Add files using upload-large-folder tool

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.AffineMonoid.UniqueSums.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["AffineAddMonoid","to_twoUniqueSums"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.920224857._hygCtx._hyg.3 : AddCancelCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.920224857._hygCtx._hyg.6 : AddMonoid.FG.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCancelCommMonoid.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.920224857._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.920224857._hygCtx._hyg.9 : IsAddTorsionFree.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M (AddCancelCommMonoid.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.920224857._hygCtx._hyg.3))], TwoUniqueSums.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M (AddCancelCommMonoid.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.920224857._hygCtx._hyg.3))))","typeFull":"∀ {M : Type u_1} [inst : AddCancelCommMonoid M] [AddMonoid.FG M] [IsAddTorsionFree M], TwoUniqueSums M","typeReadable":"∀ {M : Type u_1} [inst : AddCancelCommMonoid M] [AddMonoid.FG M] [IsAddTorsionFree M], TwoUniqueSums M","typeReferences":[["AddCancelCommMonoid"],["TwoUniqueSums"],["AddCommMonoid","toAddCommSemigroup"],["IsAddTorsionFree"],["AddMonoid","FG"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddCommMonoid","toAddMonoid"],["AddCancelCommMonoid","toAddCommMonoid"]],"valueReferences":[["instTwoUniqueSumsFreeAbelianGroup"],["AddCommGroup","toAddGroup"],["AffineAddMonoid","embedding_injective"],["FreeAbelianGroup"],["Fin"],["instAddCommGroupFreeAbelianGroup"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["TwoUniqueSums"],["AddMonoidHom","toAddHom"],["AffineAddMonoid","embedding"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["AffineAddMonoid","dim"],["inferInstance"],["TwoUniqueSums","of_injective_addHom"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["AddGroup","toSubNegMonoid"],["AddCancelCommMonoid","toAddCommMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["AffineMonoid","to_twoUniqueProds"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.2287540865._hygCtx._hyg.3 : CancelCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.2287540865._hygCtx._hyg.6 : Monoid.FG.{u_1} M (CommMonoid.toMonoid.{u_1} M (CancelCommMonoid.toCommMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.2287540865._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.2287540865._hygCtx._hyg.9 : IsMulTorsionFree.{u_1} M (CommMonoid.toMonoid.{u_1} M (CancelCommMonoid.toCommMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.2287540865._hygCtx._hyg.3))], TwoUniqueProds.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M (CancelCommMonoid.toCommMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.UniqueSums.2287540865._hygCtx._hyg.3)))))","typeFull":"∀ {M : Type u_1} [inst : CancelCommMonoid M] [Monoid.FG M] [IsMulTorsionFree M], TwoUniqueProds M","typeReadable":"∀ {M : Type u_1} [inst : CancelCommMonoid M] [Monoid.FG M] [IsMulTorsionFree M], TwoUniqueProds M","typeReferences":[["MulOneClass","toMulOne"],["IsMulTorsionFree"],["Monoid","FG"],["MulOne","toMul"],["CommMonoid","toMonoid"],["Monoid","toMulOneClass"],["TwoUniqueProds"],["CancelCommMonoid","toCommMonoid"],["CancelCommMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["Additive","add"],["AffineAddMonoid","to_twoUniqueSums"],["MulOne","toMul"],["CommMonoid","toMonoid"],["instIsAddTorsionFreeAdditiveOfIsMulTorsionFree"],["CancelCommMonoid","toCommMonoid"],["Monoid","toMulOneClass"],["Multiplicative","instTwoUniqueProdsOfTwoUniqueSums"],["Additive"],["AddMonoid","fg_of_monoid_fg"],["Additive","instAddCancelCommMonoid"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.BigOperators.Group.Finset.Indicator.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Finset","prod_mulIndicator_eq_prod_inter"],"typeFallback":"forall {ι : Type.{u_1}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6 : CommMonoid.{u_4} β] [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.9 : DecidableEq.{succ u_1} ι] (s : Finset.{u_1} ι) (t : Finset.{u_1} ι) (f : ι -> β), Eq.{succ u_4} β (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6 s (fun (i : ι) => Set.mulIndicator.{u_1, u_4} ι β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6)))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) t) f i)) (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6 (Inter.inter.{u_1} (Finset.{u_1} ι) (Finset.instInter.{u_1} ι inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.9) s t) (fun (i : ι) => f i))","typeFull":"∀ {ι : Type u_1} {β : Type u_4} [inst : CommMonoid β] [inst_1 : DecidableEq ι] (s t : Finset ι) (f : ι → β),\n  ∏ i ∈ s, (↑t).mulIndicator f i = ∏ i ∈ s ∩ t, f i","typeReadable":"∀ {ι : Type u_1} {β : Type u_4} [inst : CommMonoid β] [inst_1 : DecidableEq ι] (s t : Finset ι) (f : ι → β),\n  ∏ i ∈ s, (↑t).mulIndicator f i = ∏ i ∈ s ∩ t, f i","typeReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset"],["Set","mulIndicator"],["MulOne","toOne"],["CommMonoid","toMonoid"],["DecidableEq"],["Inter","inter"],["Finset","prod"],["SetLike","coe"],["Monoid","toMulOneClass"],["Finset","instInter"],["CommMonoid"],["Eq"]],"valueReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset"],["CommMonoid","toMonoid"],["Inter","inter"],["Membership","mem"],["Finset","decidableMem"],["Finset","prod_mulIndicator_eq_prod_filter"],["Finset","decidableMem'"],["congrArg"],["Monoid","toMulOneClass"],["Finset","instInter"],["Eq","symm"],["Eq"],["Finset","filter"],["Set","mulIndicator"],["SetLike","instMembership"],["MulOne","toOne"],["Set"],["Finset","filter_mem_eq_inter"],["Set","instMembership"],["Finset","prod"],["SetLike","coe"],["Eq","refl"],["id"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["Finset","sum_indicator_eq_sum_inter"],"typeFallback":"forall {ι : Type.{u_1}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6 : AddCommMonoid.{u_4} β] [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.9 : DecidableEq.{succ u_1} ι] (s : Finset.{u_1} ι) (t : Finset.{u_1} ι) (f : ι -> β), Eq.{succ u_4} β (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6 s (fun (i : ι) => Set.indicator.{u_1, u_4} ι β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6)))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) t) f i)) (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.6 (Inter.inter.{u_1} (Finset.{u_1} ι) (Finset.instInter.{u_1} ι inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.360664735._hygCtx._hyg.9) s t) (fun (i : ι) => f i))","typeFull":"∀ {ι : Type u_1} {β : Type u_4} [inst : AddCommMonoid β] [inst_1 : DecidableEq ι] (s t : Finset ι) (f : ι → β),\n  ∑ i ∈ s, (↑t).indicator f i = ∑ i ∈ s ∩ t, f i","typeReadable":"∀ {ι : Type u_1} {β : Type u_4} [inst : AddCommMonoid β] [inst_1 : DecidableEq ι] (s t : Finset ι) (f : ι → β),\n  ∑ i ∈ s, (↑t).indicator f i = ∑ i ∈ s ∩ t, f i","typeReferences":[["Finset","instSetLike"],["Finset"],["DecidableEq"],["Inter","inter"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","indicator"],["AddCommMonoid"],["SetLike","coe"],["Finset","instInter"],["Finset","sum"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["Inter","inter"],["Membership","mem"],["Finset","decidableMem"],["AddCommMonoid","toAddMonoid"],["Finset","decidableMem'"],["congrArg"],["Finset","instInter"],["Eq","symm"],["Finset","sum"],["Eq"],["Finset","filter"],["SetLike","instMembership"],["Set"],["AddZeroClass","toAddZero"],["Set","indicator"],["Finset","filter_mem_eq_inter"],["Set","instMembership"],["Finset","sum_indicator_eq_sum_filter"],["SetLike","coe"],["Eq","refl"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Finset","sum_indicator_subset"],"typeFallback":"forall {ι : Type.{u_1}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6 : AddCommMonoid.{u_4} β] (f : ι -> β) {s : Finset.{u_1} ι} {t : Finset.{u_1} ι}, (HasSubset.Subset.{u_1} (Finset.{u_1} ι) (Finset.instHasSubset.{u_1} ι) s t) -> (Eq.{succ u_4} β (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6 t (fun (i : ι) => Set.indicator.{u_1, u_4} ι β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6)))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) f i)) (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6 s (fun (i : ι) => f i)))","typeFull":"∀ {ι : Type u_1} {β : Type u_4} [inst : AddCommMonoid β] (f : ι → β) {s t : Finset ι},\n  s ⊆ t → ∑ i ∈ t, (↑s).indicator f i = ∑ i ∈ s, f i","typeReadable":"∀ {ι : Type u_1} {β : Type u_4} [inst : AddCommMonoid β] (f : ι → β) {s t : Finset ι},\n  s ⊆ t → ∑ i ∈ t, (↑s).indicator f i = ∑ i ∈ s, f i","typeReferences":[["Finset","instSetLike"],["Finset"],["Finset","instHasSubset"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","indicator"],["AddCommMonoid"],["SetLike","coe"],["HasSubset","Subset"],["Finset","sum"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["rfl"],["Finset","sum_indicator_subset_of_eq_zero"],["id"],["Zero","toOfNat0"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toZero"],["OfNat","ofNat"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Finset","mulIndicator_prod"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6 : CommMonoid.{u_4} β] (s : Finset.{u_1} ι) (t : Set.{u_2} κ) (f : ι -> κ -> β), Eq.{max (succ u_2) (succ u_4)} (κ -> β) (Set.mulIndicator.{u_2, u_4} κ β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)))) t (Finset.prod.{u_1, max u_2 u_4} ι (κ -> β) (Pi.commMonoid.{u_2, u_4} κ (fun (a._@._internal._hyg.0 : κ) => β) (fun (i : κ) => inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)) s (fun (i : ι) => f i))) (Finset.prod.{u_1, max u_4 u_2} ι (κ -> β) (Pi.commMonoid.{u_2, u_4} κ (fun (x : κ) => β) (fun (i : κ) => inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)) s (fun (i : ι) => Set.mulIndicator.{u_2, u_4} κ β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)))) t (f i)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (t : Set κ) (f : ι → κ → β),\n  t.mulIndicator (∏ i ∈ s, f i) = ∏ i ∈ s, t.mulIndicator (f i)","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (t : Set κ) (f : ι → κ → β),\n  t.mulIndicator (∏ i ∈ s, f i) = ∏ i ∈ s, t.mulIndicator (f i)","typeReferences":[["MulOneClass","toMulOne"],["Finset","prod"],["Set","mulIndicator"],["Finset"],["CommMonoid","toMonoid"],["MulOne","toOne"],["Set"],["Monoid","toMulOneClass"],["Pi","commMonoid"],["CommMonoid"],["Eq"]],"valueReferences":[["MulOneClass","toMulOne"],["CommMonoid","toMonoid"],["Monoid","toMulOneClass"],["MonoidHom"],["MonoidHom","instFunLike"],["Pi","commMonoid"],["Pi","mulOneClass"],["map_prod"],["MonoidHom","instMonoidHomClass"],["Set","mulIndicatorHom"]]},{"isProp":true,"kind":"theorem","name":["Finset","sum_indicator_eq_sum_filter"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6 : AddCommMonoid.{u_4} β] (s : Finset.{u_1} ι) (f : ι -> κ -> β) (t : ι -> (Set.{u_2} κ)) (g : ι -> κ) [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.23 : DecidablePred.{succ u_1} ι (fun (i : ι) => Membership.mem.{u_2, u_2} κ (Set.{u_2} κ) (Set.instMembership.{u_2} κ) (t i) (g i))], Eq.{succ u_4} β (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6 s (fun (i : ι) => Set.indicator.{u_2, u_4} κ β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6)))) (t i) (f i) (g i))) (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6 (Finset.filter.{u_1} ι (fun (i : ι) => Membership.mem.{u_2, u_2} κ (Set.{u_2} κ) (Set.instMembership.{u_2} κ) (t i) (g i)) inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.23 s) (fun (i : ι) => f i (g i)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (f : ι → κ → β) (t : ι → Set κ)\n  (g : ι → κ) [inst_1 : DecidablePred fun i => g i ∈ t i],\n  ∑ i ∈ s, (t i).indicator (f i) (g i) = ∑ i ∈ s with g i ∈ t i, f i (g i)","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (f : ι → κ → β) (t : ι → Set κ)\n  (g : ι → κ) [inst_1 : DecidablePred fun i => g i ∈ t i],\n  ∑ i ∈ s, (t i).indicator (f i) (g i) = ∑ i ∈ s with g i ∈ t i, f i (g i)","typeReferences":[["Finset"],["Set"],["Membership","mem"],["DecidablePred"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","indicator"],["Set","instMembership"],["AddCommMonoid"],["Finset","sum"],["Eq"],["AddZero","toZero"],["Finset","filter"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["Eq","trans"],["Set","indicator_of_mem"],["Set","indicator_of_notMem"],["Membership","mem"],["Iff","mp"],["Finset","sum_filter_add_sum_filter_not"],["AddCommMonoid","toAddMonoid"],["Finset","sum_congr"],["Finset","sum_eq_zero"],["Eq","symm"],["Finset","sum"],["Zero","toOfNat0"],["Finset","filter"],["Not"],["rfl"],["SetLike","instMembership"],["Set"],["instHAdd"],["And","right"],["And"],["add_zero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Set","indicator"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Finset","mem_filter"],["congr_arg₂"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["instDecidableNot"]]},{"isProp":true,"kind":"theorem","name":["Finset","mulIndicator_biUnion"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6 : CommMonoid.{u_4} β] (s : Finset.{u_1} ι) (t : ι -> (Set.{u_2} κ)) {f : κ -> β}, (Set.PairwiseDisjoint.{u_2, u_1} (Set.{u_2} κ) ι (CompleteSemilatticeInf.toPartialOrder.{u_2} (Set.{u_2} κ) (CompleteLattice.toCompleteSemilatticeInf.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ))))) (HeytingAlgebra.toOrderBot.{u_2} (Set.{u_2} κ) (Order.Frame.toHeytingAlgebra.{u_2} (Set.{u_2} κ) (CompleteDistribLattice.toFrame.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ)))))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) t) -> (Eq.{max (succ u_2) (succ u_4)} (κ -> β) (Set.mulIndicator.{u_2, u_4} κ β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6)))) (Set.iUnion.{u_2, succ u_1} κ ι (fun (i : ι) => Set.iUnion.{u_2, 0} κ (Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) (fun (h._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.37 : Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) => t i))) f) (fun (a : κ) => Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6 s (fun (i : ι) => Set.mulIndicator.