike"],["Submodule","mem_span_singleton"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddZero","toZero"],["propext"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","match_1_5"],"typeFallback":"forall {R : Type.{u_1}} {M₁ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 : AddCommMonoid.{u_2} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18 : Module.{u_1, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8] (x : M₁) (z : M₁) (motive : (Exists.{succ u_1} R (fun (a : R) => Eq.{succ u_2} M₁ (HSMul.hSMul.{u_1, u_2, u_2} R M₁ M₁ (instHSMul.{u_1, u_2} R M₁ (SMulZeroClass.toSMul.{u_1, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) a x) z)) -> Prop) (x._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx.118.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.126 : Exists.{succ u_1} R (fun (a : R) => Eq.{succ u_2} M₁ (HSMul.hSMul.{u_1, u_2, u_2} R M₁ M₁ (instHSMul.{u_1, u_2} R M₁ (SMulZeroClass.toSMul.{u_1, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) a x) z)), (forall (w : R) (hw : Eq.{succ u_2} M₁ (HSMul.hSMul.{u_1, u_2, u_2} R M₁ M₁ (instHSMul.{u_1, u_2} R M₁ (SMulZeroClass.toSMul.{u_1, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) w x) z), motive (Exists.intro.{succ u_1} R (fun (a : R) => Eq.{succ u_2} M₁ (HSMul.hSMul.{u_1, u_2, u_2} R M₁ M₁ (instHSMul.{u_1, u_2} R M₁ (SMulZeroClass.toSMul.{u_1, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_1, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_1, u_2} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) a x) z) w hw)) -> (motive x._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx.118.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.126)","typeFull":"∀ {R : Type u_1} {M₁ : Type u_2} [inst : AddCommMonoid M₁] [inst_1 : Semiring R] [inst_2 : Module R M₁] (x z : M₁)\n  (motive : (∃ a, a • x = z) → Prop) (x_1 : ∃ a, a • x = z), (∀ (w : R) (hw : w • x = z), motive ⋯) → motive x_1","typeReadable":"∀ {R : Type u_1} {M₁ : Type u_2} [inst : AddCommMonoid M₁] [inst_1 : Semiring R] [inst_2 : Module R M₁] (x z : M₁)\n  (motive : (∃ a, a • x = z) → Prop) (x_1 : ∃ a, a • x = z), (∀ (w : R) (hw : w • x = z), motive ⋯) → motive x_1","typeReferences":[["Exists"],["Module"],["DistribMulAction","toDistribSMul"],["Exists","intro"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommMonoid"],["MonoidWithZero","toMonoid"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]],"valueReferences":[["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Exists","casesOn"],["Module","toDistribMulAction"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","surjective_iff_ne_zero"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11 : DivisionSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 : Module.{u_1, u_2} R M (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5] {f : LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))}, Iff (Function.Surjective.{succ u_2, succ u_1} M R (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => R) (LinearMap.instFunLike.{u_1, u_1, u_2, u_1} R R M R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) f)) (Ne.{max (succ u_1) (succ u_2)} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) f (OfNat.ofNat.{max u_1 u_2} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) 0 (Zero.toOfNat0.{max u_1 u_2} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) (LinearMap.instZero.{u_1, u_1, u_2, u_1} R R M R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))))))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] [inst_1 : DivisionSemiring R] [inst_2 : Module R M]\n  {f : M →ₗ[R] R}, Function.Surjective ⇑f ↔ f ≠ 0","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] [inst_1 : DivisionSemiring R] [inst_2 : Module R M]\n  {f : M →ₗ[R] R}, Function.Surjective ⇑f ↔ f ≠ 0","typeReferences":[["LinearMap","instZero"],["LinearMap","instFunLike"],["Module"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DivisionSemiring","toSemiring"],["LinearMap"],["Function","Surjective"],["DFunLike","coe"],["OfNat","ofNat"],["DivisionSemiring"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Iff"],["Zero","toOfNat0"],["Ne"],["Semiring","toModule"]],"valueReferences":[["Eq","trans"],["Exists","intro"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["SMulZeroClass","toSMul"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["map_smul"],["LinearMap","semilinearMapClass"],["LinearMap","ne_zero_of_surjective"],["not_false_eq_true"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Semiring","toModule"],["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","surjective_iff_ne_zero","_simp_1_1"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["LinearMap","instZero"],["SemilinearMapClass","toMulActionSemiHomClass"],["Exists"],["InvOneClass","toInv"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Classical","not_forall","_simp_1"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["LinearMap"],["AddZeroClass","toAddZero"],["Exists","casesOn"],["MulZeroOneClass","toMulZeroClass"],["eq_false"],["HSMul","hSMul"],["id"],["instHMul"],["AddZero","toZero"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["AddMonoid","toAddZeroClass"],["inv_mul_cancel_right₀"],["LinearMap","instFunLike"],["Eq","mp"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Iff","intro"],["congrArg"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["GroupWithZero","toNontrivial"],["Not"],["Inv","inv"],["True"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["Function","Surjective"],["OfNat","ofNat"],["eq_self"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["False"],["Ne"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","surjective"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11 : DivisionSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 : Module.{u_1, u_2} R M (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5] {f : LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))}, (Ne.{max (succ u_1) (succ u_2)} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) f (OfNat.ofNat.{max u_1 u_2} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) 0 (Zero.toOfNat0.{max u_1 u_2} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) (LinearMap.instZero.{u_1, u_1, u_2, u_1} R R M R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))))))) -> (Function.Surjective.{succ u_2, succ u_1} M R (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => R) (LinearMap.instFunLike.{u_1, u_1, u_2, u_1} R R M R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.3090818165._hygCtx._hyg.11)))) f))","typeFull":"∀ {R : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] [inst_1 : DivisionSemiring R] [inst_2 : Module R M]\n  {f : M →ₗ[R] R}, f ≠ 0 → Function.Surjective ⇑f","typeReadable":"∀ {R : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] [inst_1 : DivisionSemiring R] [inst_2 : Module R M]\n  {f : M →ₗ[R] R}, f ≠ 0 → Function.Surjective ⇑f","typeReferences":[["LinearMap","instZero"],["LinearMap","instFunLike"],["Module"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DivisionSemiring","toSemiring"],["LinearMap"],["DFunLike","coe"],["Function","Surjective"],["DivisionSemiring"],["OfNat","ofNat"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Ne"],["Zero","toOfNat0"],["Semiring","toModule"]],"valueReferences":[["LinearMap","instZero"],["LinearMap","instFunLike"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DivisionSemiring","toSemiring"],["LinearMap"],["Function","Surjective"],["DFunLike","coe"],["OfNat","ofNat"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Iff","mpr"],["LinearMap","surjective_iff_ne_zero"],["Zero","toOfNat0"],["Ne"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","range_smulRight_apply"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {M₁ : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8 : AddCommMonoid.{u_3} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11 : DivisionSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 : Module.{u_1, u_2} R M (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.18 : Module.{u_1, u_3} R M₁ (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8] {f : LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))}, (Ne.{max (succ u_1) (succ u_2)} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) f (OfNat.ofNat.{max u_1 u_2} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) 0 (Zero.toOfNat0.{max u_1 u_2} (LinearMap.{u_1, u_1, u_2, u_1} R R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) (LinearMap.instZero.{u_1, u_1, u_2, u_1} R R M R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))))))) -> (forall (x : M₁), Eq.{succ u_3} (Submodule.{u_1, u_3} R M₁ (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.18) (LinearMap.range.{u_1, u_1, u_2, u_3} R R M M₁ (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.18 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) (RingHomSurjective.ids.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)) (LinearMap.smulRight.{u_1, u_1, u_3, u_2} R R M₁ M (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.14 (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) (Semiring.toModule.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11)) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.18 (IsScalarTower.left.{u_1, u_3} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) (DistribMulAction.toMulAction.{u_1, u_3} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11))) (AddCommMonoid.toAddMonoid.{u_3} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_3} R M₁ (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.18))) f x)) (Submodule.span.{u_1, u_3} R M₁ (DivisionSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.778380564._hygCtx._hyg.18 (Singleton.singleton.{u_3, u_3} M₁ (Set.{u_3} M₁) (Set.instSingletonSet.{u_3} M₁) x)))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {M₁ : Type u_3} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₁]\n  [inst_2 : DivisionSemiring R] [inst_3 : Module R M] [inst_4 : Module R M₁] {f : M →ₗ[R] R},\n  f ≠ 0 → ∀ (x : M₁), (f.smulRight x).range = R ∙ x","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {M₁ : Type u_3} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₁]\n  [inst_2 : DivisionSemiring R] [inst_3 : Module R M] [inst_4 : Module R M₁] {f : M →ₗ[R] R},\n  f ≠ 0 → ∀ (x : M₁), (f.smulRight x).range = R ∙ x","typeReferences":[["Module"],["Singleton","singleton"],["LinearMap","range"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Zero","toOfNat0"],["Semiring","toModule"],["Eq"],["LinearMap","instZero"],["LinearMap","smulRight"],["IsScalarTower","left"],["Set"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["DivisionSemiring","toSemiring"],["Set","instSingletonSet"],["LinearMap"],["OfNat","ofNat"],["DivisionSemiring"],["RingHomSurjective","ids"],["Module","toDistribMulAction"],["Ne"]],"valueReferences":[["LinearMap","surjective"],["LinearMap","range_smulRight_apply_of_surjective"],["DivisionSemiring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","_simp_1_1"],"typeFallback":"forall {R : Type.{u_1}} {R₂ : Type.{u_2}} {M : Type.{u_5}} {M₂ : Type.{u_6}} [inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 : Semiring.{u_2} R₂] [inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.20 : AddCommMonoid.{u_5} M] [inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 : AddCommMonoid.{u_6} M₂] [inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.29 : Module.{u_1, u_5} R M inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.20] [inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33 : Module.{u_2, u_6} R₂ M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23] {τ₁₂ : RingHom.{u_1, u_2} R R₂ (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11) (Semiring.toNonAssocSemiring.{u_2} R₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14)} [inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.61 : RingHomSurjective.{u_1, u_2} R R₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 τ₁₂] {f : LinearMap.{u_1, u_2, u_5, u_6} R R₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 τ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33} {x : M₂}, Eq.{1} Prop (Membership.mem.{u_6, u_6} M₂ (Submodule.{u_2, u_6} R₂ M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33) (SetLike.instMembership.{u_6, u_6} (Submodule.{u_2, u_6} R₂ M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33) M₂ (Submodule.setLike.{u_2, u_6} R₂ M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33)) (LinearMap.range.{u_1, u_2, u_5, u_6} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33 τ₁₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.61 f) x) (Exists.{succ u_5} M (fun (y : M) => Eq.{succ u_6} M₂ (DFunLike.coe.{max (succ u_5) (succ u_6), succ u_5, succ u_6} (LinearMap.{u_1, u_2, u_5, u_6} R R₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 τ₁₂ M M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₂) (LinearMap.instFunLike.{u_1, u_2, u_5, u_6} R R₂ M M₂ inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.20 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.23 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Module.Submodule.Range.2356879540._hygCtx._hyg.33 τ₁₂) f y) x))","typeFull":"∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [inst : Semiring R] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}\n  [inst_6 : RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {x : M₂}, (x ∈ f.range) = ∃ y, f y = x","typeReadable":"∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [inst : Semiring R] [inst_1 : Semiring R₂]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}\n  [inst_6 : RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {x : M₂}, (x ∈ f.range) = ∃ y, f y = x","typeReferences":[["RingHom"],["Exists"],["SetLike","instMembership"],["LinearMap","instFunLike"],["RingHomSurjective"],["Module"],["LinearMap","range"],["Membership","mem"],["LinearMap"],["DFunLike","coe"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["Submodule","setLike"],["Eq"],["Semiring"]],"valueReferences":[["Exists"],["SetLike","instMembership"],["LinearMap","instFunLike"],["Submodule","setLike"],["LinearMap","range"],["Membership","mem"],["LinearMap"],["Eq"],["LinearMap","mem_range"],["DFunLike","coe"],["propext"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","range_smulRight_apply_of_surjective"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {M₁ : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 : AddCommMonoid.{u_3} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18 : Module.{u_1, u_3} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8] {f : LinearMap.{u_1, u_1, u_2, u_1} R R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)}, (Function.Surjective.{succ u_2, succ u_1} M R (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_2, succ u_1} (LinearMap.{u_1, u_1, u_2, u_1} R R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => R) (LinearMap.instFunLike.{u_1, u_1, u_2, u_1} R R M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) f)) -> (forall (x : M₁), Eq.{succ u_3} (Submodule.{u_1, u_3} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18) (LinearMap.range.{u_1, u_1, u_2, u_3} R R M M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11) (LinearMap.smulRight.{u_1, u_1, u_3, u_2} R R M₁ M inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18 (IsScalarTower.left.{u_1, u_3} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (DistribMulAction.toMulAction.{u_1, u_3} R M₁ (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_3} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_1, u_3} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))) f x)) (Submodule.span.{u_1, u_3} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18 (Singleton.singleton.{u_3, u_3} M₁ (Set.{u_3} M₁) (Set.instSingletonSet.{u_3} M₁) x)))","typeFull":"∀ {R : Type u_1} {M : Type u_2} {M₁ : Type u_3} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₁]\n  [inst_2 : Semiring R] [inst_3 : Module R M] [inst_4 : Module R M₁] {f : M →ₗ[R] R},\n  Function.Surjective ⇑f → ∀ (x : M₁), (f.smulRight x).range = R ∙ x","typeReadable":"∀ {R : Type u_1} {M : Type u_2} {M₁ : Type u_3} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₁]\n  [inst_2 : Semiring R] [inst_3 : Module R M] [inst_4 : Module R M₁] {f : M →ₗ[R] R},\n  Function.Surjective ⇑f → ∀ (x : M₁), (f.smulRight x).range = R ∙ x","typeReferences":[["LinearMap","instFunLike"],["Module"],["LinearMap","range"],["Singleton","singleton"],["Submodule","span"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["Eq"],["Semiring","toModule"],["LinearMap","smulRight"],["IsScalarTower","left"],["Set"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["Set","instSingletonSet"],["LinearMap"],["Function","Surjective"],["RingHomSurjective","ids"],["Module","toDistribMulAction"],["Semiring"]],"valueReferences":[["Submodule","ext"],["Singleton","singleton"],["Membership","mem"],["Exists","intro"],["Submodule","span"],["SMulZeroClass","toSMul"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","match_1_3"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq","ndrec"],["Semiring","toModule"],["Eq","rec"],["DistribSMul","toSMulZeroClass"],["rfl"],["LinearMap","smulRight"],["Exists"],["SetLike","instMembership"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearMap"],["AddZeroClass","toAddZero"],["Exists","casesOn"],["Iff"],["HSMul","hSMul"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["LinearMap","instFunLike"],["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","match_1_5"],["LinearMap","range"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Submodule"],["congrArg"],["Iff","intro"],["MonoidWithZero","toMonoid"],["instHSMul"],["congrFun'"],["Eq"],["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","_simp_1_2"],["IsScalarTower","left"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["Set","instSingletonSet"],["RingHomSurjective","ids"],["Module","toDistribMulAction"],["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","_simp_1_1"],["Submodule","setLike"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Module","LinearMap","DivisionRing",0,"LinearMap","range_smulRight_apply_of_surjective","match_1_3"],"typeFallback":"forall {R : Type.{u_3}} {M : Type.{u_1}} {M₁ : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 : AddCommMonoid.{u_2} M₁] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5] [inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18 : Module.{u_3, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8] {f : LinearMap.{u_3, u_3, u_1, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)} (x : M₁) (z : M₁) (motive : (Exists.{succ u_1} M (fun (y : M) => Eq.{succ u_2} M₁ (HSMul.hSMul.{u_3, u_2, u_2} R M₁ M₁ (instHSMul.{u_3, u_2} R M₁ (SMulZeroClass.toSMul.{u_3, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} R M₁ (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_3, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) (DFunLike.coe.{max (succ u_3) (succ u_1), succ u_1, succ u_3} (LinearMap.{u_3, u_3, u_1, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => R) (LinearMap.instFunLike.{u_3, u_3, u_1, u_3} R R M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) f y) x) z)) -> Prop) (x._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx.87.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.95 : Exists.{succ u_1} M (fun (y : M) => Eq.{succ u_2} M₁ (HSMul.hSMul.{u_3, u_2, u_2} R M₁ M₁ (instHSMul.{u_3, u_2} R M₁ (SMulZeroClass.toSMul.{u_3, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} R M₁ (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_3, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) (DFunLike.coe.{max (succ u_3) (succ u_1), succ u_1, succ u_3} (LinearMap.{u_3, u_3, u_1, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => R) (LinearMap.instFunLike.{u_3, u_3, u_1, u_3} R R M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) f y) x) z)), (forall (w : M) (hw : Eq.{succ u_2} M₁ (HSMul.hSMul.{u_3, u_2, u_2} R M₁ M₁ (instHSMul.{u_3, u_2} R M₁ (SMulZeroClass.toSMul.{u_3, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} R M₁ (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_3, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) (DFunLike.coe.{max (succ u_3) (succ u_1), succ u_1, succ u_3} (LinearMap.{u_3, u_3, u_1, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => R) (LinearMap.instFunLike.{u_3, u_3, u_1, u_3} R R M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) f w) x) z), motive (Exists.intro.{succ u_1} M (fun (y : M) => Eq.{succ u_2} M₁ (HSMul.hSMul.{u_3, u_2, u_2} R M₁ M₁ (instHSMul.{u_3, u_2} R M₁ (SMulZeroClass.toSMul.{u_3, u_2} R M₁ (AddZero.toZero.{u_2} M₁ (AddZeroClass.toAddZero.{u_2} M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)))) (DistribSMul.toSMulZeroClass.{u_3, u_2} R M₁ (AddMonoid.toAddZeroClass.{u_2} M₁ (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8)) (DistribMulAction.toDistribSMul.{u_3, u_2} R M₁ (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) (AddCommMonoid.toAddMonoid.{u_2} M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8) (Module.toDistribMulAction.{u_3, u_2} R M₁ inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.18))))) (DFunLike.coe.{max (succ u_3) (succ u_1), succ u_1, succ u_3} (LinearMap.{u_3, u_3, u_1, u_3} R R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11)) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => R) (LinearMap.instFunLike.{u_3, u_3, u_1, u_3} R R M R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.5 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.14 (Semiring.toModule.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11) (RingHom.id.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.11))) f y) x) z) w hw)) -> (motive x._@.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx.87.Mathlib.Algebra.Module.LinearMap.DivisionRing.1185607289._hygCtx._hyg.95)","typeFull":"∀ {R : Type u_3} {M : Type u_1} {M₁ : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₁]\n  [inst_2 : Semiring R] [inst_3 : Module R M] [inst_4 : Module R M₁] {f : M →ₗ[R] R} (x z : M₁)\n  (motive : (∃ y, f y • x = z) → Prop) (x_1 : ∃ y, f y • x = z), (∀ (w : M) (hw : f w • x = z), motive ⋯) → motive x_1","typeReadable":"∀ {R : Type u_3} {M : Type u_1} {M₁ : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommMonoid M₁]\n  [inst_2 : Semiring R] [inst_3 : Module R M] [inst_4 : Module R M₁] {f : M →ₗ[R] R} (x z : M₁)\n  (motive : (∃ y, f y • x = z) → Prop) (x_1 : ∃ y, f y • x = z), (∀ (w : M) (hw : f w • x = z), motive ⋯) → motive x_1","typeReferences":[["LinearMap","instFunLike"],["Module"],["Exists","intro"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["MonoidWithZero","toMonoid"],["instHSMul"],["Eq"],["Semiring","toModule"],["DistribSMul","toSMulZeroClass"],["Exists"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["LinearMap"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["HSMul","hSMul"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["LinearMap","instFunLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["LinearMap"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["Exists","casesOn"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["HSMul","hSMul"],["MonoidWithZero","toMonoid"],["instHSMul"],["Semiring","toModule"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["DistribSMul","toSMulZeroClass"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LocalizedModule.Away.sym.json

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["LocalizedModule","Away"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038960._hygCtx._hyg.3 : CommSemiring.{u_1} R], R -> (forall (M : Type.{u_2}) [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038960._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038960._hygCtx._hyg.11 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038960._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038960._hygCtx._hyg.8], Type.{max u_1 u_2})","typeFull":"{R : Type u_1} →\n  [inst : CommSemiring R] → R → (M : Type u_2) → [inst_1 : AddCommMonoid M] → [Module R M] → Type (max u_1 u_2)","typeReadable":"{R : Type u_1} →\n  [inst : CommSemiring R] → R → (M : Type u_2) → [inst_1 : AddCommMonoid M] → [Module R M] → Type (max u_1 u_2)","typeReferences":[["AddCommMonoid"],["Module"],["CommSemiring","toSemiring"],["CommSemiring"]],"valueReferences":[["LocalizedModule"],["Submonoid","powers"],["CommSemiring","toSemiring"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"]]},{"isProp":false,"kind":"definition","name":["IsLocalizedModule","Away"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {M' : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.5 : CommSemiring.{u_1} R], R -> (forall [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.9 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.12 : Module.{u_1, u_2} R M (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.5) inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.16 : AddCommMonoid.{u_3} M'] [inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.19 : Module.{u_1, u_3} R M' (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.5) inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.16], (LinearMap.{u_1, u_1, u_2, u_3} R R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.5) (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.5) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.5))) M M' inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.16 inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Module.LocalizedModule.Away.3380038959._hygCtx._hyg.19) -> Prop)","typeFull":"{R : Type u_1} →\n  {M : Type u_2} →\n    {M' : Type u_3} →\n      [inst : CommSemiring R] →\n        R →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : Module R M] → [inst_3 : AddCommMonoid M'] → [inst_4 : Module R M'] → (M →ₗ[R] M') → Prop","typeReadable":"{R : Type u_1} →\n  {M : Type u_2} →\n    {M' : Type u_3} →\n      [inst : CommSemiring R] →\n        R →\n          [inst_1 : AddCommMonoid M] →\n            [inst_2 : Module R M] → [inst_3 : AddCommMonoid M'] → [inst_4 : Module R M'] → (M →ₗ[R] M') → Prop","typeReferences":[["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module"],["CommSemiring","toSemiring"],["CommSemiring"],["LinearMap"]],"valueReferences":[["IsLocalizedModule"],["Submonoid","powers"],["CommSemiring","toSemiring"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.NoZeroSMulDivisors.Pi.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Function","noZeroSMulDivisors"],"typeFallback":"forall {ι : Type.{u_1}} {α : Type.{u_2}} {β : Type.{u_3}} [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9 : Semiring.{u_2} α] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.12 : IsDomain.{u_2} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15 : AddCommGroup.{u_3} β] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.18 : Module.{u_2, u_3} α β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9 (AddCommGroup.toAddCommMonoid.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15)] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.22 : NoZeroSMulDivisors.{u_2, u_3} α β (MulZeroClass.toZero.{u_2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} α (Semiring.toNonAssocSemiring.{u_2} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9)))) (NegZeroClass.toZero.{u_3} β (SubNegZeroMonoid.toNegZeroClass.{u_3} β (SubtractionMonoid.toSubNegZeroMonoid.{u_3} β (SubtractionCommMonoid.toSubtractionMonoid.{u_3} β (AddCommGroup.toDivisionAddCommMonoid.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15))))) (SMulZeroClass.toSMul.{u_2, u_3} α β (AddZero.toZero.{u_3} β (AddZeroClass.toAddZero.{u_3} β (AddMonoid.toAddZeroClass.{u_3} β (SubNegMonoid.toAddMonoid.{u_3} β (AddGroup.toSubNegMonoid.{u_3} β (AddCommGroup.toAddGroup.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} α β (AddMonoid.toAddZeroClass.{u_3} β (SubNegMonoid.toAddMonoid.{u_3} β (AddGroup.toSubNegMonoid.{u_3} β (AddCommGroup.toAddGroup.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} α β (MonoidWithZero.toMonoid.{u_2} α (Semiring.toMonoidWithZero.{u_2} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9)) (SubNegMonoid.toAddMonoid.{u_3} β (AddGroup.toSubNegMonoid.{u_3} β (AddCommGroup.toAddGroup.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} α β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9 (AddCommGroup.toAddCommMonoid.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15) inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.18))))], NoZeroSMulDivisors.{u_2, max u_1 u_3} α (ι -> β) (MulZeroClass.toZero.{u_2} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} α (Semiring.toNonAssocSemiring.{u_2} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9)))) (Pi.instZero.{u_1, u_3} ι (fun (a._@._internal._hyg.0 : ι) => β) (fun (i : ι) => NegZeroClass.toZero.{u_3} β (SubNegZeroMonoid.toNegZeroClass.{u_3} β (SubtractionMonoid.toSubNegZeroMonoid.{u_3} β (SubtractionCommMonoid.toSubtractionMonoid.{u_3} β (AddCommGroup.toDivisionAddCommMonoid.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15)))))) (Function.hasSMul.{u_1, u_2, u_3} ι α β (SMulZeroClass.toSMul.{u_2, u_3} α β (AddZero.toZero.{u_3} β (AddZeroClass.toAddZero.{u_3} β (AddMonoid.toAddZeroClass.{u_3} β (SubNegMonoid.toAddMonoid.{u_3} β (AddGroup.toSubNegMonoid.{u_3} β (AddCommGroup.toAddGroup.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15)))))) (DistribSMul.toSMulZeroClass.{u_2, u_3} α β (AddMonoid.toAddZeroClass.{u_3} β (SubNegMonoid.toAddMonoid.{u_3} β (AddGroup.toSubNegMonoid.{u_3} β (AddCommGroup.toAddGroup.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15)))) (DistribMulAction.toDistribSMul.{u_2, u_3} α β (MonoidWithZero.toMonoid.{u_2} α (Semiring.toMonoidWithZero.{u_2} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9)) (SubNegMonoid.toAddMonoid.{u_3} β (AddGroup.toSubNegMonoid.{u_3} β (AddCommGroup.toAddGroup.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15))) (Module.toDistribMulAction.{u_2, u_3} α β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.9 (AddCommGroup.toAddCommMonoid.{u_3} β inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.15) inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.3316028918._hygCtx._hyg.18)))))","typeFull":"∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Semiring α] [IsDomain α] [inst_2 : AddCommGroup β]\n  [inst_3 : Module α β] [NoZeroSMulDivisors α β], NoZeroSMulDivisors α (ι → β)","typeReadable":"∀ {ι : Type u_1} {α : Type u_2} {β : Type u_3} [inst : Semiring α] [IsDomain α] [inst_2 : AddCommGroup β]\n  [inst_3 : Module α β] [NoZeroSMulDivisors α β], NoZeroSMulDivisors α (ι → β)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Module"],["AddCommGroup","toAddGroup"],["IsDomain"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["SubNegZeroMonoid","toNegZeroClass"],["NoZeroSMulDivisors"],["Function","hasSMul"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Pi","instZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Pi","noZeroSMulDivisors"]]},{"isProp":true,"kind":"theorem","name":["Pi","noZeroSMulDivisors"],"typeFallback":"forall {I : Type.{u}} {f : I -> Type.{v}} (α : Type.{u_1}) [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7 : Semiring.{u_1} α] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.10 : IsDomain.{u_1} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 : forall (i : I), AddCommGroup.{v} (f i)] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.22 : forall (i : I), Module.{u_1, v} α (f i) inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7 (AddCommGroup.toAddCommMonoid.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i))] [inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.32 : forall (i : I), NoZeroSMulDivisors.{u_1, v} α (f i) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} α (Semiring.toNonAssocSemiring.{u_1} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7)))) (NegZeroClass.toZero.{v} (f i) (SubNegZeroMonoid.toNegZeroClass.{v} (f i) (SubtractionMonoid.toSubNegZeroMonoid.{v} (f i) (SubtractionCommMonoid.toSubtractionMonoid.{v} (f i) (AddCommGroup.toDivisionAddCommMonoid.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i)))))) (SMulZeroClass.toSMul.{u_1, v} α (f i) (AddZero.toZero.{v} (f i) (AddZeroClass.toAddZero.{v} (f i) (AddMonoid.toAddZeroClass.{v} (f i) (SubNegMonoid.toAddMonoid.{v} (f i) (AddGroup.toSubNegMonoid.{v} (f i) (AddCommGroup.toAddGroup.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i))))))) (DistribSMul.toSMulZeroClass.{u_1, v} α (f i) (AddMonoid.toAddZeroClass.{v} (f i) (SubNegMonoid.toAddMonoid.{v} (f i) (AddGroup.toSubNegMonoid.{v} (f i) (AddCommGroup.toAddGroup.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i))))) (DistribMulAction.toDistribSMul.{u_1, v} α (f i) (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7)) (SubNegMonoid.toAddMonoid.{v} (f i) (AddGroup.toSubNegMonoid.{v} (f i) (AddCommGroup.toAddGroup.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i)))) (Module.toDistribMulAction.{u_1, v} α (f i) inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7 (AddCommGroup.toAddCommMonoid.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i)) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.22 i)))))], NoZeroSMulDivisors.{u_1, max u v} α (forall (i : I), f i) (MulZeroClass.toZero.{u_1} α (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} α (Semiring.toNonAssocSemiring.{u_1} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7)))) (Pi.instZero.{u, v} I f (fun (i : I) => NegZeroClass.toZero.{v} (f i) (SubNegZeroMonoid.toNegZeroClass.{v} (f i) (SubtractionMonoid.toSubNegZeroMonoid.{v} (f i) (SubtractionCommMonoid.toSubtractionMonoid.{v} (f i) (AddCommGroup.toDivisionAddCommMonoid.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i))))))) (Pi.instSMul.{u, u_1, v} I α f (fun (i : I) => SMulZeroClass.toSMul.{u_1, v} α (f i) (NegZeroClass.toZero.{v} (f i) (SubNegZeroMonoid.toNegZeroClass.{v} (f i) (SubtractionMonoid.toSubNegZeroMonoid.{v} (f i) (SubtractionCommMonoid.toSubtractionMonoid.{v} (f i) (AddCommGroup.toDivisionAddCommMonoid.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i)))))) (DistribSMul.toSMulZeroClass.{u_1, v} α (f i) (AddMonoid.toAddZeroClass.{v} (f i) (SubNegMonoid.toAddMonoid.{v} (f i) (AddGroup.toSubNegMonoid.{v} (f i) (AddCommGroup.toAddGroup.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i))))) (DistribMulAction.toDistribSMul.{u_1, v} α (f i) (MonoidWithZero.toMonoid.{u_1} α (Semiring.toMonoidWithZero.{u_1} α inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7)) (SubNegMonoid.toAddMonoid.{v} (f i) (AddGroup.toSubNegMonoid.{v} (f i) (AddCommGroup.toAddGroup.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i)))) (Module.toDistribMulAction.{u_1, v} α (f i) inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.7 (AddCommGroup.toAddCommMonoid.{v} (f i) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.13 i)) (inst._@.Mathlib.Algebra.NoZeroSMulDivisors.Pi.1924264250._hygCtx._hyg.22 i))))))","typeFull":"∀ {I : Type u} {f : I → Type v} (α : Type u_1) [inst : Semiring α] [IsDomain α] [inst_2 : (i : I) → AddCommGroup (f i)]\n  [inst_3 : (i : I) → Module α (f i)] [∀ (i : I), NoZeroSMulDivisors α (f i)], NoZeroSMulDivisors α ((i : I) → f i)","typeReadable":"∀ {I : Type u} {f : I → Type v} (α : Type u_1) [inst : Semiring α] [IsDomain α] [inst_2 : (i : I) → AddCommGroup (f i)]\n  [inst_3 : (i : I) → Module α (f i)] [∀ (i : I), NoZeroSMulDivisors α (f i)], NoZeroSMulDivisors α ((i : I) → f i)","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Module"],["AddCommGroup","toAddGroup"],["IsDomain"],["SubtractionCommMonoid","toSubtractionMonoid"],["SMulZeroClass","toSMul"],["SubNegZeroMonoid","toNegZeroClass"],["NoZeroSMulDivisors"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["AddGroup","toSubNegMonoid"],["DistribSMul","toSMulZeroClass"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["Pi","instSMul"],["AddZeroClass","toAddZero"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["Pi","instZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["AddCommGroup","toAddGroup"],["Iff","mp"],["SMulZeroClass","toSMul"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommMonoid","toAddMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Semiring","toNonAssocSemiring"],["Or"],["funext"],["MonoidWithZero","toMonoid"],["instHSMul"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["Not"],["congr_fun"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Or","resolve_left"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Pi","instSMul"],["NoZeroSMulDivisors","mk"],["OfNat","ofNat"],["Classical","or_iff_not_imp_left"],["Module","toDistribMulAction"],["AddCommGroup","toDivisionAddCommMonoid"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["Iff","mpr"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NegZeroClass","toZero"],["AddCommGroup","toAddCommMonoid"],["IsDomain","toIsCancelMulZero"],["AddZero","toZero"],["instIsTorsionFreeOfIsDomainOfNoZeroSMulDivisors"],["smul_eq_zero"],["Pi","instZero"],["AddMonoid","toAddZeroClass"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Notation.Defs.sym.json