{u_2, u_4} κ β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6)))) (t i) f a)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → (⋃ i ∈ s, t i).mulIndicator f = fun a => ∏ i ∈ s, (t i).mulIndicator f a","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → (⋃ i ∈ s, t i).mulIndicator f = fun a => ∏ i ∈ s, (t i).mulIndicator f a","typeReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["CommMonoid","toMonoid"],["Membership","mem"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["CompleteBooleanAlgebra","toCompleteLattice"],["Set","iUnion"],["Order","Frame","toHeytingAlgebra"],["CompleteSemilatticeInf","toPartialOrder"],["Monoid","toMulOneClass"],["Set","PairwiseDisjoint"],["HeytingAlgebra","toOrderBot"],["CommMonoid"],["Eq"],["CompleteDistribLattice","toFrame"],["Set","mulIndicator"],["SetLike","instMembership"],["MulOne","toOne"],["Set"],["Finset","prod"],["SetLike","coe"],["Set","instCompleteAtomicBooleanAlgebra"],["CompleteLattice","toCompleteSemilatticeInf"]],"valueReferences":[["PartialOrder","toPreorder"],["Finset"],["Set","mulIndicator_union_of_notMem_inter"],["Eq","trans"],["Set","iUnion_of_empty"],["Inter","inter"],["Membership","mem"],["Iff","mp"],["Classical","propDecidable"],["HMul","hMul"],["Set","iUnion"],["CompleteBooleanAlgebra","toCompleteLattice"],["Set","instInsert"],["Set","pairwiseDisjoint_insert_of_notMem"],["Union","union"],["Finset","cons"],["Order","Frame","toHeytingAlgebra"],["Set","disjoint_iUnion₂_right"],["Set","disjoint_iff"],["Disjoint"],["CompleteSemilatticeInf","toPartialOrder"],["funext"],["HeytingAlgebra","toOrderBot"],["Set","instUnion"],["Set","instInter"],["And","left"],["SemilatticeInf","toPartialOrder"],["Finset","set_biUnion_insert"],["SetLike","instMembership"],["Set","mulIndicator"],["MulOne","toOne"],["And","right"],["And"],["Insert","insert"],["Set","instMembership"],["BoundedOrder","toOrderBot"],["SetLike","coe"],["HasSubset","Subset"],["Iff","of_eq"],["Iff","not"],["Set","instCompleteAtomicBooleanAlgebra"],["Iff","mpr"],["Eq","refl"],["id"],["instHMul"],["Eq","mpr"],["Finset","cons_induction"],["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset","instInsert"],["CommMonoid","toMonoid"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["Eq","mp"],["Finset","prod_cons"],["Finset","instEmptyCollection"],["Finset","cons_eq_insert"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["EmptyCollection","emptyCollection"],["Set","instEmptyCollection"],["congrArg"],["MulOne","toMul"],["Finset","mem_coe"],["Finset","coe_cons"],["Monoid","toMulOneClass"],["Set","PairwiseDisjoint"],["Set","iUnion_congr_Prop"],["congrFun'"],["Preorder","toLE"],["Eq"],["Set","iUnion_empty"],["propext"],["Not"],["CompleteDistribLattice","toFrame"],["Lattice","toSemilatticeInf"],["True"],["Set"],["Set","instBoundedOrder"],["Finset","notMem_empty","_simp_1"],["Set","instDistribLattice"],["OfNat","ofNat"],["Set","mulIndicator_empty"],["instIsEmptyFalse"],["Set","instHasSubset"],["Finset","prod"],["eq_self"],["DistribLattice","toLattice"],["of_eq_true"],["One","toOfNat1"],["False"],["CompleteLattice","toCompleteSemilatticeInf"]]},{"isProp":true,"kind":"theorem","name":["Finset","mulIndicator_biUnion_apply"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6 : CommMonoid.{u_4} β] (s : Finset.{u_1} ι) (t : ι -> (Set.{u_2} κ)) {f : κ -> β}, (Set.PairwiseDisjoint.{u_2, u_1} (Set.{u_2} κ) ι (CompleteSemilatticeInf.toPartialOrder.{u_2} (Set.{u_2} κ) (CompleteLattice.toCompleteSemilatticeInf.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ))))) (HeytingAlgebra.toOrderBot.{u_2} (Set.{u_2} κ) (Order.Frame.toHeytingAlgebra.{u_2} (Set.{u_2} κ) (CompleteDistribLattice.toFrame.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ)))))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) t) -> (forall (x : κ), Eq.{succ u_4} β (Set.mulIndicator.{u_2, u_4} κ β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6)))) (Set.iUnion.{u_2, succ u_1} κ ι (fun (i : ι) => Set.iUnion.{u_2, 0} κ (Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) (fun (h._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.38 : Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) => t i))) f x) (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6 s (fun (i : ι) => Set.mulIndicator.{u_2, u_4} κ β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6)))) (t i) f x)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → ∀ (x : κ), (⋃ i ∈ s, t i).mulIndicator f x = ∏ i ∈ s, (t i).mulIndicator f x","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → ∀ (x : κ), (⋃ i ∈ s, t i).mulIndicator f x = ∏ i ∈ s, (t i).mulIndicator f x","typeReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["CommMonoid","toMonoid"],["Membership","mem"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["CompleteBooleanAlgebra","toCompleteLattice"],["Set","iUnion"],["Order","Frame","toHeytingAlgebra"],["CompleteSemilatticeInf","toPartialOrder"],["Monoid","toMulOneClass"],["Set","PairwiseDisjoint"],["HeytingAlgebra","toOrderBot"],["CommMonoid"],["Eq"],["CompleteDistribLattice","toFrame"],["Set","mulIndicator"],["SetLike","instMembership"],["MulOne","toOne"],["Set"],["Finset","prod"],["SetLike","coe"],["Set","instCompleteAtomicBooleanAlgebra"],["CompleteLattice","toCompleteSemilatticeInf"]],"valueReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Set","mulIndicator"],["Finset"],["SetLike","instMembership"],["MulOne","toOne"],["CommMonoid","toMonoid"],["Finset","mulIndicator_biUnion"],["Membership","mem"],["Set","iUnion"],["congrArg"],["Finset","prod"],["Eq","refl"],["Monoid","toMulOneClass"],["id"],["Eq","mpr"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["Finset","prod_mulIndicator_subset"],"typeFallback":"forall {ι : Type.{u_1}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6 : CommMonoid.{u_4} β] (f : ι -> β) {s : Finset.{u_1} ι} {t : Finset.{u_1} ι}, (HasSubset.Subset.{u_1} (Finset.{u_1} ι) (Finset.instHasSubset.{u_1} ι) s t) -> (Eq.{succ u_4} β (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6 t (fun (i : ι) => Set.mulIndicator.{u_1, u_4} ι β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6)))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) f i)) (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.2543123908._hygCtx._hyg.6 s (fun (i : ι) => f i)))","typeFull":"∀ {ι : Type u_1} {β : Type u_4} [inst : CommMonoid β] (f : ι → β) {s t : Finset ι},\n  s ⊆ t → ∏ i ∈ t, (↑s).mulIndicator f i = ∏ i ∈ s, f i","typeReadable":"∀ {ι : Type u_1} {β : Type u_4} [inst : CommMonoid β] (f : ι → β) {s t : Finset ι},\n  s ⊆ t → ∏ i ∈ t, (↑s).mulIndicator f i = ∏ i ∈ s, f i","typeReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset"],["Set","mulIndicator"],["MulOne","toOne"],["CommMonoid","toMonoid"],["Finset","instHasSubset"],["Finset","prod"],["SetLike","coe"],["HasSubset","Subset"],["Monoid","toMulOneClass"],["CommMonoid"],["Eq"]],"valueReferences":[["rfl"],["MulOneClass","toMulOne"],["One","toOfNat1"],["CommMonoid","toMonoid"],["MulOne","toOne"],["Monoid","toMulOneClass"],["id"],["Finset","prod_mulIndicator_subset_of_eq_one"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Finset","indicator_biUnion_apply"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6 : AddCommMonoid.{u_4} β] (s : Finset.{u_1} ι) (t : ι -> (Set.{u_2} κ)) {f : κ -> β}, (Set.PairwiseDisjoint.{u_2, u_1} (Set.{u_2} κ) ι (CompleteSemilatticeInf.toPartialOrder.{u_2} (Set.{u_2} κ) (CompleteLattice.toCompleteSemilatticeInf.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ))))) (HeytingAlgebra.toOrderBot.{u_2} (Set.{u_2} κ) (Order.Frame.toHeytingAlgebra.{u_2} (Set.{u_2} κ) (CompleteDistribLattice.toFrame.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ)))))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) t) -> (forall (x : κ), Eq.{succ u_4} β (Set.indicator.{u_2, u_4} κ β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6)))) (Set.iUnion.{u_2, succ u_1} κ ι (fun (i : ι) => Set.iUnion.{u_2, 0} κ (Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) (fun (h._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.38 : Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) => t i))) f x) (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6 s (fun (i : ι) => Set.indicator.{u_2, u_4} κ β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1600445366._hygCtx._hyg.6)))) (t i) f x)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → ∀ (x : κ), (⋃ i ∈ s, t i).indicator f x = ∑ i ∈ s, (t i).indicator f x","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → ∀ (x : κ), (⋃ i ∈ s, t i).indicator f x = ∑ i ∈ s, (t i).indicator f x","typeReferences":[["Finset","instSetLike"],["Finset"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["Membership","mem"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["CompleteBooleanAlgebra","toCompleteLattice"],["Set","iUnion"],["AddCommMonoid","toAddMonoid"],["Order","Frame","toHeytingAlgebra"],["AddCommMonoid"],["CompleteSemilatticeInf","toPartialOrder"],["Set","PairwiseDisjoint"],["Finset","sum"],["HeytingAlgebra","toOrderBot"],["Eq"],["CompleteDistribLattice","toFrame"],["SetLike","instMembership"],["Set"],["AddZeroClass","toAddZero"],["Set","indicator"],["SetLike","coe"],["Set","instCompleteAtomicBooleanAlgebra"],["CompleteLattice","toCompleteSemilatticeInf"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["SetLike","instMembership"],["Membership","mem"],["Set","iUnion"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","indicator"],["congrArg"],["Finset","indicator_biUnion"],["Eq","refl"],["id"],["Finset","sum"],["Eq","mpr"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Finset","indicator_sum"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6 : AddCommMonoid.{u_4} β] (s : Finset.{u_1} ι) (t : Set.{u_2} κ) (f : ι -> κ -> β), Eq.{max (succ u_2) (succ u_4)} (κ -> β) (Set.indicator.{u_2, u_4} κ β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)))) t (Finset.sum.{u_1, max u_2 u_4} ι (κ -> β) (Pi.addCommMonoid.{u_2, u_4} κ (fun (a._@._internal._hyg.0 : κ) => β) (fun (i : κ) => inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)) s (fun (i : ι) => f i))) (Finset.sum.{u_1, max u_4 u_2} ι (κ -> β) (Pi.addCommMonoid.{u_2, u_4} κ (fun (x : κ) => β) (fun (i : κ) => inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)) s (fun (i : ι) => Set.indicator.{u_2, u_4} κ β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.4142467486._hygCtx._hyg.6)))) t (f i)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (t : Set κ) (f : ι → κ → β),\n  t.indicator (∑ i ∈ s, f i) = ∑ i ∈ s, t.indicator (f i)","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (t : Set κ) (f : ι → κ → β),\n  t.indicator (∑ i ∈ s, f i) = ∑ i ∈ s, t.indicator (f i)","typeReferences":[["AddCommMonoid"],["Finset"],["Set"],["Pi","addCommMonoid"],["Finset","sum"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Eq"],["AddZero","toZero"],["Set","indicator"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Set","indicatorHom"],["AddMonoidHom"],["Pi","addCommMonoid"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["map_sum"],["AddMonoid","toAddZeroClass"],["Pi","addZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Finset","prod_mulIndicator_subset_of_eq_one"],"typeFallback":"forall {ι : Type.{u_1}} {α : Type.{u_3}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6 : CommMonoid.{u_4} β] [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.9 : One.{u_3} α] (f : ι -> α) (g : ι -> α -> β) {s : Finset.{u_1} ι} {t : Finset.{u_1} ι}, (HasSubset.Subset.{u_1} (Finset.{u_1} ι) (Finset.instHasSubset.{u_1} ι) s t) -> (forall (a : ι), Eq.{succ u_4} β (g a (OfNat.ofNat.{u_3} α 1 (One.toOfNat1.{u_3} α inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.9))) (OfNat.ofNat.{u_4} β 1 (One.toOfNat1.{u_4} β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6))))))) -> (Eq.{succ u_4} β (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6 t (fun (i : ι) => g i (Set.mulIndicator.{u_1, u_3} ι α inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.9 (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) f i))) (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6 s (fun (i : ι) => g i (f i))))","typeFull":"∀ {ι : Type u_1} {α : Type u_3} {β : Type u_4} [inst : CommMonoid β] [inst_1 : One α] (f : ι → α) (g : ι → α → β)\n  {s t : Finset ι}, s ⊆ t → (∀ (a : ι), g a 1 = 1) → ∏ i ∈ t, g i ((↑s).mulIndicator f i) = ∏ i ∈ s, g i (f i)","typeReadable":"∀ {ι : Type u_1} {α : Type u_3} {β : Type u_4} [inst : CommMonoid β] [inst_1 : One α] (f : ι → α) (g : ι → α → β)\n  {s t : Finset ι}, s ⊆ t → (∀ (a : ι), g a 1 = 1) → ∏ i ∈ t, g i ((↑s).mulIndicator f i) = ∏ i ∈ s, g i (f i)","typeReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Set","mulIndicator"],["Finset"],["MulOne","toOne"],["CommMonoid","toMonoid"],["Finset","instHasSubset"],["OfNat","ofNat"],["Finset","prod"],["SetLike","coe"],["HasSubset","Subset"],["One","toOfNat1"],["Monoid","toMulOneClass"],["One"],["CommMonoid"],["Eq"]],"valueReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Trans","trans"],["Finset"],["CommMonoid","toMonoid"],["Eq","trans"],["Membership","mem"],["congrArg"],["Set","mulIndicator_of_notMem"],["not_false_eq_true"],["Monoid","toMulOneClass"],["congrFun'"],["Eq"],["rfl"],["Not"],["instTransEq"],["SetLike","instMembership"],["Set","mulIndicator"],["MulOne","toOne"],["True"],["Set"],["Set","mulIndicator_of_mem"],["congr_arg"],["OfNat","ofNat"],["Set","instMembership"],["eq_self"],["Finset","prod"],["Finset","prod_subset"],["SetLike","coe"],["SetLike","mem_coe","_simp_1"],["of_eq_true"],["One","toOfNat1"],["Eq","refl"],["eq_false"],["Finset","prod_congr"],["id"],["False"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["Finset","prod_mulIndicator_eq_prod_filter"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6 : CommMonoid.{u_4} β] (s : Finset.{u_1} ι) (f : ι -> κ -> β) (t : ι -> (Set.{u_2} κ)) (g : ι -> κ) [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.23 : DecidablePred.{succ u_1} ι (fun (i : ι) => Membership.mem.{u_2, u_2} κ (Set.{u_2} κ) (Set.instMembership.{u_2} κ) (t i) (g i))], Eq.{succ u_4} β (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6 s (fun (i : ι) => Set.mulIndicator.{u_2, u_4} κ β (MulOne.toOne.{u_4} β (MulOneClass.toMulOne.{u_4} β (Monoid.toMulOneClass.{u_4} β (CommMonoid.toMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6)))) (t i) (f i) (g i))) (Finset.prod.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.6 (Finset.filter.{u_1} ι (fun (i : ι) => Membership.mem.{u_2, u_2} κ (Set.{u_2} κ) (Set.instMembership.