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"inductive","name":["VSub"],"typeFallback":"(outParam.{succ (succ u_1)} Type.{u_1}) -> Type.{u_2} -> Sort.{max (succ u_1) (succ u_2)}","typeFull":"outParam (Type u_1) → Type u_2 → Type (max u_1 u_2)","typeReadable":"outParam (Type u_1) → Type u_2 → Type (max u_1 u_2)","typeReferences":[["outParam"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["Zero","instNonempty"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Notation.Defs.2212486352._hygCtx._hyg.4 : Zero.{u} α], Nonempty.{succ u} α","typeFull":"∀ {α : Type u} [Zero α], Nonempty α","typeReadable":"∀ {α : Type u} [Zero α], Nonempty α","typeReferences":[["Nonempty"],["Zero"]],"valueReferences":[["Nonempty","intro"],["Zero","toOfNat0"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["instHVAdd"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Init.Prelude.4154482078._hygCtx._hyg.4 : VAdd.{u_1, u_2} α β], HVAdd.{u_1, u_2, u_2} α β β","typeFull":"{α : Type u_1} → {β : Type u_2} → [VAdd α β] → HVAdd α β β","typeReadable":"{α : Type u_1} → {β : Type u_2} → [VAdd α β] → HVAdd α β β","typeReferences":[["HVAdd"],["VAdd"]],"valueReferences":[["HVAdd","mk"],["VAdd","vadd"]]},{"isProp":false,"kind":"definition","name":["Star","noConfusion"],"typeFallback":"forall {P : Sort.{u_1}} {R : Type.{u}} {t : Star.{u} R} {R' : Type.{u}} {t' : Star.{u} R'}, (Eq.{succ (succ u)} Type.{u} R R') -> (HEq.{succ u} (Star.{u} R) t (Star.{u} R') t') -> (Star.noConfusionType.{u_1, u} P R t R' t')","typeFull":"{P : Sort u_1} →\n  {R : Type u} → {t : Star R} → {R' : Type u} → {t' : Star R'} → R = R' → t ≍ t' → Star.noConfusionType P t t'","typeReadable":"{P : Sort u_1} →\n  {R : Type u} → {t : Star R} → {R' : Type u} → {t' : Star R'} → R = R' → t ≍ t' → Star.noConfusionType P t t'","typeReferences":[["Star"],["Star","noConfusionType"],["HEq"],["Eq"]],"valueReferences":[["Star"],["HEq","refl"],["Star","noConfusionType"],["eq_of_heq"],["HEq"],["Eq","ndrec"],["Star","casesOn"]]},{"isProp":true,"kind":"theorem","name":["ite_add"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8 : Add.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.5 a b) c) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8) a c) (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8) b c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a b c : α),\n  (if P then a else b) + c = if P then a + c else b + c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a b c : α),\n  (if P then a else b) + c = if P then a + c else b + c","typeReferences":[["HAdd","hAdd"],["Decidable"],["ite"],["instHAdd"],["Add"],["Eq"]],"valueReferences":[["dite_add"]]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Notation_Defs___unexpand_HVAdd_hVAdd_1"],"typeFallback":"Lean.PrettyPrinter.Unexpander","typeFull":"Lean.PrettyPrinter.Unexpander","typeReadable":"Lean.PrettyPrinter.Unexpander","typeReferences":[["Lean","PrettyPrinter","Unexpander"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Bool","false"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","MonadQuotation","toMonadRef"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["List","cons"],["Bool","true"],["Unit","unit"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","PrettyPrinter","instMonadQuotationUnexpandM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["Bool","or"],["cond"],["Unit"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","withRef"],["Bool"],["Applicative","toPure"],["Lean","PrettyPrinter","UnexpandM"],["ReaderT","instMonadExceptOf"],["EStateM"],["Lean","Syntax","isOfKind"],["Lean","Syntax","matchesNull"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Eq"],["List","nil"],["Bind","bind"],["EStateM","instMonadExceptOfOfBacktrackable"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","node3"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"constructor","name":["VAdd","mk"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}}, (G -> P -> P) -> (VAdd.{u, v} G P)","typeFull":"{G : Type u} → {P : Type v} → (G → P → P) → VAdd G P","typeReadable":"{G : Type u} → {P : Type v} → (G → P → P) → VAdd G P","typeReferences":[["VAdd"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["VAdd","mk","noConfusion"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}} {P_1 : Sort.{u_1}} {vadd : G -> P -> P} {vadd' : G -> P -> P}, (Eq.{max (succ u) (succ v)} (VAdd.{u, v} G P) (VAdd.mk.{u, v} G P vadd) (VAdd.mk.{u, v} G P vadd')) -> ((HEq.{max (succ u) (succ v)} (G -> P -> P) vadd (G -> P -> P) vadd') -> P_1) -> P_1","typeFull":"{G : Type u} →\n  {P : Type v} →\n    {P_1 : Sort u_1} → {vadd vadd' : G → P → P} → { vadd := vadd } = { vadd := vadd' } → (vadd ≍ vadd' → P_1) → P_1","typeReadable":"{G : Type u} →\n  {P : Type v} →\n    {P_1 : Sort u_1} → {vadd vadd' : G → P → P} → { vadd := vadd } = { vadd := vadd' } → (vadd ≍ vadd' → P_1) → P_1","typeReferences":[["VAdd"],["HEq"],["Eq"],["VAdd","mk"]],"valueReferences":[["VAdd"],["Eq","refl"],["VAdd","noConfusion"],["id"],["heq_of_eq"],["VAdd","mk"]]},{"isProp":true,"kind":"theorem","name":["mul_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8 : Mul.{u_2} α] (a : α) (b : P -> α) (c : (Not P) -> α), Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8) a (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.5 (fun (h : P) => b h) (fun (h : Not P) => c h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.5 (fun (h : P) => HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8) a (b h)) (fun (h : Not P) => HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8) a (c h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a : α) (b : P → α) (c : ¬P → α),\n  (a * if h : P then b h else c h) = if h : P then a * b h else a * c h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a : α) (b : P → α) (c : ¬P → α),\n  (a * if h : P then b h else c h) = if h : P then a * b h else a * c h","typeReferences":[["Not"],["Decidable"],["Mul"],["instHMul"],["HMul","hMul"],["dite"],["Eq"]],"valueReferences":[["dif_pos"],["Eq","refl"],["congr"],["id"],["instHMul"],["HMul","hMul"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["Eq"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_mul"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8 : Mul.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : α), Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) c) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.5 (fun (h : P) => HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8) (a h) c) (fun (h : Not P) => HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8) (b h) c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) * c = if h : P then a h * c else b h * c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) * c = if h : P then a h * c else b h * c","typeReferences":[["Not"],["Decidable"],["Mul"],["instHMul"],["HMul","hMul"],["dite"],["Eq"]],"valueReferences":[["HMul","hMul"],["Decidable","casesOn"],["congrArg"],["dif_pos"],["congr"],["Eq","refl"],["id"],["instHMul"],["Eq","mpr"],["congrFun'"],["dif_neg"],["Eq"],["dite"]]},{"isProp":true,"kind":"theorem","name":["ite_div"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8 : Div.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.5 a b) c) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.5 (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8) a c) (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8) b c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a b c : α),\n  (if P then a else b) / c = if P then a / c else b / c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a b c : α),\n  (if P then a else b) / c = if P then a / c else b / c","typeReferences":[["HDiv","hDiv"],["Decidable"],["ite"],["Eq"],["instHDiv"],["Div"]],"valueReferences":[["dite_div"]]},{"isProp":true,"kind":"theorem","name":["dite_add_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8 : Add.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : P -> α) (d : (Not P) -> α), Eq.{succ u_2} α (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 (fun (h : P) => c h) (fun (h : Not P) => d h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 (fun (h : P) => HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8) (a h) (c h)) (fun (h : Not P) => HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8) (b h) (d h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) + if h : P then c h else d h) = if h : P then a h + c h else b h + d h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) + if h : P then c h else d h) = if h : P then a h + c h else b h + d h","typeReferences":[["HAdd","hAdd"],["Not"],["Decidable"],["instHAdd"],["Add"],["dite"],["Eq"]],"valueReferences":[["HAdd","hAdd"],["dif_pos"],["instHAdd"],["Eq","refl"],["congr"],["id"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["Eq"],["congrArg"]]},{"isProp":false,"kind":"recursor","name":["VAdd","rec"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}} {motive : (VAdd.{u, v} G P) -> Sort.{u_1}}, (forall (vadd : G -> P -> P), motive (VAdd.mk.{u, v} G P vadd)) -> (forall (t : VAdd.{u, v} G P), motive t)","typeFull":"{G : Type u} →\n  {P : Type v} →\n    {motive : VAdd G P → Sort u_1} → ((vadd : G → P → P) → motive { vadd := vadd }) → (t : VAdd G P) → motive t","typeReadable":"{G : Type u} →\n  {P : Type v} →\n    {motive : VAdd G P → Sort u_1} → ((vadd : G → P → P) → motive { vadd := vadd }) → (t : VAdd G P) → motive t","typeReferences":[["VAdd"],["VAdd","mk"]],"valueReferences":null},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Notation","Defs",0,"VAdd","ext","match_1"],"typeFallback":"forall {G._@.Mathlib.Algebra.Notation.Defs._hyg.173 : Type.{u_1}} {P._@.Mathlib.Algebra.Notation.Defs._hyg.174 : Type.{u_2}} (motive : (VAdd.{u_1, u_2} G._@.Mathlib.Algebra.Notation.Defs._hyg.173 P._@.Mathlib.Algebra.Notation.Defs._hyg.174) -> Prop) (h._@.Mathlib.Algebra.Notation.Defs._hyg.175.187 : VAdd.{u_1, u_2} G._@.Mathlib.Algebra.Notation.Defs._hyg.173 P._@.Mathlib.Algebra.Notation.Defs._hyg.174), (forall (vadd._@.Mathlib.Algebra.Notation.Defs._hyg.194 : G._@.Mathlib.Algebra.Notation.Defs._hyg.173 -> P._@.Mathlib.Algebra.Notation.Defs._hyg.174 -> P._@.Mathlib.Algebra.Notation.Defs._hyg.174), motive (VAdd.mk.{u_1, u_2} G._@.Mathlib.Algebra.Notation.Defs._hyg.173 P._@.Mathlib.Algebra.Notation.Defs._hyg.174 vadd._@.Mathlib.Algebra.Notation.Defs._hyg.194)) -> (motive h._@.Mathlib.Algebra.Notation.Defs._hyg.175.187)","typeFull":"∀ {G : Type u_1} {P : Type u_2} (motive : VAdd G P → Prop) (h : VAdd G P),\n  (∀ (vadd : G → P → P), motive { vadd := vadd }) → motive h","typeReadable":"∀ {G : Type u_1} {P : Type u_2} (motive : VAdd G P → Prop) (h : VAdd G P),\n  (∀ (vadd : G → P → P), motive { vadd := vadd }) → motive h","typeReferences":[["VAdd"],["VAdd","mk"]],"valueReferences":[["VAdd","casesOn"]]},{"isProp":false,"kind":"definition","name":["Star","mk","_flat_ctor"],"typeFallback":"forall {R : Type.{u}}, (R -> R) -> (Star.{u} R)","typeFull":"{R : Type u} → (R → R) → Star R","typeReadable":"{R : Type u} → (R → R) → Star R","typeReferences":[["Star"]],"valueReferences":[["Star","mk"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Notation","Defs",0,"SMul","ext","match_1"],"typeFallback":"forall {M._@.Mathlib.Algebra.Notation.Defs._hyg.7 : Type.{u_1}} {α._@.Mathlib.Algebra.Notation.Defs._hyg.8 : Type.{u_2}} (motive : (SMul.{u_1, u_2} M._@.Mathlib.Algebra.Notation.Defs._hyg.7 α._@.Mathlib.Algebra.Notation.Defs._hyg.8) -> Prop) (h._@.Mathlib.Algebra.Notation.Defs._hyg.9.21 : SMul.{u_1, u_2} M._@.Mathlib.Algebra.Notation.Defs._hyg.7 α._@.Mathlib.Algebra.Notation.Defs._hyg.8), (forall (smul._@.Mathlib.Algebra.Notation.Defs._hyg.28 : M._@.Mathlib.Algebra.Notation.Defs._hyg.7 -> α._@.Mathlib.Algebra.Notation.Defs._hyg.8 -> α._@.Mathlib.Algebra.Notation.Defs._hyg.8), motive (SMul.mk.{u_1, u_2} M._@.Mathlib.Algebra.Notation.Defs._hyg.7 α._@.Mathlib.Algebra.Notation.Defs._hyg.8 smul._@.Mathlib.Algebra.Notation.Defs._hyg.28)) -> (motive h._@.Mathlib.Algebra.Notation.Defs._hyg.9.21)","typeFull":"∀ {M : Type u_1} {α : Type u_2} (motive : SMul M α → Prop) (h : SMul M α),\n  (∀ (smul : M → α → α), motive { smul := smul }) → motive h","typeReadable":"∀ {M : Type u_1} {α : Type u_2} (motive : SMul M α → Prop) (h : SMul M α),\n  (∀ (smul : M → α → α), motive { smul := smul }) → motive h","typeReferences":[["SMul","mk"],["SMul"]],"valueReferences":[["SMul","casesOn"]]},{"isProp":false,"kind":"definition","name":["Star","noConfusionType"],"typeFallback":"Sort.{u_1} -> (forall {R : Type.{u}}, (Star.{u} R) -> (forall {R' : Type.{u}}, (Star.{u} R') -> Sort.{u_1}))","typeFull":"Sort u_1 → {R : Type u} → Star R → {R' : Type u} → Star R' → Sort u_1","typeReadable":"Sort u_1 → {R : Type u} → Star R → {R' : Type u} → Star R' → Sort u_1","typeReferences":[["Star"]],"valueReferences":[["HEq"],["Star","casesOn"]]},{"isProp":false,"kind":"definition","name":["HVAdd","mk","noConfusion"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}} {P : Sort.{u_1}} {hVAdd : α -> β -> γ} {hVAdd' : α -> β -> γ}, (Eq.{max (max (succ u) (succ v)) (succ w)} (HVAdd.{u, v, w} α β γ) (HVAdd.mk.{u, v, w} α β γ hVAdd) (HVAdd.mk.{u, v, w} α β γ hVAdd')) -> ((HEq.{max (max (succ u) (succ v)) (succ w)} (α -> β -> γ) hVAdd (α -> β -> γ) hVAdd') -> P) -> P","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : outParam (Type w)} →\n      {P : Sort u_1} → {hVAdd hVAdd' : α → β → γ} → { hVAdd := hVAdd } = { hVAdd := hVAdd' } → (hVAdd ≍ hVAdd' → P) → P","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : outParam (Type w)} →\n      {P : Sort u_1} → {hVAdd hVAdd' : α → β → γ} → { hVAdd := hVAdd } = { hVAdd := hVAdd' } → (hVAdd ≍ hVAdd' → P) → P","typeReferences":[["HVAdd"],["HVAdd","mk"],["outParam"],["HEq"],["Eq"]],"valueReferences":[["HVAdd"],["HVAdd","mk"],["outParam"],["Eq","refl"],["HVAdd","noConfusion"],["id"],["heq_of_eq"]]},{"isProp":true,"kind":"theorem","name":["ite_add_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8 : Add.{u_2} α] (a : α) (b : α) (c : α) (d : α), Eq.{succ u_2} α (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 a b) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 c d)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8) a c) (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8) b d))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a b c d : α),\n  ((if P then a else b) + if P then c else d) = if P then a + c else b + d","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a b c d : α),\n  ((if P then a else b) + if P then c else d) = if P then a + c else b + d","typeReferences":[["HAdd","hAdd"],["Decidable"],["ite"],["instHAdd"],["Add"],["Eq"]],"valueReferences":[["HAdd","hAdd"],["if_neg"],["ite"],["instHAdd"],["Eq","refl"],["congr"],["id"],["Eq","mpr"],["Decidable","casesOn"],["Eq"],["if_pos"],["congrArg"]]},{"isProp":false,"kind":"definition","name":["HVAdd","noConfusion"],"typeFallback":"forall {P : Sort.{u_1}} {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {t : HVAdd.{u, v, w} α β γ} {α' : Type.{u}} {β' : Type.{v}} {γ' : Type.{w}} {t' : HVAdd.{u, v, w} α' β' γ'}, (Eq.{succ (succ u)} Type.{u} α α') -> (Eq.{succ (succ v)} Type.{v} β β') -> (Eq.{succ (succ w)} Type.{w} γ γ') -> (HEq.{max (max (succ u) (succ v)) (succ w)} (HVAdd.{u, v, w} α β γ) t (HVAdd.{u, v, w} α' β' γ') t') -> (HVAdd.noConfusionType.{u_1, u, v, w} P α β γ t α' β' γ' t')","typeFull":"{P : Sort u_1} →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} →\n        {t : HVAdd α β γ} →\n          {α' : Type u} →\n            {β' : Type v} →\n              {γ' : Type w} → {t' : HVAdd α' β' γ'} → α = α' → β = β' → γ = γ' → t ≍ t' → HVAdd.noConfusionType P t t'","typeReadable":"{P : Sort u_1} →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} →\n        {t : HVAdd α β γ} →\n          {α' : Type u} →\n            {β' : Type v} →\n              {γ' : Type w} → {t' : HVAdd α' β' γ'} → α = α' → β = β' → γ = γ' → t ≍ t' → HVAdd.noConfusionType P t t'","typeReferences":[["HVAdd","noConfusionType"],["HVAdd"],["HEq"],["Eq"]],"valueReferences":[["HVAdd","casesOn"],["HVAdd","noConfusionType"],["HVAdd"],["HEq","refl"],["eq_of_heq"],["HEq"],["Eq","ndrec"]]},{"isProp":true,"kind":"theorem","name":["ite_mul_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8 : Mul.{u_2} α] (a : α) (b : α) (c : α) (d : α), Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 a b) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 c d)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8) a c) (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4066330426._hygCtx._hyg.8) b d))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a b c d : α),\n  ((if P then a else b) * if P then c else d) = if P then a * c else b * d","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a b c d : α),\n  ((if P then a else b) * if P then c else d) = if P then a * c else b * d","typeReferences":[["Decidable"],["ite"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"]],"valueReferences":[["if_neg"],["ite"],["Eq","refl"],["congr"],["id"],["instHMul"],["HMul","hMul"],["Eq","mpr"],["Decidable","casesOn"],["Eq"],["if_pos"],["congrArg"]]},{"isProp":false,"kind":"inductive","name":["VAdd"],"typeFallback":"Type.{u} -> Type.{v} -> Sort.{max (succ u) (succ v)}","typeFull":"Type u → Type v → Type (max u v)","typeReadable":"Type u → Type v → Type (max u v)","typeReferences":[],"valueReferences":null},{"isProp":false,"kind":"definition","name":["term_+ᵥ_"],"typeFallback":"Lean.TrailingParserDescr","typeFull":"Lean.TrailingParserDescr","typeReadable":"Lean.TrailingParserDescr","typeReferences":[["Lean","TrailingParserDescr"]],"valueReferences":[["Nat"],["Lean","ParserDescr","cat"],["Lean","Name","mkStr1"],["instOfNatNat"],["Lean","ParserDescr","trailingNode"],["Lean","ParserDescr","symbol"],["OfNat","ofNat"],["Lean","ParserDescr","binary"]]},{"isProp":false,"kind":"definition","name":["HVAdd","recOn"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {motive : (HVAdd.{u, v, w} α β γ) -> Sort.{u_1}} (t : HVAdd.{u, v, w} α β γ), (forall (hVAdd : α -> β -> γ), motive (HVAdd.mk.{u, v, w} α β γ hVAdd)) -> (motive t)","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : HVAdd α β γ → Sort u_1} →\n        (t : HVAdd α β γ) → ((hVAdd : α → β → γ) → motive { hVAdd := hVAdd }) → motive t","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : HVAdd α β γ → Sort u_1} →\n        (t : HVAdd α β γ) → ((hVAdd : α → β → γ) → motive { hVAdd := hVAdd }) → motive t","typeReferences":[["HVAdd"],["HVAdd","mk"]],"valueReferences":[["HVAdd","rec"]]},{"isProp":true,"kind":"theorem","name":["add_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8 : Add.{u_2} α] (a : α) (b : P -> α) (c : (Not P) -> α), Eq.{succ u_2} α (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8) a (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.5 (fun (h : P) => b h) (fun (h : Not P) => c h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.5 (fun (h : P) => HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8) a (b h)) (fun (h : Not P) => HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2936537523._hygCtx._hyg.8) a (c h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a : α) (b : P → α) (c : ¬P → α),\n  (a + if h : P then b h else c h) = if h : P then a + b h else a + c h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a : α) (b : P → α) (c : ¬P → α),\n  (a + if h : P then b h else c h) = if h : P then a + b h else a + c h","typeReferences":[["HAdd","hAdd"],["Not"],["Decidable"],["instHAdd"],["Add"],["dite"],["Eq"]],"valueReferences":[["HAdd","hAdd"],["dif_pos"],["instHAdd"],["Eq","refl"],["congr"],["id"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["Eq"],["congrArg"]]},{"isProp":false,"kind":"definition","name":["HVAdd","casesOn"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {motive : (HVAdd.{u, v, w} α β γ) -> Sort.{u_1}} (t : HVAdd.{u, v, w} α β γ), (forall (hVAdd : α -> β -> γ), motive (HVAdd.mk.{u, v, w} α β γ hVAdd)) -> (motive t)","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : HVAdd α β γ → Sort u_1} →\n        (t : HVAdd α β γ) → ((hVAdd : α → β → γ) → motive { hVAdd := hVAdd }) → motive t","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : HVAdd α β γ → Sort u_1} →\n        (t : HVAdd α β γ) → ((hVAdd : α → β → γ) → motive { hVAdd := hVAdd }) → motive t","typeReferences":[["HVAdd"],["HVAdd","mk"]],"valueReferences":[["HVAdd","rec"]]},{"isProp":false,"kind":"definition","name":["VSub","recOn"],"typeFallback":"forall {G : Type.{u_1}} {P : Type.{u_2}} {motive : (VSub.{u_1, u_2} G P) -> Sort.{u}} (t : VSub.{u_1, u_2} G P), (forall (vsub : P -> P -> G), motive (VSub.mk.{u_1, u_2} G P vsub)) -> (motive t)","typeFull":"{G : Type u_1} →\n  {P : Type u_2} →\n    {motive : VSub G P → Sort u} → (t : VSub G P) → ((vsub : P → P → G) → motive { vsub := vsub }) → motive t","typeReadable":"{G : Type u_1} →\n  {P : Type u_2} →\n    {motive : VSub G P → Sort u} → (t : VSub G P) → ((vsub : P → P → G) → motive { vsub := vsub }) → motive t","typeReferences":[["VSub","mk"],["VSub"]],"valueReferences":[["VSub","rec"]]},{"isProp":true,"kind":"theorem","name":["add_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8 : Add.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8) a (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.5 b c)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8) a b) (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8) a c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a b c : α),\n  (a + if P then b else c) = if P then a + b else a + c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a b c : α),\n  (a + if P then b else c) = if P then a + b else a + c","typeReferences":[["HAdd","hAdd"],["Decidable"],["ite"],["instHAdd"],["Add"],["Eq"]],"valueReferences":[["add_dite"]]},{"isProp":false,"kind":"definition","name":["VSub","noConfusionType"],"typeFallback":"Sort.{u} -> (forall {G : Type.{u_1}} {P : Type.{u_2}}, (VSub.{u_1, u_2} G P) -> (forall {G' : Type.{u_1}} {P' : Type.{u_2}}, (VSub.{u_1, u_2} G' P') -> Sort.{u}))","typeFull":"Sort u → {G : Type u_1} → {P : Type u_2} → VSub G P → {G' : Type u_1} → {P' : Type u_2} → VSub G' P' → Sort u","typeReadable":"Sort u → {G : Type u_1} → {P : Type u_2} → VSub G P → {G' : Type u_1} → {P' : Type u_2} → VSub G' P' → Sort u","typeReferences":[["VSub"]],"valueReferences":[["VSub","casesOn"],["HEq"]]},{"isProp":true,"kind":"theorem","name":["Subsingleton","eq_zero"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Notation.Defs.1536582495._hygCtx._hyg.4 : Zero.{u} α] [inst._@.Mathlib.Algebra.Notation.Defs.1536582495._hygCtx._hyg.7 : Subsingleton.{succ u} α] (a : α), Eq.{succ u} α a (OfNat.ofNat.{u} α 0 (Zero.toOfNat0.{u} α inst._@.Mathlib.Algebra.Notation.Defs.1536582495._hygCtx._hyg.4))","typeFull":"∀ {α : Type u} [inst : Zero α] [Subsingleton α] (a : α), a = 0","typeReadable":"∀ {α : Type u} [inst : Zero α] [Subsingleton α] (a : α), a = 0","typeReferences":[["Subsingleton"],["Zero","toOfNat0"],["Zero"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Subsingleton","elim"],["Zero","toOfNat0"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["VAdd","mk","_flat_ctor"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}}, (G -> P -> P) -> (VAdd.{u, v} G P)","typeFull":"{G : Type u} → {P : Type v} → (G → P → P) → VAdd G P","typeReadable":"{G : Type u} → {P : Type v} → (G → P → P) → VAdd G P","typeReferences":[["VAdd"]],"valueReferences":[["VAdd","mk"]]},{"isProp":true,"kind":"theorem","name":["dite_mul_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8 : Mul.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : P -> α) (d : (Not P) -> α), Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α �� (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 (fun (h : P) => c h) (fun (h : Not P) => d h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.5 (fun (h : P) => HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8) (a h) (c h)) (fun (h : Not P) => HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4220867991._hygCtx._hyg.8) (b h) (d h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) * if h : P then c h else d h) = if h : P then a h * c h else b h * d h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) * if h : P then c h else d h) = if h : P then a h * c h else b h * d h","typeReferences":[["Not"],["Decidable"],["Mul"],["instHMul"],["HMul","hMul"],["dite"],["Eq"]],"valueReferences":[["dif_pos"],["Eq","refl"],["congr"],["id"],["instHMul"],["HMul","hMul"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["Eq"],["congrArg"]]},{"isProp":false,"kind":"definition","name":["Star","recOn"],"typeFallback":"forall {R : Type.{u}} {motive : (Star.{u} R) -> Sort.{u_1}} (t : Star.{u} R), (forall (star : R -> R), motive (Star.mk.{u} R star)) -> (motive t)","typeFull":"{R : Type u} → {motive : Star R → Sort u_1} → (t : Star R) → ((star : R → R) → motive { star := star }) → motive t","typeReadable":"{R : Type u} → {motive : Star R → Sort u_1} → (t : Star R) → ((star : R → R) → motive { star := star }) → motive t","typeReferences":[["Star"],["Star","mk"]],"valueReferences":[["Star","rec"]]},{"isProp":false,"kind":"definition","name":["term_-ᵥ_"],"typeFallback":"Lean.TrailingParserDescr","typeFull":"Lean.TrailingParserDescr","typeReadable":"Lean.TrailingParserDescr","typeReferences":[["Lean","TrailingParserDescr"]],"valueReferences":[["Nat"],["Lean","ParserDescr","cat"],["Lean","Name","mkStr1"],["instOfNatNat"],["Lean","ParserDescr","trailingNode"],["Lean","ParserDescr","symbol"],["OfNat","ofNat"],["Lean","ParserDescr","binary"]]},{"isProp":true,"kind":"theorem","name":["sub_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8 : Sub.{u_2} α] (a : α) (b : P -> α) (c : (Not P) -> α), Eq.{succ u_2} α (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8) a (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.5 (fun (h : P) => b h) (fun (h : Not P) => c h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.5 (fun (h : P) => HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8) a (b h)) (fun (h : Not P) => HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8) a (c h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a : α) (b : P → α) (c : ¬P → α),\n  (a - if h : P then b h else c h) = if h : P then a - b h else a - c h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a : α) (b : P → α) (c : ¬P → α),\n  (a - if h : P then b h else c h) = if h : P then a - b h else a - c h","typeReferences":[["Not"],["Decidable"],["HSub","hSub"],["Sub"],["dite"],["instHSub"],["Eq"]],"valueReferences":[["dif_pos"],["Eq","refl"],["congr"],["id"],["HSub","hSub"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["instHSub"],["Eq"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_add"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8 : Add.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : α), Eq.{succ u_2} α (HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) c) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.5 (fun (h : P) => HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8) (a h) c) (fun (h : Not P) => HAdd.hAdd.{u_2, u_2, u_2} α α α (instHAdd.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.571083938._hygCtx._hyg.8) (b h) c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) + c = if h : P then a h + c else b h + c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Add α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) + c = if h : P then a h + c else b h + c","typeReferences":[["HAdd","hAdd"],["Not"],["Decidable"],["instHAdd"],["Add"],["dite"],["Eq"]],"valueReferences":[["instHAdd"],["Decidable","casesOn"],["congrArg"],["HAdd","hAdd"],["dif_pos"],["congr"],["Eq","refl"],["id"],["Eq","mpr"],["congrFun'"],["dif_neg"],["Eq"],["dite"]]},{"isProp":false,"kind":"definition","name":["VSub","noConfusion"],"typeFallback":"forall {P : Sort.{u}} {G : Type.{u_1}} {P_1 : Type.{u_2}} {t : VSub.{u_1, u_2} G P_1} {G' : Type.{u_1}} {P' : Type.{u_2}} {t' : VSub.{u_1, u_2} G' P'}, (Eq.{succ (succ u_1)} Type.{u_1} G G') -> (Eq.{succ (succ u_2)} Type.{u_2} P_1 P') -> (HEq.{max (succ u_1) (succ u_2)} (VSub.{u_1, u_2} G P_1) t (VSub.{u_1, u_2} G' P') t') -> (VSub.noConfusionType.{u, u_1, u_2} P G P_1 t G' P' t')","typeFull":"{P : Sort u} →\n  {G : Type u_1} →\n    {P_1 : Type u_2} →\n      {t : VSub G P_1} →\n        {G' : Type u_1} → {P' : Type u_2} → {t' : VSub G' P'} → G = G' → P_1 = P' → t ≍ t' → VSub.noConfusionType P t t'","typeReadable":"{P : Sort u} →\n  {G : Type u_1} →\n    {P_1 : Type u_2} →\n      {t : VSub G P_1} →\n        {G' : Type u_1} → {P' : Type u_2} → {t' : VSub G' P'} → G = G' → P_1 = P' → t ≍ t' → VSub.noConfusionType P t t'","typeReferences":[["VSub","noConfusionType"],["HEq"],["VSub"],["Eq"]],"valueReferences":[["VSub","casesOn"],["VSub","noConfusionType"],["HEq","refl"],["eq_of_heq"],["HEq"],["VSub"],["Eq","ndrec"]]},{"isProp":true,"kind":"theorem","name":["Subsingleton","eq_one"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Notation.Defs.1536582495._hygCtx._hyg.4 : One.{u} α] [inst._@.Mathlib.Algebra.Notation.Defs.1536582495._hygCtx._hyg.7 : Subsingleton.{succ u} α] (a : α), Eq.{succ u} α a (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α inst._@.Mathlib.Algebra.Notation.Defs.1536582495._hygCtx._hyg.4))","typeFull":"∀ {α : Type u} [inst : One α] [Subsingleton α] (a : α), a = 1","typeReadable":"∀ {α : Type u} [inst : One α] [Subsingleton α] (a : α), a = 1","typeReferences":[["Subsingleton"],["One","toOfNat1"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["One","toOfNat1"],["Subsingleton","elim"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["VSub","mk","_flat_ctor"],"typeFallback":"forall {G : outParam.{succ (succ u_1)} Type.{u_1}} {P : Type.{u_2}}, (P -> P -> G) -> (VSub.{u_1, u_2} G P)","typeFull":"{G : outParam (Type u_1)} → {P : Type u_2} → (P → P → G) → VSub G P","typeReadable":"{G : outParam (Type u_1)} → {P : Type u_2} → (P → P → G) → VSub G P","typeReferences":[["outParam"],["VSub"]],"valueReferences":[["VSub","mk"]]},{"isProp":true,"kind":"theorem","name":["SMul","ext_iff"],"typeFallback":"forall {M : Type.{u}} {α : Type.{v}} {x : SMul.{u, v} M α} {y : SMul.{u, v} M α}, Iff (Eq.{max (succ u) (succ v)} (SMul.{u, v} M α) x y) (Eq.{max (succ u) (succ v)} (M -> α -> α) (SMul.smul.{u, v} M α x) (SMul.smul.{u, v} M α y))","typeFull":"∀ {M : Type u} {α : Type v} {x y : SMul M α}, x = y ↔ SMul.smul = SMul.smul","typeReadable":"∀ {M : Type u} {α : Type v} {x y : SMul M α}, x = y ↔ SMul.smul = SMul.smul","typeReferences":[["SMul"],["Iff"],["SMul","smul"],["Eq"]],"valueReferences":[["SMul"],["HEq","refl"],["Eq","refl"],["Eq","symm"],["HEq"],["SMul","smul"],["SMul","ext"],["Eq","ndrec"],["Eq"],["Eq","casesOn"],["Iff","intro"]]},{"isProp":false,"kind":"definition","name":["Star","star"],"typeFallback":"forall {R : Type.{u}} [self : Star.{u} R], R -> R","typeFull":"{R : Type u} → [self : Star R] → R → R","typeReadable":"{R : Type u} → [self : Star R] → R → R","typeReferences":[["Star"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["VAdd","ctorIdx"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}}, (VAdd.{u, v} G P) -> Nat","typeFull":"{G : Type u} → {P : Type v} → VAdd G P → Nat","typeReadable":"{G : Type u} → {P : Type v} → VAdd G P → Nat","typeReferences":[["Nat"],["VAdd"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["Star","ctorIdx"],"typeFallback":"forall {R : Type.{u}}, (Star.{u} R) -> Nat","typeFull":"{R : Type u} → Star R → Nat","typeReadable":"{R : Type u} → Star R → Nat","typeReferences":[["Star"],["Nat"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["div_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8 : Div.{u_2} α] (a : α) (b : P -> α) (c : (Not P) -> α), Eq.{succ u_2} α (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8) a (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.5 (fun (h : P) => b h) (fun (h : Not P) => c h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.5 (fun (h : P) => HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8) a (b h)) (fun (h : Not P) => HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.510660181._hygCtx._hyg.8) a (c h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a : α) (b : P → α) (c : ¬P → α),\n  (a / if h : P then b h else c h) = if h : P then a / b h else a / c h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a : α) (b : P → α) (c : ¬P → α),\n  (a / if h : P then b h else c h) = if h : P then a / b h else a / c h","typeReferences":[["HDiv","hDiv"],["Not"],["Decidable"],["dite"],["Eq"],["instHDiv"],["Div"]],"valueReferences":[["HDiv","hDiv"],["dif_pos"],["Eq","refl"],["congr"],["id"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["Eq"],["instHDiv"],["congrArg"]]},{"isProp":false,"kind":"definition","name":["VSub","casesOn"],"typeFallback":"forall {G : Type.{u_1}} {P : Type.{u_2}} {motive : (VSub.{u_1, u_2} G P) -> Sort.{u}} (t : VSub.{u_1, u_2} G P), (forall (vsub : P -> P -> G), motive (VSub.mk.{u_1, u_2} G P vsub)) -> (motive t)","typeFull":"{G : Type u_1} →\n  {P : Type u_2} →\n    {motive : VSub G P → Sort u} → (t : VSub G P) → ((vsub : P → P → G) → motive { vsub := vsub }) → motive t","typeReadable":"{G : Type u_1} →\n  {P : Type u_2} →\n    {motive : VSub G P → Sort u} → (t : VSub G P) → ((vsub : P → P → G) → motive { vsub := vsub }) → motive t","typeReferences":[["VSub","mk"],["VSub"]],"valueReferences":[["VSub","rec"]]},{"isProp":true,"kind":"theorem","name":["div_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8 : Div.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8) a (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.5 b c)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.5 (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8) a b) (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8) a c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a b c : α),\n  (a / if P then b else c) = if P then a / b else a / c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a b c : α),\n  (a / if P then b else c) = if P then a / b else a / c","typeReferences":[["HDiv","hDiv"],["Decidable"],["ite"],["Eq"],["instHDiv"],["Div"]],"valueReferences":[["div_dite"]]},{"isProp":false,"kind":"definition","name":["VSub","vsub"],"typeFallback":"forall {G : outParam.{succ (succ u_1)} Type.{u_1}} {P : Type.{u_2}} [self : VSub.{u_1, u_2} G P], P -> P -> G","typeFull":"{G : outParam (Type u_1)} → {P : Type u_2} → [self : VSub G P] → P → P → G","typeReadable":"{G : outParam (Type u_1)} → {P : Type u_2} → [self : VSub G P] → P → P → G","typeReferences":[["outParam"],["VSub"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Notation_Defs___unexpand_VSub_vsub_1"],"typeFallback":"Lean.PrettyPrinter.Unexpander","typeFull":"Lean.PrettyPrinter.Unexpander","typeReadable":"Lean.PrettyPrinter.Unexpander","typeReferences":[["Lean","PrettyPrinter","Unexpander"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Bool","false"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","MonadQuotation","toMonadRef"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["List","cons"],["Bool","true"],["Unit","unit"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","PrettyPrinter","instMonadQuotationUnexpandM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["Bool","or"],["cond"],["Unit"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","withRef"],["Bool"],["Applicative","toPure"],["Lean","PrettyPrinter","UnexpandM"],["ReaderT","instMonadExceptOf"],["EStateM"],["Lean","Syntax","isOfKind"],["Lean","Syntax","matchesNull"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Eq"],["List","nil"],["Bind","bind"],["EStateM","instMonadExceptOfOfBacktrackable"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","node3"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["HVAdd","hVAdd"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}} [self : HVAdd.