{u_2} κ) (t i) (g i)) inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.3573837778._hygCtx._hyg.23 s) (fun (i : ι) => f i (g i)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (f : ι → κ → β) (t : ι → Set κ)\n  (g : ι → κ) [inst_1 : DecidablePred fun i => g i ∈ t i],\n  ∏ i ∈ s, (t i).mulIndicator (f i) (g i) = ∏ i ∈ s with g i ∈ t i, f i (g i)","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : CommMonoid β] (s : Finset ι) (f : ι → κ → β) (t : ι → Set κ)\n  (g : ι → κ) [inst_1 : DecidablePred fun i => g i ∈ t i],\n  ∏ i ∈ s, (t i).mulIndicator (f i) (g i) = ∏ i ∈ s with g i ∈ t i, f i (g i)","typeReferences":[["MulOneClass","toMulOne"],["Finset"],["Set","mulIndicator"],["MulOne","toOne"],["CommMonoid","toMonoid"],["Set"],["Membership","mem"],["DecidablePred"],["Set","instMembership"],["Finset","prod"],["Monoid","toMulOneClass"],["CommMonoid"],["Eq"],["Finset","filter"]],"valueReferences":[["MulOneClass","toMulOne"],["Finset","instSetLike"],["Finset"],["CommMonoid","toMonoid"],["Eq","trans"],["Membership","mem"],["Iff","mp"],["HMul","hMul"],["Set","mulIndicator_of_notMem"],["MulOne","toMul"],["Monoid","toMulOneClass"],["Eq","symm"],["Finset","filter"],["Not"],["rfl"],["SetLike","instMembership"],["Set","mulIndicator"],["MulOne","toOne"],["Set"],["And","right"],["Set","mulIndicator_of_mem"],["And"],["mul_one"],["OfNat","ofNat"],["Set","instMembership"],["Finset","prod"],["Finset","prod_filter_mul_prod_filter_not"],["One","toOfNat1"],["Finset","mem_filter"],["Finset","prod_congr"],["congr_arg₂"],["instHMul"],["Finset","prod_eq_one"],["instDecidableNot"]]},{"isProp":true,"kind":"theorem","name":["Finset","sum_indicator_subset_of_eq_zero"],"typeFallback":"forall {ι : Type.{u_1}} {α : Type.{u_3}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6 : AddCommMonoid.{u_4} β] [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.9 : Zero.{u_3} α] (f : ι -> α) (g : ι -> α -> β) {s : Finset.{u_1} ι} {t : Finset.{u_1} ι}, (HasSubset.Subset.{u_1} (Finset.{u_1} ι) (Finset.instHasSubset.{u_1} ι) s t) -> (forall (a : ι), Eq.{succ u_4} β (g a (OfNat.ofNat.{u_3} α 0 (Zero.toOfNat0.{u_3} α inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.9))) (OfNat.ofNat.{u_4} β 0 (Zero.toOfNat0.{u_4} β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6))))))) -> (Eq.{succ u_4} β (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6 t (fun (i : ι) => g i (Set.indicator.{u_1, u_3} ι α inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.9 (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) f i))) (Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1503594173._hygCtx._hyg.6 s (fun (i : ι) => g i (f i))))","typeFull":"∀ {ι : Type u_1} {α : Type u_3} {β : Type u_4} [inst : AddCommMonoid β] [inst_1 : Zero α] (f : ι → α) (g : ι → α → β)\n  {s t : Finset ι}, s ⊆ t → (∀ (a : ι), g a 0 = 0) → ∑ i ∈ t, g i ((↑s).indicator f i) = ∑ i ∈ s, g i (f i)","typeReadable":"∀ {ι : Type u_1} {α : Type u_3} {β : Type u_4} [inst : AddCommMonoid β] [inst_1 : Zero α] (f : ι → α) (g : ι → α → β)\n  {s t : Finset ι}, s ⊆ t → (∀ (a : ι), g a 0 = 0) → ∑ i ∈ t, g i ((↑s).indicator f i) = ∑ i ∈ s, g i (f i)","typeReferences":[["Finset","instSetLike"],["Finset"],["Finset","instHasSubset"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["Set","indicator"],["OfNat","ofNat"],["AddCommMonoid"],["SetLike","coe"],["HasSubset","Subset"],["Finset","sum"],["Zero","toOfNat0"],["Zero"],["Eq"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Finset","instSetLike"],["Trans","trans"],["Finset"],["Set","indicator_of_mem"],["Eq","trans"],["Set","indicator_of_notMem"],["Membership","mem"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Finset","sum_congr"],["SetLike","mem_coe"],["not_false_eq_true"],["Finset","sum"],["Zero","toOfNat0"],["congrFun'"],["Eq"],["propext"],["rfl"],["Not"],["instTransEq"],["SetLike","instMembership"],["True"],["Set"],["congr_arg"],["AddZeroClass","toAddZero"],["Set","indicator"],["OfNat","ofNat"],["Set","instMembership"],["eq_self"],["SetLike","coe"],["of_eq_true"],["Eq","refl"],["eq_false"],["Finset","sum_subset"],["id"],["False"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Finset","indicator_biUnion"],"typeFallback":"forall {ι : Type.{u_1}} {κ : Type.{u_2}} {β : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6 : AddCommMonoid.{u_4} β] (s : Finset.{u_1} ι) (t : ι -> (Set.{u_2} κ)) {f : κ -> β}, (Set.PairwiseDisjoint.{u_2, u_1} (Set.{u_2} κ) ι (CompleteSemilatticeInf.toPartialOrder.{u_2} (Set.{u_2} κ) (CompleteLattice.toCompleteSemilatticeInf.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ))))) (HeytingAlgebra.toOrderBot.{u_2} (Set.{u_2} κ) (Order.Frame.toHeytingAlgebra.{u_2} (Set.{u_2} κ) (CompleteDistribLattice.toFrame.{u_2} (Set.{u_2} κ) (CompleteBooleanAlgebra.toCompleteDistribLattice.{u_2} (Set.{u_2} κ) (CompleteAtomicBooleanAlgebra.toCompleteBooleanAlgebra.{u_2} (Set.{u_2} κ) (Set.instCompleteAtomicBooleanAlgebra.{u_2} κ)))))) (SetLike.coe.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι) s) t) -> (Eq.{max (succ u_2) (succ u_4)} (κ -> β) (Set.indicator.{u_2, u_4} κ β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6)))) (Set.iUnion.{u_2, succ u_1} κ ι (fun (i : ι) => Set.iUnion.{u_2, 0} κ (Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) (fun (h._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.37 : Membership.mem.{u_1, u_1} ι (Finset.{u_1} ι) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} ι) ι (Finset.instSetLike.{u_1} ι)) s i) => t i))) f) (fun (a : κ) => Finset.sum.{u_1, u_4} ι β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6 s (fun (i : ι) => Set.indicator.{u_2, u_4} κ β (AddZero.toZero.{u_4} β (AddZeroClass.toAddZero.{u_4} β (AddMonoid.toAddZeroClass.{u_4} β (AddCommMonoid.toAddMonoid.{u_4} β inst._@.Mathlib.Algebra.BigOperators.Group.Finset.Indicator.1224773839._hygCtx._hyg.6)))) (t i) f a)))","typeFull":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → (⋃ i ∈ s, t i).indicator f = fun a => ∑ i ∈ s, (t i).indicator f a","typeReadable":"∀ {ι : Type u_1} {κ : Type u_2} {β : Type u_4} [inst : AddCommMonoid β] (s : Finset ι) (t : ι → Set κ) {f : κ → β},\n  (↑s).PairwiseDisjoint t → (⋃ i ∈ s, t i).indicator f = fun a => ∑ i ∈ s, (t i).indicator f a","typeReferences":[["Finset","instSetLike"],["Finset"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["Membership","mem"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["CompleteBooleanAlgebra","toCompleteLattice"],["Set","iUnion"],["AddCommMonoid","toAddMonoid"],["Order","Frame","toHeytingAlgebra"],["AddCommMonoid"],["CompleteSemilatticeInf","toPartialOrder"],["Set","PairwiseDisjoint"],["Finset","sum"],["HeytingAlgebra","toOrderBot"],["Eq"],["CompleteDistribLattice","toFrame"],["SetLike","instMembership"],["Set"],["AddZeroClass","toAddZero"],["Set","indicator"],["SetLike","coe"],["Set","instCompleteAtomicBooleanAlgebra"],["CompleteLattice","toCompleteSemilatticeInf"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["PartialOrder","toPreorder"],["Finset"],["Eq","trans"],["Set","iUnion_of_empty"],["Inter","inter"],["Membership","mem"],["Iff","mp"],["Classical","propDecidable"],["Set","iUnion"],["CompleteBooleanAlgebra","toCompleteLattice"],["Set","instInsert"],["Set","pairwiseDisjoint_insert_of_notMem"],["Union","union"],["Finset","cons"],["Order","Frame","toHeytingAlgebra"],["Set","disjoint_iUnion₂_right"],["Set","disjoint_iff"],["Disjoint"],["CompleteSemilatticeInf","toPartialOrder"],["funext"],["Finset","sum"],["HeytingAlgebra","toOrderBot"],["Set","instUnion"],["Set","instInter"],["And","left"],["SemilatticeInf","toPartialOrder"],["Finset","set_biUnion_insert"],["SetLike","instMembership"],["Finset","sum_cons"],["And","right"],["And"],["Insert","insert"],["AddZeroClass","toAddZero"],["Set","indicator"],["Set","instMembership"],["Finset","notMem_empty"],["BoundedOrder","toOrderBot"],["SetLike","coe"],["HasSubset","Subset"],["Iff","of_eq"],["Iff","not"],["Set","instCompleteAtomicBooleanAlgebra"],["Iff","mpr"],["eq_false"],["Eq","refl"],["id"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Finset","cons_induction"],["Finset","instSetLike"],["Set","indicator_empty"],["Set","indicator_union_of_notMem_inter"],["Finset","instInsert"],["CompleteAtomicBooleanAlgebra","toCompleteBooleanAlgebra"],["Eq","mp"],["Finset","instEmptyCollection"],["Finset","cons_eq_insert"],["CompleteBooleanAlgebra","toCompleteDistribLattice"],["AddCommMonoid","toAddMonoid"],["EmptyCollection","emptyCollection"],["Set","instEmptyCollection"],["congrArg"],["Finset","mem_coe"],["Finset","coe_cons"],["Set","PairwiseDisjoint"],["Set","iUnion_congr_Prop"],["Zero","toOfNat0"],["congrFun'"],["Preorder","toLE"],["Eq"],["Set","iUnion_empty"],["propext"],["Not"],["CompleteDistribLattice","toFrame"],["Lattice","toSemilatticeInf"],["True"],["instHAdd"],["Set"],["Set","instBoundedOrder"],["Set","instDistribLattice"],["AddZero","toAdd"],["OfNat","ofNat"],["instIsEmptyFalse"],["Set","instHasSubset"],["HAdd","hAdd"],["eq_self"],["DistribLattice","toLattice"],["of_eq_true"],["False"],["CompleteLattice","toCompleteSemilatticeInf"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.CharZero.Infinite.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["CharZero","infinite"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.CharZero.Infinite.1289300613._hygCtx._hyg.3 : AddMonoidWithOne.{u_1} M] [inst._@.Mathlib.Algebra.CharZero.Infinite.1289300613._hygCtx._hyg.6 : CharZero.{u_1} M inst._@.Mathlib.Algebra.CharZero.Infinite.1289300613._hygCtx._hyg.3], Infinite.{succ u_1} M","typeFull":"∀ (M : Type u_1) [inst : AddMonoidWithOne M] [CharZero M], Infinite M","typeReadable":"∀ (M : Type u_1) [inst : AddMonoidWithOne M] [CharZero M], Infinite M","typeReferences":[["CharZero"],["Infinite"],["AddMonoidWithOne"]],"valueReferences":[["Nat"],["Nat","cast_injective"],["AddMonoidWithOne","toNatCast"],["Nat","cast"],["Infinite","of_injective"],["instInfiniteNat"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.ContinuedFractions.Computation.Basic.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","mk","sizeOf_spec"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3317515927._hygCtx._hyg.13 : SizeOf.{succ u_1} K] (b : Int) (fr : K), Eq.{1} Nat (SizeOf.sizeOf.{succ u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair._sizeOf_inst.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3317515927._hygCtx._hyg.13) (GenContFract.IntFractPair.mk.{u_1} K b fr)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (SizeOf.sizeOf.{1} Int Int._sizeOf_inst b)) (SizeOf.sizeOf.{succ u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3317515927._hygCtx._hyg.13 fr))","typeFull":"∀ {K : Type u_1} [inst : SizeOf K] (b : ℤ) (fr : K), sizeOf { b := b, fr := fr } = 1 + sizeOf b + sizeOf fr","typeReadable":"∀ {K : Type u_1} [inst : SizeOf K] (b : ℤ) (fr : K), sizeOf { b := b, fr := fr } = 1 + sizeOf b + sizeOf fr","typeReferences":[["instAddNat"],["instHAdd"],["GenContFract","IntFractPair","mk"],["SizeOf"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["Int"],["Int","_sizeOf_inst"],["HAdd","hAdd"],["Nat"],["SizeOf","sizeOf"],["instOfNatNat"],["GenContFract","IntFractPair","_sizeOf_inst"],["Eq"]],"valueReferences":[["Int","_sizeOf_inst"],["instAddNat"],["HAdd","hAdd"],["SizeOf","sizeOf"],["Nat"],["instOfNatNat"],["instHAdd"],["Eq","refl"],["OfNat","ofNat"],["Int"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","stream","_unsafe_rec"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6], K -> (Stream'.{u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)))","typeFull":"{K : Type u_1} →\n  [inst : DivisionRing K] →\n    [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream' (Option (GenContFract.IntFractPair K))","typeReadable":"{K : Type u_1} →\n  [inst : DivisionRing K] →\n    [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream' (Option (GenContFract.IntFractPair K))","typeReferences":[["Option"],["DivisionRing","toRing"],["LinearOrder"],["Stream'"],["DivisionRing"],["GenContFract","IntFractPair"],["FloorRing"]],"valueReferences":[["Ring","toNonAssocRing"],["Option","some"],["GenContFract","IntFractPair","of"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["GenContFract","IntFractPair","stream","match_1"],["DivisionRing","toDivisionSemiring"],["Zero","toOfNat0"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Inv","inv"],["InvOneClass","toInv"],["ite"],["DivisionRing","toRing"],["Option","bind"],["DivisionSemiring","toGroupWithZero"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["LinearOrder","toDecidableEq"],["GenContFract","IntFractPair","stream","_unsafe_rec"],["Option","none"],["DivInvOneMonoid","toInvOneClass"],["MulZeroClass","toZero"],["Option"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["GenContFract","IntFractPair","fr"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","mk","_flat_ctor"],"typeFallback":"forall {K : Type.{u_1}}, Int -> K -> (GenContFract.IntFractPair.{u_1} K)","typeFull":"{K : Type u_1} → ℤ → K → GenContFract.IntFractPair K","typeReadable":"{K : Type u_1} → ℤ → K → GenContFract.IntFractPair K","typeReferences":[["GenContFract","IntFractPair"],["Int"]],"valueReferences":[["GenContFract","IntFractPair","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","instRepr"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.437227469._hygCtx._hyg.3 : Repr.{u_1} K], Repr.{u_1} (GenContFract.IntFractPair.{u_1} K)","typeFull":"{K : Type u_1} → [Repr K] → Repr (GenContFract.IntFractPair K)","typeReadable":"{K : Type u_1} → [Repr K] → Repr (GenContFract.IntFractPair K)","typeReferences":[["Repr"],["GenContFract","IntFractPair"]],"valueReferences":[["repr"],["instHAppendOfAppend"],["HAppend","hAppend"],["instReprInt"],["Std","Format","text"],["Std","Format"],["Std","Format","instAppend"],["Repr","mk"],["GenContFract","IntFractPair","fr"],["GenContFract","IntFractPair","b"],["GenContFract","IntFractPair"],["Int"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","stream","_sunfold"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6], K -> (Stream'.{u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)))","typeFull":"{K : Type u_1} →\n  [inst : DivisionRing K] →\n    [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream' (Option (GenContFract.IntFractPair K))","typeReadable":"{K : Type u_1} →\n  [inst : DivisionRing K] →\n    [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream' (Option (GenContFract.IntFractPair K))","typeReferences":[["Option"],["DivisionRing","toRing"],["LinearOrder"],["Stream'"],["DivisionRing"],["GenContFract","IntFractPair"],["FloorRing"]],"valueReferences":[["Ring","toNonAssocRing"],["GenContFract","IntFractPair","stream"],["Option","some"],["GenContFract","IntFractPair","of"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["GenContFract","IntFractPair","stream","match_1"],["DivisionRing","toDivisionSemiring"],["Zero","toOfNat0"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Inv","inv"],["InvOneClass","toInv"],["ite"],["DivisionRing","toRing"],["Option","bind"],["DivisionSemiring","toGroupWithZero"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["LinearOrder","toDecidableEq"],["Option","none"],["DivInvOneMonoid","toInvOneClass"],["MulZeroClass","toZero"],["Option"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["GenContFract","IntFractPair","fr"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","noConfusion"],"typeFallback":"forall {P : Sort.