{u, v, w} α β γ], α -> β -> γ","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HVAdd α β γ] → α → β → γ","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HVAdd α β γ] → α → β → γ","typeReferences":[["HVAdd"],["outParam"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Notation_Defs___macroRules_term_+ᵥ__1"],"typeFallback":"Lean.Macro","typeFull":"Lean.Macro","typeReadable":"Lean.Macro","typeReferences":[["Lean","Macro"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Lean","Macro","State"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","Syntax","ident"],["String"],["Lean","Macro","Context"],["Lean","Macro","Exception","unsupportedSyntax"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["Lean","Syntax","Preresolved"],["List","cons"],["Bool","true"],["Lean","Syntax","Preresolved","decl"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","Macro","instMonadQuotationMacroM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Bool"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["EStateM"],["String","toRawSubstring'"],["Lean","Syntax","isOfKind"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Lean","addMacroScope"],["Lean","MacroM"],["Eq"],["Lean","Name","mkStr2"],["Lean","Syntax","node2"],["EStateM","instMonadExceptOfOfBacktrackable"],["Bind","bind"],["List","nil"],["Lean","Name","mkStr1"],["ite"],["Lean","Macro","instMonadRefMacroM"],["Lean","Macro","Exception"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["VAdd","casesOn"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}} {motive : (VAdd.{u, v} G P) -> Sort.{u_1}} (t : VAdd.{u, v} G P), (forall (vadd : G -> P -> P), motive (VAdd.mk.{u, v} G P vadd)) -> (motive t)","typeFull":"{G : Type u} →\n  {P : Type v} →\n    {motive : VAdd G P → Sort u_1} → (t : VAdd G P) → ((vadd : G → P → P) → motive { vadd := vadd }) → motive t","typeReadable":"{G : Type u} →\n  {P : Type v} →\n    {motive : VAdd G P → Sort u_1} → (t : VAdd G P) → ((vadd : G → P → P) → motive { vadd := vadd }) → motive t","typeReferences":[["VAdd"],["VAdd","mk"]],"valueReferences":[["VAdd","rec"]]},{"isProp":true,"kind":"theorem","name":["One","instNonempty"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Notation.Defs.2212486352._hygCtx._hyg.4 : One.{u} α], Nonempty.{succ u} α","typeFull":"∀ {α : Type u} [One α], Nonempty α","typeReadable":"∀ {α : Type u} [One α], Nonempty α","typeReferences":[["One"],["Nonempty"]],"valueReferences":[["Nonempty","intro"],["One","toOfNat1"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["Star","mk","noConfusion"],"typeFallback":"forall {R : Type.{u}} {P : Sort.{u_1}} {star : R -> R} {star' : R -> R}, (Eq.{succ u} (Star.{u} R) (Star.mk.{u} R star) (Star.mk.{u} R star')) -> ((HEq.{succ u} (R -> R) star (R -> R) star') -> P) -> P","typeFull":"{R : Type u} → {P : Sort u_1} → {star star' : R → R} → { star := star } = { star := star' } → (star ≍ star' → P) → P","typeReadable":"{R : Type u} → {P : Sort u_1} → {star star' : R → R} → { star := star } = { star := star' } → (star ≍ star' → P) → P","typeReferences":[["Star"],["HEq"],["Star","mk"],["Eq"]],"valueReferences":[["Star"],["Eq","refl"],["id"],["Star","noConfusion"],["heq_of_eq"],["Star","mk"]]},{"isProp":false,"kind":"definition","name":["HVAdd","ctorIdx"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}}, (HVAdd.{u, v, w} α β γ) -> Nat","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → HVAdd α β γ → Nat","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → HVAdd α β γ → Nat","typeReferences":[["HVAdd"],["Nat"],["outParam"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["VAdd","noConfusion"],"typeFallback":"forall {P : Sort.{u_1}} {G : Type.{u}} {P_1 : Type.{v}} {t : VAdd.{u, v} G P_1} {G' : Type.{u}} {P' : Type.{v}} {t' : VAdd.{u, v} G' P'}, (Eq.{succ (succ u)} Type.{u} G G') -> (Eq.{succ (succ v)} Type.{v} P_1 P') -> (HEq.{max (succ u) (succ v)} (VAdd.{u, v} G P_1) t (VAdd.{u, v} G' P') t') -> (VAdd.noConfusionType.{u_1, u, v} P G P_1 t G' P' t')","typeFull":"{P : Sort u_1} →\n  {G : Type u} →\n    {P_1 : Type v} →\n      {t : VAdd G P_1} →\n        {G' : Type u} → {P' : Type v} → {t' : VAdd G' P'} → G = G' → P_1 = P' → t ≍ t' → VAdd.noConfusionType P t t'","typeReadable":"{P : Sort u_1} →\n  {G : Type u} →\n    {P_1 : Type v} →\n      {t : VAdd G P_1} →\n        {G' : Type u} → {P' : Type v} → {t' : VAdd G' P'} → G = G' → P_1 = P' → t ≍ t' → VAdd.noConfusionType P t t'","typeReferences":[["VAdd"],["HEq"],["Eq"],["VAdd","noConfusionType"]],"valueReferences":[["VAdd"],["HEq","refl"],["eq_of_heq"],["HEq"],["VAdd","casesOn"],["Eq","ndrec"],["VAdd","noConfusionType"]]},{"isProp":true,"kind":"theorem","name":["VAdd","ext"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}} {x : VAdd.{u, v} G P} {y : VAdd.{u, v} G P}, (Eq.{max (succ u) (succ v)} (G -> P -> P) (VAdd.vadd.{u, v} G P x) (VAdd.vadd.{u, v} G P y)) -> (Eq.{max (succ u) (succ v)} (VAdd.{u, v} G P) x y)","typeFull":"∀ {G : Type u} {P : Type v} {x y : VAdd G P}, VAdd.vadd = VAdd.vadd → x = y","typeReadable":"∀ {G : Type u} {P : Type v} {x y : VAdd G P}, VAdd.vadd = VAdd.vadd → x = y","typeReferences":[["VAdd"],["VAdd","vadd"],["Eq"]],"valueReferences":[["VAdd"],["HEq","refl"],["VAdd","vadd"],["Eq","refl"],["Eq","symm"],["HEq"],["Eq","ndrec"],["Eq"],["Eq","casesOn"],["VAdd","mk"],["_private","Mathlib","Algebra","Notation","Defs",0,"VAdd","ext","match_1"]]},{"isProp":false,"kind":"recursor","name":["Star","rec"],"typeFallback":"forall {R : Type.{u}} {motive : (Star.{u} R) -> Sort.{u_1}}, (forall (star : R -> R), motive (Star.mk.{u} R star)) -> (forall (t : Star.{u} R), motive t)","typeFull":"{R : Type u} → {motive : Star R → Sort u_1} → ((star : R → R) → motive { star := star }) → (t : Star R) → motive t","typeReadable":"{R : Type u} → {motive : Star R → Sort u_1} → ((star : R → R) → motive { star := star }) → (t : Star R) → motive t","typeReferences":[["Star"],["Star","mk"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["VAdd","vadd"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}} [self : VAdd.{u, v} G P], G -> P -> P","typeFull":"{G : Type u} → {P : Type v} → [self : VAdd G P] → G → P → P","typeReadable":"{G : Type u} → {P : Type v} → [self : VAdd G P] → G → P → P","typeReferences":[["VAdd"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["VSub","ctorIdx"],"typeFallback":"forall {G : outParam.{succ (succ u_1)} Type.{u_1}} {P : Type.{u_2}}, (VSub.{u_1, u_2} G P) -> Nat","typeFull":"{G : outParam (Type u_1)} → {P : Type u_2} → VSub G P → Nat","typeReadable":"{G : outParam (Type u_1)} → {P : Type u_2} → VSub G P → Nat","typeReferences":[["Nat"],["outParam"],["VSub"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["VAdd","ext_iff"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}} {x : VAdd.{u, v} G P} {y : VAdd.{u, v} G P}, Iff (Eq.{max (succ u) (succ v)} (VAdd.{u, v} G P) x y) (Eq.{max (succ u) (succ v)} (G -> P -> P) (VAdd.vadd.{u, v} G P x) (VAdd.vadd.{u, v} G P y))","typeFull":"∀ {G : Type u} {P : Type v} {x y : VAdd G P}, x = y ↔ VAdd.vadd = VAdd.vadd","typeReadable":"∀ {G : Type u} {P : Type v} {x y : VAdd G P}, x = y ↔ VAdd.vadd = VAdd.vadd","typeReferences":[["VAdd"],["VAdd","vadd"],["Iff"],["Eq"]],"valueReferences":[["VAdd"],["HEq","refl"],["VAdd","vadd"],["Eq","refl"],["Eq","symm"],["HEq"],["VAdd","ext"],["Eq","ndrec"],["Eq"],["Eq","casesOn"],["Iff","intro"]]},{"isProp":false,"kind":"constructor","name":["VSub","mk"],"typeFallback":"forall {G : outParam.{succ (succ u_1)} Type.{u_1}} {P : Type.{u_2}}, (P -> P -> G) -> (VSub.{u_1, u_2} G P)","typeFull":"{G : outParam (Type u_1)} → {P : Type u_2} → (P → P → G) → VSub G P","typeReadable":"{G : outParam (Type u_1)} → {P : Type u_2} → (P → P → G) → VSub G P","typeReferences":[["outParam"],["VSub"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["dite_div_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8 : Div.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : P -> α) (d : (Not P) -> α), Eq.{succ u_2} α (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 (fun (h : P) => c h) (fun (h : Not P) => d h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 (fun (h : P) => HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8) (a h) (c h)) (fun (h : Not P) => HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8) (b h) (d h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) / if h : P then c h else d h) = if h : P then a h / c h else b h / d h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) / if h : P then c h else d h) = if h : P then a h / c h else b h / d h","typeReferences":[["HDiv","hDiv"],["Not"],["Decidable"],["dite"],["Eq"],["instHDiv"],["Div"]],"valueReferences":[["HDiv","hDiv"],["dif_pos"],["Eq","refl"],["congr"],["id"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["Eq"],["instHDiv"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["ite_div_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8 : Div.{u_2} α] (a : α) (b : α) (c : α) (d : α), Eq.{succ u_2} α (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 a b) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 c d)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8) a c) (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8) b d))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a b c d : α),\n  ((if P then a else b) / if P then c else d) = if P then a / c else b / d","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a b c d : α),\n  ((if P then a else b) / if P then c else d) = if P then a / c else b / d","typeReferences":[["HDiv","hDiv"],["Decidable"],["ite"],["Eq"],["instHDiv"],["Div"]],"valueReferences":[["dite_div_dite"]]},{"isProp":false,"kind":"inductive","name":["HVAdd"],"typeFallback":"Type.{u} -> Type.{v} -> (outParam.{succ (succ w)} Type.{w}) -> Sort.{max (max (succ u) (succ v)) (succ w)}","typeFull":"Type u → Type v → outParam (Type w) → Type (max (max u v) w)","typeReadable":"Type u → Type v → outParam (Type w) → Type (max (max u v) w)","typeReferences":[["outParam"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["ite_sub"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8 : Sub.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.5 a b) c) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.5 (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8) a c) (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.665606850._hygCtx._hyg.8) b c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a b c : α),\n  (if P then a else b) - c = if P then a - c else b - c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a b c : α),\n  (if P then a else b) - c = if P then a - c else b - c","typeReferences":[["Decidable"],["ite"],["HSub","hSub"],["Sub"],["instHSub"],["Eq"]],"valueReferences":[["dite_sub"]]},{"isProp":true,"kind":"theorem","name":["ite_mul"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8 : Mul.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.5 a b) c) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8) a c) (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2615697279._hygCtx._hyg.8) b c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a b c : α),\n  (if P then a else b) * c = if P then a * c else b * c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a b c : α),\n  (if P then a else b) * c = if P then a * c else b * c","typeReferences":[["Decidable"],["ite"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"]],"valueReferences":[["dite_mul"]]},{"isProp":false,"kind":"recursor","name":["VSub","rec"],"typeFallback":"forall {G : Type.{u_1}} {P : Type.{u_2}} {motive : (VSub.{u_1, u_2} G P) -> Sort.{u}}, (forall (vsub : P -> P -> G), motive (VSub.mk.{u_1, u_2} G P vsub)) -> (forall (t : VSub.{u_1, u_2} G P), motive t)","typeFull":"{G : Type u_1} →\n  {P : Type u_2} →\n    {motive : VSub G P → Sort u} → ((vsub : P → P → G) → motive { vsub := vsub }) → (t : VSub G P) → motive t","typeReadable":"{G : Type u_1} →\n  {P : Type u_2} →\n    {motive : VSub G P → Sort u} → ((vsub : P → P → G) → motive { vsub := vsub }) → (t : VSub G P) → motive t","typeReferences":[["VSub","mk"],["VSub"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["instHVAdd","eq_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Init.Prelude.4154482078._hygCtx._hyg.4 : VAdd.{u_1, u_2} α β], Eq.{max (succ u_1) (succ u_2)} (HVAdd.{u_1, u_2, u_2} α β β) (instHVAdd.{u_1, u_2} α β inst._@.Init.Prelude.4154482078._hygCtx._hyg.4) (HVAdd.mk.{u_1, u_2, u_2} α β β (VAdd.vadd.{u_1, u_2} α β inst._@.Init.Prelude.4154482078._hygCtx._hyg.4))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : VAdd α β], instHVAdd = { hVAdd := VAdd.vadd }","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : VAdd α β], instHVAdd = { hVAdd := VAdd.vadd }","typeReferences":[["HVAdd"],["VAdd"],["HVAdd","mk"],["instHVAdd"],["VAdd","vadd"],["Eq"]],"valueReferences":[["HVAdd"],["instHVAdd"],["Eq","refl"]]},{"isProp":false,"kind":"definition","name":["HVAdd","noConfusionType"],"typeFallback":"Sort.{u_1} -> (forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}}, (HVAdd.{u, v, w} α β γ) -> (forall {α' : Type.{u}} {β' : Type.{v}} {γ' : Type.{w}}, (HVAdd.{u, v, w} α' β' γ') -> Sort.{u_1}))","typeFull":"Sort u_1 →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} → HVAdd α β γ → {α' : Type u} → {β' : Type v} → {γ' : Type w} → HVAdd α' β' γ' → Sort u_1","typeReadable":"Sort u_1 →\n  {α : Type u} →\n    {β : Type v} →\n      {γ : Type w} → HVAdd α β γ → {α' : Type u} → {β' : Type v} → {γ' : Type w} → HVAdd α' β' γ' → Sort u_1","typeReferences":[["HVAdd"]],"valueReferences":[["HVAdd","casesOn"],["HEq"]]},{"isProp":true,"kind":"theorem","name":["dite_div"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8 : Div.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : α), Eq.{succ u_2} α (HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) c) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.5 (fun (h : P) => HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8) (a h) c) (fun (h : Not P) => HDiv.hDiv.{u_2, u_2, u_2} α α α (instHDiv.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8) (b h) c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) / c = if h : P then a h / c else b h / c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Div α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) / c = if h : P then a h / c else b h / c","typeReferences":[["HDiv","hDiv"],["Not"],["Decidable"],["dite"],["Eq"],["instHDiv"],["Div"]],"valueReferences":[["Decidable","casesOn"],["instHDiv"],["congrArg"],["HDiv","hDiv"],["dif_pos"],["congr"],["Eq","refl"],["id"],["Eq","mpr"],["congrFun'"],["dif_neg"],["Eq"],["dite"]]},{"isProp":true,"kind":"theorem","name":["sub_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8 : Sub.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8) a (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.5 b c)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.5 (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8) a b) (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.2011596237._hygCtx._hyg.8) a c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a b c : α),\n  (a - if P then b else c) = if P then a - b else a - c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a b c : α),\n  (a - if P then b else c) = if P then a - b else a - c","typeReferences":[["Decidable"],["ite"],["HSub","hSub"],["Sub"],["instHSub"],["Eq"]],"valueReferences":[["sub_dite"]]},{"isProp":false,"kind":"recursor","name":["HVAdd","rec"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : Type.{w}} {motive : (HVAdd.{u, v, w} α β γ) -> Sort.{u_1}}, (forall (hVAdd : α -> β -> γ), motive (HVAdd.mk.{u, v, w} α β γ hVAdd)) -> (forall (t : HVAdd.{u, v, w} α β γ), motive t)","typeFull":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : HVAdd α β γ → Sort u_1} →\n        ((hVAdd : α → β → γ) → motive { hVAdd := hVAdd }) → (t : HVAdd α β γ) → motive t","typeReadable":"{α : Type u} →\n  {β : Type v} →\n    {γ : Type w} →\n      {motive : HVAdd α β γ → Sort u_1} →\n        ((hVAdd : α → β → γ) → motive { hVAdd := hVAdd }) → (t : HVAdd α β γ) → motive t","typeReferences":[["HVAdd"],["HVAdd","mk"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["VAdd","noConfusionType"],"typeFallback":"Sort.{u_1} -> (forall {G : Type.{u}} {P : Type.{v}}, (VAdd.{u, v} G P) -> (forall {G' : Type.{u}} {P' : Type.{v}}, (VAdd.{u, v} G' P') -> Sort.{u_1}))","typeFull":"Sort u_1 → {G : Type u} → {P : Type v} → VAdd G P → {G' : Type u} → {P' : Type v} → VAdd G' P' → Sort u_1","typeReadable":"Sort u_1 → {G : Type u} → {P : Type v} → VAdd G P → {G' : Type u} → {P' : Type v} → VAdd G' P' → Sort u_1","typeReferences":[["VAdd"]],"valueReferences":[["HEq"],["VAdd","casesOn"]]},{"isProp":false,"kind":"definition","name":["VSub","mk","noConfusion"],"typeFallback":"forall {G : outParam.{succ (succ u_1)} Type.{u_1}} {P : Type.{u_2}} {P_1 : Sort.{u}} {vsub : P -> P -> G} {vsub' : P -> P -> G}, (Eq.{max (succ u_1) (succ u_2)} (VSub.{u_1, u_2} G P) (VSub.mk.{u_1, u_2} G P vsub) (VSub.mk.{u_1, u_2} G P vsub')) -> ((HEq.{max (succ u_1) (succ u_2)} (P -> P -> G) vsub (P -> P -> G) vsub') -> P_1) -> P_1","typeFull":"{G : outParam (Type u_1)} →\n  {P : Type u_2} →\n    {P_1 : Sort u} → {vsub vsub' : P → P → G} → { vsub := vsub } = { vsub := vsub' } → (vsub ≍ vsub' → P_1) → P_1","typeReadable":"{G : outParam (Type u_1)} →\n  {P : Type u_2} →\n    {P_1 : Sort u} → {vsub vsub' : P → P → G} → { vsub := vsub } = { vsub := vsub' } → (vsub ≍ vsub' → P_1) → P_1","typeReferences":[["outParam"],["HEq"],["VSub","mk"],["VSub"],["Eq"]],"valueReferences":[["outParam"],["Eq","refl"],["VSub","mk"],["id"],["VSub"],["heq_of_eq"],["VSub","noConfusion"]]},{"isProp":true,"kind":"theorem","name":["instHSMul","eq_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Init.Prelude.4154482078._hygCtx._hyg.4 : SMul.{u_1, u_2} α β], Eq.{max (succ u_1) (succ u_2)} (HSMul.{u_1, u_2, u_2} α β β) (instHSMul.{u_1, u_2} α β inst._@.Init.Prelude.4154482078._hygCtx._hyg.4) (HSMul.mk.{u_1, u_2, u_2} α β β (SMul.smul.{u_1, u_2} α β inst._@.Init.Prelude.4154482078._hygCtx._hyg.4))","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β], instHSMul = { hSMul := SMul.smul }","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : SMul α β], instHSMul = { hSMul := SMul.smul }","typeReferences":[["SMul"],["HSMul"],["instHSMul"],["SMul","smul"],["HSMul","mk"],["Eq"]],"valueReferences":[["Eq","refl"],["HSMul"],["instHSMul"]]},{"isProp":true,"kind":"theorem","name":["mul_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8 : Mul.{u_2} α] (a : α) (b : α) (c : α), Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8) a (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.5 b c)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8) a b) (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.1225709769._hygCtx._hyg.8) a c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a b c : α),\n  (a * if P then b else c) = if P then a * b else a * c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Mul α] (a b c : α),\n  (a * if P then b else c) = if P then a * b else a * c","typeReferences":[["Decidable"],["ite"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"]],"valueReferences":[["mul_dite"]]},{"isProp":true,"kind":"theorem","name":["dite_sub"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8 : Sub.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : α), Eq.{succ u_2} α (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) c) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.5 (fun (h : P) => HSub.hSub.{u_2, u_2, u_2} α α �� (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8) (a h) c) (fun (h : Not P) => HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.873242624._hygCtx._hyg.8) (b h) c))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) - c = if h : P then a h - c else b h - c","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a : P → α) (b : ¬P → α) (c : α),\n  (if h : P then a h else b h) - c = if h : P then a h - c else b h - c","typeReferences":[["Not"],["Decidable"],["HSub","hSub"],["Sub"],["dite"],["instHSub"],["Eq"]],"valueReferences":[["Decidable","casesOn"],["congrArg"],["dif_pos"],["congr"],["Eq","refl"],["HSub","hSub"],["id"],["Eq","mpr"],["congrFun'"],["dif_neg"],["Eq"],["instHSub"],["dite"]]},{"isProp":true,"kind":"theorem","name":["SMul","ext"],"typeFallback":"forall {M : Type.{u}} {α : Type.{v}} {x : SMul.{u, v} M α} {y : SMul.{u, v} M α}, (Eq.{max (succ u) (succ v)} (M -> α -> α) (SMul.smul.{u, v} M α x) (SMul.smul.{u, v} M α y)) -> (Eq.{max (succ u) (succ v)} (SMul.{u, v} M α) x y)","typeFull":"∀ {M : Type u} {α : Type v} {x y : SMul M α}, SMul.smul = SMul.smul → x = y","typeReadable":"∀ {M : Type u} {α : Type v} {x y : SMul M α}, SMul.smul = SMul.smul → x = y","typeReferences":[["SMul"],["SMul","smul"],["Eq"]],"valueReferences":[["SMul","mk"],["SMul"],["HEq","refl"],["Eq","refl"],["_private","Mathlib","Algebra","Notation","Defs",0,"SMul","ext","match_1"],["Eq","symm"],["HEq"],["SMul","smul"],["Eq","ndrec"],["Eq"],["Eq","casesOn"]]},{"isProp":false,"kind":"definition","name":["VAdd","recOn"],"typeFallback":"forall {G : Type.{u}} {P : Type.{v}} {motive : (VAdd.{u, v} G P) -> Sort.{u_1}} (t : VAdd.{u, v} G P), (forall (vadd : G -> P -> P), motive (VAdd.mk.{u, v} G P vadd)) -> (motive t)","typeFull":"{G : Type u} →\n  {P : Type v} →\n    {motive : VAdd G P → Sort u_1} → (t : VAdd G P) → ((vadd : G → P → P) → motive { vadd := vadd }) → motive t","typeReadable":"{G : Type u} →\n  {P : Type v} →\n    {motive : VAdd G P → Sort u_1} → (t : VAdd G P) → ((vadd : G → P → P) → motive { vadd := vadd }) → motive t","typeReferences":[["VAdd"],["VAdd","mk"]],"valueReferences":[["VAdd","rec"]]},{"isProp":true,"kind":"theorem","name":["ite_sub_ite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8 : Sub.{u_2} α] (a : α) (b : α) (c : α) (d : α), Eq.{succ u_2} α (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 a b) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 c d)) (ite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.5 (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8) a c) (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.3086771933._hygCtx._hyg.8) b d))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a b c d : α),\n  ((if P then a else b) - if P then c else d) = if P then a - c else b - d","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a b c d : α),\n  ((if P then a else b) - if P then c else d) = if P then a - c else b - d","typeReferences":[["Decidable"],["ite"],["HSub","hSub"],["Sub"],["instHSub"],["Eq"]],"valueReferences":[["dite_sub_dite"]]},{"isProp":true,"kind":"theorem","name":["dite_sub_dite"],"typeFallback":"forall {α : Type.{u_2}} (P : Prop) [inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 : Decidable P] [inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8 : Sub.{u_2} α] (a : P -> α) (b : (Not P) -> α) (c : P -> α) (d : (Not P) -> α), Eq.{succ u_2} α (HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 (fun (h : P) => a h) (fun (h : Not P) => b h)) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 (fun (h : P) => c h) (fun (h : Not P) => d h))) (dite.{succ u_2} α P inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.5 (fun (h : P) => HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8) (a h) (c h)) (fun (h : Not P) => HSub.hSub.{u_2, u_2, u_2} α α α (instHSub.{u_2} α inst._@.Mathlib.Algebra.Notation.Defs.4276398922._hygCtx._hyg.8) (b h) (d h)))","typeFull":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) - if h : P then c h else d h) = if h : P then a h - c h else b h - d h","typeReadable":"∀ {α : Type u_2} (P : Prop) [inst : Decidable P] [inst_1 : Sub α] (a : P → α) (b : ¬P → α) (c : P → α) (d : ¬P → α),\n  ((if h : P then a h else b h) - if h : P then c h else d h) = if h : P then a h - c h else b h - d h","typeReferences":[["Not"],["Decidable"],["HSub","hSub"],["Sub"],["dite"],["instHSub"],["Eq"]],"valueReferences":[["dif_pos"],["Eq","refl"],["congr"],["id"],["HSub","hSub"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["dite"],["instHSub"],["Eq"],["congrArg"]]},{"isProp":false,"kind":"constructor","name":["HVAdd","mk"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}}, (α -> β -> γ) -> (HVAdd.{u, v, w} α β γ)","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → (α → β → γ) → HVAdd α β γ","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → (α → β → γ) → HVAdd α β γ","typeReferences":[["HVAdd"],["outParam"]],"valueReferences":null},{"isProp":false,"kind":"inductive","name":["Star"],"typeFallback":"Type.{u} -> Type.{u}","typeFull":"Type u → Type u","typeReadable":"Type u → Type u","typeReferences":[],"valueReferences":null},{"isProp":false,"kind":"definition","name":["Star","casesOn"],"typeFallback":"forall {R : Type.{u}} {motive : (Star.{u} R) -> Sort.{u_1}} (t : Star.{u} R), (forall (star : R -> R), motive (Star.mk.{u} R star)) -> (motive t)","typeFull":"{R : Type u} → {motive : Star R → Sort u_1} → (t : Star R) → ((star : R → R) → motive { star := star }) → motive t","typeReadable":"{R : Type u} → {motive : Star R → Sort u_1} → (t : Star R) → ((star : R → R) → motive { star := star }) → motive t","typeReferences":[["Star"],["Star","mk"]],"valueReferences":[["Star","rec"]]},{"isProp":false,"kind":"definition","name":["HVAdd","mk","_flat_ctor"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} {γ : outParam.{succ (succ w)} Type.{w}}, (α -> β -> γ) -> (HVAdd.{u, v, w} α β γ)","typeFull":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → (α → β → γ) → HVAdd α β γ","typeReadable":"{α : Type u} → {β : Type v} → {γ : outParam (Type w)} → (α → β → γ) → HVAdd α β γ","typeReferences":[["HVAdd"],["outParam"]],"valueReferences":[["HVAdd","mk"]]},{"isProp":false,"kind":"constructor","name":["Star","mk"],"typeFallback":"forall {R : Type.{u}}, (R -> R) -> (Star.{u} R)","typeFull":"{R : Type u} → (R → R) → Star R","typeReadable":"{R : Type u} → (R → R) → Star R","typeReferences":[["Star"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Notation_Defs___macroRules_term_-ᵥ__1"],"typeFallback":"Lean.Macro","typeFull":"Lean.Macro","typeReadable":"Lean.Macro","typeReferences":[["Lean","Macro"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Lean","Macro","State"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","Syntax","ident"],["String"],["Lean","Macro","Context"],["Lean","Macro","Exception","unsupportedSyntax"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["Lean","Syntax","Preresolved"],["List","cons"],["Bool","true"],["Lean","Syntax","Preresolved","decl"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","Macro","instMonadQuotationMacroM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Bool"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["EStateM"],["String","toRawSubstring'"],["Lean","Syntax","isOfKind"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Lean","addMacroScope"],["Lean","MacroM"],["Eq"],["Lean","Name","mkStr2"],["Lean","Syntax","node2"],["EStateM","instMonadExceptOfOfBacktrackable"],["Bind","bind"],["List","nil"],["Lean","Name","mkStr1"],["ite"],["Lean","Macro","instMonadRefMacroM"],["Lean","Macro","Exception"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Group.Action.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["smul_le_smul_left"],"typeFallback":"forall {M : Type.{u_2}} {α : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.5 : SMul.{u_2, u_3} M α] [inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.9 : Preorder.{u_3} α] [inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.12 : CovariantClass.{u_2, u_3} M α (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.5)) (LE.le.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.9))] (m : M) {a : α} {b : α}, (LE.le.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.9) a b) -> (LE.le.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.9) (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.5) m a) (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.759613997._hygCtx._hyg.5) m b))","typeFull":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : Preorder α] [CovariantClass M α HSMul.hSMul LE.le] (m : M)\n  {a b : α}, a ≤ b → m • a ≤ m • b","typeReadable":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : Preorder α] [CovariantClass M α HSMul.hSMul LE.le] (m : M)\n  {a b : α}, a ≤ b → m • a ≤ m • b","typeReferences":[["Preorder"],["SMul"],["LE","le"],["HSMul","hSMul"],["instHSMul"],["CovariantClass"],["Preorder","toLE"]],"valueReferences":[["smul_mono_right"]]},{"isProp":true,"kind":"theorem","name":["smul_inf_le"],"typeFallback":"forall {M : Type.{u_2}} {α : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.5 : SMul.{u_2, u_3} M α] [inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.9 : SemilatticeInf.{u_3} α] [inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.12 : CovariantClass.{u_2, u_3} M α (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.5)) (LE.le.{u_3} α (Preorder.toLE.{u_3} α (PartialOrder.toPreorder.{u_3} α (SemilatticeInf.toPartialOrder.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.9))))] (m : M) (a₁ : α) (a₂ : α), LE.le.{u_3} α (Preorder.toLE.{u_3} α (PartialOrder.toPreorder.{u_3} α (SemilatticeInf.toPartialOrder.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.9))) (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.5) m (Min.min.{u_3} α (SemilatticeInf.toMin.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.9) a₁ a₂)) (Min.min.{u_3} α (SemilatticeInf.toMin.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.9) (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.5) m a₁) (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.1199600213._hygCtx._hyg.5) m a₂))","typeFull":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : SemilatticeInf α] [CovariantClass M α HSMul.hSMul LE.le]\n  (m : M) (a₁ a₂ : α), m • (a₁ ⊓ a₂) ≤ m • a₁ ⊓ m • a₂","typeReadable":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : SemilatticeInf α] [CovariantClass M α HSMul.hSMul LE.le]\n  (m : M) (a₁ a₂ : α), m • (a₁ ⊓ a₂) ≤ m • a₁ ⊓ m • a₂","typeReferences":[["SemilatticeInf"],["SemilatticeInf","toMin"],["SMul"],["PartialOrder","toPreorder"],["Min","min"],["LE","le"],["HSMul","hSMul"],["instHSMul"],["CovariantClass"],["Preorder","toLE"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["PartialOrder","toPreorder"],["HSMul","hSMul"],["instHSMul"],["Monotone","map_inf_le"],["SemilatticeInf","toPartialOrder"],["smul_mono_right"]]},{"isProp":true,"kind":"theorem","name":["smul_mono_right"],"typeFallback":"forall {M : Type.{u_2}} {α : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.5 : SMul.{u_2, u_3} M α] [inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.9 : Preorder.{u_3} α] [inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.12 : CovariantClass.{u_2, u_3} M α (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.5)) (LE.le.{u_3} α (Preorder.toLE.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.9))] (m : M), Monotone.{u_3, u_3} α α inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.9 (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.292009661._hygCtx._hyg.5) m)","typeFull":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : Preorder α] [CovariantClass M α HSMul.hSMul LE.le] (m : M),\n  Monotone (HSMul.hSMul m)","typeReadable":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : Preorder α] [CovariantClass M α HSMul.hSMul LE.le] (m : M),\n  Monotone (HSMul.hSMul m)","typeReferences":[["Preorder"],["SMul"],["Monotone"],["LE","le"],["HSMul","hSMul"],["instHSMul"],["CovariantClass"],["Preorder","toLE"]],"valueReferences":[["LE","le"],["HSMul","hSMul"],["instHSMul"],["Preorder","toLE"],["CovariantClass","elim"]]},{"isProp":true,"kind":"theorem","name":["le_pow_smul"],"typeFallback":"forall {G : Type.{u_4}} [inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6 : Monoid.{u_4} G] {α : Type.{u_5}} [inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.10 : Preorder.{u_5} α] {g : G} {a : α} [inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.15 : MulAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6] [inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.19 : CovariantClass.{u_4, u_5} G α (HSMul.hSMul.{u_4, u_5, u_5} G α α (instHSMul.{u_4, u_5} G α (SemigroupAction.toSMul.{u_4, u_5} G α (Monoid.toSemigroup.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6) (MulAction.toSemigroupAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.15)))) (LE.le.{u_5} α (Preorder.toLE.{u_5} α inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.10))], (LE.le.{u_5} α (Preorder.toLE.{u_5} α inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.10) a (HSMul.hSMul.{u_4, u_5, u_5} G α α (instHSMul.{u_4, u_5} G α (SemigroupAction.toSMul.{u_4, u_5} G α (Monoid.toSemigroup.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6) (MulAction.toSemigroupAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.15))) g a)) -> (forall (n : Nat), LE.le.{u_5} α (Preorder.toLE.{u_5} α inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.10) a (HSMul.hSMul.{u_4, u_5, u_5} G α α (instHSMul.{u_4, u_5} G α (SemigroupAction.toSMul.{u_4, u_5} G α (Monoid.toSemigroup.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6) (MulAction.toSemigroupAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.15))) (HPow.hPow.{u_4, 0, u_4} G Nat G (instHPow.{u_4, 0} G Nat (Monoid.toPow.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.3094995653._hygCtx._hyg.6)) g n) a))","typeFull":"∀ {G : Type u_4} [inst : Monoid G] {α : Type u_5} [inst_1 : Preorder α] {g : G} {a : α} [inst_2 : MulAction G α]\n  [CovariantClass G α HSMul.hSMul LE.le], a ≤ g • a → ∀ (n : ℕ), a ≤ g ^ n • a","typeReadable":"∀ {G : Type u_4} [inst : Monoid G] {α : Type u_5} [inst_1 : Preorder α] {g : G} {a : α} [inst_2 : MulAction G α]\n  [CovariantClass G α HSMul.hSMul LE.le], a ≤ g • a → ∀ (n : ℕ), a ≤ g ^ n • a","typeReferences":[["instHPow"],["SemigroupAction","toSMul"],["CovariantClass"],["HPow","hPow"],["Preorder"],["Nat"],["Monoid","toPow"],["MulAction"],["HSMul","hSMul"],["LE","le"],["Monoid"],["instHSMul"],["Monoid","toSemigroup"],["Preorder","toLE"],["MulAction","toSemigroupAction"]],"valueReferences":[["instAddNat"],["MulOneClass","toMulOne"],["le_refl"],["pow_succ'"],["HMul","hMul"],["SemigroupAction","toSMul"],["LE","le","trans"],["Semigroup","toMul"],["congrArg"],["MulOne","toMul"],["Monoid","toPow"],["instOfNatNat"],["Monoid","toMulOneClass"],["instHSMul"],["Monoid","toSemigroup"],["Preorder","toLE"],["Eq"],["SemigroupAction","mul_smul"],["instHPow"],["MulOne","toOne"],["instHAdd"],["Nat","recAux"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["One","toOfNat1"],["HSMul","hSMul"],["LE","le"],["one_smul"],["id"],["instHMul"],["Eq","mpr"],["MulAction","toSemigroupAction"],["pow_zero"],["smul_mono_right"]]},{"isProp":true,"kind":"theorem","name":["smul_iInf_le"],"typeFallback":"forall {ι : Sort.{u_1}} {M : Type.{u_2}} {α : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.5 : SMul.{u_2, u_3} M α] [inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.9 : CompleteLattice.{u_3} α] [inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.12 : CovariantClass.{u_2, u_3} M α (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.5)) (LE.le.{u_3} α (Preorder.toLE.{u_3} α (PartialOrder.toPreorder.{u_3} α (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_3} α (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} α (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} α (CompleteLattice.toConditionallyCompleteLattice.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.9)))))))] {m : M} {t : ι -> α}, LE.le.{u_3} α (Preorder.toLE.{u_3} α (PartialOrder.toPreorder.{u_3} α (ConditionallyCompletePartialOrderSup.toPartialOrder.{u_3} α (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup.{u_3} α (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} α (CompleteLattice.toConditionallyCompleteLattice.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.9)))))) (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.5) m (iInf.{u_3, u_1} α ι (ConditionallyCompletePartialOrderInf.toInfSet.{u_3} α (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{u_3} α (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} α (CompleteLattice.toConditionallyCompleteLattice.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.9)))) t)) (iInf.{u_3, u_1} α ι (ConditionallyCompletePartialOrderInf.toInfSet.{u_3} α (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{u_3} α (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{u_3} α (CompleteLattice.toConditionallyCompleteLattice.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.9)))) (fun (i : ι) => HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.3023235747._hygCtx._hyg.5) m (t i)))","typeFull":"∀ {ι : Sort u_1} {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : CompleteLattice α]\n  [CovariantClass M α HSMul.hSMul LE.le] {m : M} {t : ι → α}, m • iInf t ≤ ⨅ i, m • t i","typeReadable":"∀ {ι : Sort u_1} {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : CompleteLattice α]\n  [CovariantClass M α HSMul.hSMul LE.le] {m : M} {t : ι → α}, m • iInf t ≤ ⨅ i, m • t i","typeReferences":[["PartialOrder","toPreorder"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["CovariantClass"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["SMul"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["HSMul","hSMul"],["LE","le"],["instHSMul"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["Preorder","toLE"],["CompleteLattice"],["CompleteLattice","toConditionallyCompleteLattice"]],"valueReferences":[["PartialOrder","toPreorder"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["ConditionallyCompletePartialOrderSup","toPartialOrder"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderSup"],["HSMul","hSMul"],["le_iInf"],["iInf_le"],["instHSMul"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["CompleteLattice","toConditionallyCompleteLattice"],["smul_mono_right"]]},{"isProp":true,"kind":"theorem","name":["smul_strictMono_right"],"typeFallback":"forall {M : Type.{u_2}} {α : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.5 : SMul.{u_2, u_3} M α] [inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.9 : Preorder.{u_3} α] [inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.12 : CovariantClass.{u_2, u_3} M α (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.5)) (LT.lt.{u_3} α (Preorder.toLT.{u_3} α inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.9))] (m : M), StrictMono.{u_3, u_3} α α inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.9 (HSMul.hSMul.{u_2, u_3, u_3} M α α (instHSMul.{u_2, u_3} M α inst._@.Mathlib.Algebra.Order.Group.Action.530935288._hygCtx._hyg.5) m)","typeFull":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : Preorder α] [CovariantClass M α HSMul.hSMul LT.lt] (m : M),\n  StrictMono (HSMul.hSMul m)","typeReadable":"∀ {M : Type u_2} {α : Type u_3} [inst : SMul M α] [inst_1 : Preorder α] [CovariantClass M α HSMul.hSMul LT.lt] (m : M),\n  StrictMono (HSMul.hSMul m)","typeReferences":[["LT","lt"],["Preorder"],["SMul"],["HSMul","hSMul"],["Preorder","toLT"],["instHSMul"],["CovariantClass"],["StrictMono"]],"valueReferences":[["LT","lt"],["HSMul","hSMul"],["Preorder","toLT"],["instHSMul"],["CovariantClass","elim"]]},{"isProp":true,"kind":"theorem","name":["pow_smul_le"],"typeFallback":"forall {G : Type.{u_4}} [inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6 : Monoid.{u_4} G] {α : Type.{u_5}} [inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.10 : Preorder.{u_5} α] {g : G} {a : α} [inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.15 : MulAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6] [inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.19 : CovariantClass.{u_4, u_5} G α (HSMul.hSMul.{u_4, u_5, u_5} G α α (instHSMul.{u_4, u_5} G α (SemigroupAction.toSMul.{u_4, u_5} G α (Monoid.toSemigroup.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6) (MulAction.toSemigroupAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.15)))) (LE.le.{u_5} α (Preorder.toLE.{u_5} α inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.10))], (LE.le.{u_5} α (Preorder.toLE.{u_5} α inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.10) (HSMul.hSMul.{u_4, u_5, u_5} G α α (instHSMul.{u_4, u_5} G α (SemigroupAction.toSMul.{u_4, u_5} G α (Monoid.toSemigroup.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6) (MulAction.toSemigroupAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.15))) g a) a) -> (forall (n : Nat), LE.le.{u_5} α (Preorder.toLE.{u_5} α inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.10) (HSMul.hSMul.{u_4, u_5, u_5} G α α (instHSMul.{u_4, u_5} G α (SemigroupAction.toSMul.{u_4, u_5} G α (Monoid.toSemigroup.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6) (MulAction.toSemigroupAction.{u_4, u_5} G α inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.15))) (HPow.hPow.{u_4, 0, u_4} G Nat G (instHPow.{u_4, 0} G Nat (Monoid.toPow.{u_4} G inst._@.Mathlib.Algebra.Order.Group.Action.103008296._hygCtx._hyg.6)) g n) a) a)","typeFull":"∀ {G : Type u_4} [inst : Monoid G] {α : Type u_5} [inst_1 : Preorder α] {g : G} {a : α} [inst_2 : MulAction G α]\n  [CovariantClass G α HSMul.hSMul LE.le], g • a ≤ a → ∀ (n : ℕ), g ^ n • a ≤ a","typeReadable":"∀ {G : Type u_4} [inst : Monoid G] {α : Type u_5} [inst_1 : Preorder α] {g : G} {a : α} [inst_2 : MulAction G α]\n  [CovariantClass G α HSMul.hSMul LE.le], g • a ≤ a → ∀ (n : ℕ), g ^ n • a ≤ a","typeReferences":[["instHPow"],["SemigroupAction","toSMul"],["CovariantClass"],["HPow","hPow"],["Preorder"],["Nat"],["Monoid","toPow"],["MulAction"],["HSMul","hSMul"],["LE","le"],["Monoid"],["instHSMul"],["Monoid","toSemigroup"],["Preorder","toLE"],["MulAction","toSemigroupAction"]],"valueReferences":[["instAddNat"],["MulOneClass","toMulOne"],["le_refl"],["pow_succ'"],["HMul","hMul"],["SemigroupAction","toSMul"],["LE","le","trans"],["Semigroup","toMul"],["congrArg"],["MulOne","toMul"],["Monoid","toPow"],["instOfNatNat"],["Monoid","toMulOneClass"],["instHSMul"],["Monoid","toSemigroup"],["Preorder","toLE"],["Eq"],["SemigroupAction","mul_smul"],["instHPow"],["MulOne","toOne"],["instHAdd"],["Nat","recAux"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["One","toOfNat1"],["HSMul","hSMul"],["LE","le"],["one_smul"],["id"],["instHMul"],["Eq","mpr"],["MulAction","toSemigroupAction"],["pow_zero"],["smul_mono_right"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Group.Int.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Int","instIsOrderedAddMonoid"],"typeFallback":"IsOrderedAddMonoid.{0} Int Int.instAddCommMonoid (PartialOrder.toPreorder.{0} Int (SemilatticeInf.toPartialOrder.{0} Int (Lattice.toSemilatticeInf.{0} Int instLatticeInt)))","typeFull":"IsOrderedAddMonoid ℤ","typeReadable":"IsOrderedAddMonoid ℤ","typeReferences":[["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["IsOrderedAddMonoid"],["Int","instAddCommMonoid"],["instLatticeInt"],["SemilatticeInf","toPartialOrder"],["Int"]],"valueReferences":[["Int","add_le_add_right"],["IsOrderedAddMonoid","mk"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["IsOrderedAddMonoid","_proof_1"],["Int","instAddCommMonoid"],["instLatticeInt"],["SemilatticeInf","toPartialOrder"],["Int"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Monoid.PNat.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat","_proof_6"],"typeFallback":"forall (x : PNat) (y : PNat), Eq.{1} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (Subtype.mk.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (PNat.val (HMul.hMul.{0, 0, 0} PNat PNat PNat (instHMul.{0} PNat instMulPNat) x y)) (PNat.equivNonZeroDivisorsNat._proof_1 (HMul.hMul.{0, 0, 0} PNat PNat PNat (instHMul.{0} PNat instMulPNat) x y))) (HMul.hMul.{0, 0, 0} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (instHMul.{0} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero))) (Subtype.mk.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (PNat.val x) (PNat.equivNonZeroDivisorsNat._proof_1 x)) (Subtype.mk.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (PNat.val y) (PNat.equivNonZeroDivisorsNat._proof_1 y)))","typeFull":"∀ (x y : ℕ+), ⟨↑(x * y), ⋯⟩ = ⟨↑x, ⋯⟩ * ⟨↑y, ⋯⟩","typeReadable":"∀ (x y : ℕ+), ⟨↑(x * y), ⋯⟩ = ⟨↑x, ⋯⟩ * ⟨↑y, ⋯⟩","typeReferences":[["PNat"],["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["instMulPNat"],["Membership","mem"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Nat","instMonoidWithZero"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["Submonoid"],["PNat","val"],["Nat"],["Submonoid","mul"],["nonZeroDivisors"],["instHMul"],["Subtype","mk"],["Eq"]],"valueReferences":[["Subtype"],["Eq","trans"],["instMulPNat"],["Membership","mem"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Nat","instMonoidWithZero"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["PNat","val"],["forall_congr"],["instMulNat"],["Eq"],["Eq","ndrec"],["SetLike","instMembership"],["PNat"],["True"],["Submonoid","instSetLike"],["Submonoid"],["implies_true"],["eq_self"],["Nat"],["of_eq_true"],["Eq","refl"],["nonZeroDivisors"],["id"],["instHMul"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Order","Monoid","PNat",0,"PNat","equivNonZeroDivisorsNat","_simp_2"],"typeFallback":"forall {n : Nat}, Eq.{1} Prop (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))","typeFull":"∀ {n : ℕ}, (0 < n) = (n ≠ 0)","typeReadable":"∀ {n : ℕ}, (0 < n) = (n ≠ 0)","typeReferences":[["LT","lt"],["instLTNat"],["Nat"],["instOfNatNat"],["Ne"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["instLTNat"],["Nat"],["instOfNatNat"],["Nat","pos_iff_ne_zero"],["Ne"],["OfNat","ofNat"],["propext"]]},{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat","_proof_5"],"typeFallback":"Function.RightInverse.{1, 1} PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (fun (x : Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) => Subtype.mk.{1} Nat (fun (n : Nat) => LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) x) (PNat.equivNonZeroDivisorsNat._proof_3 x)) (fun (x : PNat) => Subtype.mk.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (PNat.val x) (PNat.equivNonZeroDivisorsNat._proof_1 x))","typeFull":"Function.RightInverse (fun x => ⟨↑x, ⋯⟩) fun x => ⟨↑x, ⋯⟩","typeReadable":"Function.RightInverse (fun x => ⟨↑x, ⋯⟩) fun x => ⟨↑x, ⋯⟩","typeReferences":[["instLTNat"],["PNat"],["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["PNat","equivNonZeroDivisorsNat","_proof_3"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Subtype","val"],["Nat","instMonoidWithZero"],["OfNat","ofNat"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["LT","lt"],["Submonoid"],["Function","RightInverse"],["PNat","val"],["Nat"],["instOfNatNat"],["nonZeroDivisors"],["Subtype","mk"]],"valueReferences":[["instLTNat"],["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["PNat","equivNonZeroDivisorsNat","_proof_3"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Subtype","val"],["Nat","instMonoidWithZero"],["OfNat","ofNat"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["LT","lt"],["Submonoid"],["Nat"],["PNat","val"],["instOfNatNat"],["Eq","refl"],["nonZeroDivisors"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat_apply_coe"],"typeFallback":"forall (x : PNat), Eq.{1} Nat (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (DFunLike.coe.{1, 1, 1} (OrderMonoidIso.{0, 0} PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) instMulPNat (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero))) PNat (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : PNat) => Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (EquivLike.toFunLike.{1, 1, 1} (OrderMonoidIso.{0, 0} PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) instMulPNat (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero))) PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (OrderMonoidIso.instEquivLike.{0, 0} PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) instMulPNat (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero)))) PNat.equivNonZeroDivisorsNat x)) (PNat.val x)","typeFull":"∀ (x : ℕ+), ↑(PNat.equivNonZeroDivisorsNat x) = ↑x","typeReadable":"∀ (x : ℕ+), ↑(PNat.equivNonZeroDivisorsNat x) = ↑x","typeReferences":[["instLinearOrderPNat"],["Subtype"],["PartialOrder","toPreorder"],["instMulPNat"],["Membership","mem"],["Subtype","preorder"],["PNat","equivNonZeroDivisorsNat"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["DFunLike","coe"],["Nat","instMonoidWithZero"],["Subtype","val"],["Nat","instPreorder"],["instDistribLatticeOfLinearOrder"],["PNat","val"],["OrderMonoidIso","instEquivLike"],["EquivLike","toFunLike"],["Eq"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["PNat"],["Lattice","toSemilatticeInf"],["Submonoid","instSetLike"],["Submonoid"],["OrderMonoidIso"],["Nat"],["DistribLattice","toLattice"],["Submonoid","mul"],["nonZeroDivisors"]],"valueReferences":[["instLinearOrderPNat"],["Subtype"],["PartialOrder","toPreorder"],["instMulPNat"],["Membership","mem"],["Subtype","preorder"],["PNat","equivNonZeroDivisorsNat"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["DFunLike","coe"],["Nat","instMonoidWithZero"],["Subtype","val"],["Nat","instPreorder"],["instDistribLatticeOfLinearOrder"],["OrderMonoidIso","instEquivLike"],["EquivLike","toFunLike"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["PNat"],["Lattice","toSemilatticeInf"],["Submonoid","instSetLike"],["OrderMonoidIso"],["Submonoid"],["Nat"],["DistribLattice","toLattice"],["Submonoid","mul"],["Eq","refl"],["nonZeroDivisors"]]},{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat_symm_apply_coe"],"typeFallback":"forall (x : Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)), Eq.{1} Nat (Subtype.val.{1} Nat (fun (n : Nat) => LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) (DFunLike.coe.{1, 1, 1} (OrderMonoidIso.{0, 0} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) PNat (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero)) instMulPNat) (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) => PNat) (EquivLike.toFunLike.{1, 1, 1} (OrderMonoidIso.{0, 0} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) PNat (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero)) instMulPNat) (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) PNat (OrderMonoidIso.instEquivLike.{0, 0} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) PNat (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero)) instMulPNat)) (OrderMonoidIso.symm.{0, 0} PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) instMulPNat (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero)) PNat.equivNonZeroDivisorsNat) x)) (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) x)","typeFull":"∀ (x : ↥(nonZeroDivisors ℕ)), ↑(PNat.equivNonZeroDivisorsNat.symm x) = ↑x","typeReadable":"∀ (x : ↥(nonZeroDivisors ℕ)), ↑(PNat.equivNonZeroDivisorsNat.symm x) = ↑x","typeReferences":[["instLinearOrderPNat"],["Subtype"],["PartialOrder","toPreorder"],["instMulPNat"],["Membership","mem"],["Subtype","preorder"],["PNat","equivNonZeroDivisorsNat"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["Nat","instMonoidWithZero"],["DFunLike","coe"],["Nat","instPreorder"],["instDistribLatticeOfLinearOrder"],["OrderMonoidIso","instEquivLike"],["instOfNatNat"],["EquivLike","toFunLike"],["Eq"],["SemilatticeInf","toPartialOrder"],["instLTNat"],["SetLike","instMembership"],["PNat"],["Lattice","toSemilatticeInf"],["Submonoid","instSetLike"],["OfNat","ofNat"],["LT","lt"],["Submonoid"],["OrderMonoidIso"],["Nat"],["DistribLattice","toLattice"],["Submonoid","mul"],["nonZeroDivisors"],["OrderMonoidIso","symm"]],"valueReferences":[["instLinearOrderPNat"],["Subtype"],["PartialOrder","toPreorder"],["instMulPNat"],["Membership","mem"],["Subtype","preorder"],["PNat","equivNonZeroDivisorsNat"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["DFunLike","coe"],["Subtype","val"],["Nat","instMonoidWithZero"],["Nat","instPreorder"],["instDistribLatticeOfLinearOrder"],["OrderMonoidIso","instEquivLike"],["instOfNatNat"],["EquivLike","toFunLike"],["SemilatticeInf","toPartialOrder"],["instLTNat"],["SetLike","instMembership"],["PNat"],["Lattice","toSemilatticeInf"],["Submonoid","instSetLike"],["OfNat","ofNat"],["Submonoid"],["OrderMonoidIso"],["LT","lt"],["Nat"],["DistribLattice","toLattice"],["Submonoid","mul"],["Eq","refl"],["nonZeroDivisors"],["OrderMonoidIso","symm"]]},{"isProp":false,"kind":"definition","name":["PNat","equivNonZeroDivisorsNat"],"typeFallback":"OrderMonoidIso.{0, 0} PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat))))) (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) instMulPNat (Submonoid.mul.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero))","typeFull":"ℕ+ ≃*o ↥(nonZeroDivisors ℕ)","typeReadable":"ℕ+ ≃*o ↥(nonZeroDivisors ℕ)","typeReferences":[["instLinearOrderPNat"],["Lattice","toSemilatticeInf"],["PNat"],["Subtype"],["SetLike","instMembership"],["PartialOrder","toPreorder"],["Submonoid","instSetLike"],["instMulPNat"],["Membership","mem"],["Subtype","preorder"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Nat","instMonoidWithZero"],["Nat","instPreorder"],["OrderMonoidIso"],["Submonoid"],["instDistribLatticeOfLinearOrder"],["Nat"],["DistribLattice","toLattice"],["Submonoid","mul"],["nonZeroDivisors"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["instLinearOrderPNat"],["PartialOrder","toPreorder"],["Subtype"],["Equiv","mk"],["instMulPNat"],["PNat","equivNonZeroDivisorsNat","_proof_5"],["PNat","equivNonZeroDivisorsNat","_proof_3"],["Membership","mem"],["Subtype","preorder"],["MulEquiv","mk"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Subtype","val"],["Nat","instMonoidWithZero"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["Nat","instPreorder"],["instDistribLatticeOfLinearOrder"],["PNat","val"],["instOfNatNat"],["PNat","equivNonZeroDivisorsNat","_proof_4"],["OrderMonoidIso","mk"],["SemilatticeInf","toPartialOrder"],["instLTNat"],["SetLike","instMembership"],["PNat"],["Lattice","toSemilatticeInf"],["Submonoid","instSetLike"],["OfNat","ofNat"],["PNat","equivNonZeroDivisorsNat","_proof_7"],["LT","lt"],["Submonoid"],["Nat"],["PNat","equivNonZeroDivisorsNat","_proof_6"],["DistribLattice","toLattice"],["Submonoid","mul"],["nonZeroDivisors"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat","_proof_3"],"typeFallback":"forall (x : Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)), LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) x)","typeFull":"∀ (x : ↥(nonZeroDivisors ℕ)), 0 < ↑x","typeReadable":"∀ (x : ↥(nonZeroDivisors ℕ)), 0 < ↑x","typeReferences":[["instLTNat"],["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Subtype","val"],["Nat","instMonoidWithZero"],["OfNat","ofNat"],["Submonoid"],["LT","lt"],["Nat"],["instOfNatNat"],["nonZeroDivisors"]],"valueReferences":[["Eq","trans"],["Membership","mem"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Nat","instMonoidWithZero"],["Subtype","val"],["congrArg"],["not_false_eq_true"],["instOfNatNat"],["Eq"],["Not"],["instLTNat"],["SetLike","instMembership"],["True"],["Submonoid","instSetLike"],["nonZeroDivisors","coe_ne_zero","_simp_1"],["Nat","instNontrivial"],["OfNat","ofNat"],["Submonoid"],["LT","lt"],["Nat"],["of_eq_true"],["nonZeroDivisors"],["False"],["Ne"],["_private","Mathlib","Algebra","Order","Monoid","PNat",0,"PNat","equivNonZeroDivisorsNat","_simp_2"]]},{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat","_proof_1"],"typeFallback":"forall (x : PNat), Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) (PNat.val x)","typeFull":"∀ (x : ℕ+), ↑x ∈ nonZeroDivisors ℕ","typeReadable":"∀ (x : ℕ+), ↑x ∈ nonZeroDivisors ℕ","typeReferences":[["Submonoid"],["PNat","val"],["Nat"],["SetLike","instMembership"],["PNat"],["Submonoid","instSetLike"],["Membership","mem"],["nonZeroDivisors"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Nat","instMonoidWithZero"]],"valueReferences":[["IsDomain","to_noZeroDivisors"],["Eq","trans"],["Membership","mem"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Nat","instMonoidWithZero"],["congrArg"],["PNat","val"],["not_false_eq_true"],["mem_nonZeroDivisors_iff_ne_zero","_simp_1"],["Nat","instSemiring"],["Zero","toOfNat0"],["Eq"],["Not"],["SetLike","instMembership"],["True"],["Submonoid","instSetLike"],["Nat","instNontrivial"],["PNat","ne_zero","_simp_1"],["OfNat","ofNat"],["Submonoid"],["Nat"],["of_eq_true"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["nonZeroDivisors"],["False"],["Ne"],["Nat","instIsDomain"]]},{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat","_proof_4"],"typeFallback":"Function.LeftInverse.{1, 1} PNat (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (fun (x : Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) => Subtype.mk.{1} Nat (fun (n : Nat) => LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) x) (PNat.equivNonZeroDivisorsNat._proof_3 x)) (fun (x : PNat) => Subtype.mk.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (PNat.val x) (PNat.equivNonZeroDivisorsNat._proof_1 x))","typeFull":"Function.LeftInverse (fun x => ⟨↑x, ⋯⟩) fun x => ⟨↑x, ⋯⟩","typeReadable":"Function.LeftInverse (fun x => ⟨↑x, ⋯⟩) fun x => ⟨↑x, ⋯⟩","typeReferences":[["instLTNat"],["PNat"],["Subtype"],["SetLike","instMembership"],["Submonoid","instSetLike"],["PNat","equivNonZeroDivisorsNat","_proof_3"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Subtype","val"],["Nat","instMonoidWithZero"],["OfNat","ofNat"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["LT","lt"],["Submonoid"],["PNat","val"],["Nat"],["instOfNatNat"],["Function","LeftInverse"],["nonZeroDivisors"],["Subtype","mk"]],"valueReferences":[["instLTNat"],["SetLike","instMembership"],["PNat"],["Submonoid","instSetLike"],["PNat","equivNonZeroDivisorsNat","_proof_3"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZero","toMulZeroOneClass"],["Nat","instMonoidWithZero"],["Subtype","val"],["OfNat","ofNat"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["LT","lt"],["Submonoid"],["PNat","val"],["Nat"],["instOfNatNat"],["Eq","refl"],["nonZeroDivisors"],["Subtype","mk"]]},{"isProp":true,"kind":"theorem","name":["PNat","equivNonZeroDivisorsNat","_proof_7"],"typeFallback":"forall {a : PNat} {b : PNat}, Iff (LE.le.{0} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (Preorder.toLE.{0} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x)) (Subtype.preorder.{0} Nat Nat.instPreorder (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x))) (Subtype.mk.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (PNat.val a) (PNat.equivNonZeroDivisorsNat._proof_1 a)) (Subtype.mk.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) (SetLike.instMembership.{0, 0} (Submonoid.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero))) Nat (Submonoid.instSetLike.{0} Nat (MulZeroOneClass.toMulOneClass.{0} Nat (MonoidWithZero.toMulZeroOneClass.{0} Nat Nat.instMonoidWithZero)))) (nonZeroDivisors.{0} Nat Nat.instMonoidWithZero) x) (PNat.val b) (PNat.equivNonZeroDivisorsNat._proof_1 b))) (LE.le.{0} PNat (Preorder.toLE.{0} PNat (PartialOrder.toPreorder.{0} PNat (SemilatticeInf.toPartialOrder.{0} PNat (Lattice.toSemilatticeInf.{0} PNat (DistribLattice.toLattice.{0} PNat (instDistribLatticeOfLinearOrder.{0} PNat instLinearOrderPNat)))))) a b)","typeFull":"∀ {a b : ℕ+}, ⟨↑a, ⋯⟩ ≤ ⟨↑b, ⋯⟩ ↔ a ≤ b","typeReadable":"∀ {a b : ℕ+}, ⟨↑a, ⋯⟩ ≤ ⟨↑b, ⋯⟩ ↔ a ≤ b","typeReferences":[["instLinearOrderPNat"],["Subtype"],["PartialOrder","toPreorder"],["Membership","mem"],["Subtype","preorder"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Nat","instMonoidWithZero"],["Nat","instPreorder"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["instDistribLatticeOfLinearOrder"],["PNat","val"],["Preorder","toLE"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["PNat"],["Lattice","toSemilatticeInf"],["Submonoid","instSetLike"],["Submonoid"],["Nat"],["DistribLattice","toLattice"],["Iff"],["LE","le"],["nonZeroDivisors"],["Subtype","mk"]],"valueReferences":[["instLinearOrderPNat"],["Subtype"],["PartialOrder","toPreorder"],["Eq","trans"],["Subtype","instLE"],["Membership","mem"],["Subtype","preorder"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Nat","instMonoidWithZero"],["congrArg"],["Nat","instPreorder"],["PNat","equivNonZeroDivisorsNat","_proof_1"],["Subtype","mk_le_mk","_simp_1"],["instDistribLatticeOfLinearOrder"],["PNat","val"],["iff_self"],["forall_congr"],["PNat","coe_le_coe","_simp_1"],["congrFun'"],["Preorder","toLE"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["PNat"],["Lattice","toSemilatticeInf"],["True"],["Submonoid","instSetLike"],["Submonoid"],["implies_true"],["LinearOrder","toPartialOrder"],["Nat"],["DistribLattice","toLattice"],["of_eq_true"],["Iff"],["LE","le"],["nonZeroDivisors"],["Subtype","mk"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Ring.Pow.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["one_add_mul_le_pow"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3 : Ring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.6))))] {a : R}, (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.6)))))) (Neg.neg.{u_1} R (NegZeroClass.toNeg.{u_1} R (SubNegZeroMonoid.toNegZeroClass.{u_1} R (SubtractionMonoid.toSubNegZeroMonoid.{u_1} R (SubtractionCommMonoid.toSubtractionMonoid.{u_1} R (AddCommGroup.toDivisionAddCommMonoid.{u_1} R (Ring.toAddCommGroup.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))))) a) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.6)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonAssocRing.toNonUnitalNonAssocRing.{u_1} R (Ring.toNonAssocRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonAssocRing.toNonUnitalNonAssocRing.{u_1} R (Ring.toNonAssocRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3)))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3))) n) a)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonAssocRing.toNonUnitalNonAssocRing.{u_1} R (Ring.toNonAssocRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1611226582._hygCtx._hyg.3))))) a) n))","typeFull":"∀ {R : Type u_1} [inst : Ring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {a : R},\n  -2 ≤ a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReadable":"∀ {R : Type u_1} [inst : Ring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {a : R},\n  -2 ≤ a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Nat","cast"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["Nat","instNeZeroSucc"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["Monoid","toPow"],["Ring","toAddGroupWithOne"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["instHPow"],["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Neg","neg"],["Distrib","toMul"],["LinearOrder"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["Ring","toSemiring"],["HAdd","hAdd"],["Ring","toAddCommGroup"],["Nat"],["NegZeroClass","toNeg"],["DistribLattice","toLattice"],["AddMonoidWithOne","toNatCast"],["One","toOfNat1"],["AddCommGroup","toDivisionAddCommMonoid"],["AddMonoidWithOne","toOne"],["LE","le"],["instHMul"],["Nat","instAtLeastTwoHAddOfNat"],["Ring"]],"valueReferences":[["one_add_le_pow_of_two_add_nonneg"],["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["Iff","mp"],["AddGroupWithOne","toAddMonoidWithOne"],["IsOrderedRing","toIsOrderedAddMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["AddGroup","toSubtractionMonoid"],["Nat","instNeZeroSucc"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["Ring","toAddGroupWithOne"],["instOfNatNat"],["neg_le_iff_add_nonneg'"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["AddGroup","existsAddOfLE"],["SemilatticeInf","toPartialOrder"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing","toIsOrderedRing"],["instHAdd"],["Neg","neg"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["instOfNatAtLeastTwo"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["OfNat","ofNat"],["Ring","toSemiring"],["HAdd","hAdd"],["Nat"],["AddGroupWithOne","toAddGroup"],["DistribLattice","toLattice"],["NegZeroClass","toNeg"],["AddMonoidWithOne","toNatCast"],["SubNegMonoid","toAddMonoid"],["LE","le"],["NegZeroClass","toZero"],["Nat","instAtLeastTwoHAddOfNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["one_add_mul_le_pow_of_sq_nonneg"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.6 : PartialOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.9 : IsOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.6] {a : R}, (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))) a (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))))) a) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) a)) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.6)) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))) n) a)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2841932628._hygCtx._hyg.3)))))) a) n))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a : R},\n  0 ≤ a ^ 2 → 0 ≤ (1 + a) ^ 2 → 0 ≤ 2 + a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a : R},\n  0 ≤ a ^ 2 → 0 ≤ (1 + a) ^ 2 → 0 ≤ 2 + a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReferences":[["Nat","cast"],["PartialOrder","toPreorder"],["HMul","hMul"],["Nat","instNeZeroSucc"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["PartialOrder"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["instHPow"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsOrderedRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["One","toOfNat1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["AddMonoidWithOne","toOne"],["instHMul"],["Nat","instAtLeastTwoHAddOfNat"],["Semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["HMul","hMul"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["HSub","hSub"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["instHPow"],["IsOrderedRing","toZeroLEOneClass"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Nat"],["AddMonoidWithOne","toNatCast"],["id"],["instHMul"],["Eq","mpr"],["Commute","one_left"],["MulOneClass","toMulOne"],["one_pow"],["Nat","cast"],["Eq","mp"],["MulZeroOneClass","toMulOneClass"],["congrArg"],["Nat","instNeZeroSucc"],["instOfNatNat"],["congr"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["congrFun'"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Commute","pow_add_mul_le_add_pow_of_sq_nonneg"],["Eq"],["Preorder","toLE"],["Distrib","toAdd"],["instHAdd"],["Distrib","toMul"],["zero_le_one"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["mul_one"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["One","toOfNat1"],["instSubNat"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Order","Ring","Pow",0,"Commute","pow_add_mul_le_add_pow_of_sq_nonneg","match_1_1"],"typeFallback":"forall (motive : Nat -> Prop) (x._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx.79.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.91 : Nat), (Unit -> (motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (Unit -> (motive (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> (Unit -> (motive (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) -> (forall (n : Nat), motive (Nat.succ (Nat.succ (Nat.succ n)))) -> (motive x._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx.79.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.91)","typeFull":"∀ (motive : ℕ → Prop) (x : ℕ),\n  (∀ (a : Unit), motive 0) →\n    (∀ (a : Unit), motive 1) → (∀ (a : Unit), motive 2) → (∀ (n : ℕ), motive n.succ.succ.succ) → motive x","typeReadable":"∀ (motive : ℕ → Prop) (x : ℕ),\n  (∀ (a : Unit), motive 0) →\n    (∀ (a : Unit), motive 1) → (∀ (a : Unit), motive 2) → (∀ (n : ℕ), motive n.succ.succ.succ) → motive x","typeReferences":[["Nat"],["Nat","succ"],["instOfNatNat"],["OfNat","ofNat"],["Unit"]],"valueReferences":[["Nat","casesOn"],["Nat","succ"],["Unit","unit"]]},{"isProp":true,"kind":"theorem","name":["one_add_mul_le_pow'"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.6 : PartialOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.9 : IsOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.6] {a : R}, (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) a a)) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))))) a) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))))) a))) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) a)) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.6)) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))) n) a)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1615424382._hygCtx._hyg.3)))))) a) n))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a : R},\n  0 ≤ a * a → 0 ≤ (1 + a) * (1 + a) → 0 ≤ 2 + a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a : R},\n  0 ≤ a * a → 0 ≤ (1 + a) * (1 + a) → 0 ≤ 2 + a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReferences":[["Nat","cast"],["PartialOrder","toPreorder"],["HMul","hMul"],["Nat","instNeZeroSucc"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["PartialOrder"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["instHPow"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsOrderedRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["One","toOfNat1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["AddMonoidWithOne","toOne"],["instHMul"],["Nat","instAtLeastTwoHAddOfNat"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["PartialOrder","toPreorder"],["HMul","hMul"],["congrArg"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["instHPow"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["one_add_mul_le_pow_of_sq_nonneg"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["One","toOfNat1"],["MulZeroClass","toZero"],["AddMonoidWithOne","toOne"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["sq"],["id"],["instHMul"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["pow_add_mul_le_add_pow"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.9 : IsOrderedRing.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.12 : ExistsAddOfLE.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))) (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.6))))))] {a : R} {b : R}, (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))))) a) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) a) b)) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.6)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3)))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))) a n) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3)))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))) n) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))) a (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) b)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133368._hygCtx._hyg.3)))))) a b) n))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {a b : R},\n  0 ≤ a → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {a b : R},\n  0 ≤ a → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReferences":[["Nat","cast"],["PartialOrder","toPreorder"],["CommSemiring"],["HMul","hMul"],["Nat","instNeZeroSucc"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["ExistsAddOfLE"],["SemilatticeInf","toPartialOrder"],["instHPow"],["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["Distrib","toMul"],["instOfNatAtLeastTwo"],["LinearOrder"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsOrderedRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["DistribLattice","toLattice"],["instSubNat"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHMul"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["NonUnitalCommSemiring","toNonUnitalNonAssocCommSemiring"],["CommSemiring","toNonUnitalCommSemiring"],["Commute","all"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["CommSemiring","toSemiring"],["Commute","pow_add_mul_le_add_pow"]]},{"isProp":true,"kind":"theorem","name":["pow_add_mul_le_add_pow_of_sq_nonneg"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3 : CommSemiring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.6 : PartialOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.9 : IsOrderedRing.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.6] {a : R} {b : R}, (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))))) a) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))) b (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3)))))) a b) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) a) b)) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.6)) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3)))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))) a n) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3)))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))) n) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))) a (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) b)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528707._hygCtx._hyg.3)))))) a b) n))","typeFull":"∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a b : R},\n  0 ≤ a → 0 ≤ b ^ 2 → 0 ≤ (a + b) ^ 2 → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReadable":"∀ {R : Type u_1} [inst : CommSemiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a b : R},\n  0 ≤ a → 0 ≤ b ^ 2 → 0 ≤ (a + b) ^ 2 → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReferences":[["Nat","cast"],["PartialOrder","toPreorder"],["CommSemiring"],["HMul","hMul"],["Nat","instNeZeroSucc"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["PartialOrder"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["instHPow"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toSemiring"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsOrderedRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["instSubNat"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["instHMul"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"]],"valueReferences":[["NonUnitalCommSemiring","toNonUnitalNonAssocCommSemiring"],["CommSemiring","toNonUnitalCommSemiring"],["Commute","all"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["CommSemiring","toSemiring"],["Commute","pow_add_mul_le_add_pow_of_sq_nonneg"]]},{"isProp":true,"kind":"theorem","name":["one_add_mul_sub_le_pow"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3 : Ring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.6))))] {a : R}, (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.6)))))) (Neg.neg.