{u}} {K : Type.{u_1}} {t : GenContFract.IntFractPair.{u_1} K} {K' : Type.{u_1}} {t' : GenContFract.IntFractPair.{u_1} K'}, (Eq.{succ (succ u_1)} Type.{u_1} K K') -> (HEq.{succ u_1} (GenContFract.IntFractPair.{u_1} K) t (GenContFract.IntFractPair.{u_1} K') t') -> (GenContFract.IntFractPair.noConfusionType.{u, u_1} P K t K' t')","typeFull":"{P : Sort u} →\n  {K : Type u_1} →\n    {t : GenContFract.IntFractPair K} →\n      {K' : Type u_1} →\n        {t' : GenContFract.IntFractPair K'} → K = K' → t ≍ t' → GenContFract.IntFractPair.noConfusionType P t t'","typeReadable":"{P : Sort u} →\n  {K : Type u_1} →\n    {t : GenContFract.IntFractPair K} →\n      {K' : Type u_1} →\n        {t' : GenContFract.IntFractPair K'} → K = K' → t ≍ t' → GenContFract.IntFractPair.noConfusionType P t t'","typeReferences":[["GenContFract","IntFractPair","noConfusionType"],["HEq"],["Eq"],["GenContFract","IntFractPair"]],"valueReferences":[["HEq","refl"],["Eq","refl"],["eq_of_heq"],["GenContFract","IntFractPair","noConfusionType"],["HEq"],["Eq","ndrec"],["GenContFract","IntFractPair"],["Int"],["GenContFract","IntFractPair","casesOn"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","fr"],"typeFallback":"forall {K : Type.{u_1}}, (GenContFract.IntFractPair.{u_1} K) -> K","typeFull":"{K : Type u_1} → GenContFract.IntFractPair K → K","typeReadable":"{K : Type u_1} → GenContFract.IntFractPair K → K","typeReferences":[["GenContFract","IntFractPair"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","stream"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6], K -> (Stream'.{u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)))","typeFull":"{K : Type u_1} →\n  [inst : DivisionRing K] →\n    [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream' (Option (GenContFract.IntFractPair K))","typeReadable":"{K : Type u_1} →\n  [inst : DivisionRing K] →\n    [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream' (Option (GenContFract.IntFractPair K))","typeReferences":[["Option"],["DivisionRing","toRing"],["LinearOrder"],["Stream'"],["DivisionRing"],["GenContFract","IntFractPair"],["FloorRing"]],"valueReferences":[["Ring","toNonAssocRing"],["Option","some"],["GenContFract","IntFractPair","of"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["GenContFract","IntFractPair","stream","match_1"],["DivisionRing","toDivisionSemiring"],["Zero","toOfNat0"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Inv","inv"],["InvOneClass","toInv"],["ite"],["DivisionRing","toRing"],["Option","bind"],["DivisionSemiring","toGroupWithZero"],["Nat","brecOn"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["LinearOrder","toDecidableEq"],["Option","none"],["DivInvOneMonoid","toInvOneClass"],["MulZeroClass","toZero"],["Option"],["Nat","below"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["GenContFract","IntFractPair","fr"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","ContinuedFractions","Computation","Basic",0,"GenContFract","IntFractPair","stream","match_1","eq_2"],"typeFallback":"forall (motive : Nat -> Sort.{u_1}) (n : Nat) (h_1 : Unit -> (motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) (h_2 : forall (n : Nat), motive (Nat.succ n)), Eq.{u_1} (motive (Nat.succ n)) (GenContFract.IntFractPair.stream.match_1.{u_1} motive (Nat.succ n) h_1 h_2) (h_2 n)","typeFull":"∀ (motive : ℕ → Sort u_1) (n : ℕ) (h_1 : Unit → motive 0) (h_2 : (n : ℕ) → motive n.succ),\n  (match n.succ with\n    | 0 => h_1 ()\n    | n.succ => h_2 n) =\n    h_2 n","typeReadable":"∀ (motive : ℕ → Sort u_1) (n : ℕ) (h_1 : Unit → motive 0) (h_2 : (n : ℕ) → motive n.succ),\n  (match n.succ with\n    | 0 => h_1 ()\n    | n.succ => h_2 n) =\n    h_2 n","typeReferences":[["Nat"],["GenContFract","IntFractPair","stream","match_1"],["Nat","succ"],["instOfNatNat"],["Eq"],["OfNat","ofNat"],["Unit"]],"valueReferences":[["GenContFract","IntFractPair","stream","match_1"],["Nat","succ"],["Eq","refl"]]},{"isProp":false,"kind":"definition","name":["_private","Mathlib","Algebra","ContinuedFractions","Computation","Basic",0,"GenContFract","IntFractPair","stream","match_1","splitter"],"typeFallback":"forall (motive : Nat -> Sort.{u_1}) (x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx.21.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.33 : Nat), (Unit -> (motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (forall (n : Nat), motive (Nat.succ n)) -> (motive x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx.21.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.33)","typeFull":"(motive : ℕ → Sort u_1) → (x : ℕ) → (Unit → motive 0) → ((n : ℕ) → motive n.succ) → motive x","typeReadable":"(motive : ℕ → Sort u_1) → (x : ℕ) → (Unit → motive 0) → ((n : ℕ) → motive n.succ) → motive x","typeReferences":[["Nat"],["Nat","succ"],["instOfNatNat"],["OfNat","ofNat"],["Unit"]],"valueReferences":[["GenContFract","IntFractPair","stream","match_1"]]},{"isProp":false,"kind":"constructor","name":["GenContFract","IntFractPair","mk"],"typeFallback":"forall {K : Type.{u_1}}, Int -> K -> (GenContFract.IntFractPair.{u_1} K)","typeFull":"{K : Type u_1} → ℤ → K → GenContFract.IntFractPair K","typeReadable":"{K : Type u_1} → ℤ → K → GenContFract.IntFractPair K","typeReferences":[["GenContFract","IntFractPair"],["Int"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","stream","eq_2"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6] (v : K) (n : Nat), Eq.{succ u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)) (GenContFract.IntFractPair.stream.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 v (Nat.succ n)) (Option.bind.{u_1, u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.stream.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 v n) (fun (ap_n : GenContFract.IntFractPair.{u_1} K) => ite.{succ u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)) (Eq.{succ u_1} K (GenContFract.IntFractPair.fr.{u_1} K ap_n) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonAssocRing.toNonUnitalNonAssocRing.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3))))))))) (LinearOrder.toDecidableEq.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 (GenContFract.IntFractPair.fr.{u_1} K ap_n) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonAssocRing.toNonUnitalNonAssocRing.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3))))))))) (Option.none.{u_1} (GenContFract.IntFractPair.{u_1} K)) (Option.some.{u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.of.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (GroupWithZero.toDivisionMonoid.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (DivisionRing.toDivisionSemiring.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3)))))) (GenContFract.IntFractPair.fr.{u_1} K ap_n))))))","typeFull":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K) (n : ℕ),\n  GenContFract.IntFractPair.stream v n.succ =\n    (GenContFract.IntFractPair.stream v n).bind fun ap_n =>\n      if ap_n.fr = 0 then none else some (GenContFract.IntFractPair.of ap_n.fr⁻¹)","typeReadable":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K) (n : ℕ),\n  GenContFract.IntFractPair.stream v n.succ =\n    (GenContFract.IntFractPair.stream v n).bind fun ap_n =>\n      if ap_n.fr = 0 then none else some (GenContFract.IntFractPair.of ap_n.fr⁻¹)","typeReferences":[["Ring","toNonAssocRing"],["GenContFract","IntFractPair","stream"],["Option","some"],["GenContFract","IntFractPair","of"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["DivisionRing","toDivisionSemiring"],["Zero","toOfNat0"],["Eq"],["DivisionRing"],["NonAssocRing","toNonUnitalNonAssocRing"],["Inv","inv"],["InvOneClass","toInv"],["ite"],["DivisionRing","toRing"],["LinearOrder"],["Option","bind"],["DivisionSemiring","toGroupWithZero"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["Nat"],["LinearOrder","toDecidableEq"],["DivInvOneMonoid","toInvOneClass"],["Option","none"],["Nat","succ"],["MulZeroClass","toZero"],["Option"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["GenContFract","IntFractPair","fr"],["FloorRing"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Nat","succ"],["Option"],["Eq","refl"],["GenContFract","IntFractPair","stream"],["GenContFract","IntFractPair"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","stream","match_1"],"typeFallback":"forall (motive : Nat -> Sort.{u_1}) (x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx.21.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.33 : Nat), (Unit -> (motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) -> (forall (n : Nat), motive (Nat.succ n)) -> (motive x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx.21.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.33)","typeFull":"(motive : ℕ → Sort u_1) → (x : ℕ) → (Unit → motive 0) → ((n : ℕ) → motive n.succ) → motive x","typeReadable":"(motive : ℕ → Sort u_1) → (x : ℕ) → (Unit → motive 0) → ((n : ℕ) → motive n.succ) → motive x","typeReferences":[["Nat"],["Nat","succ"],["instOfNatNat"],["OfNat","ofNat"],["Unit"]],"valueReferences":[["Nat","casesOn"],["Unit","unit"]]},{"isProp":false,"kind":"inductive","name":["GenContFract","IntFractPair"],"typeFallback":"Type.{u_1} -> Type.{u_1}","typeFull":"Type u_1 → Type u_1","typeReadable":"Type u_1 → Type u_1","typeReferences":[],"valueReferences":null},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","ctorIdx"],"typeFallback":"forall {K : Type.{u_1}}, (GenContFract.IntFractPair.{u_1} K) -> Nat","typeFull":"{K : Type u_1} → GenContFract.IntFractPair K → ℕ","typeReadable":"{K : Type u_1} → GenContFract.IntFractPair K → ℕ","typeReferences":[["Nat"],["GenContFract","IntFractPair"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["GenContFract","of","match_1"],"typeFallback":"forall {K : Type.{u_1}} (motive : (Stream'.Seq1.{u_1} (GenContFract.IntFractPair.{u_1} K)) -> Sort.{u_2}) (x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723276._hygCtx._hyg.26 : Stream'.Seq1.{u_1} (GenContFract.IntFractPair.{u_1} K)), (forall (h : GenContFract.IntFractPair.{u_1} K) (s : Stream'.Seq.{u_1} (GenContFract.IntFractPair.{u_1} K)), motive (Prod.mk.{u_1, u_1} (GenContFract.IntFractPair.{u_1} K) (Stream'.Seq.{u_1} (GenContFract.IntFractPair.{u_1} K)) h s)) -> (motive x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723276._hygCtx._hyg.26)","typeFull":"{K : Type u_1} →\n  (motive : Stream'.Seq1 (GenContFract.IntFractPair K) → Sort u_2) →\n    (x : Stream'.Seq1 (GenContFract.IntFractPair K)) →\n      ((h : GenContFract.IntFractPair K) → (s : Stream'.Seq (GenContFract.IntFractPair K)) → motive (h, s)) → motive x","typeReadable":"{K : Type u_1} →\n  (motive : Stream'.Seq1 (GenContFract.IntFractPair K) → Sort u_2) →\n    (x : Stream'.Seq1 (GenContFract.IntFractPair K)) →\n      ((h : GenContFract.IntFractPair K) → (s : Stream'.Seq (GenContFract.IntFractPair K)) → motive (h, s)) → motive x","typeReferences":[["Stream'","Seq"],["Stream'","Seq1"],["Prod","mk"],["GenContFract","IntFractPair"]],"valueReferences":[["Prod","casesOn"],["Stream'","Seq"],["GenContFract","IntFractPair"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","mk","injEq"],"typeFallback":"forall {K : Type.{u_1}} (b : Int) (fr : K) (b_1 : Int) (fr_1 : K), Eq.{1} Prop (Eq.{succ u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.mk.{u_1} K b fr) (GenContFract.IntFractPair.mk.{u_1} K b_1 fr_1)) (And (Eq.{1} Int b b_1) (Eq.{succ u_1} K fr fr_1))","typeFull":"∀ {K : Type u_1} (b : ℤ) (fr : K) (b_1 : ℤ) (fr_1 : K),\n  ({ b := b, fr := fr } = { b := b_1, fr := fr_1 }) = (b = b_1 ∧ fr = fr_1)","typeReadable":"∀ {K : Type u_1} (b : ℤ) (fr : K) (b_1 : ℤ) (fr_1 : K),\n  ({ b := b, fr := fr } = { b := b_1, fr := fr_1 }) = (b = b_1 ∧ fr = fr_1)","typeReferences":[["GenContFract","IntFractPair","mk"],["And"],["Eq"],["GenContFract","IntFractPair"],["Int"]],"valueReferences":[["Eq","propIntro"],["Eq","refl"],["GenContFract","IntFractPair","mk"],["And"],["Eq","ndrec"],["Lean","injEq_helper"],["Eq"],["GenContFract","IntFractPair"],["GenContFract","IntFractPair","mk","inj"],["Int"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","coeFn"],"typeFallback":"forall {K : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1174359230._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} K β], (GenContFract.IntFractPair.{u_1} K) -> (GenContFract.IntFractPair.{u_2} β)","typeFull":"{K : Type u_1} → {β : Type u_2} → [Coe K β] → GenContFract.IntFractPair K → GenContFract.IntFractPair β","typeReadable":"{K : Type u_1} → {β : Type u_2} → [Coe K β] → GenContFract.IntFractPair K → GenContFract.IntFractPair β","typeReferences":[["Coe"],["GenContFract","IntFractPair"]],"valueReferences":[["Coe","coe"],["GenContFract","IntFractPair","mapFr"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","mapFr"],"typeFallback":"forall {K : Type.{u_1}} {β : Type.{u_2}}, (K -> β) -> (GenContFract.IntFractPair.{u_1} K) -> (GenContFract.IntFractPair.{u_2} β)","typeFull":"{K : Type u_1} → {β : Type u_2} → (K → β) → GenContFract.IntFractPair K → GenContFract.IntFractPair β","typeReadable":"{K : Type u_1} → {β : Type u_2} → (K → β) → GenContFract.IntFractPair K → GenContFract.IntFractPair β","typeReferences":[["GenContFract","IntFractPair"]],"valueReferences":[["GenContFract","IntFractPair","mk"],["GenContFract","IntFractPair","fr"],["GenContFract","IntFractPair","b"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","ContinuedFractions","Computation","Basic",0,"GenContFract","IntFractPair","stream","match_1","eq_1"],"typeFallback":"forall (motive : Nat -> Sort.{u_1}) (h_1 : Unit -> (motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) (h_2 : forall (n : Nat), motive (Nat.succ n)), Eq.{u_1} (motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (GenContFract.IntFractPair.stream.match_1.{u_1} motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) h_1 h_2) (h_1 Unit.unit)","typeFull":"∀ (motive : ℕ → Sort u_1) (h_1 : Unit → motive 0) (h_2 : (n : ℕ) → motive n.succ),\n  (match 0 with\n    | 0 => h_1 ()\n    | n.succ => h_2 n) =\n    h_1 ()","typeReadable":"∀ (motive : ℕ → Sort u_1) (h_1 : Unit → motive 0) (h_2 : (n : ℕ) → motive n.succ),\n  (match 0 with\n    | 0 => h_1 ()\n    | n.succ => h_2 n) =\n    h_1 ()","typeReferences":[["Nat"],["GenContFract","IntFractPair","stream","match_1"],["Nat","succ"],["instOfNatNat"],["Eq"],["Unit","unit"],["OfNat","ofNat"],["Unit"]],"valueReferences":[["Nat"],["GenContFract","IntFractPair","stream","match_1"],["instOfNatNat"],["Eq","refl"],["OfNat","ofNat"]]},{"isProp":false,"kind":"recursor","name":["GenContFract","IntFractPair","rec"],"typeFallback":"forall {K : Type.{u_1}} {motive : (GenContFract.IntFractPair.{u_1} K) -> Sort.