{u_1} R (NegZeroClass.toNeg.{u_1} R (SubNegZeroMonoid.toNegZeroClass.{u_1} R (SubtractionMonoid.toSubNegZeroMonoid.{u_1} R (SubtractionCommMonoid.toSubtractionMonoid.{u_1} R (AddCommGroup.toDivisionAddCommMonoid.{u_1} R (Ring.toAddCommGroup.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3)))))) a) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.6)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonAssocRing.toNonUnitalNonAssocRing.{u_1} R (Ring.toNonAssocRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3)))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonAssocRing.toNonUnitalNonAssocRing.{u_1} R (Ring.toNonAssocRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3)))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3))) n) (HSub.hSub.{u_1, u_1, u_1} R R R (instHSub.{u_1} R (SubNegMonoid.toSub.{u_1} R (AddGroup.toSubNegMonoid.{u_1} R (AddGroupWithOne.toAddGroup.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3))))) a (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3)))))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2129248316._hygCtx._hyg.3))))) a n))","typeFull":"∀ {R : Type u_1} [inst : Ring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {a : R},\n  -1 ≤ a → ∀ (n : ℕ), 1 + ↑n * (a - 1) ≤ a ^ n","typeReadable":"∀ {R : Type u_1} [inst : Ring R] [inst_1 : LinearOrder R] [IsStrictOrderedRing R] {a : R},\n  -1 ≤ a → ∀ (n : ℕ), 1 + ↑n * (a - 1) ≤ a ^ n","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Nat","cast"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["SubtractionCommMonoid","toSubtractionMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["instDistribLatticeOfLinearOrder"],["Monoid","toPow"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Preorder","toLE"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["instHPow"],["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Neg","neg"],["Distrib","toMul"],["LinearOrder"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["HAdd","hAdd"],["Nat"],["AddGroupWithOne","toAddGroup"],["AddMonoidWithOne","toNatCast"],["NegZeroClass","toNeg"],["DistribLattice","toLattice"],["AddCommGroup","toDivisionAddCommMonoid"],["One","toOfNat1"],["LE","le"],["AddMonoidWithOne","toOne"],["instHMul"],["instHSub"],["Ring"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Ring","toNonAssocRing"],["PartialOrder","toPreorder"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["AddMonoidWithOne","toAddMonoid"],["sub_eq_add_neg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Monoid","toPow"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["HSub","hSub"],["Eq","symm"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["AddSemigroup","toAdd"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["Nat"],["AddMonoidWithOne","toNatCast"],["SubtractionMonoid","toSubNegMonoid"],["AddMonoid","toAddSemigroup"],["AddMonoidWithOne","toOne"],["id"],["instHMul"],["Eq","mpr"],["covariant_swap_add_of_covariant_add"],["AddMonoid","toAddZeroClass"],["Nat","cast"],["Eq","mp"],["one_add_mul_le_pow"],["SubtractionCommMonoid","toSubtractionMonoid"],["neg_add"],["IsOrderedRing","toIsOrderedAddMonoid"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["Nat","instNeZeroSucc"],["add_sub_cancel"],["instDistribLatticeOfLinearOrder"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["congrFun'"],["Preorder","toLE"],["Eq"],["propext"],["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["IsStrictOrderedRing","toIsOrderedRing"],["instHAdd"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["SubNegMonoid","toNeg"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["AddZero","toAdd"],["OfNat","ofNat"],["Ring","toAddCommGroup"],["HAdd","hAdd"],["DistribLattice","toLattice"],["NegZeroClass","toNeg"],["AddGroupWithOne","toAddGroup"],["AddCommMonoid","toAddCommSemigroup"],["AddCommGroup","toDivisionAddCommMonoid"],["One","toOfNat1"],["SubNegMonoid","toAddMonoid"],["LE","le"],["one_add_one_eq_two"],["instHSub"],["sub_le_sub_iff_right"],["Nat","instAtLeastTwoHAddOfNat"]]},{"isProp":true,"kind":"theorem","name":["Commute","pow_add_mul_le_add_pow"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.9 : IsOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.12 : ExistsAddOfLE.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))) (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.6))))))] {a : R} {b : R}, (Commute.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))) a b) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))))) a) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) a) b)) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.6)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))) a n) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))) n) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))) a (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) b)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3)))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2369133367._hygCtx._hyg.3))))) a b) n))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {a b : R},\n  Commute a b → 0 ≤ a → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {a b : R},\n  Commute a b → 0 ≤ a → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReferences":[["Nat","cast"],["PartialOrder","toPreorder"],["HMul","hMul"],["Nat","instNeZeroSucc"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["Commute"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["ExistsAddOfLE"],["SemilatticeInf","toPartialOrder"],["instHPow"],["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["instOfNatAtLeastTwo"],["LinearOrder"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsOrderedRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["DistribLattice","toLattice"],["instSubNat"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHMul"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"],["Semiring"]],"valueReferences":[["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["IsOrderedAddMonoid","toAddLeftMono"],["sq_nonneg"],["IsOrderedRing","toIsOrderedAddMonoid"],["HAdd","hAdd"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Semiring","toNonAssocSemiring"],["Commute","pow_add_mul_le_add_pow_of_sq_nonneg"],["IsOrderedRing","toPosMulMono"],["SemilatticeInf","toPartialOrder"]]},{"isProp":true,"kind":"theorem","name":["Commute","pow_add_mul_le_add_pow_of_sq_nonneg"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.6 : PartialOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.9 : IsOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.6] {a : R} {b : R}, (Commute.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) a b) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))))) a) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) b (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3))))) a b) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) -> (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) a) b)) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.6)) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3))))) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) a n) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) n) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) a (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) b)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3)))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.2719528706._hygCtx._hyg.3))))) a b) n))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a b : R},\n  Commute a b →\n    0 ≤ a → 0 ≤ b ^ 2 → 0 ≤ (a + b) ^ 2 → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : PartialOrder R] [IsOrderedRing R] {a b : R},\n  Commute a b →\n    0 ≤ a → 0 ≤ b ^ 2 → 0 ≤ (a + b) ^ 2 → 0 ≤ 2 * a + b → ∀ (n : ℕ), a ^ n + ↑n * a ^ (n - 1) * b ≤ (a + b) ^ n","typeReferences":[["Nat","cast"],["PartialOrder","toPreorder"],["HMul","hMul"],["Nat","instNeZeroSucc"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Commute"],["instOfNatNat"],["PartialOrder"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["instHPow"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsOrderedRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["instSubNat"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHMul"],["instHSub"],["Nat","instAtLeastTwoHAddOfNat"],["Semiring"]],"valueReferences":[["instAddNat"],["Eq","trans"],["Commute","left_comm"],["Eq","symm"],["pow_nonneg"],["NonAssocSemiring","toAddCommMonoidWithOne"],["IsOrderedRing","toZeroLEOneClass"],["Nat","cast_commute"],["Commute","symm"],["Nat","brecOn"],["MulZeroClass","zero_mul"],["Eq","refl"],["AddMonoid","toAddSemigroup"],["AddMonoidWithOne","toOne"],["Eq","mpr"],["pow_zero"],["one_mul"],["IsOrderedRing","toPosMulMono"],["AddMonoid","toAddZeroClass"],["Commute","pow_left"],["MulOneClass","toMulOne"],["MulZeroOneClass","toMulOneClass"],["instTransLE"],["AddCommMonoid","toAddMonoid"],["IsOrderedRing","toIsOrderedAddMonoid"],["Nat","instAddMonoidWithOne"],["Semigroup","toMul"],["add_mul"],["pow_one"],["Nat","instNeZeroSucc"],["MulOne","toMul"],["instOfNatNat"],["congr"],["AddCommMagma","toAdd"],["Std","le_refl","_simp_1"],["Preorder","toLE"],["Eq"],["Distrib","toAdd"],["two_mul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["mul_nonneg"],["Mathlib","Tactic","Abel","term_atom"],["instOfNatAtLeastTwo"],["Mathlib","Tactic","Abel","term_add_const"],["mul_one"],["HPow","hPow"],["AddZero","toAdd"],["OfNat","ofNat"],["le_add_of_nonneg_right"],["Nat","cast_add"],["HAdd","hAdd"],["eq_self"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Nat","below"],["instHSub"],["Distrib","leftDistribClass"],["Nat","instAtLeastTwoHAddOfNat"],["Nat","cast_one"],["pow_succ'"],["Trans","trans"],["PartialOrder","toPreorder"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["AddMonoidWithOne","toAddMonoid"],["Nat","instAddCommMonoid"],["Mathlib","Meta","NormNum","IsNat","to_eq"],["Distrib","rightDistribClass"],["Mathlib","Meta","NormNum","isNat_add"],["Mathlib","Tactic","Abel","term"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Nat","instOrderedSub"],["HSub","hSub"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toMulZeroOneClass"],["AddSemigroup","toAdd"],["instHPow"],["add_nonneg"],["Mathlib","Tactic","Abel","const_add_term"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["Nat","instPartialOrder"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["mul_add"],["mul_le_mul_of_nonneg_left"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["add_zero"],["Nat","cast_nonneg"],["Nat","instCanonicallyOrderedAdd"],["zero_add"],["Nat"],["AddMonoidWithOne","toNatCast"],["Nat","cast_zero"],["id"],["instHMul"],["_private","Mathlib","Algebra","Order","Ring","Pow",0,"Commute","pow_add_mul_le_add_pow_of_sq_nonneg","match_1_1"],["Mathlib","Meta","NormNum","isNat_ofNat"],["instTransEq_1"],["AddZero","toZero"],["Mathlib","Tactic","Abel","subst_into_add"],["Nat","cast"],["mul_assoc"],["Mathlib","Tactic","Abel","term_add_term"],["congrArg"],["Commute","eq"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["AddCommSemigroup","toAddCommMagma"],["congrFun'"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["True"],["instHAdd"],["Distrib","toMul"],["zero_le_one"],["Semiring","toMonoidWithZero"],["instReflLe"],["tsub_self"],["AddCommMonoid","toAddCommSemigroup"],["One","toOfNat1"],["of_eq_true"],["instSubNat"],["zero_tsub"],["add_assoc"],["LE","le"],["sq"],["SemigroupWithZero","toSemigroup"],["two_add_one_eq_three"]]},{"isProp":true,"kind":"theorem","name":["one_add_le_pow_of_two_add_nonneg"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.9 : IsOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.6))))] [inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.12 : ExistsAddOfLE.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3)))) (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.6))))))] {a : R}, (LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.6)))))) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 2 (instOfNatAtLeastTwo.{u_1} R 2 (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3)))) (Nat.instAtLeastTwoHAddOfNat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Nat.instNeZeroSucc (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))))) a)) -> (forall (n : Nat), LE.le.{u_1} R (Preorder.toLE.{u_1} R (PartialOrder.toPreorder.{u_1} R (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.6)))))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3)))))) (HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R (Distrib.toMul.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3)))) n) a)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3)))) (HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3))))) (OfNat.ofNat.{u_1} R 1 (One.toOfNat1.{u_1} R (AddMonoidWithOne.toOne.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Pow.1153509797._hygCtx._hyg.3)))))) a) n))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {a : R},\n  0 ≤ 2 + a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : LinearOrder R] [IsOrderedRing R] [ExistsAddOfLE R] {a : R},\n  0 ≤ 2 + a → ∀ (n : ℕ), 1 + ↑n * a ≤ (1 + a) ^ n","typeReferences":[["Nat","cast"],["PartialOrder","toPreorder"],["HMul","hMul"],["Nat","instNeZeroSucc"],["instDistribLatticeOfLinearOrder"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Preorder","toLE"],["NonAssocSemiring","toAddCommMonoidWithOne"],["ExistsAddOfLE"],["SemilatticeInf","toPartialOrder"],["instHPow"],["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearOrder"],["instOfNatAtLeastTwo"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsOrderedRing"],["Nat"],["DistribLattice","toLattice"],["AddMonoidWithOne","toNatCast"],["One","toOfNat1"],["MulZeroClass","toZero"],["AddMonoidWithOne","toOne"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHMul"],["Nat","instAtLeastTwoHAddOfNat"],["Semiring"]],"valueReferences":[["Distrib","toAdd"],["Lattice","toSemilatticeInf"],["PartialOrder","toPreorder"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["one_add_mul_le_pow_of_sq_nonneg"],["IsOrderedAddMonoid","toAddLeftMono"],["sq_nonneg"],["IsOrderedRing","toIsOrderedAddMonoid"],["OfNat","ofNat"],["HAdd","hAdd"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["Semiring","toNonAssocSemiring"],["One","toOfNat1"],["AddMonoidWithOne","toOne"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["IsOrderedRing","toPosMulMono"],["SemilatticeInf","toPartialOrder"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Ring.Units.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Units","index_posSubgroup"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.3 : Ring.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.3) (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.6))))], Eq.{1} Nat (Subgroup.index.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.3)))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.3)))) (Units.posSubgroup.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.Ring.Units.4252074170._hygCtx._hyg.9)) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))","typeFull":"∀ (R : Type u_1) [inst : Ring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R],\n  (Units.posSubgroup R).index = 2","typeReadable":"∀ (R : Type u_1) [inst : Ring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R],\n  (Units.posSubgroup R).index = 2","typeReferences":[["Units","instGroup"],["IsStrictOrderedRing"],["Lattice","toSemilatticeInf"],["LinearOrder"],["Semiring","toMonoidWithZero"],["OfNat","ofNat"],["Ring","toSemiring"],["instDistribLatticeOfLinearOrder"],["Nat"],["DistribLattice","toLattice"],["instOfNatNat"],["Subgroup","index"],["MonoidWithZero","toMonoid"],["Units","posSubgroup"],["Eq"],["Units"],["Ring"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["Units","instHasDistribNeg"],["SubtractionMonoid","toSubNegZeroMonoid"],["not_lt","_simp_1"],["Ring","toNonAssocRing"],["Eq","trans"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["eq_true"],["MonoidWithZero","toMulZeroOneClass"],["AddGroup","toSubtractionMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["congrFun"],["Units","posSubgroup"],["Group","toDivInvMonoid"],["Units"],["Exists"],["Units","instMulOneClass"],["Ring","toSemiring"],["MulZeroOneClass","toMulZeroClass"],["DivInvMonoid","toMonoid"],["Units","val"],["instIsLeftCancelAddOfAddLeftReflectLE"],["IsOrderedCancelAddMonoid","toAddLeftReflectLE"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Units","instNeg"],["Units","instGroup"],["IsOrderedRing","toIsOrderedAddMonoid"],["MulOne","toMul"],["instOfNatNat"],["congr"],["Preorder","toLE"],["Eq"],["Subgroup","index_eq_two_iff"],["propext"],["IsStrictOrderedRing","toIsOrderedCancelAddMonoid"],["IsStrictOrderedRing","toIsOrderedRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["mul_one"],["AddZero","toAdd"],["OfNat","ofNat"],["LinearOrder","toPartialOrder"],["AddGroupWithOne","toAddGroup"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Subgroup"],["IsLeftCancelAdd","addLeftStrictMono_of_addLeftMono"],["Units","instMul"],["PartialOrder","toPreorder"],["Membership","mem"],["instNontrivialOfCharZero"],["lt_or_gt_of_ne"],["Preorder","toLT"],["HMul","hMul"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["mul_neg"],["NonUnitalNonAssocRing","toHasDistribNeg"],["InvolutiveNeg","toNeg"],["AddGroup","toSubNegMonoid"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["SetLike","instMembership"],["Neg","neg"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["xor_comm"],["IsOrderedAddMonoid","toAddLeftMono"],["AddZeroClass","toAddZero"],["Units","instOne"],["Nat"],["id"],["NegZeroClass","toZero"],["instHMul"],["Units","ne_zero"],["Subgroup","instSetLike"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Subgroup","index"],["Monoid","toMulOneClass"],["Left","neg_pos_iff","_simp_1"],["Zero","toOfNat0"],["LT","lt","le"],["IsStrictOrderedRing","toCharZero"],["Not"],["Lattice","toSemilatticeInf"],["True"],["Semiring","toMonoidWithZero"],["Xor'"],["LT","lt"],["Or","casesOn"],["xor_true"],["DistribLattice","toLattice"],["NegZeroClass","toNeg"],["One","toOfNat1"],["of_eq_true"],["HasDistribNeg","toInvolutiveNeg"],["SubNegMonoid","toAddMonoid"],["LE","le"],["Units","mem_posSubgroup","_simp_1"]]},{"isProp":true,"kind":"theorem","name":["Units","posSubgroup","congr_simp"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 : LinearOrder.{u_1} R] [inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9 : IsStrictOrderedRing.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 (SemilatticeInf.toPartialOrder.{u_1} R (Lattice.toSemilatticeInf.{u_1} R (DistribLattice.toLattice.{u_1} R (instDistribLatticeOfLinearOrder.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6))))], Eq.{succ u_1} (Subgroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3)))) (Units.posSubgroup.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9) (Units.posSubgroup.{u_1} R inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Ring.Subring.Units.3369667631._hygCtx._hyg.9)","typeFull":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R],\n  Units.posSubgroup R = Units.posSubgroup R","typeReadable":"∀ (R : Type u_1) [inst : Semiring R] [inst_1 : LinearOrder R] [inst_2 : IsStrictOrderedRing R],\n  Units.posSubgroup R = Units.posSubgroup R","typeReferences":[["IsStrictOrderedRing"],["Lattice","toSemilatticeInf"],["Units","instGroup"],["LinearOrder"],["Semiring","toMonoidWithZero"],["instDistribLatticeOfLinearOrder"],["DistribLattice","toLattice"],["MonoidWithZero","toMonoid"],["Units","posSubgroup"],["Subgroup"],["Eq"],["Units"],["Semiring"],["SemilatticeInf","toPartialOrder"]],"valueReferences":[["Units","instGroup"],["Eq","refl"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["Units","posSubgroup"],["Subgroup"],["Units"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Star.BigOperators.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["star_prod"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.3 : CommMonoid.{u_1} R] [inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.6 : StarMul.{u_1} R (MulOne.toMul.{u_1} R (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (CommMonoid.toMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.3))))] {α : Type.{u_2}} (s : Finset.{u_2} α) (f : α -> R), Eq.{succ u_1} R (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarMul.toInvolutiveStar.{u_1} R (MulOne.toMul.{u_1} R (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (CommMonoid.toMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.3)))) inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.6)) (Finset.prod.{u_2, u_1} α R inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.3 s (fun (x : α) => f x))) (Finset.prod.{u_2, u_1} α R inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.3 s (fun (x : α) => Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarMul.toInvolutiveStar.{u_1} R (MulOne.toMul.{u_1} R (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (CommMonoid.toMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.3)))) inst._@.Mathlib.Algebra.Star.BigOperators.952080030._hygCtx._hyg.6)) (f x)))","typeFull":"∀ {R : Type u_1} [inst : CommMonoid R] [inst_1 : StarMul R] {α : Type u_2} (s : Finset α) (f : α → R),\n  star (∏ x ∈ s, f x) = ∏ x ∈ s, star (f x)","typeReadable":"∀ {R : Type u_1} [inst : CommMonoid R] [inst_1 : StarMul R] {α : Type u_2} (s : Finset α) (f : α → R),\n  star (∏ x ∈ s, f x) = ∏ x ∈ s, star (f x)","typeReferences":[["Star","star"],["MulOneClass","toMulOne"],["Finset","prod"],["MulOne","toMul"],["Finset"],["CommMonoid","toMonoid"],["StarMul"],["InvolutiveStar","toStar"],["Monoid","toMulOneClass"],["StarMul","toInvolutiveStar"],["CommMonoid"],["Eq"]],"valueReferences":[["CommMonoid","toMonoid"],["CommMagma","toMul"],["starMulAut"],["CommMonoid","toCommSemigroup"],["MulAut"],["Monoid","toMulOneClass"],["EquivLike","toFunLike"],["MulEquivClass","instMonoidHomClass"],["MulEquiv","instEquivLike"],["map_prod"],["MulEquiv","instMulEquivClass"],["CommSemigroup","toCommMagma"]]},{"isProp":true,"kind":"theorem","name":["star_finsuppProd"],"typeFallback":"forall {R : Type.{u_1}} {ι : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.5 : Zero.{u_3} M] [inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.8 : CommMonoid.{u_1} R] [inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.11 : StarMul.{u_1} R (MulOne.toMul.{u_1} R (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (CommMonoid.toMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.8))))] (s : Finsupp.{u_2, u_3} ι M inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.5) (f : ι -> M -> R), Eq.{succ u_1} R (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarMul.toInvolutiveStar.{u_1} R (MulOne.toMul.{u_1} R (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (CommMonoid.toMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.8)))) inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.11)) (Finsupp.prod.{u_2, u_3, u_1} ι M R inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.8 s f)) (Finsupp.prod.{u_2, u_3, u_1} ι M R inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.8 s (fun (i : ι) (m : M) => Star.star.{max (max u_1 u_2) u_3} (ι -> M -> R) (Pi.instStarForall.{u_2, max u_1 u_3} ι (fun (a._@._internal._hyg.0 : ι) => M -> R) (fun (i : ι) => Pi.instStarForall.{u_3, u_1} M (fun (a._@._internal._hyg.0 : M) => R) (fun (i : M) => InvolutiveStar.toStar.{u_1} R (StarMul.toInvolutiveStar.{u_1} R (MulOne.toMul.{u_1} R (MulOneClass.toMulOne.{u_1} R (Monoid.toMulOneClass.{u_1} R (CommMonoid.toMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.8)))) inst._@.Mathlib.Algebra.Star.BigOperators.2364695642._hygCtx._hyg.11)))) f i m))","typeFull":"∀ {R : Type u_1} {ι : Type u_2} {M : Type u_3} [inst : Zero M] [inst_1 : CommMonoid R] [inst_2 : StarMul R] (s : ι →₀ M)\n  (f : ι → M → R), star (s.prod f) = s.prod fun i m => star f i m","typeReadable":"∀ {R : Type u_1} {ι : Type u_2} {M : Type u_3} [inst : Zero M] [inst_1 : CommMonoid R] [inst_2 : StarMul R] (s : ι →₀ M)\n  (f : ι → M → R), star (s.prod f) = s.prod fun i m => star f i m","typeReferences":[["MulOneClass","toMulOne"],["Star","star"],["CommMonoid","toMonoid"],["StarMul"],["StarMul","toInvolutiveStar"],["MulOne","toMul"],["Finsupp","prod"],["Pi","instStarForall"],["Monoid","toMulOneClass"],["InvolutiveStar","toStar"],["Zero"],["CommMonoid"],["Eq"],["Finsupp"]],"valueReferences":[["MulOneClass","toMulOne"],["Finset"],["Eq","trans"],["CommMonoid","toMonoid"],["star_prod"],["StarMul","toInvolutiveStar"],["DFunLike","coe"],["Finsupp","support"],["congrArg"],["MulOne","toMul"],["Pi","instStarForall"],["congr"],["Monoid","toMulOneClass"],["Eq"],["Star","star"],["Finsupp","instFunLike"],["True"],["Finset","prod"],["eq_self"],["Finsupp","prod"],["of_eq_true"],["Eq","refl"],["InvolutiveStar","toStar"],["Finset","prod_congr"],["Finsupp"]]},{"isProp":true,"kind":"theorem","name":["star_finsuppSum"],"typeFallback":"forall {R : Type.{u_1}} {ι : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.5 : Zero.{u_3} M] [inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.8 : AddCommMonoid.{u_1} R] [inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.11 : StarAddMonoid.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.8)] (s : Finsupp.{u_2, u_3} ι M inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.5) (f : ι -> M -> R), Eq.{succ u_1} R (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.11)) (Finsupp.sum.{u_2, u_3, u_1} ι M R inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.8 s f)) (Finsupp.sum.{u_2, u_3, u_1} ι M R inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.8 s (fun (i : ι) (m : M) => Star.star.{max (max u_1 u_2) u_3} (ι -> M -> R) (Pi.instStarForall.{u_2, max u_1 u_3} ι (fun (a._@._internal._hyg.0 : ι) => M -> R) (fun (i : ι) => Pi.instStarForall.{u_3, u_1} M (fun (a._@._internal._hyg.0 : M) => R) (fun (i : M) => InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.8) inst._@.Mathlib.Algebra.Star.BigOperators.2397670787._hygCtx._hyg.11)))) f i m))","typeFull":"∀ {R : Type u_1} {ι : Type u_2} {M : Type u_3} [inst : Zero M] [inst_1 : AddCommMonoid R] [inst_2 : StarAddMonoid R]\n  (s : ι →₀ M) (f : ι → M → R), star (s.sum f) = s.sum fun i m => star f i m","typeReadable":"∀ {R : Type u_1} {ι : Type u_2} {M : Type u_3} [inst : Zero M] [inst_1 : AddCommMonoid R] [inst_2 : StarAddMonoid R]\n  (s : ι →₀ M) (f : ι → M → R), star (s.sum f) = s.sum fun i m => star f i m","typeReferences":[["Star","star"],["StarAddMonoid"],["AddCommMonoid"],["Pi","instStarForall"],["StarAddMonoid","toInvolutiveStar"],["InvolutiveStar","toStar"],["Finsupp","sum"],["AddCommMonoid","toAddMonoid"],["Zero"],["Eq"],["Finsupp"]],"valueReferences":[["Star","star"],["Finsupp","instFunLike"],["Finset"],["True"],["Eq","trans"],["StarAddMonoid","toInvolutiveStar"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Finsupp","support"],["congrArg"],["Finset","sum_congr"],["eq_self"],["star_sum"],["of_eq_true"],["Eq","refl"],["congr"],["Pi","instStarForall"],["InvolutiveStar","toStar"],["Finsupp","sum"],["Finset","sum"],["Eq"],["Finsupp"]]},{"isProp":true,"kind":"theorem","name":["isSelfAdjoint_sum"],"typeFallback":"forall {R : Type.{u_1}} {ι : Type.{u_2}} [inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.4 : AddCommMonoid.{u_1} R] [inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.7 : StarAddMonoid.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.4)] (s : Finset.{u_2} ι) {x : ι -> R}, (forall (i : ι), (Membership.mem.{u_2, u_2} ι (Finset.{u_2} ι) (SetLike.instMembership.{u_2, u_2} (Finset.{u_2} ι) ι (Finset.instSetLike.{u_2} ι)) s i) -> (IsSelfAdjoint.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.7)) (x i))) -> (IsSelfAdjoint.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.4) inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.7)) (Finset.sum.{u_2, u_1} ι R inst._@.Mathlib.Algebra.Star.BigOperators.626757307._hygCtx._hyg.4 s (fun (i : ι) => x i)))","typeFull":"∀ {R : Type u_1} {ι : Type u_2} [inst : AddCommMonoid R] [inst_1 : StarAddMonoid R] (s : Finset ι) {x : ι → R},\n  (∀ i ∈ s, IsSelfAdjoint (x i)) → IsSelfAdjoint (∑ i ∈ s, x i)","typeReadable":"∀ {R : Type u_1} {ι : Type u_2} [inst : AddCommMonoid R] [inst_1 : StarAddMonoid R] (s : Finset ι) {x : ι → R},\n  (∀ i ∈ s, IsSelfAdjoint (x i)) → IsSelfAdjoint (∑ i ∈ s, x i)","typeReferences":[["Finset","instSetLike"],["StarAddMonoid"],["AddCommMonoid"],["SetLike","instMembership"],["Finset"],["StarAddMonoid","toInvolutiveStar"],["InvolutiveStar","toStar"],["Membership","mem"],["Finset","sum"],["AddCommMonoid","toAddMonoid"],["IsSelfAdjoint"]],"valueReferences":[["rfl"],["Star","star"],["Finset"],["StarAddMonoid","toInvolutiveStar"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Finset","sum_congr"],["star_sum"],["InvolutiveStar","toStar"],["id"],["Finset","sum"],["Eq","mpr"],["congrFun'"],["Eq"],["IsSelfAdjoint"]]},{"isProp":true,"kind":"theorem","name":["star_sum"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.3 : AddCommMonoid.{u_1} R] [inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.6 : StarAddMonoid.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.3)] {α : Type.{u_2}} (s : Finset.{u_2} α) (f : α -> R), Eq.{succ u_1} R (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.6)) (Finset.sum.{u_2, u_1} α R inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.3 s (fun (x : α) => f x))) (Finset.sum.{u_2, u_1} α R inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.3 s (fun (x : α) => Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Star.BigOperators.4013683521._hygCtx._hyg.6)) (f x)))","typeFull":"∀ {R : Type u_1} [inst : AddCommMonoid R] [inst_1 : StarAddMonoid R] {α : Type u_2} (s : Finset α) (f : α → R),\n  star (∑ x ∈ s, f x) = ∑ x ∈ s, star (f x)","typeReadable":"∀ {R : Type u_1} [inst : AddCommMonoid R] [inst_1 : StarAddMonoid R] {α : Type u_2} (s : Finset α) (f : α → R),\n  star (∑ x ∈ s, f x) = ∑ x ∈ s, star (f x)","typeReferences":[["Star","star"],["StarAddMonoid"],["AddCommMonoid"],["Finset"],["StarAddMonoid","toInvolutiveStar"],["InvolutiveStar","toStar"],["Finset","sum"],["AddCommMonoid","toAddMonoid"],["Eq"]],"valueReferences":[["AddMonoid","toAddSemigroup"],["AddEquivClass","instAddMonoidHomClass"],["EquivLike","toFunLike"],["AddEquiv","instAddEquivClass"],["AddCommMonoid","toAddMonoid"],["map_sum"],["AddEquiv"],["starAddEquiv"],["AddEquiv","instEquivLike"],["AddMonoid","toAddZeroClass"],["AddSemigroup","toAdd"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Star.RingQuot.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","starRing","_proof_2"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 : StarRing.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15))] (r : R -> R -> Prop) (hr : forall (a : R) (b : R), (r a b) -> (r (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) a) (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) b))) (r._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.68 : RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (s : RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r), Eq.{succ u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr (HMul.hMul.{u_1, u_1, u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (instHMul.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Distrib.toMul.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonUnitalNonAssocSemiring.toDistrib.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Semiring.toNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.instSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r)))))) r._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.68 s)) (HMul.hMul.{u_1, u_1, u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (instHMul.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Distrib.toMul.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonUnitalNonAssocSemiring.toDistrib.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Semiring.toNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.instSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r)))))) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr s) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr r._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.68))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : StarRing R] (r : R → R → Prop)\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (r_1 s : RingQuot r),\n  RingQuot.star'✝ r hr (r_1 * s) = RingQuot.star'✝¹ r hr s * RingQuot.star'✝² r hr r_1","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : StarRing R] (r : R → R → Prop)\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (r_1 s : RingQuot r),\n  RingQuot.star'✝ r hr (r_1 * s) = RingQuot.star'✝¹ r hr s * RingQuot.star'✝² r hr r_1","typeReferences":[["Star","star"],["StarRing"],["NonUnitalNonAssocSemiring","toDistrib"],["StarAddMonoid","toInvolutiveStar"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["HMul","hMul"],["AddCommMonoid","toAddMonoid"],["StarRing","toStarAddMonoid"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["RingQuot","instSemiring"],["InvolutiveStar","toStar"],["instHMul"],["Eq"],["RingQuot"],["Semiring"]],"valueReferences":[["StarMul","star_mul"],["Eq","trans"],["HMul","hMul"],["Quot","ind"],["StarMul","toInvolutiveStar"],["AddCommMonoid","toAddMonoid"],["RingQuot","mk"],["StarRing","toStarAddMonoid"],["congrArg"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'","congr_simp"],["Semiring","toNonAssocSemiring"],["RingQuot","instSemiring"],["congr"],["Eq"],["RingQuot"],["Quot","mk"],["Star","star"],["True"],["NonUnitalNonAssocSemiring","toDistrib"],["StarAddMonoid","toInvolutiveStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["RingQuot","casesOn"],["Quot"],["eq_self"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["of_eq_true"],["StarRing","toStarMul"],["InvolutiveStar","toStar"],["instHMul"],["RingQuot","mul_quot"],["RingQuot","Rel"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'","congr_simp"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 : Semiring.{u} R] (r : R -> R -> Prop) [inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11 : StarRing.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3))] (hr : forall (a : R) (b : R), (r a b) -> (r (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11))) a) (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11))) b))) (a._@._internal._hyg.0 : RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r) (a_1._@._internal._hyg.0 : RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r), (Eq.{succ u} (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r) a._@._internal._hyg.0 a_1._@._internal._hyg.0) -> (Eq.{succ u} (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11 hr a._@._internal._hyg.0) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11 hr a_1._@._internal._hyg.0))","typeFull":"∀ {R : Type u} [inst : Semiring R] (r : R → R → Prop) [inst_1 : StarRing R]\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (a a_1 : RingQuot r),\n  a = a_1 → RingQuot.star'✝ r hr a = RingQuot.star'✝¹ r hr a_1","typeReadable":"∀ {R : Type u} [inst : Semiring R] (r : R → R → Prop) [inst_1 : StarRing R]\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (a a_1 : RingQuot r),\n  a = a_1 → RingQuot.star'✝ r hr a = RingQuot.star'✝¹ r hr a_1","typeReferences":[["Star","star"],["StarRing"],["StarAddMonoid","toInvolutiveStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommMonoid","toAddMonoid"],["StarRing","toStarAddMonoid"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["InvolutiveStar","toStar"],["Eq"],["RingQuot"],["Semiring"]],"valueReferences":[["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["Eq","refl"],["Eq"],["Eq","rec"],["RingQuot"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","starRing","_proof_3"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 : StarRing.