{u}}, (forall (b : Int) (fr : K), motive (GenContFract.IntFractPair.mk.{u_1} K b fr)) -> (forall (t : GenContFract.IntFractPair.{u_1} K), motive t)","typeFull":"{K : Type u_1} →\n  {motive : GenContFract.IntFractPair K → Sort u} →\n    ((b : ℤ) → (fr : K) → motive { b := b, fr := fr }) → (t : GenContFract.IntFractPair K) → motive t","typeReadable":"{K : Type u_1} →\n  {motive : GenContFract.IntFractPair K → Sort u} →\n    ((b : ℤ) → (fr : K) → motive { b := b, fr := fr }) → (t : GenContFract.IntFractPair K) → motive t","typeReferences":[["GenContFract","IntFractPair","mk"],["GenContFract","IntFractPair"],["Int"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","coe_to_intFractPair"],"typeFallback":"forall {K : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3633255818._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} K β] {b : Int} {fr : K}, Eq.{succ u_2} (GenContFract.IntFractPair.{u_2} β) (GenContFract.IntFractPair.coeFn.{u_1, u_2} K β inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3633255818._hygCtx._hyg.4 (GenContFract.IntFractPair.mk.{u_1} K b fr)) (GenContFract.IntFractPair.mk.{u_2} β b (Coe.coe.{succ u_1, succ u_2} K β inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3633255818._hygCtx._hyg.4 fr))","typeFull":"∀ {K : Type u_1} {β : Type u_2} [inst : Coe K β] {b : ℤ} {fr : K}, ↑{ b := b, fr := fr } = { b := b, fr := Coe.coe fr }","typeReadable":"∀ {K : Type u_1} {β : Type u_2} [inst : Coe K β] {b : ℤ} {fr : K}, ↑{ b := b, fr := fr } = { b := b, fr := Coe.coe fr }","typeReferences":[["GenContFract","IntFractPair","mk"],["Coe"],["Coe","coe"],["GenContFract","IntFractPair","coeFn"],["Eq"],["GenContFract","IntFractPair"],["Int"]],"valueReferences":[["rfl"],["GenContFract","IntFractPair","mk"],["GenContFract","IntFractPair","coeFn"],["GenContFract","IntFractPair"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","stream","eq_def"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6] (v : K) (x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.21 : Nat), Eq.{succ u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)) (GenContFract.IntFractPair.stream.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 v x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.21) (GenContFract.IntFractPair.stream.match_1.{succ u_1} (fun (x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx.21.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.33 : Nat) => Option.{u_1} (GenContFract.IntFractPair.{u_1} K)) x._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.21 (fun (_ : Unit) => Option.some.{u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.of.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 v)) (fun (n : Nat) => Option.bind.{u_1, u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.stream.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 v n) (fun (ap_n : GenContFract.IntFractPair.{u_1} K) => ite.{succ u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)) (Eq.{succ u_1} K (GenContFract.IntFractPair.fr.{u_1} K ap_n) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonAssocRing.toNonUnitalNonAssocRing.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3))))))))) (LinearOrder.toDecidableEq.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 (GenContFract.IntFractPair.fr.{u_1} K ap_n) (OfNat.ofNat.{u_1} K 0 (Zero.toOfNat0.{u_1} K (MulZeroClass.toZero.{u_1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} K (NonAssocRing.toNonUnitalNonAssocRing.{u_1} K (Ring.toNonAssocRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3))))))))) (Option.none.{u_1} (GenContFract.IntFractPair.{u_1} K)) (Option.some.{u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.of.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 (Inv.inv.{u_1} K (InvOneClass.toInv.{u_1} K (DivInvOneMonoid.toInvOneClass.{u_1} K (DivisionMonoid.toDivInvOneMonoid.{u_1} K (GroupWithZero.toDivisionMonoid.{u_1} K (DivisionSemiring.toGroupWithZero.{u_1} K (DivisionRing.toDivisionSemiring.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3)))))) (GenContFract.IntFractPair.fr.{u_1} K ap_n)))))))","typeFull":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K) (x : ℕ),\n  GenContFract.IntFractPair.stream v x =\n    match x with\n    | 0 => some (GenContFract.IntFractPair.of v)\n    | n.succ =>\n      (GenContFract.IntFractPair.stream v n).bind fun ap_n =>\n        if ap_n.fr = 0 then none else some (GenContFract.IntFractPair.of ap_n.fr⁻¹)","typeReadable":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K) (x : ℕ),\n  GenContFract.IntFractPair.stream v x =\n    match x with\n    | 0 => some (GenContFract.IntFractPair.of v)\n    | n.succ =>\n      (GenContFract.IntFractPair.stream v n).bind fun ap_n =>\n        if ap_n.fr = 0 then none else some (GenContFract.IntFractPair.of ap_n.fr⁻¹)","typeReferences":[["Ring","toNonAssocRing"],["GenContFract","IntFractPair","stream"],["Option","some"],["GenContFract","IntFractPair","of"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["GenContFract","IntFractPair","stream","match_1"],["DivisionRing","toDivisionSemiring"],["Zero","toOfNat0"],["Eq"],["DivisionRing"],["NonAssocRing","toNonUnitalNonAssocRing"],["Inv","inv"],["InvOneClass","toInv"],["ite"],["DivisionRing","toRing"],["LinearOrder"],["Option","bind"],["DivisionSemiring","toGroupWithZero"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["Nat"],["LinearOrder","toDecidableEq"],["DivInvOneMonoid","toInvOneClass"],["Option","none"],["MulZeroClass","toZero"],["Option"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["GenContFract","IntFractPair","fr"],["FloorRing"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]],"valueReferences":[["Ring","toNonAssocRing"],["GenContFract","IntFractPair","stream"],["Option","some"],["GenContFract","IntFractPair","of"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["PUnit"],["GenContFract","IntFractPair","stream","match_1"],["instOfNatNat"],["DivisionRing","toDivisionSemiring"],["_private","Mathlib","Algebra","ContinuedFractions","Computation","Basic",0,"GenContFract","IntFractPair","stream","match_1","splitter"],["Eq","symm"],["Zero","toOfNat0"],["Eq","ndrec"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Inv","inv"],["InvOneClass","toInv"],["ite"],["Nat","brecOn","eq"],["DivisionRing","toRing"],["PProd"],["Option","bind"],["Nat","brecOn"],["DivisionSemiring","toGroupWithZero"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["Nat"],["LinearOrder","toDecidableEq"],["Option","none"],["DivInvOneMonoid","toInvOneClass"],["Nat","brecOn","go"],["Nat","succ"],["MulZeroClass","toZero"],["Eq","refl"],["Nat","rec"],["Option"],["Nat","below"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["id"],["GenContFract","IntFractPair","fr"],["Eq","mpr"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","_sizeOf_inst"],"typeFallback":"forall (K : Type.{u_1}) [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3317515927._hygCtx._hyg.12 : SizeOf.{succ u_1} K], SizeOf.{succ u_1} (GenContFract.IntFractPair.{u_1} K)","typeFull":"(K : Type u_1) → [SizeOf K] → SizeOf (GenContFract.IntFractPair K)","typeReadable":"(K : Type u_1) → [SizeOf K] → SizeOf (GenContFract.IntFractPair K)","typeReferences":[["GenContFract","IntFractPair"],["SizeOf"]],"valueReferences":[["GenContFract","IntFractPair","_sizeOf_1"],["GenContFract","IntFractPair"],["SizeOf","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","mk","noConfusion"],"typeFallback":"forall {K : Type.{u_1}} {P : Sort.{u}} {b : Int} {fr : K} {b' : Int} {fr' : K}, (Eq.{succ u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.mk.{u_1} K b fr) (GenContFract.IntFractPair.mk.{u_1} K b' fr')) -> ((Eq.{1} Int b b') -> (HEq.{succ u_1} K fr K fr') -> P) -> P","typeFull":"{K : Type u_1} →\n  {P : Sort u} →\n    {b : ℤ} →\n      {fr : K} → {b' : ℤ} → {fr' : K} → { b := b, fr := fr } = { b := b', fr := fr' } → (b = b' → fr ≍ fr' → P) → P","typeReadable":"{K : Type u_1} →\n  {P : Sort u} →\n    {b : ℤ} →\n      {fr : K} → {b' : ℤ} → {fr' : K} → { b := b, fr := fr } = { b := b', fr := fr' } → (b = b' → fr ≍ fr' → P) → P","typeReferences":[["GenContFract","IntFractPair","mk"],["HEq"],["Eq"],["GenContFract","IntFractPair"],["Int"]],"valueReferences":[["Eq","refl"],["GenContFract","IntFractPair","mk"],["id"],["heq_of_eq"],["GenContFract","IntFractPair"],["GenContFract","IntFractPair","noConfusion"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","coe"],"typeFallback":"forall {K : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3397239003._hygCtx._hyg.4 : Coe.{succ u_1, succ u_2} K β], Coe.{succ u_1, succ u_2} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.{u_2} β)","typeFull":"{K : Type u_1} → {β : Type u_2} → [Coe K β] → Coe (GenContFract.IntFractPair K) (GenContFract.IntFractPair β)","typeReadable":"{K : Type u_1} → {β : Type u_2} → [Coe K β] → Coe (GenContFract.IntFractPair K) (GenContFract.IntFractPair β)","typeReferences":[["Coe"],["GenContFract","IntFractPair"]],"valueReferences":[["Coe","mk"],["GenContFract","IntFractPair","coeFn"],["GenContFract","IntFractPair"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","inhabited"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.2802996212._hygCtx._hyg.3 : Inhabited.{succ u_1} K], Inhabited.{succ u_1} (GenContFract.IntFractPair.{u_1} K)","typeFull":"{K : Type u_1} → [Inhabited K] → Inhabited (GenContFract.IntFractPair K)","typeReadable":"{K : Type u_1} → [Inhabited K] → Inhabited (GenContFract.IntFractPair K)","typeReferences":[["Inhabited"],["GenContFract","IntFractPair"]],"valueReferences":[["Inhabited","mk"],["instOfNat"],["GenContFract","IntFractPair","mk"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["Inhabited","default"],["Int"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","of"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723275._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723275._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723275._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723275._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723275._hygCtx._hyg.6], K -> (GenContFract.IntFractPair.{u_1} K)","typeFull":"{K : Type u_1} → [inst : DivisionRing K] → [inst_1 : LinearOrder K] → [FloorRing K] → K → GenContFract.IntFractPair K","typeReadable":"{K : Type u_1} → [inst : DivisionRing K] → [inst_1 : LinearOrder K] → [FloorRing K] → K → GenContFract.IntFractPair K","typeReferences":[["DivisionRing","toRing"],["LinearOrder"],["DivisionRing"],["GenContFract","IntFractPair"],["FloorRing"]],"valueReferences":[["GenContFract","IntFractPair","mk"],["DivisionRing","toRing"],["Int","floor"],["Int","fract"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","_sizeOf_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.3317515927._hygCtx._hyg.11 : SizeOf.{succ u_1} K], (GenContFract.IntFractPair.{u_1} K) -> Nat","typeFull":"{K : Type u_1} → [SizeOf K] → GenContFract.IntFractPair K → ℕ","typeReadable":"{K : Type u_1} → [SizeOf K] → GenContFract.IntFractPair K → ℕ","typeReferences":[["Nat"],["GenContFract","IntFractPair"],["SizeOf"]],"valueReferences":[["Int","_sizeOf_inst"],["instAddNat"],["HAdd","hAdd"],["GenContFract","IntFractPair","rec"],["SizeOf","sizeOf"],["Nat"],["instOfNatNat"],["instHAdd"],["OfNat","ofNat"],["Int"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","recOn"],"typeFallback":"forall {K : Type.{u_1}} {motive : (GenContFract.IntFractPair.{u_1} K) -> Sort.{u}} (t : GenContFract.IntFractPair.{u_1} K), (forall (b : Int) (fr : K), motive (GenContFract.IntFractPair.mk.{u_1} K b fr)) -> (motive t)","typeFull":"{K : Type u_1} →\n  {motive : GenContFract.IntFractPair K → Sort u} →\n    (t : GenContFract.IntFractPair K) → ((b : ℤ) → (fr : K) → motive { b := b, fr := fr }) → motive t","typeReadable":"{K : Type u_1} →\n  {motive : GenContFract.IntFractPair K → Sort u} →\n    (t : GenContFract.IntFractPair K) → ((b : ℤ) → (fr : K) → motive { b := b, fr := fr }) → motive t","typeReferences":[["GenContFract","IntFractPair","mk"],["GenContFract","IntFractPair"],["Int"]],"valueReferences":[["GenContFract","IntFractPair","rec"]]},{"isProp":false,"kind":"definition","name":["GenContFract","of"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723276._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723276._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723276._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723276._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.432723276._hygCtx._hyg.6], K -> (GenContFract.{u_1} K)","typeFull":"{K : Type u_1} → [inst : DivisionRing K] → [inst_1 : LinearOrder K] → [FloorRing K] → K → GenContFract K","typeReadable":"{K : Type u_1} → [inst : DivisionRing K] → [inst_1 : LinearOrder K] → [FloorRing K] → K → GenContFract K","typeReferences":[["DivisionRing","toRing"],["LinearOrder"],["GenContFract"],["DivisionRing"],["FloorRing"]],"valueReferences":[["DivisionRing","toRing"],["AddGroupWithOne","toAddMonoidWithOne"],["GenContFract","IntFractPair","b"],["GenContFract","of","match_1"],["GenContFract"],["GenContFract","mk"],["Int","cast"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["One","toOfNat1"],["Ring","toAddGroupWithOne"],["GenContFract","IntFractPair","seq1"],["AddMonoidWithOne","toOne"],["AddGroupWithOne","toIntCast"],["GenContFract","Pair"],["Stream'","Seq","map"],["GenContFract","Pair","mk"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","casesOn"],"typeFallback":"forall {K : Type.{u_1}} {motive : (GenContFract.IntFractPair.{u_1} K) -> Sort.{u}} (t : GenContFract.IntFractPair.{u_1} K), (forall (b : Int) (fr : K), motive (GenContFract.IntFractPair.mk.{u_1} K b fr)) -> (motive t)","typeFull":"{K : Type u_1} →\n  {motive : GenContFract.IntFractPair K → Sort u} →\n    (t : GenContFract.IntFractPair K) → ((b : ℤ) → (fr : K) → motive { b := b, fr := fr }) → motive t","typeReadable":"{K : Type u_1} →\n  {motive : GenContFract.IntFractPair K → Sort u} →\n    (t : GenContFract.IntFractPair K) → ((b : ℤ) → (fr : K) → motive { b := b, fr := fr }) → motive t","typeReferences":[["GenContFract","IntFractPair","mk"],["GenContFract","IntFractPair"],["Int"]],"valueReferences":[["GenContFract","IntFractPair","rec"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","stream","eq_1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6] (v : K), Eq.{succ u_1} (Option.{u_1} (GenContFract.IntFractPair.{u_1} K)) (GenContFract.IntFractPair.stream.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 v (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Option.some.{u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.of.