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15))] (r : R -> R -> Prop) (hr : forall (a : R) (b : R), (r a b) -> (r (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) a) (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) b))) (r._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.78 : RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (s : RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r), Eq.{succ u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr (HAdd.hAdd.{u_1, u_1, u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (instHAdd.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Distrib.toAdd.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonUnitalNonAssocSemiring.toDistrib.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Semiring.toNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.instSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r)))))) r._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.78 s)) (HAdd.hAdd.{u_1, u_1, u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (instHAdd.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Distrib.toAdd.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonUnitalNonAssocSemiring.toDistrib.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Semiring.toNonAssocSemiring.{u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.instSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r)))))) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr r._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.78) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr s))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : StarRing R] (r : R → R → Prop)\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (r_1 s : RingQuot r),\n  RingQuot.star'✝ r hr (r_1 + s) = RingQuot.star'✝¹ r hr r_1 + RingQuot.star'✝² r hr s","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : StarRing R] (r : R → R → Prop)\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (r_1 s : RingQuot r),\n  RingQuot.star'✝ r hr (r_1 + s) = RingQuot.star'✝¹ r hr r_1 + RingQuot.star'✝² r hr s","typeReferences":[["Star","star"],["Distrib","toAdd"],["StarRing"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["StarAddMonoid","toInvolutiveStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommMonoid","toAddMonoid"],["StarRing","toStarAddMonoid"],["HAdd","hAdd"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["RingQuot","instSemiring"],["InvolutiveStar","toStar"],["Eq"],["RingQuot"],["Semiring"]],"valueReferences":[["RingQuot","add_quot"],["Eq","trans"],["AddCommMonoid","toAddMonoid"],["Quot","ind"],["RingQuot","mk"],["AddMonoidWithOne","toAddMonoid"],["StarRing","toStarAddMonoid"],["congrArg"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'","congr_simp"],["Semiring","toNonAssocSemiring"],["RingQuot","instSemiring"],["congr"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["StarAddMonoid","star_add"],["NonAssocSemiring","toAddCommMonoidWithOne"],["RingQuot"],["Quot","mk"],["AddSemigroup","toAdd"],["Star","star"],["Distrib","toAdd"],["True"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["StarAddMonoid","toInvolutiveStar"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["RingQuot","casesOn"],["Quot"],["HAdd","hAdd"],["eq_self"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["of_eq_true"],["AddMonoid","toAddSemigroup"],["InvolutiveStar","toStar"],["RingQuot","Rel"]]},{"isProp":false,"kind":"definition","name":["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'","match_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 : Semiring.{u_1} R] (r : R -> R -> Prop) (motive : (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r) -> Sort.{u_2}) (x._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx.36.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.48 : RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r), (forall (a : Quot.{succ u_1} R (RingQuot.Rel.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r)), motive (RingQuot.mk.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r a)) -> (motive x._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx.36.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.48)","typeFull":"{R : Type u_1} →\n  [inst : Semiring R] →\n    (r : R → R → Prop) →\n      (motive : RingQuot r → Sort u_2) →\n        (x : RingQuot r) → ((a : Quot (RingQuot.Rel r)) → motive { toQuot := a }) → motive x","typeReadable":"{R : Type u_1} →\n  [inst : Semiring R] →\n    (r : R → R → Prop) →\n      (motive : RingQuot r → Sort u_2) →\n        (x : RingQuot r) → ((a : Quot (RingQuot.Rel r)) → motive { toQuot := a }) → motive x","typeReferences":[["RingQuot","mk"],["RingQuot","Rel"],["Quot"],["RingQuot"],["Semiring"]],"valueReferences":[["RingQuot","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'_quot"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3 : Semiring.{u} R] (r : R -> R -> Prop) [inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.11 : StarRing.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3))] (hr : forall (a : R) (b : R), (r a b) -> (r (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.11))) a) (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.11))) b))) {a : R}, Eq.{succ u} (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3 r) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3 r inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.11 hr (RingQuot.mk.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3 r (Quot.mk.{succ u} R (RingQuot.Rel.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3 r) a))) (RingQuot.mk.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3 r (Quot.mk.{succ u} R (RingQuot.Rel.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3 r) (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.2370717908._hygCtx._hyg.11))) a)))","typeFull":"∀ {R : Type u} [inst : Semiring R] (r : R → R → Prop) [inst_1 : StarRing R]\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) {a : R},\n  RingQuot.star'✝ r hr { toQuot := Quot.mk (RingQuot.Rel r) a } = { toQuot := Quot.mk (RingQuot.Rel r) (star a) }","typeReadable":"∀ {R : Type u} [inst : Semiring R] (r : R → R → Prop) [inst_1 : StarRing R]\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) {a : R},\n  RingQuot.star'✝ r hr { toQuot := Quot.mk (RingQuot.Rel r) a } = { toQuot := Quot.mk (RingQuot.Rel r) (star a) }","typeReferences":[["Star","star"],["StarRing"],["StarAddMonoid","toInvolutiveStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommMonoid","toAddMonoid"],["RingQuot","mk"],["StarRing","toStarAddMonoid"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["InvolutiveStar","toStar"],["Eq"],["RingQuot","Rel"],["Quot","mk"],["RingQuot"],["Semiring"]],"valueReferences":[["rfl"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["RingQuot","mk"],["RingQuot","Rel"],["Quot","mk"],["RingQuot"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","starRing","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 : StarRing.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15))] (r : R -> R -> Prop) (hr : forall (a : R) (b : R), (r a b) -> (r (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) a) (Star.star.{u_1} R (InvolutiveStar.toStar.{u_1} R (StarAddMonoid.toInvolutiveStar.{u_1} R (AddCommMonoid.toAddMonoid.{u_1} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) b))) (x._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.61 : RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r), Eq.{succ u_1} (RingQuot.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr (_private.Mathlib.Algebra.Star.RingQuot.0.RingQuot.star'.{u_1} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 hr x._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.61)) x._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.61","typeFull":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : StarRing R] (r : R → R → Prop)\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (x : RingQuot r), RingQuot.star'✝ r hr (RingQuot.star'✝¹ r hr x) = x","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] [inst_1 : StarRing R] (r : R → R → Prop)\n  (hr : ∀ (a b : R), r a b → r (star a) (star b)) (x : RingQuot r), RingQuot.star'✝ r hr (RingQuot.star'✝¹ r hr x) = x","typeReferences":[["Star","star"],["StarRing"],["StarAddMonoid","toInvolutiveStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["AddCommMonoid","toAddMonoid"],["StarRing","toStarAddMonoid"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["InvolutiveStar","toStar"],["Eq"],["RingQuot"],["Semiring"]],"valueReferences":[["Star","star"],["Eq","trans"],["True"],["StarAddMonoid","toInvolutiveStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["RingQuot","casesOn"],["AddCommMonoid","toAddMonoid"],["Quot","ind"],["star_star"],["RingQuot","mk"],["StarRing","toStarAddMonoid"],["Quot"],["congrArg"],["eq_self"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["of_eq_true"],["InvolutiveStar","toStar"],["congrFun'"],["Eq"],["RingQuot"],["RingQuot","Rel"],["Quot","mk"]]},{"isProp":false,"kind":"definition","name":["RingQuot","starRing"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18 : StarRing.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15))] (r : R -> R -> Prop), (forall (a : R) (b : R), (r a b) -> (r (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) a) (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15)) inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.18))) b))) -> (StarRing.{u} (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (Semiring.toNonAssocSemiring.{u} (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r) (RingQuot.instSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.3139772356._hygCtx._hyg.15 r))))","typeFull":"{R : Type u} →\n  [inst : Semiring R] →\n    [inst_1 : StarRing R] → (r : R → R → Prop) → (∀ (a b : R), r a b → r (star a) (star b)) → StarRing (RingQuot r)","typeReadable":"{R : Type u} →\n  [inst : Semiring R] →\n    [inst_1 : StarRing R] → (r : R → R → Prop) → (∀ (a b : R), r a b → r (star a) (star b)) → StarRing (RingQuot r)","typeReferences":[["Star","star"],["Semiring","toNonAssocSemiring"],["StarRing"],["RingQuot","instSemiring"],["StarAddMonoid","toInvolutiveStar"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["InvolutiveStar","toStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddCommMonoid","toAddMonoid"],["RingQuot"],["StarRing","toStarAddMonoid"],["Semiring"]],"valueReferences":[["InvolutiveStar","mk"],["NonUnitalNonAssocSemiring","toDistrib"],["StarRing","mk"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["StarMul","mk"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","starRing","_proof_2"],["Star","mk"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","starRing","_proof_3"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","starRing","_proof_1"],["Semiring","toNonAssocSemiring"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],["RingQuot","instSemiring"],["RingQuot"]]},{"isProp":false,"kind":"definition","name":["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 : Semiring.{u} R] (r : R -> R -> Prop) [inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11 : StarRing.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3))], (forall (a : R) (b : R), (r a b) -> (r (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11))) a) (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.11))) b))) -> (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r) -> (RingQuot.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.788165188._hygCtx._hyg.3 r)","typeFull":"{R : Type u} →\n  [inst : Semiring R] →\n    (r : R → R → Prop) → [inst_1 : StarRing R] → (∀ (a b : R), r a b → r (star a) (star b)) → RingQuot r → RingQuot r","typeReadable":"{R : Type u} →\n  [inst : Semiring R] →\n    (r : R → R → Prop) → [inst_1 : StarRing R] → (∀ (a b : R), r a b → r (star a) (star b)) → RingQuot r → RingQuot r","typeReferences":[["Star","star"],["Semiring","toNonAssocSemiring"],["StarRing"],["StarAddMonoid","toInvolutiveStar"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["InvolutiveStar","toStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddCommMonoid","toAddMonoid"],["RingQuot"],["StarRing","toStarAddMonoid"],["Semiring"]],"valueReferences":[["Star","star"],["StarAddMonoid","toInvolutiveStar"],["_private","Mathlib","Algebra","Star","RingQuot",0,"RingQuot","star'","match_1"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddCommMonoid","toAddMonoid"],["RingQuot","mk"],["StarRing","toStarAddMonoid"],["Semiring","toNonAssocSemiring"],["RingQuot","Rel","star"],["InvolutiveStar","toStar"],["Quot","map"],["RingQuot"],["RingQuot","Rel"]]},{"isProp":true,"kind":"theorem","name":["RingQuot","Rel","star"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3 : Semiring.{u} R] (r : R -> R -> Prop) [inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.11 : StarRing.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3))], (forall (a : R) (b : R), (r a b) -> (r (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.11))) a) (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.11))) b))) -> (forall {{a : R}} {{b : R}}, (RingQuot.Rel.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3 r a b) -> (RingQuot.Rel.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3 r (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.11))) a) (Star.star.{u} R (InvolutiveStar.toStar.{u} R (StarAddMonoid.toInvolutiveStar.{u} R (AddCommMonoid.toAddMonoid.{u} R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)))) (StarRing.toStarAddMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.3)) inst._@.Mathlib.Algebra.Star.RingQuot.212405414._hygCtx._hyg.11))) b)))","typeFull":"∀ {R : Type u} [inst : Semiring R] (r : R → R → Prop) [inst_1 : StarRing R],\n  (∀ (a b : R), r a b → r (star a) (star b)) → ∀ ⦃a b : R⦄, RingQuot.Rel r a b → RingQuot.Rel r (star a) (star b)","typeReadable":"∀ {R : Type u} [inst : Semiring R] (r : R → R → Prop) [inst_1 : StarRing R],\n  (∀ (a b : R), r a b → r (star a) (star b)) → ∀ ⦃a b : R⦄, RingQuot.Rel r a b → RingQuot.Rel r (star a) (star b)","typeReferences":[["Star","star"],["Semiring","toNonAssocSemiring"],["StarRing"],["StarAddMonoid","toInvolutiveStar"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["InvolutiveStar","toStar"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddCommMonoid","toAddMonoid"],["RingQuot","Rel"],["StarRing","toStarAddMonoid"],["Semiring"]],"valueReferences":[["StarMul","star_mul"],["HMul","hMul"],["StarMul","toInvolutiveStar"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["StarRing","toStarAddMonoid"],["congrArg"],["Semiring","toNonAssocSemiring"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["RingQuot","Rel","of"],["Eq"],["RingQuot","Rel","mul_left"],["NonAssocSemiring","toAddCommMonoidWithOne"],["StarAddMonoid","star_add"],["AddSemigroup","toAdd"],["RingQuot","Rel","add_left"],["Star","star"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["StarAddMonoid","toInvolutiveStar"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["RingQuot","Rel","rec"],["HAdd","hAdd"],["StarRing","toStarMul"],["AddMonoid","toAddSemigroup"],["InvolutiveStar","toStar"],["RingQuot","Rel","mul_right"],["id"],["instHMul"],["Eq","mpr"],["RingQuot","Rel"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.BoundedVariation.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem_pi"],"typeFallback":"forall {ι : Type.{u_2}} [inst._@.Mathlib.Analysis.BoundedVariation.3275597788._hygCtx._hyg.19 : Fintype.{u_2} ι] {f : Real -> ι -> Real} {s : Set.{0} Real}, (LocallyBoundedVariationOn.{0, u_2} Real Real.linearOrder (ι -> Real) (pseudoEMetricSpacePi.{u_2, 0} ι (fun (a._@._internal._hyg.0 : ι) => Real) inst._@.Mathlib.Analysis.BoundedVariation.3275597788._hygCtx._hyg.19 (fun (b : ι) => EMetricSpace.toPseudoEMetricSpace.{0} Real (MetricSpace.toEMetricSpace.{0} Real Real.metricSpace))) f s) -> (Filter.Eventually.{0} Real (fun (x : Real) => (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembership.{0} Real) s x) -> (DifferentiableWithinAt.{0, 0, u_2} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (ι -> Real) (Pi.addCommGroup.{u_2, 0} ι (fun (a._@._internal._hyg.0 : ι) => Real) (fun (i : ι) => Real.instAddCommGroup)) (Pi.Function.module.{u_2, 0, 0} ι Real Real (DivisionSemiring.toSemiring.{0} Real (Semifield.toDivisionSemiring.{0} Real (Field.toSemifield.{0} Real (NormedField.toField.{0} Real (NontriviallyNormedField.toNormedField.{0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField)))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Real (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{0} Real (Semiring.toNonUnitalSemiring.{0} Real (Ring.toSemiring.{0} Real Real.instRing)))) (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike)))) (Pi.topologicalSpace.{0, u_2} ι (fun (a._@._internal._hyg.0 : ι) => Real) (fun (i : ι) => UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace))) f s x)) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {ι : Type u_2} [inst : Fintype ι] {f : ℝ → ι → ℝ} {s : Set ℝ},\n  LocallyBoundedVariationOn f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReadable":"∀ {ι : Type u_2} [inst : Fintype ι] {f : ℝ → ι → ℝ} {s : Set ℝ},\n  LocallyBoundedVariationOn f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReferences":[["Filter","Eventually"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["Membership","mem"],["Real","instRCLike"],["DenselyNormedField","toNontriviallyNormedField"],["Semifield","toDivisionSemiring"],["Semiring","toNonUnitalSemiring"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["DifferentiableWithinAt"],["DivisionSemiring","toSemiring"],["Fintype"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["Ring","toSemiring"],["Real","pseudoMetricSpace"],["Real","linearOrder"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["pseudoEMetricSpacePi"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["UniformSpace","toTopologicalSpace"],["LocallyBoundedVariationOn"],["Real","instRing"],["Pi","Function","module"],["NontriviallyNormedField","toNormedField"],["MeasureTheory","Measure"],["Real","measureSpace"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["NormedField","toField"],["MetricSpace","toEMetricSpace"],["MeasureTheory","ae"],["Real","metricSpace"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","instAddCommGroup"],["Real","denselyNormedField"],["Field","toSemifield"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Pi","topologicalSpace"],["NormedCommRing","toSeminormedCommRing"],["EMetricSpace","toPseudoEMetricSpace"],["InnerProductSpace","toNormedSpace"],["Pi","addCommGroup"]],"valueReferences":[["Filter","Eventually"],["PseudoMetricSpace","toUniformSpace"],["Membership","mem"],["Real","normedCommRing"],["Filter","univ_mem'"],["MeasureTheory","ae_all_iff"],["NormedAddCommGroup","toAddCommGroup"],["Finite","of_fintype"],["Pi","module"],["NNReal","instOne"],["Real","instRCLike"],["DenselyNormedField","toNontriviallyNormedField"],["Semifield","toDivisionSemiring"],["Semiring","toNonUnitalSemiring"],["Real","normedAddCommGroup"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NNReal"],["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem_real"],["DifferentiableWithinAt"],["DivisionSemiring","toSemiring"],["MeasureTheory","Measure","instFunLike"],["differentiableWithinAt_pi"],["Ring","toSemiring"],["Set","instMembership"],["Real","pseudoMetricSpace"],["Iff","mpr"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["id"],["Real","linearOrder"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["setOf"],["pseudoEMetricSpacePi"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["Filter","mp_mem"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["LipschitzWith","comp_locallyBoundedVariationOn"],["Pi","Function","module"],["NontriviallyNormedField","toNormedField"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["NormedField","toField"],["MetricSpace","toEMetricSpace"],["MeasureTheory","ae"],["Real","metricSpace"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Finite","to_countable"],["OfNat","ofNat"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["One","toOfNat1"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["Field","toSemifield"],["Pi","topologicalSpace"],["LipschitzWith","eval"],["EMetricSpace","toPseudoEMetricSpace"],["NormedCommRing","toSeminormedCommRing"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["InnerProductSpace","toNormedSpace"],["Pi","addCommGroup"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","BoundedVariation",0,"BoundedVariationOn","ae_differentiableAt_of_mem_uIcc","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {F : Type.{u_3}} [inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3787073176._hygCtx._hyg.5 : FunLike.{succ u_3, succ u_1, 1} F (Set.{u_1} α) ENNReal] [inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3787073176._hygCtx._hyg.15 : MeasureTheory.OuterMeasureClass.{u_3, u_1} F α inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3787073176._hygCtx._hyg.5] {μ : F} {p : α -> Prop}, Eq.{1} Prop (Filter.Eventually.{u_1} α (fun (a : α) => p a) (MeasureTheory.ae.{u_1, u_3} α F inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3787073176._hygCtx._hyg.5 inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3787073176._hygCtx._hyg.15 μ)) (Eq.{1} ENNReal (DFunLike.coe.{succ u_3, succ u_1, 1} F (Set.{u_1} α) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Set.{u_1} α) => ENNReal) inst._@.Mathlib.MeasureTheory.OuterMeasure.AE.3787073176._hygCtx._hyg.5 μ (setOf.{u_1} α (fun (a : α) => Not (p a)))) (OfNat.ofNat.{0} ENNReal 0 (Zero.toOfNat0.{0} ENNReal instZeroENNReal)))","typeFull":"∀ {α : Type u_1} {F : Type u_3} [inst : FunLike F (Set α) ENNReal] [inst_1 : MeasureTheory.OuterMeasureClass F α]\n  {μ : F} {p : α → Prop}, (∀ᵐ (a : α) ∂μ, p a) = (μ {a | ¬p a} = 0)","typeReadable":"∀ {α : Type u_1} {F : Type u_3} [inst : FunLike F (Set α) ENNReal] [inst_1 : MeasureTheory.OuterMeasureClass F α]\n  {μ : F} {p : α → Prop}, (∀ᵐ (a : α) ∂μ, p a) = (μ {a | ¬p a} = 0)","typeReferences":[["Filter","Eventually"],["Not"],["FunLike"],["Set"],["instZeroENNReal"],["DFunLike","coe"],["OfNat","ofNat"],["ENNReal"],["MeasureTheory","OuterMeasureClass"],["Zero","toOfNat0"],["Eq"],["MeasureTheory","ae"],["setOf"]],"valueReferences":[["Filter","Eventually"],["Not"],["MeasureTheory","ae_iff"],["Set"],["instZeroENNReal"],["DFunLike","coe"],["OfNat","ofNat"],["ENNReal"],["Zero","toOfNat0"],["Eq"],["MeasureTheory","ae"],["propext"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.12 : FiniteDimensional.{0, u_1} Real V Real.instDivisionRing (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.6)] {f : Real -> V} {s : Set.{0} Real}, (LocallyBoundedVariationOn.{0, u_1} Real Real.linearOrder V (EMetricSpace.toPseudoEMetricSpace.{u_1} V (MetricSpace.toEMetricSpace.{u_1} V (NormedAddCommGroup.toMetricSpace.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3))) f s) -> (Filter.Eventually.{0} Real (fun (x : Real) => (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembership.{0} Real) s x) -> (DifferentiableWithinAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) V (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2616315869._hygCtx._hyg.3)))) f s x)) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V}\n  {s : Set ℝ}, LocallyBoundedVariationOn f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReadable":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V}\n  {s : Set ℝ}, LocallyBoundedVariationOn f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReferences":[["Filter","Eventually"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["FiniteDimensional"],["UniformSpace","toTopologicalSpace"],["Real","normedCommRing"],["Membership","mem"],["LocallyBoundedVariationOn"],["NormedAddCommGroup","toAddCommGroup"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["Real","instDivisionRing"],["MetricSpace","toEMetricSpace"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["NormedAddCommGroup","toMetricSpace"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DifferentiableWithinAt"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["NormedCommRing","toSeminormedCommRing"],["EMetricSpace","toPseudoEMetricSpace"],["Real","linearOrder"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["PseudoMetricSpace","toUniformSpace"],["AddCommGroup","toAddGroup"],["Real","normedCommRing"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["Module","Basis","ofVectorSpace"],["Filter","univ_mem'"],["Set","Elem"],["Finite","of_fintype"],["RingHom","id"],["Pi","topologicalAddGroup"],["Monoid","toSemigroup"],["NormedField","toNormedCommRing"],["Real","normedAddCommGroup"],["NormedAddCommGroup","toENormedAddCommMonoid"],["NormedSpace","toModule"],["Real","instCommSemiring"],["Real","instCompleteSpace"],["EMetricSpace","metrizableSpace"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Pi","normedAddCommGroup"],["DifferentiableWithinAt"],["DivisionSemiring","toSemiring"],["TopologicalSpace","t2Space_of_metrizableSpace"],["SeminormedCommRing","toSeminormedRing"],["MeasureTheory","Measure","instFunLike"],["Ring","toSemiring"],["Set","instMembership"],["Pi","seminormedAddCommGroup"],["instIsTopologicalAddGroupReal"],["AddCommGroup","toAddCommMonoid"],["RingHomInvPair","ids"],["RingHomIsometric","ids"],["setOf"],["AddMonoid","toAddZeroClass"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["Pi","t2Space"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommMonoid","toAddMonoid"],["LinearEquiv","toContinuousLinearEquiv"],["Pi","Function","module"],["NontriviallyNormedField","toNormedField"],["ContinuousLinearEquiv","comp_differentiableWithinAt_iff"],["SeminormedAddCommGroup","toIsTopologicalAddGroup"],["SeminormedRing","toPseudoMetricSpace"],["DivisionRing","toDivisionSemiring"],["EquivLike","toFunLike"],["MeasureTheory","ae"],["SeminormedAddCommGroup","toAddCommGroup"],["IsTopologicalSemiring","toIsModuleTopology"],["Set"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Real","instAdd"],["SeminormedAddCommGroup","toSeminormedAddGroup"],["SeminormedAddGroup","toNNNorm"],["RCLike","toInnerProductSpaceReal"],["Module","toDistribMulAction"],["ContinuousLinearMap","toSeminormedAddCommGroup"],["Real","normedField"],["Real","instZero"],["AddCommGroup","toDivisionAddCommMonoid"],["ContinuousLinearEquiv"],["Pi","normedSpace"],["Real","denselyNormedField"],["Pi","topologicalSpace"],["ESeminormedAddCommMonoid","toAddCommMonoid"],["Pi","addCommGroup"],["Membership","mem"],["Iff","mp"],["NormedAddCommGroup","toAddCommGroup"],["Algebra","id"],["NNNorm","nnnorm"],["Real","instRCLike"],["Semiring","toNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["instContinuousSMulForall"],["DenselyNormedField","toNontriviallyNormedField"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["DistribMulAction","toMulAction"],["IsBoundedSMul","continuousSMul"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["FiniteDimensional","instFintypeElemOfVectorSpaceIndex"],["Semiring","toModule"],["Semiring","toNonUnitalSemiring"],["DistribSMul","toSMulZeroClass"],["Module","Basis","equivFun"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["Real"],["NormedAddCommGroup","toMetricSpace"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["Real","pseudoMetricSpace"],["ENormedAddCommMonoid","toESeminormedAddCommMonoid"],["id"],["NegZeroClass","toZero"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["AddZero","toZero"],["ContinuousLinearEquiv","toContinuousLinearMap"],["PseudoMetricSpace","toPseudoEMetricSpace"],["Filter","mp_mem"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["LipschitzWith","comp_locallyBoundedVariationOn"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["ContinuousLinearMap"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["IsModuleTopology","toContinuousSMul"],["NormedField","toField"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["Real","instDivisionRing"],["MetricSpace","toEMetricSpace"],["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem_pi"],["ContinuousLinearEquiv","lipschitz"],["Real","metricSpace"],["ContinuousLinearEquiv","equivLike"],["Semiring","toMonoidWithZero"],["Function","comp"],["Real","semiring"],["MeasureTheory","Measure","instOuterMeasureClass"],["SubNegMonoid","toAddMonoid"],["MeasureTheory","MeasureSpace","volume"],["Module","Basis","ofVectorSpaceIndex"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Real","instAddCommGroup"],["Field","toSemifield"],["NormedCommRing","toSeminormedCommRing"],["NormedSpace","toIsBoundedSMul"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["MulAction","toSemigroupAction"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["LocallyBoundedVariationOn","ae_differentiableAt"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.12 : FiniteDimensional.{0, u_1} Real V Real.instDivisionRing (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.6)] {f : Real -> V}, (LocallyBoundedVariationOn.{0, u_1} Real Real.linearOrder V (EMetricSpace.toPseudoEMetricSpace.{u_1} V (MetricSpace.toEMetricSpace.{u_1} V (NormedAddCommGroup.toMetricSpace.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3))) f (Set.univ.{0} Real)) -> (Filter.Eventually.{0} Real (fun (x : Real) => DifferentiableAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) V (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.1527171599._hygCtx._hyg.3)))) f x) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V},\n  LocallyBoundedVariationOn f Set.univ → ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x","typeReadable":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V},\n  LocallyBoundedVariationOn f Set.univ → ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x","typeReferences":[["Filter","Eventually"],["DifferentiableAt"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["FiniteDimensional"],["UniformSpace","toTopologicalSpace"],["Real","normedCommRing"],["LocallyBoundedVariationOn"],["NormedAddCommGroup","toAddCommGroup"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["MetricSpace","toEMetricSpace"],["Real","instDivisionRing"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["NormedAddCommGroup","toMetricSpace"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["MeasureTheory","Measure","instFunLike"],["Set","univ"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["EMetricSpace","toPseudoEMetricSpace"],["Real","linearOrder"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]],"valueReferences":[["DifferentiableAt"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["Eq","mp"],["Filter","mp_mem"],["UniformSpace","toTopologicalSpace"],["Membership","mem"],["Real","normedCommRing"],["Filter","univ_mem'"],["NormedAddCommGroup","toAddCommGroup"],["Set","mem_univ"],["congrArg"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["DenselyNormedField","toNontriviallyNormedField"],["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem"],["MeasureTheory","ae"],["propext"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DifferentiableWithinAt"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["Set","univ"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["Real","pseudoMetricSpace"],["MeasureTheory","MeasureSpace","volume"],["id"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["InnerProductSpace","toNormedSpace"],["differentiableWithinAt_univ"],["setOf"]]},{"isProp":true,"kind":"theorem","name":["LipschitzOnWith","ae_differentiableWithinAt_real"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.12 : FiniteDimensional.{0, u_1} Real V Real.instDivisionRing (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.6)] {C : NNReal} {f : Real -> V} {s : Set.{0} Real}, (LipschitzOnWith.{0, u_1} Real V (EMetricSpace.toPseudoEMetricSpace.{0} Real (MetricSpace.toEMetricSpace.{0} Real Real.metricSpace)) (EMetricSpace.toPseudoEMetricSpace.{u_1} V (MetricSpace.toEMetricSpace.{u_1} V (NormedAddCommGroup.toMetricSpace.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3))) C f s) -> (MeasurableSet.{0} Real Real.measurableSpace s) -> (Filter.Eventually.{0} Real (fun (x : Real) => DifferentiableWithinAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) V (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2920643620._hygCtx._hyg.3)))) f s x) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.restrict.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace) s)))","typeFull":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : NNReal}\n  {f : ℝ → V} {s : Set ℝ},\n  LipschitzOnWith C f s → MeasurableSet s → ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x","typeReadable":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : NNReal}\n  {f : ℝ → V} {s : Set ℝ},\n  LipschitzOnWith C f s → MeasurableSet s → ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x","typeReferences":[["Filter","Eventually"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["FiniteDimensional"],["UniformSpace","toTopologicalSpace"],["Real","normedCommRing"],["NormedAddCommGroup","toAddCommGroup"],["Real","measureSpace"],["MeasureTheory","Measure"],["MeasureTheory","Measure","restrict"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["Real","instDivisionRing"],["MetricSpace","toEMetricSpace"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["Real","metricSpace"],["NormedAddCommGroup","toMetricSpace"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NNReal"],["DifferentiableWithinAt"],["LipschitzOnWith"],["MeasureTheory","Measure","instFunLike"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","measurableSpace"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["NormedCommRing","toSeminormedCommRing"],["EMetricSpace","toPseudoEMetricSpace"],["MeasurableSet"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]],"valueReferences":[["LipschitzOnWith","locallyBoundedVariationOn"],["NormedAddCommGroup","toMetricSpace"],["EMetricSpace","toPseudoEMetricSpace"],["MetricSpace","toEMetricSpace"],["LocallyBoundedVariationOn","ae_differentiableWithinAt"]]},{"isProp":true,"kind":"theorem","name":["BoundedVariationOn","ae_differentiableAt_of_mem_uIcc"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.12 : FiniteDimensional.{0, u_1} Real V Real.instDivisionRing (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.6)] {f : Real -> V} {a : Real} {b : Real}, (BoundedVariationOn.{0, u_1} Real Real.linearOrder V (EMetricSpace.toPseudoEMetricSpace.{u_1} V (MetricSpace.toEMetricSpace.{u_1} V (NormedAddCommGroup.toMetricSpace.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3))) f (Set.uIcc.{0} Real Real.lattice a b)) -> (Filter.Eventually.{0} Real (fun (x : Real) => (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembership.{0} Real) (Set.uIcc.