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.4121521724._hygCtx._hyg.9 v))","typeFull":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K),\n  GenContFract.IntFractPair.stream v 0 = some (GenContFract.IntFractPair.of v)","typeReadable":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K),\n  GenContFract.IntFractPair.stream v 0 = some (GenContFract.IntFractPair.of v)","typeReferences":[["DivisionRing","toRing"],["GenContFract","IntFractPair","stream"],["LinearOrder"],["Option","some"],["GenContFract","IntFractPair"],["GenContFract","IntFractPair","of"],["OfNat","ofNat"],["Nat"],["instOfNatNat"],["Option"],["DivisionRing"],["Eq"],["FloorRing"]],"valueReferences":[["Nat"],["instOfNatNat"],["Option"],["Eq","refl"],["GenContFract","IntFractPair","stream"],["GenContFract","IntFractPair"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","b"],"typeFallback":"forall {K : Type.{u_1}}, (GenContFract.IntFractPair.{u_1} K) -> Int","typeFull":"{K : Type u_1} → GenContFract.IntFractPair K → ℤ","typeReadable":"{K : Type u_1} → GenContFract.IntFractPair K → ℤ","typeReferences":[["GenContFract","IntFractPair"],["Int"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","stream_isSeq"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.6] (v : K), Stream'.IsSeq.{u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.stream.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.3 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.6 inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.1083217005._hygCtx._hyg.9 v)","typeFull":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K),\n  (GenContFract.IntFractPair.stream v).IsSeq","typeReadable":"∀ {K : Type u_1} [inst : DivisionRing K] [inst_1 : LinearOrder K] [inst_2 : FloorRing K] (v : K),\n  (GenContFract.IntFractPair.stream v).IsSeq","typeReferences":[["GenContFract","IntFractPair","stream"],["DivisionRing","toRing"],["LinearOrder"],["Stream'","IsSeq"],["DivisionRing"],["GenContFract","IntFractPair"],["FloorRing"]],"valueReferences":[["instAddNat"],["Ring","toNonAssocRing"],["Eq","trans"],["GenContFract","IntFractPair","stream"],["Option","some"],["GenContFract","IntFractPair","of"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["DivisionRing","toDivisionSemiring"],["instOfNatNat"],["congrFun'"],["Option","bind_congr'"],["Zero","toOfNat0"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Inv","inv"],["InvOneClass","toInv"],["True"],["instHAdd"],["ite"],["DivisionRing","toRing"],["DivisionSemiring","toGroupWithZero"],["GenContFract","IntFractPair"],["OfNat","ofNat"],["HAdd","hAdd"],["eq_self"],["LinearOrder","toDecidableEq"],["Nat"],["DivInvOneMonoid","toInvOneClass"],["Option","none"],["of_eq_true"],["MulZeroClass","toZero"],["Eq","refl"],["Option"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["GenContFract","IntFractPair","fr"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","noConfusionType"],"typeFallback":"Sort.{u} -> (forall {K : Type.{u_1}}, (GenContFract.IntFractPair.{u_1} K) -> (forall {K' : Type.{u_1}}, (GenContFract.IntFractPair.{u_1} K') -> Sort.{u}))","typeFull":"Sort u → {K : Type u_1} → GenContFract.IntFractPair K → {K' : Type u_1} → GenContFract.IntFractPair K' → Sort u","typeReadable":"Sort u → {K : Type u_1} → GenContFract.IntFractPair K → {K' : Type u_1} → GenContFract.IntFractPair K' → Sort u","typeReferences":[["GenContFract","IntFractPair"]],"valueReferences":[["HEq"],["Eq"],["Int"],["GenContFract","IntFractPair","casesOn"]]},{"isProp":false,"kind":"definition","name":["GenContFract","IntFractPair","seq1"],"typeFallback":"forall {K : Type.{u_1}} [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.2566793764._hygCtx._hyg.3 : DivisionRing.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.2566793764._hygCtx._hyg.6 : LinearOrder.{u_1} K] [inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.2566793764._hygCtx._hyg.9 : FloorRing.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.2566793764._hygCtx._hyg.3) inst._@.Mathlib.Algebra.ContinuedFractions.Computation.Basic.2566793764._hygCtx._hyg.6], K -> (Stream'.Seq1.{u_1} (GenContFract.IntFractPair.{u_1} K))","typeFull":"{K : Type u_1} →\n  [inst : DivisionRing K] → [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream'.Seq1 (GenContFract.IntFractPair K)","typeReadable":"{K : Type u_1} →\n  [inst : DivisionRing K] → [inst_1 : LinearOrder K] → [FloorRing K] → K → Stream'.Seq1 (GenContFract.IntFractPair K)","typeReferences":[["Stream'","Seq1"],["DivisionRing","toRing"],["LinearOrder"],["DivisionRing"],["GenContFract","IntFractPair"],["FloorRing"]],"valueReferences":[["Stream'","Seq"],["Prod","mk"],["Option"],["GenContFract","IntFractPair","stream_isSeq"],["GenContFract","IntFractPair","stream"],["Stream'","IsSeq"],["Stream'","Seq","tail"],["Subtype","mk"],["Stream'"],["GenContFract","IntFractPair","of"],["GenContFract","IntFractPair"]]},{"isProp":true,"kind":"theorem","name":["GenContFract","IntFractPair","mk","inj"],"typeFallback":"forall {K : Type.{u_1}} {b : Int} {fr : K} {b_1 : Int} {fr_1 : K}, (Eq.{succ u_1} (GenContFract.IntFractPair.{u_1} K) (GenContFract.IntFractPair.mk.{u_1} K b fr) (GenContFract.IntFractPair.mk.{u_1} K b_1 fr_1)) -> (And (Eq.{1} Int b b_1) (Eq.{succ u_1} K fr fr_1))","typeFull":"∀ {K : Type u_1} {b : ℤ} {fr : K} {b_1 : ℤ} {fr_1 : K},\n  { b := b, fr := fr } = { b := b_1, fr := fr_1 } → b = b_1 ∧ fr = fr_1","typeReadable":"∀ {K : Type u_1} {b : ℤ} {fr : K} {b_1 : ℤ} {fr_1 : K},\n  { b := b, fr := fr } = { b := b_1, fr := fr_1 } → b = b_1 ∧ fr = fr_1","typeReferences":[["GenContFract","IntFractPair","mk"],["And"],["Eq"],["GenContFract","IntFractPair"],["Int"]],"valueReferences":[["And","intro"],["GenContFract","IntFractPair","mk","noConfusion"],["eq_of_heq"],["And"],["Eq"],["Int"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Int","cast_negOnePow"],"typeFallback":"forall (K : Type.{u_1}) (n : Int) [inst._@.Mathlib.Algebra.Field.NegOnePow.3146386386._hygCtx._hyg.6 : DivisionRing.{u_1} K], Eq.{succ u_1} K (Int.cast.{u_1} K (AddGroupWithOne.toIntCast.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.Field.NegOnePow.3146386386._hygCtx._hyg.6))) (Units.val.{0} Int Int.instMonoid (Int.negOnePow n))) (HPow.hPow.{u_1, 0, u_1} K Int K (instHPow.{u_1, 0} K Int (DivInvMonoid.toZPow.{u_1} K (DivisionRing.toDivInvMonoid.{u_1} K inst._@.Mathlib.Algebra.Field.NegOnePow.3146386386._hygCtx._hyg.6))) (Neg.neg.{u_1} K (NegZeroClass.toNeg.{u_1} K (SubNegZeroMonoid.toNegZeroClass.{u_1} K (SubtractionMonoid.toSubNegZeroMonoid.{u_1} K (SubtractionCommMonoid.toSubtractionMonoid.{u_1} K (AddCommGroup.toDivisionAddCommMonoid.{u_1} K (Ring.toAddCommGroup.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.Field.NegOnePow.3146386386._hygCtx._hyg.6))))))) (OfNat.ofNat.{u_1} K 1 (One.toOfNat1.{u_1} K (AddMonoidWithOne.toOne.{u_1} K (AddGroupWithOne.toAddMonoidWithOne.{u_1} K (Ring.toAddGroupWithOne.{u_1} K (DivisionRing.toRing.{u_1} K inst._@.Mathlib.Algebra.Field.NegOnePow.3146386386._hygCtx._hyg.6))))))) n)","typeFull":"∀ (K : Type u_1) (n : ℤ) [inst : DivisionRing K], ↑↑n.negOnePow = (-1) ^ n","typeReadable":"∀ (K : Type u_1) (n : ℤ) [inst : DivisionRing K], ↑↑n.negOnePow = (-1) ^ n","typeReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Int","negOnePow"],["Int","instMonoid"],["DivInvMonoid","toZPow"],["AddGroupWithOne","toAddMonoidWithOne"],["SubtractionCommMonoid","toSubtractionMonoid"],["Int","cast"],["SubNegZeroMonoid","toNegZeroClass"],["Ring","toAddGroupWithOne"],["Eq"],["DivisionRing"],["instHPow"],["Neg","neg"],["DivisionRing","toRing"],["HPow","hPow"],["DivisionRing","toDivInvMonoid"],["OfNat","ofNat"],["Int"],["Ring","toAddCommGroup"],["NegZeroClass","toNeg"],["AddCommGroup","toDivisionAddCommMonoid"],["One","toOfNat1"],["Units","val"],["AddMonoidWithOne","toOne"],["AddGroupWithOne","toIntCast"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["Int","instMonoid"],["Ring","toNonAssocRing"],["Eq","trans"],["MulZeroClass","toMul"],["AddGroupWithOne","toAddMonoidWithOne"],["MonoidWithZero","toMulZeroOneClass"],["AddGroup","toSubtractionMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["DivisionMonoid","toDivInvMonoid"],["one_ne_zero","_simp_1"],["Even"],["Eq","symm"],["Eq","ndrec"],["Units"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","toSemiring"],["instOfNat"],["DivInvMonoid","toMonoid"],["MulZeroOneClass","toMulZeroClass"],["Units","val"],["AddMonoid","toAddSemigroup"],["AddMonoidWithOne","toOne"],["Eq","mpr"],["MulOneClass","toMulOne"],["one_pow"],["Units","instNeg"],["MulZeroOneClass","toMulOneClass"],["SubtractionCommMonoid","toSubtractionMonoid"],["Int","negOnePow_two_mul_add_one"],["zpow_mul"],["Int","cast"],["Int","instMul"],["Nat","instAddMonoidWithOne"],["Nat","instNeZeroSucc"],["MulOne","toMul"],["DivisionRing","toDivisionSemiring"],["instOfNatNat"],["congr"],["Int","instAdd"],["Eq"],["DivisionRing","toRing"],["instOfNatAtLeastTwo"],["mul_one"],["HPow","hPow"],["Int","cast_neg"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["eq_self"],["AddGroupWithOne","toAddGroup"],["Int","even_or_odd'"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["zpow_add_one₀"],["AddGroupWithOne","toIntCast"],["Nat","instAtLeastTwoHAddOfNat"],["Even","neg_pow"],["Int","instCommRing"],["Int","negOnePow"],["DivInvMonoid","toZPow"],["GroupWithZero","toDivInvMonoid"],["HMul","hMul"],["AddMonoidWithOne","toAddMonoid"],["not_false_eq_true"],["Semiring","toNonAssocSemiring"],["Or"],["Ring","toAddGroupWithOne"],["Monoid","toPow"],["mul_neg"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocRing","toHasDistribNeg"],["InvolutiveNeg","toNeg"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["zpow_ofNat"],["AddSemigroup","toAdd"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["MulOne","toOne"],["Neg","neg"],["InvOneClass","toOne"],["DivisionRing","toDivInvMonoid"],["Units","instOne"],["Exists","casesOn"],["Nat"],["AddMonoidWithOne","toNatCast"],["NegZeroClass","toZero"],["id"],["instHMul"],["DivisionMonoid","toDivInvOneMonoid"],["GroupWithZero","toDivisionMonoid"],["even_two","_simp_1"],["one_zpow"],["Int","negOnePow_two_mul"],["NeZero","one"],["CommRing","toNonUnitalCommRing"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["neg_eq_zero","_simp_1"],["Int","cast_one"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["congrFun'"],["Zero","toOfNat0"],["Not"],["True"],["instHAdd"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["Or","casesOn"],["Ring","toAddCommGroup"],["DivInvOneMonoid","toInvOneClass"],["NegZeroClass","toNeg"],["of_eq_true"],["One","toOfNat1"],["HasDistribNeg","toInvolutiveNeg"],["False"],["DivisionRing","toNontrivial"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_2"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_2} α] (a : ULift.{u_1, u_2} α) (b : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HDiv.hDiv.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)))) a b) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))) b))","typeFull":"∀ {α : Type u_2} [inst : Semifield α] (a b : ULift α), a / b = a * b⁻¹","typeReadable":"∀ {α : Type u_2} [inst : Semifield α] (a b : ULift α), a / b = a * b⁻¹","typeReferences":[["Inv","inv"],["NonUnitalNonAssocSemiring","toMul"],["ULift"],["HMul","hMul"],["DivisionSemiring","toSemiring"],["ULift","divisionSemiring"],["DivisionSemiring","toInv"],["instHDiv"],["DivisionSemiring","toDiv"],["HDiv","hDiv"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Semifield"],["instHMul"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["DivisionSemiring","div_eq_mul_inv"],["ULift"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":false,"kind":"definition","name":["ULift","divisionSemiring"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u} α], DivisionSemiring.{max u u_1} (ULift.{u_1, u} α)","typeFull":"{α : Type u} → [DivisionSemiring α] → DivisionSemiring (ULift α)","typeReadable":"{α : Type u} → [DivisionSemiring α] → DivisionSemiring (ULift α)","typeReferences":[["ULift"],["DivisionSemiring"]],"valueReferences":[["ULift","down"],["ULift","divisionSemiring","_proof_8"],["DivisionSemiring","toNNRatCast"],["ULift","semiring"],["ULift"],["ULift","divisionSemiring","_proof_1"],["ULift","instNNRatCast"],["GroupWithZero","toDiv"],["ULift","divisionSemiring","_proof_2"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["ULift","up"],["DivisionSemiring","nnqsmul"],["ULift","divisionSemiring","_proof_3"],["ULift","divisionSemiring","_proof_9"],["DivisionSemiring","mk"],["GroupWithZero","zpow"],["ULift","divisionSemiring","_proof_7"],["GroupWithZero","toInv"],["ULift","divisionSemiring","_proof_5"],["ULift","divisionSemiring","_proof_4"],["ULift","divisionSemiring","_proof_6"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","up_ratCast"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.1050624187._hygCtx._hyg.3 : RatCast.{u} α] (q : Rat), Eq.{max (succ u) (succ u_1)} (ULift.{u_1, u} α) (ULift.up.{u_1, u} α (Rat.cast.{u} α inst._@.Mathlib.Algebra.Field.ULift.1050624187._hygCtx._hyg.3 q)) (Rat.cast.{max u u_1} (ULift.{u_1, u} α) (ULift.instRatCast.{u, u_1} α inst._@.Mathlib.Algebra.Field.ULift.1050624187._hygCtx._hyg.3) q)","typeFull":"∀ {α : Type u} [inst : RatCast α] (q : ℚ), { down := ↑q } = ↑q","typeReadable":"∀ {α : Type u} [inst : RatCast α] (q : ℚ), { down := ↑q } = ↑q","typeReferences":[["RatCast"],["ULift"],["Rat","cast"],["Rat"],["ULift","instRatCast"],["Eq"],["ULift","up"]],"valueReferences":[["rfl"],["ULift"],["Rat","cast"],["ULift","up"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_8"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_1} α] (q : NNRat), Eq.{succ (max u_1 u_2)} (ULift.{u_2, u_1} α) (NNRat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toNNRatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) q) (HDiv.hDiv.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.{u_2, u_1} α) (ULift.{u_2, u_1} α) (instHDiv.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toDiv.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))) (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toRing.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))))) (NNRat.num q)) (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toRing.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))))) (NNRat.den q)))","typeFull":"∀ {α : Type u_1} [inst : Field α] (q : ℚ≥0), ↑q = ↑q.num / ↑q.den","typeReadable":"∀ {α : Type u_1} [inst : Field α] (q : ℚ≥0), ↑q = ↑q.num / ↑q.den","typeReferences":[["DivisionRing","toDiv"],["NNRat","den"],["Nat","cast"],["ULift","divisionRing"],["Field"],["ULift"],["DivisionRing","toRing"],["DivisionRing","toNNRatCast"],["Field","toDivisionRing"],["instHDiv"],["Ring","toSemiring"],["HDiv","hDiv"],["NNRat"],["Semiring","toNatCast"],["NNRat","num"],["NNRat","cast"],["Eq"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"],["DivisionRing","nnratCast_def"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_9"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_2} α] (x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.40 : NNRat) (x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.42 : ULift.