{0} Real Real.lattice a b) x) -> (DifferentiableAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) V (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.4126001278._hygCtx._hyg.3)))) f x)) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V} {a b : ℝ},\n  BoundedVariationOn f (Set.uIcc a b) → ∀ᵐ (x : ℝ), x ∈ Set.uIcc a b → DifferentiableAt ℝ f x","typeReadable":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V} {a b : ℝ},\n  BoundedVariationOn f (Set.uIcc a b) → ∀ᵐ (x : ℝ), x ∈ Set.uIcc a b → DifferentiableAt ℝ f x","typeReferences":[["Filter","Eventually"],["DifferentiableAt"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["FiniteDimensional"],["UniformSpace","toTopologicalSpace"],["Real","normedCommRing"],["Membership","mem"],["NormedAddCommGroup","toAddCommGroup"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["MetricSpace","toEMetricSpace"],["Real","instDivisionRing"],["Real","lattice"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["NormedAddCommGroup","toMetricSpace"],["Set","uIcc"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["BoundedVariationOn"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["NormedCommRing","toSeminormedCommRing"],["EMetricSpace","toPseudoEMetricSpace"],["Real","linearOrder"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]],"valueReferences":[["Filter","Eventually"],["DifferentiableAt"],["Lattice","toSemilatticeSup"],["Decidable","not_not","_simp_1"],["PartialOrder","toPreorder"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["instZeroENNReal"],["Singleton","singleton"],["Real","noAtoms_volume"],["Real","normedCommRing"],["Membership","mem"],["Filter","univ_mem'"],["NormedAddCommGroup","toAddCommGroup"],["BoundedVariationOn","locallyBoundedVariationOn"],["Real","instRCLike"],["DenselyNormedField","toNontriviallyNormedField"],["funext"],["Icc_mem_nhds"],["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem"],["Real","lattice"],["And","left"],["SemilatticeInf","toPartialOrder"],["Real","normedAddCommGroup"],["MeasureTheory","NoAtoms","measure_singleton"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["Set","mem_Icc"],["NormedSpace","toModule"],["NormedAddCommGroup","toMetricSpace"],["Real"],["Set","uIcc"],["Real","instIsOrderedAddMonoid"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["Real","instMin"],["And","right"],["And"],["DifferentiableWithinAt"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["Real","decidableEq"],["Real","pseudoMetricSpace"],["DifferentiableWithinAt","differentiableAt"],["id"],["Real","linearOrder"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["setOf"],["SemilatticeInf","toMin"],["Ne","symm"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["Set","Icc"],["Eq","mp"],["Filter","mp_mem"],["UniformSpace","toTopologicalSpace"],["lt_of_le_of_ne"],["DFunLike","coe"],["congrArg"],["HasSolidNorm","orderClosedTopology"],["MeasureTheory","Measure"],["Real","measureSpace"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["congrFun'"],["Zero","toOfNat0"],["MetricSpace","toEMetricSpace"],["Set","uIcc","eq_1"],["Preorder","toLE"],["Eq"],["MeasureTheory","ae"],["Real","partialOrder"],["propext"],["Not"],["Real","instMax"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Set","instSingletonSet"],["_private","Mathlib","Analysis","BoundedVariation",0,"BoundedVariationOn","ae_differentiableAt_of_mem_uIcc","_simp_1_1"],["OfNat","ofNat"],["instHasSolidNormReal"],["RCLike","toInnerProductSpaceReal"],["eq_self"],["ENNReal"],["Max","max"],["MeasureTheory","Measure","instOuterMeasureClass"],["of_eq_true"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Min","min"],["LE","le"],["SemilatticeSup","toMax"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["EMetricSpace","toPseudoEMetricSpace"],["NormedCommRing","toSeminormedCommRing"],["Ne"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["InnerProductSpace","toNormedSpace"]]},{"isProp":true,"kind":"theorem","name":["LipschitzWith","ae_differentiableAt_real"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.12 : FiniteDimensional.{0, u_1} Real V Real.instDivisionRing (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.6)] {C : NNReal} {f : Real -> V}, (LipschitzWith.{0, u_1} Real V (EMetricSpace.toPseudoEMetricSpace.{0} Real (MetricSpace.toEMetricSpace.{0} Real Real.metricSpace)) (EMetricSpace.toPseudoEMetricSpace.{u_1} V (MetricSpace.toEMetricSpace.{u_1} V (NormedAddCommGroup.toMetricSpace.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3))) C f) -> (Filter.Eventually.{0} Real (fun (x : Real) => DifferentiableAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) V (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.608323612._hygCtx._hyg.3)))) f x) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : NNReal}\n  {f : ℝ → V}, LipschitzWith C f → ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x","typeReadable":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : NNReal}\n  {f : ℝ → V}, LipschitzWith C f → ∀ᵐ (x : ℝ), DifferentiableAt ℝ f x","typeReferences":[["Filter","Eventually"],["DifferentiableAt"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["FiniteDimensional"],["UniformSpace","toTopologicalSpace"],["Real","normedCommRing"],["NormedAddCommGroup","toAddCommGroup"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["MetricSpace","toEMetricSpace"],["Real","instDivisionRing"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["Real","metricSpace"],["NormedAddCommGroup","toMetricSpace"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NNReal"],["MeasureTheory","Measure","instFunLike"],["LipschitzWith"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["EMetricSpace","toPseudoEMetricSpace"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]],"valueReferences":[["Set","univ"],["Real"],["NormedAddCommGroup","toMetricSpace"],["EMetricSpace","toPseudoEMetricSpace"],["MetricSpace","toEMetricSpace"],["LocallyBoundedVariationOn","ae_differentiableAt"],["LipschitzWith","locallyBoundedVariationOn"]]},{"isProp":true,"kind":"theorem","name":["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem_real"],"typeFallback":"forall {f : Real -> Real} {s : Set.{0} Real}, (LocallyBoundedVariationOn.{0, 0} Real Real.linearOrder Real (EMetricSpace.toPseudoEMetricSpace.{0} Real (MetricSpace.toEMetricSpace.{0} Real Real.metricSpace)) f s) -> (Filter.Eventually.{0} Real (fun (x : Real) => (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembership.{0} Real) s x) -> (DifferentiableWithinAt.{0, 0, 0} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) f s x)) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {f : ℝ → ℝ} {s : Set ℝ}, LocallyBoundedVariationOn f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReadable":"∀ {f : ℝ → ℝ} {s : Set ℝ}, LocallyBoundedVariationOn f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReferences":[["Filter","Eventually"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["Membership","mem"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["LocallyBoundedVariationOn"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["DenselyNormedField","toNontriviallyNormedField"],["MetricSpace","toEMetricSpace"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real","metricSpace"],["Real"],["Set"],["DifferentiableWithinAt"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["EMetricSpace","toPseudoEMetricSpace"],["Real","linearOrder"],["NormedCommRing","toSeminormedCommRing"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"]],"valueReferences":[["Filter","Eventually"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Membership","mem"],["Real","normedCommRing"],["Filter","univ_mem'"],["Real","instRCLike"],["DenselyNormedField","toNontriviallyNormedField"],["LocallyBoundedVariationOn","exists_monotoneOn_sub_monotoneOn"],["Eq","symm"],["HSub","hSub"],["Eq","ndrec"],["Real","normedAddCommGroup"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Exists"],["Real"],["And"],["DifferentiableWithinAt"],["Real","instSub"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["Exists","casesOn"],["Real","pseudoMetricSpace"],["id"],["MonotoneOn","ae_differentiableWithinAt_of_mem"],["Real","linearOrder"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["setOf"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["Filter","mp_mem"],["UniformSpace","toTopologicalSpace"],["LocallyBoundedVariationOn"],["DifferentiableWithinAt","sub"],["MeasureTheory","Measure"],["Real","measureSpace"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["MetricSpace","toEMetricSpace"],["Eq"],["MeasureTheory","ae"],["Pi","instSub"],["Real","metricSpace"],["Set"],["MonotoneOn"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","instAddCommGroup"],["Real","denselyNormedField"],["NormedCommRing","toSeminormedCommRing"],["EMetricSpace","toPseudoEMetricSpace"],["instHSub"],["NormedAddCommGroup","toSeminormedAddCommGroup"],["InnerProductSpace","toNormedSpace"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["LocallyBoundedVariationOn","ae_differentiableWithinAt"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.12 : FiniteDimensional.{0, u_1} Real V Real.instDivisionRing (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.6)] {f : Real -> V} {s : Set.{0} Real}, (LocallyBoundedVariationOn.{0, u_1} Real Real.linearOrder V (EMetricSpace.toPseudoEMetricSpace.{u_1} V (MetricSpace.toEMetricSpace.{u_1} V (NormedAddCommGroup.toMetricSpace.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3))) f s) -> (MeasurableSet.{0} Real Real.measurableSpace s) -> (Filter.Eventually.{0} Real (fun (x : Real) => DifferentiableWithinAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) V (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.2415563987._hygCtx._hyg.3)))) f s x) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.restrict.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace) s)))","typeFull":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V}\n  {s : Set ℝ},\n  LocallyBoundedVariationOn f s →\n    MeasurableSet s → ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x","typeReadable":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V}\n  {s : Set ℝ},\n  LocallyBoundedVariationOn f s →\n    MeasurableSet s → ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x","typeReferences":[["Filter","Eventually"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["FiniteDimensional"],["UniformSpace","toTopologicalSpace"],["Real","normedCommRing"],["LocallyBoundedVariationOn"],["NormedAddCommGroup","toAddCommGroup"],["Real","measureSpace"],["MeasureTheory","Measure"],["MeasureTheory","Measure","restrict"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["Real","instDivisionRing"],["MetricSpace","toEMetricSpace"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["NormedAddCommGroup","toMetricSpace"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DifferentiableWithinAt"],["MeasureTheory","Measure","instFunLike"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","measurableSpace"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["NormedCommRing","toSeminormedCommRing"],["EMetricSpace","toPseudoEMetricSpace"],["Real","linearOrder"],["MeasurableSet"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]],"valueReferences":[["Filter","Eventually"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["MeasureTheory","ae_restrict_iff'"],["Membership","mem"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["NormedAddCommGroup","toAddCommGroup"],["congrArg"],["Real","measureSpace"],["MeasureTheory","Measure"],["MeasureTheory","Measure","restrict"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["DenselyNormedField","toNontriviallyNormedField"],["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem"],["Eq"],["MeasureTheory","ae"],["propext"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["DifferentiableWithinAt"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","measurableSpace"],["id"],["Real","instAddCommGroup"],["Real","denselyNormedField"],["NormedCommRing","toSeminormedCommRing"],["Eq","mpr"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]]},{"isProp":true,"kind":"theorem","name":["LipschitzOnWith","ae_differentiableWithinAt_of_mem_real"],"typeFallback":"forall {V : Type.{u_1}} [inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3 : NormedAddCommGroup.{u_1} V] [inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.6 : NormedSpace.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.12 : FiniteDimensional.{0, u_1} Real V Real.instDivisionRing (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.6)] {C : NNReal} {f : Real -> V} {s : Set.{0} Real}, (LipschitzOnWith.{0, u_1} Real V (EMetricSpace.toPseudoEMetricSpace.{0} Real (MetricSpace.toEMetricSpace.{0} Real Real.metricSpace)) (EMetricSpace.toPseudoEMetricSpace.{u_1} V (MetricSpace.toEMetricSpace.{u_1} V (NormedAddCommGroup.toMetricSpace.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3))) C f s) -> (Filter.Eventually.{0} Real (fun (x : Real) => (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembership.{0} Real) s x) -> (DifferentiableWithinAt.{0, 0, u_1} Real (DenselyNormedField.toNontriviallyNormedField.{0} Real Real.denselyNormedField) Real Real.instAddCommGroup (NormedSpace.toModule.{0, 0} Real Real Real.normedField (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (InnerProductSpace.toNormedSpace.{0, 0} Real Real Real.instRCLike (NonUnitalSeminormedRing.toSeminormedAddCommGroup.{0} Real (NonUnitalSeminormedCommRing.toNonUnitalSeminormedRing.{0} Real (SeminormedCommRing.toNonUnitalSeminormedCommRing.{0} Real (NormedCommRing.toSeminormedCommRing.{0} Real Real.normedCommRing)))) (RCLike.toInnerProductSpaceReal.{0} Real Real.instRCLike))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) V (NormedAddCommGroup.toAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3) (NormedSpace.toModule.{0, u_1} Real V Real.normedField (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3) inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.6) (UniformSpace.toTopologicalSpace.{u_1} V (PseudoMetricSpace.toUniformSpace.{u_1} V (SeminormedAddCommGroup.toPseudoMetricSpace.{u_1} V (NormedAddCommGroup.toSeminormedAddCommGroup.{u_1} V inst._@.Mathlib.Analysis.BoundedVariation.246554684._hygCtx._hyg.3)))) f s x)) (MeasureTheory.ae.{0, 0} Real (MeasureTheory.Measure.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instFunLike.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.Measure.instOuterMeasureClass.{0} Real (MeasureTheory.MeasureSpace.toMeasurableSpace.{0} Real Real.measureSpace)) (MeasureTheory.MeasureSpace.volume.{0} Real Real.measureSpace)))","typeFull":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : NNReal}\n  {f : ℝ → V} {s : Set ℝ}, LipschitzOnWith C f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReadable":"∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : NNReal}\n  {f : ℝ → V} {s : Set ℝ}, LipschitzOnWith C f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s x","typeReferences":[["Filter","Eventually"],["NonUnitalSeminormedCommRing","toNonUnitalSeminormedRing"],["PseudoMetricSpace","toUniformSpace"],["FiniteDimensional"],["UniformSpace","toTopologicalSpace"],["Real","normedCommRing"],["Membership","mem"],["NormedAddCommGroup","toAddCommGroup"],["Real","measureSpace"],["MeasureTheory","Measure"],["SeminormedCommRing","toNonUnitalSeminormedCommRing"],["Real","instRCLike"],["NormedSpace"],["DenselyNormedField","toNontriviallyNormedField"],["NormedAddCommGroup"],["Real","instDivisionRing"],["MetricSpace","toEMetricSpace"],["MeasureTheory","ae"],["NonUnitalSeminormedRing","toSeminormedAddCommGroup"],["NormedSpace","toModule"],["Real"],["Real","metricSpace"],["NormedAddCommGroup","toMetricSpace"],["Set"],["SeminormedAddCommGroup","toPseudoMetricSpace"],["NNReal"],["DifferentiableWithinAt"],["LipschitzOnWith"],["MeasureTheory","Measure","instFunLike"],["Set","instMembership"],["RCLike","toInnerProductSpaceReal"],["MeasureTheory","Measure","instOuterMeasureClass"],["Real","normedField"],["MeasureTheory","MeasureSpace","volume"],["Real","pseudoMetricSpace"],["Real","denselyNormedField"],["Real","instAddCommGroup"],["NormedCommRing","toSeminormedCommRing"],["EMetricSpace","toPseudoEMetricSpace"],["MeasureTheory","MeasureSpace","toMeasurableSpace"],["InnerProductSpace","toNormedSpace"],["NormedAddCommGroup","toSeminormedAddCommGroup"]],"valueReferences":[["LipschitzOnWith","locallyBoundedVariationOn"],["NormedAddCommGroup","toMetricSpace"],["LocallyBoundedVariationOn","ae_differentiableWithinAt_of_mem"],["EMetricSpace","toPseudoEMetricSpace"],["MetricSpace","toEMetricSpace"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Analysis.Calculus.TangentCone.Real.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ico","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3 : Preorder.{u_1} α] {a : α} {b : α} [inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.14 : DenselyOrdered.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3)], Eq.{1} Prop (Set.Nonempty.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3 a b)) (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3) a b)","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], (Set.Ioo a b).Nonempty = (a < b)","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], (Set.Ioo a b).Nonempty = (a < b)","typeReferences":[["LT","lt"],["Set","Nonempty"],["Preorder"],["DenselyOrdered"],["Preorder","toLT"],["Eq"],["Set","Ioo"]],"valueReferences":[["LT","lt"],["Set","Nonempty"],["Preorder","toLT"],["Set","nonempty_Ioo"],["Set","Ioo"],["propext"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Iio"],"typeFallback":"forall (a : Real), UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Iio.{0} Real Real.instPreorder a)","typeFull":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Iio a)","typeReadable":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Iio a)","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Set","Iio"],["Semiring","toModule"],["UniqueDiffOn"],["Real","semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["instClosedIciTopology"],["IsTopologicalSemiring","toContinuousMul"],["Set","Iio"],["Real","instField"],["IsOpen","uniqueDiffOn"],["instOrderTopologyReal"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["IsSemitopologicalSemiring","toContinuousAdd"],["DenselyNormedField","toNontriviallyNormedField"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["SemilatticeInf","toPartialOrder"],["Real","instMul"],["Lattice","toSemilatticeInf"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["instIsTopologicalRingReal"],["DivisionSemiring","toSemiring"],["NormedField","nhdsNE_neBot"],["Real","semiring"],["OfNat","ofNat"],["isOpen_Iio"],["DistribLattice","toLattice"],["OrderTopology","to_orderClosedTopology"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Real","pseudoMetricSpace"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["IsTopologicalRing","toIsSemitopologicalRing"],["Real","instAddCommGroup"],["Field","toSemifield"],["Real","denselyNormedField"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffWithinAt_Ioi"],"typeFallback":"forall (a : Real), UniqueDiffWithinAt.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ioi.{0} Real Real.instPreorder a) a","typeFull":"∀ (a : ℝ), UniqueDiffWithinAt ℝ (Set.Ioi a) a","typeReadable":"∀ (a : ℝ), UniqueDiffWithinAt ℝ (Set.Ioi a) a","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["UniqueDiffWithinAt"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Set","Ioi"],["Semiring","toModule"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["MulZeroClass","toMul"],["Membership","mem"],["Real","normedCommRing"],["Real","instNontrivial"],["IsTopologicalSemiring","toContinuousMul"],["SMulZeroClass","toSMul"],["Real","instIsOrderedCancelAddMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Set","Ioi"],["Real","instAddCommMonoid"],["Semiring","toModule"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["IsStrictOrderedRing","toPosMulStrictMono"],["closure_Ioi"],["Real"],["Set","nonempty_Ioi","_simp_1"],["interior"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsStrictOrderedModule","toPosSMulStrictMono"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearOrderedSemiField","toDenselyOrdered"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["Set","instMembership"],["OrderTopology","to_orderClosedTopology"],["instIsTopologicalAddGroupReal"],["Real","pseudoMetricSpace"],["instNoMaxOrderOfNontrivial"],["convex_Ioi"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Real","instIsStrictOrderedRing"],["uniqueDiffWithinAt_convex"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["AddCommMonoid","toAddMonoid"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instIsOrderedRing"],["Set","mem_Ici","_simp_1"],["Real","instField"],["congrArg"],["instOrderTopologyReal"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["Std","le_refl","_simp_1"],["congrFun'"],["closure"],["Preorder","toLE"],["Real","partialOrder"],["Real","instMul"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["instClosedIicTopology"],["True"],["Set"],["Semiring","toMonoidWithZero"],["IsStrictOrderedRing","toIsStrictOrderedModule"],["instReflLe"],["Real","semiring"],["Set","Ici"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["of_eq_true"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["interior_Ioi"],["Real","instAddCommGroup"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffWithinAt_Ici"],"typeFallback":"forall (x : Real), UniqueDiffWithinAt.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ici.{0} Real Real.instPreorder x) x","typeFull":"∀ (x : ℝ), UniqueDiffWithinAt ℝ (Set.Ici x) x","typeReadable":"∀ (x : ℝ), UniqueDiffWithinAt ℝ (Set.Ici x) x","typeReferences":[["Set","Ici"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["UniqueDiffWithinAt"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Semiring","toModule"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["uniqueDiffWithinAt_Ioi"],["Real"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["Real","semiring"],["Set","Ioi_subset_Ici_self"],["Set","Ici"],["UniqueDiffWithinAt","mono"],["Real","pseudoMetricSpace"],["Real","instAddCommGroup"],["Set","Ioi"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Icc_zero_one"],"typeFallback":"UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Icc.{0} Real Real.instPreorder (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) (OfNat.ofNat.{0} Real 1 (One.toOfNat1.{0} Real Real.instOne)))","typeFull":"UniqueDiffOn ℝ (Set.Icc 0 1)","typeReadable":"UniqueDiffOn ℝ (Set.Icc 0 1)","typeReferences":[["Real","instPreorder"],["Real"],["PseudoMetricSpace","toUniformSpace"],["Set","Icc"],["UniformSpace","toTopologicalSpace"],["UniqueDiffOn"],["Real","semiring"],["OfNat","ofNat"],["One","toOfNat1"],["Real","instZero"],["Real","pseudoMetricSpace"],["Real","instAddCommGroup"],["Real","instOne"],["Zero","toOfNat0"],["Semiring","toModule"]],"valueReferences":[["IsStrictOrderedRing","toCharZero"],["Real","instIsStrictOrderedRing"],["Real"],["Real","instZeroLEOneClass"],["AddGroupWithOne","toAddMonoidWithOne"],["Real","instRing"],["zero_lt_one"],["OfNat","ofNat"],["Ring","toSemiring"],["Real","instZero"],["One","toOfNat1"],["Ring","toAddGroupWithOne"],["uniqueDiffOn_Icc"],["Zero","toOfNat0"],["Real","instOne"],["NeZero","charZero_one"],["Real","partialOrder"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ico","_simp_1_2"],"typeFallback":"forall {𝕜 : Type.{u_1}} {E : Type.{u_2}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.4 : Semiring.{u_1} 𝕜] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.7 : AddCommGroup.{u_2} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.10 : Module.{u_1, u_2} 𝕜 E inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.4 (AddCommGroup.toAddCommMonoid.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.7)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.14 : TopologicalSpace.{u_2} E], Eq.{1} Prop (UniqueDiffOn.{u_1, u_2} 𝕜 E inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.7 inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.10 inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.14 (EmptyCollection.emptyCollection.{u_2} (Set.{u_2} E) (Set.instEmptyCollection.{u_2} E))) True","typeFull":"∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]\n  [inst_3 : TopologicalSpace E], UniqueDiffOn 𝕜 ∅ = True","typeReadable":"∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]\n  [inst_3 : TopologicalSpace E], UniqueDiffOn 𝕜 ∅ = True","typeReferences":[["TopologicalSpace"],["True"],["Set"],["Module"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["UniqueDiffOn"],["Eq"],["Semiring"]],"valueReferences":[["uniqueDiffOn_empty"],["Set"],["eq_true"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["UniqueDiffOn"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffWithinAt_convex"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.15 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.12] {s : Set.{u_1} E} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.25 : IsTopologicalAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.12 (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3)], (Convex.{0, u_1} Real E Real.semiring Real.partialOrder (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3) (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.6)))) s) -> (Set.Nonempty.{u_1} E (interior.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.12 s)) -> (forall {x : E}, (Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (closure.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.12 s) x) -> (UniqueDiffWithinAt.{0, u_1} Real E Real.semiring inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.6 inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2439796940._hygCtx._hyg.12 s x))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} [IsTopologicalAddGroup E],\n  Convex ℝ s → (interior s).Nonempty → ∀ {x : E}, x ∈ closure s → UniqueDiffWithinAt ℝ s x","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} [IsTopologicalAddGroup E],\n  Convex ℝ s → (interior s).Nonempty → ∀ {x : E}, x ∈ closure s → UniqueDiffWithinAt ℝ s x","typeReferences":[["UniqueDiffWithinAt"],["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["IsTopologicalAddGroup"],["ContinuousSMul"],["closure"],["AddGroup","toSubNegMonoid"],["Real","partialOrder"],["DistribSMul","toSMulZeroClass"],["Set","Nonempty"],["Real"],["interior"],["Set"],["Convex"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Real","semiring"],["Set","instMembership"],["TopologicalSpace"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Eq","trans"],["UniqueDiffWithinAt"],["AddCommGroup","toAddGroup"],["Membership","mem"],["eq_true"],["Submodule","span"],["SMulZeroClass","toSMul"],["congrArg"],["Submodule"],["Dense"],["congr"],["MonoidWithZero","toMonoid"],["closure"],["AddGroup","toSubNegMonoid"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffWithinAt_convex","_simp_1_1"],["Convex","span_tangentConeAt"],["DistribSMul","toSMulZeroClass"],["True"],["Real"],["Set"],["Submodule","instTop"],["And"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Real","semiring"],["Set","instMembership"],["Set","univ"],["and_self"],["Module","toDistribMulAction"],["SetLike","coe"],["of_eq_true"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["dense_univ","_simp_1"],["Top","top"],["tangentConeAt"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ici","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.3 : Preorder.{u_1} α] {a : α} [inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.14 : NoMaxOrder.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.3)], Eq.{1} Prop (Set.Nonempty.{u_1} α (Set.Ioi.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.3 a)) True","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α} [NoMaxOrder α], (Set.Ioi a).Nonempty = True","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α} [NoMaxOrder α], (Set.Ioi a).Nonempty = True","typeReferences":[["NoMaxOrder"],["Set","Nonempty"],["Preorder"],["True"],["Preorder","toLT"],["Set","Ioi"],["Eq"]],"valueReferences":[["Set","nonempty_Ioi"],["Set","Nonempty"],["Set","Ioi"],["eq_true"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffWithinAt_convex","_simp_1_1"],"typeFallback":"forall (R : Type.{u}) {E : Type.{v}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19 : Semiring.{u} R] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22 : AddCommGroup.{v} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.25 : Module.{u, v} R E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19 (AddCommGroup.toAddCommMonoid.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.29 : TopologicalSpace.{v} E] (s : Set.{v} E) (x : E), Eq.{1} Prop (UniqueDiffWithinAt.{u, v} R E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19 inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22 inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.25 inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.29 s x) (And (Dense.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.29 (SetLike.coe.{v, v} (Submodule.{u, v} R E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19 (AddCommGroup.toAddCommMonoid.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22) inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.25) E (Submodule.setLike.{u, v} R E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19 (AddCommGroup.toAddCommMonoid.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22) inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.25) (Submodule.span.{u, v} R E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19 (AddCommGroup.toAddCommMonoid.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22) inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.25 (tangentConeAt.{u, v} R E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22 (SMulZeroClass.toSMul.{u, v} R E (AddZero.toZero.{v} E (AddZeroClass.toAddZero.{v} E (AddMonoid.toAddZeroClass.{v} E (SubNegMonoid.toAddMonoid.{v} E (AddGroup.toSubNegMonoid.{v} E (AddCommGroup.toAddGroup.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22)))))) (DistribSMul.toSMulZeroClass.{u, v} R E (AddMonoid.toAddZeroClass.{v} E (SubNegMonoid.toAddMonoid.{v} E (AddGroup.toSubNegMonoid.{v} E (AddCommGroup.toAddGroup.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22)))) (DistribMulAction.toDistribSMul.{u, v} R E (MonoidWithZero.toMonoid.{u} R (Semiring.toMonoidWithZero.{u} R inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19)) (SubNegMonoid.toAddMonoid.{v} E (AddGroup.toSubNegMonoid.{v} E (AddCommGroup.toAddGroup.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22))) (Module.toDistribMulAction.{u, v} R E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.19 (AddCommGroup.toAddCommMonoid.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.22) inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.25)))) inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.29 s x)))) (Membership.mem.{v, v} E (Set.{v} E) (Set.instMembership.{v} E) (closure.{v} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Defs.271849881._hygCtx._hyg.29 s) x))","typeFull":"∀ (R : Type u) {E : Type v} [inst : Semiring R] [inst_1 : AddCommGroup E] [inst_2 : Module R E]\n  [inst_3 : TopologicalSpace E] (s : Set E) (x : E),\n  UniqueDiffWithinAt R s x = (Dense ↑(Submodule.span R (tangentConeAt R s x)) ∧ x ∈ closure s)","typeReadable":"∀ (R : Type u) {E : Type v} [inst : Semiring R] [inst_1 : AddCommGroup E] [inst_2 : Module R E]\n  [inst_3 : TopologicalSpace E] (s : Set E) (x : E),\n  UniqueDiffWithinAt R s x = (Dense ↑(Submodule.span R (tangentConeAt R s x)) ∧ x ∈ closure s)","typeReferences":[["UniqueDiffWithinAt"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Submodule","span"],["SMulZeroClass","toSMul"],["Submodule"],["Dense"],["MonoidWithZero","toMonoid"],["closure"],["AddGroup","toSubNegMonoid"],["Eq"],["DistribSMul","toSMulZeroClass"],["Set"],["And"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Set","instMembership"],["TopologicalSpace"],["Module","toDistribMulAction"],["SetLike","coe"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["AddCommGroup","toAddCommMonoid"],["tangentConeAt"],["AddZero","toZero"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["UniqueDiffWithinAt"],["AddCommGroup","toAddGroup"],["Membership","mem"],["SMulZeroClass","toSMul"],["Submodule","span"],["uniqueDiffWithinAt_iff"],["Submodule"],["Dense"],["MonoidWithZero","toMonoid"],["closure"],["AddGroup","toSubNegMonoid"],["propext"],["DistribSMul","toSMulZeroClass"],["Set"],["And"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Module","toDistribMulAction"],["SetLike","coe"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["tangentConeAt"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffWithinAt_Iio"],"typeFallback":"forall (a : Real), UniqueDiffWithinAt.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Iio.{0} Real Real.instPreorder a) a","typeFull":"∀ (a : ℝ), UniqueDiffWithinAt ℝ (Set.Iio a) a","typeReadable":"∀ (a : ℝ), UniqueDiffWithinAt ℝ (Set.Iio a) a","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["UniqueDiffWithinAt"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Set","Iio"],["Semiring","toModule"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["PseudoMetricSpace","toUniformSpace"],["Eq","trans"],["MulZeroClass","toMul"],["Membership","mem"],["Real","normedCommRing"],["instClosedIciTopology"],["Real","instNontrivial"],["IsTopologicalSemiring","toContinuousMul"],["SMulZeroClass","toSMul"],["Set","Iio"],["Real","instIsOrderedCancelAddMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Real","instAddCommMonoid"],["Semiring","toModule"],["DistribSMul","toSMulZeroClass"],["SemilatticeInf","toPartialOrder"],["IsStrictOrderedRing","toPosMulStrictMono"],["Real"],["interior"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsStrictOrderedModule","toPosSMulStrictMono"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearOrderedSemiField","toDenselyOrdered"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["Set","Iic"],["Set","mem_Iic","_simp_2"],["Set","instMembership"],["OrderTopology","to_orderClosedTopology"],["instIsTopologicalAddGroupReal"],["Real","pseudoMetricSpace"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Real","instIsStrictOrderedRing"],["uniqueDiffWithinAt_convex"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["AddCommMonoid","toAddMonoid"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instIsOrderedRing"],["Real","instField"],["congrArg"],["instOrderTopologyReal"],["instDistribLatticeOfLinearOrder"],["convex_Iio"],["closure_Iio"],["MonoidWithZero","toMonoid"],["instNoMinOrderOfNontrivial"],["Std","le_refl","_simp_1"],["congrFun'"],["closure"],["Preorder","toLE"],["Real","partialOrder"],["Real","instMul"],["Set","nonempty_Iio","_simp_2"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Semiring","toMonoidWithZero"],["IsStrictOrderedRing","toIsStrictOrderedModule"],["instReflLe"],["Real","semiring"],["interior_Iio"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["of_eq_true"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Real","instAddCommGroup"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Ioo"],"typeFallback":"forall (a : Real) (b : Real), UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ioo.{0} Real Real.instPreorder a b)","typeFull":"∀ (a b : ℝ), UniqueDiffOn ℝ (Set.Ioo a b)","typeReadable":"∀ (a b : ℝ), UniqueDiffOn ℝ (Set.Ioo a b)","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Semiring","toModule"],["UniqueDiffOn"],["Set","Ioo"],["Real","semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["IsTopologicalSemiring","toContinuousMul"],["Set","Ioo"],["Real","instField"],["IsOpen","uniqueDiffOn"],["instOrderTopologyReal"],["isOpen_Ioo"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["IsSemitopologicalSemiring","toContinuousAdd"],["DenselyNormedField","toNontriviallyNormedField"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["SemilatticeInf","toPartialOrder"],["Real","instMul"],["Lattice","toSemilatticeInf"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["instIsTopologicalRingReal"],["DivisionSemiring","toSemiring"],["NormedField","nhdsNE_neBot"],["Real","semiring"],["OfNat","ofNat"],["DistribLattice","toLattice"],["OrderTopology","to_orderClosedTopology"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Real","pseudoMetricSpace"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["IsTopologicalRing","toIsSemitopologicalRing"],["Real","instAddCommGroup"],["Field","toSemifield"],["Real","denselyNormedField"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["sub_mem_posTangentConeAt_of_openSegment_subset"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.15 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.12] {s : Set.{u_1} E} {x : E} {y : E}, (HasSubset.Subset.{u_1} (Set.{u_1} E) (Set.instHasSubset.{u_1} E) (openSegment.{0, u_1} Real E Real.semiring Real.partialOrder (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3) (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.6)))) x y) s) -> (Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (tangentConeAt.{0, u_1} NNReal E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3 (SMulZeroClass.toSMul.{0, u_1} NNReal E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} NNReal E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} NNReal E (MonoidWithZero.toMonoid.{0} NNReal (Semiring.toMonoidWithZero.{0} NNReal NNReal.instSemiring)) (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3))) (NNReal.instDistribMulActionOfReal.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.6))))) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.12 s x) (HSub.hSub.{u_1, u_1, u_1} E E E (instHSub.{u_1} E (SubNegMonoid.toSub.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.450485339._hygCtx._hyg.3)))) y x))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x y : E}, openSegment ℝ x y ⊆ s → y - x ∈ tangentConeAt NNReal s x","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x y : E}, openSegment ℝ x y ⊆ s → y - x ∈ tangentConeAt NNReal s x","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["ContinuousSMul"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Real","partialOrder"],["DistribSMul","toSMulZeroClass"],["Real"],["Set"],["NNReal"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddCommGroup"],["AddZeroClass","toAddZero"],["NNReal","instDistribMulActionOfReal"],["Real","semiring"],["Set","instMembership"],["TopologicalSpace"],["Set","instHasSubset"],["Module","toDistribMulAction"],["openSegment"],["SubNegMonoid","toAddMonoid"],["HasSubset","Subset"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["tangentConeAt"],["AddCommGroup","toAddCommMonoid"],["NNReal","instSemiring"],["instHSub"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Real","instPreorder"],["Ring","toNonAssocRing"],["Eq","trans"],["Singleton","singleton"],["AddCommGroup","toAddGroup"],["instClosedIciTopology"],["SemigroupAction","toSMul"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["SMulZeroClass","toSMul"],["Filter","univ_mem'"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["NNReal","instLinearOrder"],["NNReal","instLinearOrderedCommGroupWithZero"],["Filter","tendsto_id'"],["SubNegMonoid","toSub"],["Eq","symm"],["nhdsGT_neBot"],["Monoid","toSemigroup"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Real","instAddCommSemigroup"],["NNReal","instContinuousSMulOfReal"],["NNReal"],["nhdsGT_le_nhdsNE"],["DivisionSemiring","toSemiring"],["Filter"],["Ring","toSemiring"],["Set","instMembership"],["OrderTopology","to_orderClosedTopology"],["AddMonoidWithOne","toOne"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["covariant_swap_add_of_covariant_add"],["setOf"],["AddMonoid","toAddZeroClass"],["Real","instNatCast"],["AddCommMonoid","toAddMonoid"],["NNReal","instIsStrictOrderedRing"],["instOfNatNat"],["congr"],["AddCommMagma","toAdd"],["Eq"],["Preorder","toLE"],["LinearOrderedCommMonoidWithZero","toZeroLeOneClass"],["cast"],["Set"],["OfNat","ofNat"],["HAdd","hAdd"],["eq_self"],["Module","toDistribMulAction"],["AddGroupWithOne","toAddGroup"],["MulZeroClass","toZero"],["IsStrictOrderedRing","toNoMaxOrder"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHSub"],["AffineMap","lineMap"],["AffineMap","instFunLike"],["Nat","cast_one"],["NNReal","instTopologicalSpace"],["PartialOrder","toPreorder"],["Membership","mem"],["AddMonoidWithOne","toAddMonoid"],["Set","Ioo"],["And","intro"],["Semiring","toNonAssocSemiring"],["NNReal","instOne"],["Ring","toAddGroupWithOne"],["NNReal","instModuleOfReal"],["one_pos"],["DistribMulAction","toMulAction"],["HSub","hSub"],["Set","Ioi"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Real","instAddCommMonoid"],["Semiring","toModule"],["add_comm"],["Filter","Tendsto"],["NonAssocRing","toNonUnitalNonAssocRing"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["Real"],["LinearOrderedCommGroupWithZero","toLinearOrderedCommMonoidWithZero"],["Real","instIsOrderedAddMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["DistribMulAction","toDistribSMul"],["IsOrderedAddMonoid","toAddLeftMono"],["AffineMap"],["AddZeroClass","toAddZero"],["addGroupIsAddTorsor"],["nhdsWithin"],["Nat"],["openSegment"],["Set","image"],["AddMonoidWithOne","toNatCast"],["Nat","cast_zero"],["Iff","mpr"],["NNReal","instOrderTopology"],["Real","instMonoid"],["HSMul","hSMul"],["id"],["NeZero","charZero_one"],["AddZero","toZero"],["Nat","cast"],["Filter","mp_mem"],["Compl","compl"],["Ioo_mem_nhdsGT"],["NNReal","toReal"],["NNReal","instPartialOrder"],["Real","instRing"],["DFunLike","coe"],["congrArg"],["instDistribLatticeOfLinearOrder"],["MonoidWithZero","toMonoid"],["mem_tangentConeAt_of_add_smul_mem"],["instHSMul"],["congrFun'"],["openSegment_eq_image_lineMap"],["Zero","toOfNat0"],["AddCommSemigroup","toAddCommMagma"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Real","partialOrder"],["IsStrictOrderedRing","toCharZero"],["Lattice","toSemilatticeInf"],["True"],["instHAdd"],["NNReal","instSemifield"],["Filter","instPartialOrder"],["Semiring","toMonoidWithZero"],["Set","instSingletonSet"],["Real","semiring"],["Set","instCompl"],["Ring","toAddCommGroup"],["DistribLattice","toLattice"],["of_eq_true"],["One","toOfNat1"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["LE","le"],["NNReal","instSemiring"],["MulAction","toSemigroupAction"],["NNReal","instDenselyOrdered"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Icc"],"typeFallback":"forall {a : Real} {b : Real}, (LT.lt.{0} Real Real.instLT a b) -> (UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Icc.{0} Real Real.instPreorder a b))","typeFull":"∀ {a b : ℝ}, a < b → UniqueDiffOn ℝ (Set.Icc a b)","typeReadable":"∀ {a b : ℝ}, a < b → UniqueDiffOn ℝ (Set.Icc a b)","typeReferences":[["LT","lt"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Set","Icc"],["Real","pseudoMetricSpace"],["Real","instLT"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Semiring","toModule"],["UniqueDiffOn"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["MulZeroClass","toMul"],["Real","normedCommRing"],["Preorder","toLT"],["eq_true"],["Real","instNontrivial"],["IsTopologicalSemiring","toContinuousMul"],["SMulZeroClass","toSMul"],["MulZeroClass","toSMulWithZero"],["Set","Ioo"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["interior_Icc"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["convex_Icc"],["Real","instAddCommMonoid"],["Semiring","toModule"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["IsStrictOrderedRing","toPosMulStrictMono"],["Real"],["interior"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Real","instIsOrderedAddMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearOrderedSemiField","toDenselyOrdered"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["instIsTopologicalAddGroupReal"],["Real","pseudoMetricSpace"],["instNoMaxOrderOfNontrivial"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Real","instIsStrictOrderedRing"],["Set","Icc"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["AddCommMonoid","toAddMonoid"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Icc","_simp_1_1"],["Real","instIsOrderedRing"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instField"],["congrArg"],["instOrderTopologyReal"],["uniqueDiffOn_convex"],["instDistribLatticeOfLinearOrder"],["Real","instLT"],["IsOrderedModule","toPosSMulMono"],["MonoidWithZero","toMonoid"],["instNoMinOrderOfNontrivial"],["Real","partialOrder"],["Real","instMul"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Semiring","toMonoidWithZero"],["IsStrictOrderedRing","toIsStrictOrderedModule"],["Real","semiring"],["LT","lt"],["IsStrictOrderedModule","toIsOrderedModule"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Real","instAddCommGroup"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Ioc"],"typeFallback":"forall (a : Real) (b : Real), UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ioc.{0} Real Real.instPreorder a b)","typeFull":"∀ (a b : ℝ), UniqueDiffOn ℝ (Set.Ioc a b)","typeReadable":"∀ (a b : ℝ), UniqueDiffOn ℝ (Set.Ioc a b)","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Set","Ioc"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Semiring","toModule"],["UniqueDiffOn"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["Set","Ioc_eq_empty"],["MulZeroClass","toMul"],["Real","normedCommRing"],["Preorder","toLT"],["eq_true"],["Real","instNontrivial"],["IsTopologicalSemiring","toContinuousMul"],["SMulZeroClass","toSMul"],["Set","Ioo"],["Real","decidableLT"],["Real","instIsOrderedCancelAddMonoid"],["interior_Ioc"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semiring","toModule"],["Real","instAddCommMonoid"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["IsStrictOrderedRing","toPosMulStrictMono"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ioc","_simp_1_2"],["Real"],["interior"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsStrictOrderedModule","toPosSMulStrictMono"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearOrderedSemiField","toDenselyOrdered"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["instIsTopologicalAddGroupReal"],["Real","pseudoMetricSpace"],["Set","Ioc"],["instNoMaxOrderOfNontrivial"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Real","instIsStrictOrderedRing"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["AddCommMonoid","toAddMonoid"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["Real","instIsOrderedRing"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instField"],["congrArg"],["instOrderTopologyReal"],["uniqueDiffOn_convex"],["instDistribLatticeOfLinearOrder"],["convex_Ioc"],["Real","instLT"],["MonoidWithZero","toMonoid"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ioc","_simp_1_1"],["Real","partialOrder"],["Real","instMul"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Semiring","toMonoidWithZero"],["IsStrictOrderedRing","toIsStrictOrderedModule"],["UniqueDiffOn"],["Real","semiring"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Real","instAddCommGroup"],["dite"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Iic"],"typeFallback":"forall (a : Real), UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Iic.{0} Real Real.instPreorder a)","typeFull":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Iic a)","typeReadable":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Iic a)","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Semiring","toModule"],["UniqueDiffOn"],["Set","Iic"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["MulZeroClass","toMul"],["Real","normedCommRing"],["Real","instNontrivial"],["IsTopologicalSemiring","toContinuousMul"],["SMulZeroClass","toSMul"],["Set","Iio"],["MulZeroClass","toSMulWithZero"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Real","instAddCommMonoid"],["Semiring","toModule"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["IsStrictOrderedRing","toPosMulStrictMono"],["Real"],["interior"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Real","instIsOrderedAddMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearOrderedSemiField","toDenselyOrdered"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["Set","Iic"],["instIsTopologicalAddGroupReal"],["Real","pseudoMetricSpace"],["instNoMaxOrderOfNontrivial"],["convex_Iic"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Real","instIsStrictOrderedRing"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["AddCommMonoid","toAddMonoid"],["Real","instIsOrderedRing"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instField"],["congrArg"],["instOrderTopologyReal"],["uniqueDiffOn_convex"],["instDistribLatticeOfLinearOrder"],["IsOrderedModule","toPosSMulMono"],["MonoidWithZero","toMonoid"],["instNoMinOrderOfNontrivial"],["interior_Iic"],["Real","partialOrder"],["Real","instMul"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Semiring","toMonoidWithZero"],["IsStrictOrderedRing","toIsStrictOrderedModule"],["Real","semiring"],["IsStrictOrderedModule","toIsOrderedModule"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Iic","_simp_1_1"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Real","instAddCommGroup"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Icc","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3 : Preorder.{u_1} α] {a : α} {b : α} [inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.14 : DenselyOrdered.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3)], Eq.{1} Prop (Set.Nonempty.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3 a b)) (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3) a b)","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], (Set.Ioo a b).Nonempty = (a < b)","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], (Set.Ioo a b).Nonempty = (a < b)","typeReferences":[["LT","lt"],["Set","Nonempty"],["Preorder"],["DenselyOrdered"],["Preorder","toLT"],["Eq"],["Set","Ioo"]],"valueReferences":[["LT","lt"],["Set","Nonempty"],["Preorder","toLT"],["Set","nonempty_Ioo"],["Set","Ioo"],["propext"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Ico"],"typeFallback":"forall (a : Real) (b : Real), UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ico.{0} Real Real.instPreorder a b)","typeFull":"∀ (a b : ℝ), UniqueDiffOn ℝ (Set.Ico a b)","typeReadable":"∀ (a b : ℝ), UniqueDiffOn ℝ (Set.Ico a b)","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Set","Ico"],["Semiring","toModule"],["UniqueDiffOn"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["MulZeroClass","toMul"],["Real","normedCommRing"],["Preorder","toLT"],["eq_true"],["Real","instNontrivial"],["IsTopologicalSemiring","toContinuousMul"],["SMulZeroClass","toSMul"],["Set","Ioo"],["Real","decidableLT"],["Real","instIsOrderedCancelAddMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semiring","toModule"],["Set","Ico"],["Real","instAddCommMonoid"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["IsStrictOrderedRing","toPosMulStrictMono"],["Real"],["interior"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["IsStrictOrderedModule","toPosSMulStrictMono"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearOrderedSemiField","toDenselyOrdered"],["instIsTopologicalRingReal"],["convex_Ico"],["AddZeroClass","toAddZero"],["instIsTopologicalAddGroupReal"],["Real","pseudoMetricSpace"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ico","_simp_1_2"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Real","instIsStrictOrderedRing"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["AddCommMonoid","toAddMonoid"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ico","_simp_1_1"],["interior_Ico"],["Real","instIsOrderedRing"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instField"],["congrArg"],["instOrderTopologyReal"],["uniqueDiffOn_convex"],["instDistribLatticeOfLinearOrder"],["Real","instLT"],["MonoidWithZero","toMonoid"],["instNoMinOrderOfNontrivial"],["Set","Ico_eq_empty"],["Real","partialOrder"],["Real","instMul"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Semiring","toMonoidWithZero"],["IsStrictOrderedRing","toIsStrictOrderedModule"],["UniqueDiffOn"],["Real","semiring"],["LT","lt"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Real","instAddCommGroup"],["dite"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_convex"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.15 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.12] {s : Set.{u_1} E} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.25 : IsTopologicalAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.12 (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3)], (Convex.{0, u_1} Real E Real.semiring Real.partialOrder (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3) (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.6)))) s) -> (Set.Nonempty.{u_1} E (interior.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.12 s)) -> (UniqueDiffOn.{0, u_1} Real E Real.semiring inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.3 inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.6 inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.4191691927._hygCtx._hyg.12 s)","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} [IsTopologicalAddGroup E], Convex ℝ s → (interior s).Nonempty → UniqueDiffOn ℝ s","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} [IsTopologicalAddGroup E], Convex ℝ s → (interior s).Nonempty → UniqueDiffOn ℝ s","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["IsTopologicalAddGroup"],["ContinuousSMul"],["AddGroup","toSubNegMonoid"],["Real","partialOrder"],["DistribSMul","toSMulZeroClass"],["Set","Nonempty"],["Real"],["interior"],["Set"],["Convex"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["UniqueDiffOn"],["AddZeroClass","toAddZero"],["Real","semiring"],["TopologicalSpace"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["uniqueDiffWithinAt_convex"],["subset_closure"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ioc","_simp_1_2"],"typeFallback":"forall {𝕜 : Type.{u_1}} {E : Type.{u_2}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.4 : Semiring.{u_1} 𝕜] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.7 : AddCommGroup.{u_2} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.10 : Module.{u_1, u_2} 𝕜 E inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.4 (AddCommGroup.toAddCommMonoid.{u_2} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.7)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.14 : TopologicalSpace.{u_2} E], Eq.{1} Prop (UniqueDiffOn.{u_1, u_2} 𝕜 E inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.4 inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.7 inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.10 inst._@.Mathlib.Analysis.Calculus.TangentCone.Basic.3510134759._hygCtx._hyg.14 (EmptyCollection.emptyCollection.{u_2} (Set.{u_2} E) (Set.instEmptyCollection.{u_2} E))) True","typeFull":"∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]\n  [inst_3 : TopologicalSpace E], UniqueDiffOn 𝕜 ∅ = True","typeReadable":"∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]\n  [inst_3 : TopologicalSpace E], UniqueDiffOn 𝕜 ∅ = True","typeReferences":[["TopologicalSpace"],["True"],["Set"],["Module"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["UniqueDiffOn"],["Eq"],["Semiring"]],"valueReferences":[["uniqueDiffOn_empty"],["Set"],["eq_true"],["Set","instEmptyCollection"],["EmptyCollection","emptyCollection"],["UniqueDiffOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Iic","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.3 : Preorder.{u_1} α] {a : α} [inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.14 : NoMinOrder.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.3)], Eq.{1} Prop (Set.Nonempty.{u_1} α (Set.Iio.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.2624579663._hygCtx._hyg.3 a)) True","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a : α} [NoMinOrder α], (Set.Iio a).Nonempty = True","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a : α} [NoMinOrder α], (Set.Iio a).Nonempty = True","typeReferences":[["Set","Nonempty"],["Preorder"],["True"],["Preorder","toLT"],["NoMinOrder"],["Set","Iio"],["Eq"]],"valueReferences":[["Set","Nonempty"],["Set","nonempty_Iio"],["eq_true"],["Set","Iio"]]},{"isProp":true,"kind":"theorem","name":["mem_tangentConeAt_of_openSegment_subset"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.15 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.12] {s : Set.{u_1} E} {x : E} {y : E}, (HasSubset.Subset.{u_1} (Set.{u_1} E) (Set.instHasSubset.{u_1} E) (openSegment.{0, u_1} Real E Real.semiring Real.partialOrder (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3) (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.6)))) x y) s) -> (Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (tangentConeAt.{0, u_1} Real E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3 (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.6)))) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.12 s x) (HSub.hSub.{u_1, u_1, u_1} E E E (instHSub.{u_1} E (SubNegMonoid.toSub.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.309841371._hygCtx._hyg.3)))) y x))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x y : E}, openSegment ℝ x y ⊆ s → y - x ∈ tangentConeAt ℝ s x","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x y : E}, openSegment ℝ x y ⊆ s → y - x ∈ tangentConeAt ℝ s x","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["ContinuousSMul"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Real","partialOrder"],["DistribSMul","toSMulZeroClass"],["Real"],["Set"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Real","semiring"],["Set","instMembership"],["TopologicalSpace"],["Set","instHasSubset"],["Module","toDistribMulAction"],["openSegment"],["SubNegMonoid","toAddMonoid"],["HasSubset","Subset"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["tangentConeAt"],["AddCommGroup","toAddCommMonoid"],["instHSub"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["NNReal","instAlgebraOfReal"],["AddCommGroup","toAddGroup"],["sub_mem_posTangentConeAt_of_openSegment_subset"],["SemigroupAction","toSMul"],["SMulZeroClass","toSMul"],["AddCommMonoid","toAddMonoid"],["Algebra","id"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["Algebra","toSMul"],["HSub","hSub"],["DistribMulAction","toMulAction"],["Monoid","toSemigroup"],["AddGroup","toSubNegMonoid"],["Semiring","toModule"],["Real","instAddCommMonoid"],["NNReal","instCommSemiring"],["DistribSMul","toSMulZeroClass"],["Real","instCommSemiring"],["IsScalarTower","left"],["Real"],["NNReal"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["NNReal","instDistribMulActionOfReal"],["Real","semiring"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["NNReal","instIsScalarTowerOfReal"],["tangentConeAt_mono_field"],["Real","instMonoid"],["AddCommGroup","toAddCommMonoid"],["NNReal","instSemiring"],["instHSub"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffWithinAt_Iic"],"typeFallback":"forall (x : Real), UniqueDiffWithinAt.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Iic.{0} Real Real.instPreorder x) x","typeFull":"∀ (x : ℝ), UniqueDiffWithinAt ℝ (Set.Iic x) x","typeReadable":"∀ (x : ℝ), UniqueDiffWithinAt ℝ (Set.Iic x) x","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["UniqueDiffWithinAt"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Semiring","toModule"],["Set","Iic"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["Real"],["PseudoMetricSpace","toUniformSpace"],["UniformSpace","toTopologicalSpace"],["Set","Iio_subset_Iic_self"],["Set","Iio"],["Set","Iic"],["Real","semiring"],["UniqueDiffWithinAt","mono"],["uniqueDiffWithinAt_Iio"],["Real","pseudoMetricSpace"],["Real","instAddCommGroup"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffWithinAt_Ioo"],"typeFallback":"forall {a : Real} {b : Real} {t : Real}, (Membership.mem.{0, 0} Real (Set.{0} Real) (Set.instMembership.{0} Real) (Set.Ioo.{0} Real Real.instPreorder a b) t) -> (UniqueDiffWithinAt.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ioo.{0} Real Real.instPreorder a b) t)","typeFull":"∀ {a b t : ℝ}, t ∈ Set.Ioo a b → UniqueDiffWithinAt ℝ (Set.Ioo a b) t","typeReadable":"∀ {a b t : ℝ}, t ∈ Set.Ioo a b → UniqueDiffWithinAt ℝ (Set.Ioo a b) t","typeReferences":[["Real","instPreorder"],["Real"],["UniqueDiffWithinAt"],["PseudoMetricSpace","toUniformSpace"],["Set"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["Set","Ioo"],["Real","semiring"],["Set","instMembership"],["Real","pseudoMetricSpace"],["Real","instAddCommGroup"],["Semiring","toModule"]],"valueReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["IsTopologicalSemiring","toContinuousMul"],["Set","Ioo"],["Real","instField"],["instOrderTopologyReal"],["isOpen_Ioo"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["IsSemitopologicalSemiring","toContinuousAdd"],["DenselyNormedField","toNontriviallyNormedField"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["Real","instMul"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["instIsTopologicalRingReal"],["DivisionSemiring","toSemiring"],["NormedField","nhdsNE_neBot"],["Real","semiring"],["OfNat","ofNat"],["OrderTopology","to_orderClosedTopology"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Real","pseudoMetricSpace"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["IsTopologicalRing","toIsSemitopologicalRing"],["Real","instAddCommGroup"],["Field","toSemifield"],["Real","denselyNormedField"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["IsOpen","uniqueDiffWithinAt"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ioc","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3 : Preorder.{u_1} α] {a : α} {b : α} [inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.14 : DenselyOrdered.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3)], Eq.{1} Prop (Set.Nonempty.{u_1} α (Set.Ioo.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3 a b)) (LT.lt.{u_1} α (Preorder.toLT.{u_1} α inst._@.Mathlib.Order.Interval.Set.Basic.3782882697._hygCtx._hyg.3) a b)","typeFull":"∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], (Set.Ioo a b).Nonempty = (a < b)","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] {a b : α} [DenselyOrdered α], (Set.Ioo a b).Nonempty = (a < b)","typeReferences":[["LT","lt"],["Set","Nonempty"],["Preorder"],["DenselyOrdered"],["Preorder","toLT"],["Eq"],["Set","Ioo"]],"valueReferences":[["LT","lt"],["Set","Nonempty"],["Preorder","toLT"],["Set","nonempty_Ioo"],["Set","Ioo"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Convex","span_tangentConeAt"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.15 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.12] {s : Set.{u_1} E} {x : E} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.25 : IsTopologicalAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.12 (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)], (Convex.{0, u_1} Real E Real.semiring Real.partialOrder (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6)))) s) -> (Set.Nonempty.{u_1} E (interior.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.12 s)) -> (Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (closure.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.12 s) x) -> (Eq.{succ u_1} (Submodule.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6) (Submodule.span.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6 (tangentConeAt.{0, u_1} Real E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3 (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6)))) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.12 s x)) (Top.top.{u_1} (Submodule.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6) (Submodule.instTop.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.2252452614._hygCtx._hyg.6)))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x : E} [IsTopologicalAddGroup E],\n  Convex ℝ s → (interior s).Nonempty → x ∈ closure s → Submodule.span ℝ (tangentConeAt ℝ s x) = ⊤","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x : E} [IsTopologicalAddGroup E],\n  Convex ℝ s → (interior s).Nonempty → x ∈ closure s → Submodule.span ℝ (tangentConeAt ℝ s x) = ⊤","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["Submodule","span"],["SMulZeroClass","toSMul"],["Submodule"],["IsTopologicalAddGroup"],["ContinuousSMul"],["closure"],["AddGroup","toSubNegMonoid"],["Eq"],["Real","partialOrder"],["DistribSMul","toSMulZeroClass"],["Set","Nonempty"],["Real"],["interior"],["Set"],["Submodule","instTop"],["Convex"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Real","semiring"],["Set","instMembership"],["TopologicalSpace"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["Top","top"],["tangentConeAt"],["AddCommGroup","toAddCommMonoid"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["mem_tangentConeAt_of_openSegment_subset"],["PseudoMetricSpace","toUniformSpace"],["AddCommGroup","toAddGroup"],["Membership","mem"],["Exists","intro"],["Submodule","span"],["SMulZeroClass","toSMul"],["IsOpenMap","image_mem_nhds"],["Set","Subset","trans"],["DenselyNormedField","toNontriviallyNormedField"],["SubNegMonoid","toSub"],["isOpen_interior"],["HSub","hSub"],["Semifield","toDivisionSemiring"],["AddGroup","toSubNegMonoid"],["Eq","ndrec"],["nhds"],["DistribSMul","toSMulZeroClass"],["Real"],["interior"],["DistribMulAction","toDistribSMul"],["And"],["DivisionSemiring","toSemiring"],["AddZeroClass","toAddZero"],["Filter"],["Ring","toSemiring"],["Set","instMembership"],["interior_subset"],["Exists","casesOn"],["Set","image"],["openSegment"],["SetLike","coe"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["Convex","openSegment_closure_interior_subset_interior"],["interior_mono"],["Top","top"],["AddCommGroup","toAddCommMonoid"],["tangentConeAt"],["id"],["Eq","mpr"],["AddZero","toZero"],["Submodule","subset_span"],["AddMonoid","toAddZeroClass"],["ContinuousSMul","continuousConstSMul"],["UniformSpace","toTopologicalSpace"],["Submodule","eq_top_of_nonempty_interior'"],["IsOpen","mem_nhds"],["Real","instRing"],["Real","instField"],["Submodule"],["congrArg"],["MonoidWithZero","toMonoid"],["mem_interior_iff_mem_nhds"],["Eq"],["NormedField","nhdsWithin_isUnit_neBot"],["propext"],["Real","partialOrder"],["Filter","instMembership"],["Set"],["Submodule","instTop"],["isOpenMap_sub_right"],["Semiring","toMonoidWithZero"],["Real","semiring"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["Submodule","setLike"],["Filter","mem_of_superset"],["Real","denselyNormedField"],["Field","toSemifield"],["IsTopologicalAddGroup","toContinuousAdd"],["instHSub"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Ici"],"typeFallback":"forall (a : Real), UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ici.{0} Real Real.instPreorder a)","typeFull":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Ici a)","typeReadable":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Ici a)","typeReferences":[["Set","Ici"],["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Semiring","toModule"],["UniqueDiffOn"],["Real","semiring"]],"valueReferences":[["Real","instPreorder"],["PartialOrder","toPreorder"],["Eq","trans"],["PseudoMetricSpace","toUniformSpace"],["MulZeroClass","toMul"],["Real","normedCommRing"],["Real","instNontrivial"],["IsTopologicalSemiring","toContinuousMul"],["SMulZeroClass","toSMul"],["MulZeroClass","toSMulWithZero"],["convex_Ici"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["interior_Ici"],["Set","Ioi"],["Real","instAddCommMonoid"],["Semiring","toModule"],["SemilatticeInf","toPartialOrder"],["DistribSMul","toSMulZeroClass"],["IsStrictOrderedRing","toPosMulStrictMono"],["Real"],["interior"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Real","instIsOrderedAddMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["LinearOrderedSemiField","toDenselyOrdered"],["instIsTopologicalRingReal"],["AddZeroClass","toAddZero"],["instIsTopologicalAddGroupReal"],["Real","pseudoMetricSpace"],["instNoMaxOrderOfNontrivial"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["PosMulStrictMono","toPosMulReflectLE"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Real","instIsStrictOrderedRing"],["UniformSpace","toTopologicalSpace"],["Real","instRing"],["AddCommMonoid","toAddMonoid"],["Real","instIsOrderedRing"],["PosMulReflectLE","toPosMulReflectLT"],["Real","instField"],["congrArg"],["instOrderTopologyReal"],["uniqueDiffOn_convex"],["instDistribLatticeOfLinearOrder"],["IsOrderedModule","toPosSMulMono"],["MonoidWithZero","toMonoid"],["instNoMinOrderOfNontrivial"],["Real","partialOrder"],["Real","instMul"],["Set","Nonempty"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Semiring","toMonoidWithZero"],["_private","Mathlib","Analysis","Calculus","TangentCone","Real",0,"uniqueDiffOn_Ici","_simp_1_1"],["IsStrictOrderedRing","toIsStrictOrderedModule"],["Real","semiring"],["Set","Ici"],["IsStrictOrderedModule","toIsOrderedModule"],["Module","toDistribMulAction"],["DistribLattice","toLattice"],["of_eq_true"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Field","toSemifield"],["Real","instAddCommGroup"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["uniqueDiffOn_Ioi"],"typeFallback":"forall (a : Real), UniqueDiffOn.{0, 0} Real Real Real.semiring Real.instAddCommGroup (Semiring.toModule.{0} Real Real.semiring) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (Set.Ioi.{0} Real Real.instPreorder a)","typeFull":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Ioi a)","typeReadable":"∀ (a : ℝ), UniqueDiffOn ℝ (Set.Ioi a)","typeReferences":[["Real","instPreorder"],["PseudoMetricSpace","toUniformSpace"],["Real"],["Real","pseudoMetricSpace"],["UniformSpace","toTopologicalSpace"],["Real","instAddCommGroup"],["Set","Ioi"],["Semiring","toModule"],["UniqueDiffOn"],["Real","semiring"]],"valueReferences":[["PartialOrder","toPreorder"],["PseudoMetricSpace","toUniformSpace"],["isOpen_Ioi"],["Real","normedCommRing"],["UniformSpace","toTopologicalSpace"],["IsTopologicalSemiring","toContinuousMul"],["Real","instField"],["IsOpen","uniqueDiffOn"],["instOrderTopologyReal"],["instDistribLatticeOfLinearOrder"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["IsTopologicalRing","toIsTopologicalSemiring"],["Semiring","toNonAssocSemiring"],["IsSemitopologicalSemiring","toContinuousAdd"],["DenselyNormedField","toNontriviallyNormedField"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Set","Ioi"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Semiring","toModule"],["SemilatticeInf","toPartialOrder"],["Real","instMul"],["Lattice","toSemilatticeInf"],["instClosedIicTopology"],["Real"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["instIsTopologicalRingReal"],["DivisionSemiring","toSemiring"],["NormedField","nhdsNE_neBot"],["Real","semiring"],["OfNat","ofNat"],["DistribLattice","toLattice"],["OrderTopology","to_orderClosedTopology"],["IsSemitopologicalRing","toIsSemitopologicalSemiring"],["Real","pseudoMetricSpace"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["IsTopologicalRing","toIsSemitopologicalRing"],["Real","instAddCommGroup"],["Field","toSemifield"],["Real","denselyNormedField"],["ContinuousMul","to_continuousSMul"],["Real","linearOrder"],["NonUnitalNormedCommRing","toNonUnitalCommRing"],["NormedCommRing","toNonUnitalNormedCommRing"]]},{"isProp":true,"kind":"theorem","name":["mem_tangentConeAt_of_segment_subset"],"typeFallback":"forall {E : Type.{u_1}} [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3 : AddCommGroup.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.6 : Module.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.12 : TopologicalSpace.{u_1} E] [inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.15 : ContinuousSMul.{0, u_1} Real E (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.6)))) (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.12] {s : Set.{u_1} E} {x : E} {y : E}, (HasSubset.Subset.{u_1} (Set.{u_1} E) (Set.instHasSubset.{u_1} E) (segment.{0, u_1} Real E Real.semiring Real.partialOrder (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3) (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.6)))) x y) s) -> (Membership.mem.{u_1, u_1} E (Set.{u_1} E) (Set.instMembership.{u_1} E) (tangentConeAt.{0, u_1} Real E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3 (SMulZeroClass.toSMul.{0, u_1} Real E (AddZero.toZero.{u_1} E (AddZeroClass.toAddZero.{u_1} E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)))))) (DistribSMul.toSMulZeroClass.{0, u_1} Real E (AddMonoid.toAddZeroClass.{u_1} E (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)))) (DistribMulAction.toDistribSMul.{0, u_1} Real E Real.instMonoid (SubNegMonoid.toAddMonoid.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3))) (Module.toDistribMulAction.{0, u_1} Real E Real.semiring (AddCommGroup.toAddCommMonoid.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.6)))) inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.12 s x) (HSub.hSub.{u_1, u_1, u_1} E E E (instHSub.{u_1} E (SubNegMonoid.toSub.{u_1} E (AddGroup.toSubNegMonoid.{u_1} E (AddCommGroup.toAddGroup.{u_1} E inst._@.Mathlib.Analysis.Calculus.TangentCone.Real.1979329352._hygCtx._hyg.3)))) y x))","typeFull":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x y : E}, segment ℝ x y ⊆ s → y - x ∈ tangentConeAt ℝ s x","typeReadable":"∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]\n  {s : Set E} {x y : E}, segment ℝ x y ⊆ s → y - x ∈ tangentConeAt ℝ s x","typeReferences":[["PseudoMetricSpace","toUniformSpace"],["Module"],["AddCommGroup","toAddGroup"],["Membership","mem"],["UniformSpace","toTopologicalSpace"],["SMulZeroClass","toSMul"],["ContinuousSMul"],["SubNegMonoid","toSub"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Real","partialOrder"],["DistribSMul","toSMulZeroClass"],["Real"],["Set"],["segment"],["DistribMulAction","toDistribSMul"],["AddCommGroup"],["AddZeroClass","toAddZero"],["Real","semiring"],["Set","instMembership"],["TopologicalSpace"],["Set","instHasSubset"],["Module","toDistribMulAction"],["SubNegMonoid","toAddMonoid"],["HasSubset","Subset"],["Real","pseudoMetricSpace"],["Real","instMonoid"],["tangentConeAt"],["AddCommGroup","toAddCommMonoid"],["instHSub"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["openSegment_subset_segment"],["mem_tangentConeAt_of_openSegment_subset"],["Real"],["Set"],["AddCommGroup","toAddGroup"],["Set","instIsTransSubset"],["segment"],["DistribMulAction","toDistribSMul"],["SMulZeroClass","toSMul"],["HasSubset","Subset","trans"],["AddZeroClass","toAddZero"],["Real","semiring"],["Set","instHasSubset"],["Module","toDistribMulAction"],["openSegment"],["SubNegMonoid","toAddMonoid"],["Real","instMonoid"],["AddCommGroup","toAddCommMonoid"],["AddGroup","toSubNegMonoid"],["AddZero","toZero"],["Real","partialOrder"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]}]