{u_1, u_2} α), Eq.{max (succ u_2) (succ u_1)} (ULift.{u_1, u_2} α) (ULift.up.{u_1, u_2} α (DivisionSemiring.nnqsmul.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.40 (ULift.down.{u_1, u_2} α x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.42))) (ULift.up.{u_1, u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α (Distrib.toMul.{u_2} α (NonUnitalNonAssocSemiring.toDistrib.{u_2} α (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_2} α (Semiring.toNonUnitalSemiring.{u_2} α (DivisionSemiring.toSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)))))) (ULift.down.{u_1, u_2} α (NNRat.cast.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.instNNRatCast.{u_2, u_1} α (DivisionSemiring.toNNRatCast.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)) x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.40)) (ULift.down.{u_1, u_2} α x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.42)))","typeFull":"∀ {α : Type u_2} [inst : DivisionSemiring α] (x : ℚ≥0) (x_1 : ULift α),\n  { down := DivisionSemiring.nnqsmul x x_1.down } = { down := (↑x).down * x_1.down }","typeReadable":"∀ {α : Type u_2} [inst : DivisionSemiring α] (x : ℚ≥0) (x_1 : ULift α),\n  { down := DivisionSemiring.nnqsmul x x_1.down } = { down := (↑x).down * x_1.down }","typeReferences":[["ULift","down"],["DivisionSemiring","toNNRatCast"],["NonUnitalNonAssocSemiring","toDistrib"],["ULift"],["Distrib","toMul"],["ULift","instNNRatCast"],["HMul","hMul"],["DivisionSemiring","toSemiring"],["ULift","up"],["DivisionSemiring"],["DivisionSemiring","nnqsmul"],["NNRat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NNRat","cast"],["instHMul"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["ULift","down"],["DivisionSemiring","toNNRatCast"],["NonUnitalNonAssocSemiring","toDistrib"],["ULift"],["Distrib","toMul"],["ULift","instNNRatCast"],["HMul","hMul"],["DivisionSemiring","toSemiring"],["ULift","up"],["DivisionSemiring","nnqsmul"],["congrArg"],["DivisionSemiring","nnqsmul_def"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["NNRat","cast"],["Semiring","toNonUnitalSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_8"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_1} α] (x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.20 : NNRat), Eq.{max (succ u_1) (succ u_2)} (ULift.{u_2, u_1} α) (ULift.up.{u_2, u_1} α (NNRat.cast.{u_1} α (DivisionSemiring.toNNRatCast.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3) x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.20)) (ULift.up.{u_2, u_1} α (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 (DivisionSemiring.toGroupWithZero.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)))) (ULift.down.{u_2, u_1} α (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.semiring.{u_1, u_2} α (DivisionSemiring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))) (NNRat.num x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.20))) (ULift.down.{u_2, u_1} α (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.semiring.{u_1, u_2} α (DivisionSemiring.toSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))) (NNRat.den x._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.20)))))","typeFull":"∀ {α : Type u_1} [inst : DivisionSemiring α] (x : ℚ≥0), { down := ↑x } = { down := (↑x.num).down / (↑x.den).down }","typeReadable":"∀ {α : Type u_1} [inst : DivisionSemiring α] (x : ℚ≥0), { down := ↑x } = { down := (↑x.num).down / (↑x.den).down }","typeReferences":[["ULift","down"],["NNRat","den"],["ULift","semiring"],["Nat","cast"],["DivisionSemiring","toNNRatCast"],["ULift"],["GroupWithZero","toDivInvMonoid"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["ULift","up"],["instHDiv"],["DivisionSemiring"],["DivInvMonoid","toDiv"],["HDiv","hDiv"],["NNRat"],["Semiring","toNatCast"],["NNRat","num"],["NNRat","cast"],["Eq"]],"valueReferences":[["ULift","down"],["NNRat","den"],["DivisionSemiring","toNNRatCast"],["Nat","cast"],["ULift","semiring"],["ULift"],["GroupWithZero","toDivInvMonoid"],["DivisionSemiring","toSemiring"],["DivisionSemiring","toGroupWithZero"],["ULift","up"],["instHDiv"],["DivInvMonoid","toDiv"],["congrArg"],["HDiv","hDiv"],["Semiring","toNatCast"],["DivisionSemiring","nnratCast_def"],["NNRat","num"],["NNRat","cast"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_11"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_2} α] (x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.80 : Rat) (x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.82 : ULift.{u_1, u_2} α), Eq.{max (succ u_2) (succ u_1)} (ULift.{u_1, u_2} α) (ULift.up.{u_1, u_2} α (DivisionRing.qsmul.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.80 (ULift.down.{u_1, u_2} α x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.82))) (ULift.up.{u_1, u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α (Distrib.toMul.{u_2} α (NonUnitalNonAssocSemiring.toDistrib.{u_2} α (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_2} α (Semiring.toNonUnitalSemiring.{u_2} α (Ring.toSemiring.{u_2} α (DivisionRing.toRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))))))) (ULift.down.{u_1, u_2} α (Rat.cast.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.instRatCast.{u_2, u_1} α (DivisionRing.toRatCast.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)) x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.80)) (ULift.down.{u_1, u_2} α x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.82)))","typeFull":"∀ {α : Type u_2} [inst : DivisionRing α] (x : ℚ) (x_1 : ULift α),\n  { down := DivisionRing.qsmul x x_1.down } = { down := (↑x).down * x_1.down }","typeReadable":"∀ {α : Type u_2} [inst : DivisionRing α] (x : ℚ) (x_1 : ULift α),\n  { down := DivisionRing.qsmul x x_1.down } = { down := (↑x).down * x_1.down }","typeReferences":[["ULift","down"],["NonUnitalNonAssocSemiring","toDistrib"],["ULift"],["DivisionRing","qsmul"],["DivisionRing","toRing"],["Distrib","toMul"],["HMul","hMul"],["ULift","up"],["Ring","toSemiring"],["DivisionRing","toRatCast"],["Rat","cast"],["Rat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["ULift","instRatCast"],["DivisionRing"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["ULift","down"],["NonUnitalNonAssocSemiring","toDistrib"],["ULift"],["DivisionRing","qsmul"],["Distrib","toMul"],["DivisionRing","toRing"],["HMul","hMul"],["ULift","up"],["congrArg"],["Ring","toSemiring"],["DivisionRing","toRatCast"],["Rat","cast"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["DivisionRing","qsmul_def"],["ULift","instRatCast"],["Semiring","toNonUnitalSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_6"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_2} α] (a : ULift.{u_1, u_2} α), (Ne.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) a (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 0 (Zero.toOfNat0.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))))))) -> (Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))) a)) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))))))))","typeFull":"∀ {α : Type u_2} [inst : DivisionRing α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReadable":"∀ {α : Type u_2} [inst : DivisionRing α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReferences":[["Inv","inv"],["ULift"],["Monoid","toOne"],["HMul","hMul"],["DivisionSemiring","toGroupWithZero"],["OfNat","ofNat"],["Semigroup","toMul"],["One","toOfNat1"],["DivisionRing","toDivisionSemiring"],["GroupWithZero","toInv"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Ne"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["DivisionRing"],["MonoidWithZero","toZero"],["Eq"],["ULift","groupWithZero"]],"valueReferences":[["GroupWithZero","mul_inv_cancel"],["DivisionRing","toDivisionSemiring"],["ULift"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_1"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_2} α] (a : ULift.{u_1, u_2} α) (b : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HDiv.hDiv.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))) a b) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) b))","typeFull":"∀ {α : Type u_2} [inst : Field α] (a b : ULift α), a / b = a * b⁻¹","typeReadable":"∀ {α : Type u_2} [inst : Field α] (a b : ULift α), a / b = a * b⁻¹","typeReferences":[["Inv","inv"],["DivisionRing","toDiv"],["ULift","divisionRing"],["NonUnitalNonAssocSemiring","toMul"],["Field"],["ULift"],["DivisionRing","toRing"],["Field","toDivisionRing"],["HMul","hMul"],["DivisionRing","toInv"],["instHDiv"],["Ring","toSemiring"],["HDiv","hDiv"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"],["DivisionRing","div_eq_mul_inv"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_6"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_1} α], Nontrivial.{max u_1 u_2} (ULift.{u_2, u_1} α)","typeFull":"∀ {α : Type u_1} [Semifield α], Nontrivial (ULift α)","typeReadable":"∀ {α : Type u_1} [Semifield α], Nontrivial (ULift α)","typeReferences":[["ULift"],["Nontrivial"],["Semifield"]],"valueReferences":[["ULift"],["DivisionSemiring","toNontrivial"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_8"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_1} α] (x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.27 : NNRat), Eq.{max (succ u_1) (succ u_2)} (ULift.{u_2, u_1} α) (ULift.up.{u_2, u_1} α (NNRat.cast.{u_1} α (DivisionRing.toNNRatCast.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3) x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.27)) (ULift.up.{u_2, u_1} α (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 (DivisionSemiring.toGroupWithZero.{u_1} α (DivisionRing.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))))) (ULift.down.{u_2, u_1} α (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.ring.{u_1, u_2} α (DivisionRing.toRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))) (NNRat.num x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.27))) (ULift.down.{u_2, u_1} α (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.ring.{u_1, u_2} α (DivisionRing.toRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))) (NNRat.den x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.27)))))","typeFull":"∀ {α : Type u_1} [inst : DivisionRing α] (x : ℚ≥0), { down := ↑x } = { down := (↑x.num).down / (↑x.den).down }","typeReadable":"∀ {α : Type u_1} [inst : DivisionRing α] (x : ℚ≥0), { down := ↑x } = { down := (↑x.num).down / (↑x.den).down }","typeReferences":[["ULift","down"],["NNRat","den"],["Nat","cast"],["ULift"],["GroupWithZero","toDivInvMonoid"],["DivisionRing","toRing"],["DivisionRing","toNNRatCast"],["ULift","ring"],["DivisionSemiring","toGroupWithZero"],["ULift","up"],["instHDiv"],["DivInvMonoid","toDiv"],["Ring","toSemiring"],["HDiv","hDiv"],["NNRat"],["Semiring","toNatCast"],["DivisionRing","toDivisionSemiring"],["NNRat","num"],["NNRat","cast"],["DivisionRing"],["Eq"]],"valueReferences":[["ULift","down"],["NNRat","den"],["Nat","cast"],["ULift"],["GroupWithZero","toDivInvMonoid"],["DivisionRing","toRing"],["DivisionRing","toNNRatCast"],["ULift","ring"],["DivisionSemiring","toGroupWithZero"],["ULift","up"],["instHDiv"],["DivInvMonoid","toDiv"],["Ring","toSemiring"],["congrArg"],["HDiv","hDiv"],["Semiring","toNatCast"],["DivisionSemiring","nnratCast_def"],["DivisionRing","toDivisionSemiring"],["NNRat","num"],["NNRat","cast"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_3"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))))))) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))) (Nat.cast.{0} Int instNatCastInt n) a) a)","typeFull":"∀ {α : Type u_2} [inst : DivisionRing α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (↑n.succ) a = GroupWithZero.zpow (↑n) a * a","typeReadable":"∀ {α : Type u_2} [inst : DivisionRing α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (↑n.succ) a = GroupWithZero.zpow (↑n) a * a","typeReferences":[["Nat","cast"],["ULift"],["HMul","hMul"],["DivisionSemiring","toGroupWithZero"],["Semigroup","toMul"],["Int"],["Nat"],["GroupWithZero","zpow"],["DivisionRing","toDivisionSemiring"],["Nat","succ"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["instHMul"],["Monoid","toSemigroup"],["DivisionRing"],["Eq"],["ULift","groupWithZero"],["instNatCastInt"]],"valueReferences":[["DivisionRing","toDivisionSemiring"],["ULift"],["DivisionSemiring","toGroupWithZero"],["GroupWithZero","zpow_succ'"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_9"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_2} α] (x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.47 : NNRat) (x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.49 : ULift.{u_1, u_2} α), Eq.{max (succ u_2) (succ u_1)} (ULift.{u_1, u_2} α) (ULift.up.{u_1, u_2} α (DivisionSemiring.nnqsmul.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3) x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.47 (ULift.down.{u_1, u_2} α x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.49))) (ULift.up.{u_1, u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α (Distrib.toMul.{u_2} α (NonUnitalNonAssocSemiring.toDistrib.{u_2} α (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{u_2} α (Semiring.toNonUnitalSemiring.{u_2} α (Ring.toSemiring.{u_2} α (DivisionRing.toRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))))))) (ULift.down.{u_1, u_2} α (NNRat.cast.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.instNNRatCast.{u_2, u_1} α (DivisionRing.toNNRatCast.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)) x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.47)) (ULift.down.{u_1, u_2} α x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.49)))","typeFull":"∀ {α : Type u_2} [inst : DivisionRing α] (x : ℚ≥0) (x_1 : ULift α),\n  { down := DivisionSemiring.nnqsmul x x_1.down } = { down := (↑x).down * x_1.down }","typeReadable":"∀ {α : Type u_2} [inst : DivisionRing α] (x : ℚ≥0) (x_1 : ULift α),\n  { down := DivisionSemiring.nnqsmul x x_1.down } = { down := (↑x).down * x_1.down }","typeReferences":[["ULift","down"],["NonUnitalNonAssocSemiring","toDistrib"],["ULift"],["DivisionRing","toNNRatCast"],["DivisionRing","toRing"],["Distrib","toMul"],["ULift","instNNRatCast"],["HMul","hMul"],["ULift","up"],["DivisionSemiring","nnqsmul"],["Ring","toSemiring"],["NNRat"],["DivisionRing","toDivisionSemiring"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NNRat","cast"],["instHMul"],["DivisionRing"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["ULift","down"],["NonUnitalNonAssocSemiring","toDistrib"],["ULift"],["DivisionRing","toNNRatCast"],["Distrib","toMul"],["DivisionRing","toRing"],["ULift","instNNRatCast"],["HMul","hMul"],["ULift","up"],["DivisionSemiring","nnqsmul"],["congrArg"],["Ring","toSemiring"],["DivisionSemiring","nnqsmul_def"],["DivisionRing","toDivisionSemiring"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NNRat","cast"],["instHMul"],["Semiring","toNonUnitalSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_3"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionRing.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))) (DivisionRing.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt n) a) a)","typeFull":"∀ {α : Type u_2} [inst : Field α] (n : ℕ) (a : ULift α), DivisionRing.zpow (↑n.succ) a = DivisionRing.zpow (↑n) a * a","typeReadable":"∀ {α : Type u_2} [inst : Field α] (n : ℕ) (a : ULift α), DivisionRing.zpow (↑n.succ) a = DivisionRing.zpow (↑n) a * a","typeReferences":[["Nat","cast"],["NonUnitalNonAssocSemiring","toMul"],["ULift","divisionRing"],["Field"],["ULift"],["DivisionRing","toRing"],["Field","toDivisionRing"],["HMul","hMul"],["Ring","toSemiring"],["Int"],["Nat"],["Nat","succ"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["Eq"],["DivisionRing","zpow"],["Semiring","toNonUnitalSemiring"],["instNatCastInt"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"],["DivisionRing","zpow_succ'"]]},{"isProp":false,"kind":"definition","name":["ULift","instRatCast"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.1542641068._hygCtx._hyg.3 : RatCast.{u} α], RatCast.{max u u_1} (ULift.{u_1, u} α)","typeFull":"{α : Type u} → [RatCast α] → RatCast (ULift α)","typeReadable":"{α : Type u} → [RatCast α] → RatCast (ULift α)","typeReferences":[["RatCast"],["ULift"]],"valueReferences":[["ULift"],["Rat","cast"],["RatCast","mk"],["ULift","up"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_4"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionSemiring.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))) (DivisionSemiring.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt n) a) a)","typeFull":"∀ {α : Type u_2} [inst : Semifield α] (n : ℕ) (a : ULift α),\n  DivisionSemiring.zpow (↑n.succ) a = DivisionSemiring.zpow (↑n) a * a","typeReadable":"∀ {α : Type u_2} [inst : Semifield α] (n : ℕ) (a : ULift α),\n  DivisionSemiring.zpow (↑n.succ) a = DivisionSemiring.zpow (↑n) a * a","typeReferences":[["Nat","cast"],["NonUnitalNonAssocSemiring","toMul"],["ULift"],["DivisionSemiring","zpow"],["HMul","hMul"],["DivisionSemiring","toSemiring"],["ULift","divisionSemiring"],["Int"],["Nat"],["Nat","succ"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Semifield"],["instHMul"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toNonUnitalSemiring"],["instNatCastInt"]],"valueReferences":[["ULift"],["DivisionSemiring","zpow_succ'"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_2"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_2} α] (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)) (OfNat.ofNat.{0} Int 0 (instOfNat 0)) a) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)))))))","typeFull":"∀ {α : Type u_2} [inst : DivisionSemiring α] (a : ULift α), GroupWithZero.zpow 0 a = 1","typeReadable":"∀ {α : Type u_2} [inst : DivisionSemiring α] (a : ULift α), GroupWithZero.zpow 0 a = 1","typeReferences":[["ULift"],["Monoid","toOne"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring"],["OfNat","ofNat"],["Int"],["GroupWithZero","zpow"],["instOfNat"],["One","toOfNat1"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["Eq"],["ULift","groupWithZero"]],"valueReferences":[["ULift"],["GroupWithZero","zpow_zero'"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_5"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_1} α], Nontrivial.{max u_1 u_2} (ULift.{u_2, u_1} α)","typeFull":"∀ {α : Type u_1} [DivisionRing α], Nontrivial (ULift α)","typeReadable":"∀ {α : Type u_1} [DivisionRing α], Nontrivial (ULift α)","typeReferences":[["ULift"],["Nontrivial"],["DivisionRing"]],"valueReferences":[["DivisionRing","toDivisionSemiring"],["ULift"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"],["GroupWithZero","toNontrivial"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_4"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)) (Int.negSucc n) a) (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a))","typeFull":"∀ {α : Type u_2} [inst : DivisionSemiring α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (Int.negSucc n) a = (GroupWithZero.zpow (↑n.succ) a)⁻¹","typeReadable":"∀ {α : Type u_2} [inst : DivisionSemiring α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (Int.negSucc n) a = (GroupWithZero.zpow (↑n.succ) a)⁻¹","typeReferences":[["Inv","inv"],["Nat","cast"],["ULift"],["Int","negSucc"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring"],["Int"],["Nat"],["GroupWithZero","zpow"],["Nat","succ"],["GroupWithZero","toInv"],["Eq"],["ULift","groupWithZero"],["instNatCastInt"]],"valueReferences":[["ULift"],["DivisionSemiring","toGroupWithZero"],["GroupWithZero","zpow_neg'"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_8"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_2} α] (a : ULift.{u_1, u_2} α), (Ne.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) a (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 0 (Zero.toOfNat0.{max u_2 u_1} (ULift.{u_1, u_2} α) (AddMonoid.toZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (AddCommMonoid.toAddMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))))))) -> (Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))) a)) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)))))))","typeFull":"∀ {α : Type u_2} [inst : Semifield α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReadable":"∀ {α : Type u_2} [inst : Semifield α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReferences":[["AddMonoid","toZero"],["Inv","inv"],["Semiring","toOne"],["NonUnitalNonAssocSemiring","toMul"],["ULift"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["HMul","hMul"],["AddCommMonoid","toAddMonoid"],["DivisionSemiring","toSemiring"],["ULift","divisionSemiring"],["DivisionSemiring","toInv"],["OfNat","ofNat"],["One","toOfNat1"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["Semifield"],["Ne"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["DivisionSemiring","mul_inv_cancel"],["ULift"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_3"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))))))) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt n) a) a)","typeFull":"∀ {α : Type u_2} [inst : DivisionSemiring α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (↑n.succ) a = GroupWithZero.zpow (↑n) a * a","typeReadable":"∀ {α : Type u_2} [inst : DivisionSemiring α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (↑n.succ) a = GroupWithZero.zpow (↑n) a * a","typeReferences":[["Nat","cast"],["ULift"],["HMul","hMul"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring"],["Semigroup","toMul"],["Int"],["Nat"],["GroupWithZero","zpow"],["Nat","succ"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["instHMul"],["Monoid","toSemigroup"],["Eq"],["ULift","groupWithZero"],["instNatCastInt"]],"valueReferences":[["ULift"],["DivisionSemiring","toGroupWithZero"],["GroupWithZero","zpow_succ'"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_7"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_2} α] (a : ULift.{u_1, u_2} α), (Ne.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) a (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 0 (Zero.toOfNat0.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))))))) -> (Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))) a)) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))))))))","typeFull":"∀ {α : Type u_2} [inst : DivisionSemiring α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReadable":"∀ {α : Type u_2} [inst : DivisionSemiring α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReferences":[["Inv","inv"],["ULift"],["Monoid","toOne"],["HMul","hMul"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring"],["OfNat","ofNat"],["Semigroup","toMul"],["One","toOfNat1"],["GroupWithZero","toInv"],["MonoidWithZero","toMonoid"],["GroupWithZero","toMonoidWithZero"],["instHMul"],["Ne"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["MonoidWithZero","toZero"],["Eq"],["ULift","groupWithZero"]],"valueReferences":[["GroupWithZero","mul_inv_cancel"],["ULift"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_7"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_1} α], Eq.{succ (max u_1 u_2)} (ULift.{u_2, u_1} α) (Inv.inv.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionSemiring.toInv.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionSemiring.{u_1, u_2} α (Semifield.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddMonoid.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionSemiring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionSemiring.{u_1, u_2} α (Semifield.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))))))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddMonoid.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionSemiring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionSemiring.{u_1, u_2} α (Semifield.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))))))","typeFull":"∀ {α : Type u_1} [inst : Semifield α], 0⁻¹ = 0","typeReadable":"∀ {α : Type u_1} [inst : Semifield α], 0⁻¹ = 0","typeReferences":[["AddMonoid","toZero"],["Inv","inv"],["ULift"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DivisionSemiring","toSemiring"],["ULift","divisionSemiring"],["AddCommMonoid","toAddMonoid"],["DivisionSemiring","toInv"],["OfNat","ofNat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Semifield"],["Zero","toOfNat0"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["DivisionSemiring","inv_zero"],["ULift"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_10"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_2} α] (q : NNRat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionSemiring.nnqsmul.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)) q a) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))) (NNRat.cast.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toNNRatCast.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))) q) a)","typeFull":"∀ {α : Type u_2} [inst : Semifield α] (q : ℚ≥0) (a : ULift α), DivisionSemiring.nnqsmul q a = ↑q * a","typeReadable":"∀ {α : Type u_2} [inst : Semifield α] (q : ℚ≥0) (a : ULift α), DivisionSemiring.nnqsmul q a = ↑q * a","typeReferences":[["DivisionSemiring","toNNRatCast"],["NonUnitalNonAssocSemiring","toMul"],["ULift"],["HMul","hMul"],["ULift","divisionSemiring"],["DivisionSemiring","toSemiring"],["DivisionSemiring","nnqsmul"],["NNRat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NNRat","cast"],["Semifield"],["instHMul"],["Semifield","toDivisionSemiring"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["DivisionSemiring","nnqsmul_def"],["ULift"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_6"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_1} α], Eq.{succ (max u_1 u_2)} (ULift.{u_2, u_1} α) (Inv.inv.{max u_1 u_2} (ULift.{u_2, u_1} α) (GroupWithZero.toInv.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.groupWithZero.{u_1, u_2} α (DivisionSemiring.toGroupWithZero.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (MonoidWithZero.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (GroupWithZero.toMonoidWithZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.groupWithZero.{u_1, u_2} α (DivisionSemiring.toGroupWithZero.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))))))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (MonoidWithZero.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (GroupWithZero.toMonoidWithZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.groupWithZero.{u_1, u_2} α (DivisionSemiring.toGroupWithZero.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u_1} [inst : DivisionSemiring α], 0⁻¹ = 0","typeReadable":"∀ {α : Type u_1} [inst : DivisionSemiring α], 0⁻¹ = 0","typeReferences":[["Inv","inv"],["ULift"],["GroupWithZero","toInv"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["DivisionSemiring","toGroupWithZero"],["MonoidWithZero","toZero"],["Eq"],["OfNat","ofNat"],["DivisionSemiring"],["ULift","groupWithZero"]],"valueReferences":[["ULift"],["GroupWithZero","inv_zero"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_5"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_1} α], Nontrivial.{max u_1 u_2} (ULift.{u_2, u_1} α)","typeFull":"∀ {α : Type u_1} [Field α], Nontrivial (ULift α)","typeReadable":"∀ {α : Type u_1} [Field α], Nontrivial (ULift α)","typeReferences":[["ULift"],["Field"],["Nontrivial"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"],["DivisionRing","toNontrivial"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_7"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_1} α], Eq.{succ (max u_1 u_2)} (ULift.{u_2, u_1} α) (Inv.inv.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toInv.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddMonoid.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toRing.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))))))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddMonoid.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_1 u_2} (ULift.{u_2, u_1} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNonUnitalSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toRing.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))))))","typeFull":"∀ {α : Type u_1} [inst : Field α], 0⁻¹ = 0","typeReadable":"∀ {α : Type u_1} [inst : Field α], 0⁻¹ = 0","typeReferences":[["AddMonoid","toZero"],["Inv","inv"],["ULift","divisionRing"],["Field"],["ULift"],["DivisionRing","toRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Field","toDivisionRing"],["AddCommMonoid","toAddMonoid"],["DivisionRing","toInv"],["OfNat","ofNat"],["Ring","toSemiring"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["Zero","toOfNat0"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["DivisionRing","inv_zero"],["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"]]},{"isProp":true,"kind":"theorem","name":["ULift","down_nnratCast"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.1989271912._hygCtx._hyg.3 : NNRatCast.{u} α] (q : NNRat), Eq.{succ u} α (ULift.down.{u_1, u} α (NNRat.cast.{max u u_1} (ULift.{u_1, u} α) (ULift.instNNRatCast.{u, u_1} α inst._@.Mathlib.Algebra.Field.ULift.1989271912._hygCtx._hyg.3) q)) (NNRat.cast.{u} α inst._@.Mathlib.Algebra.Field.ULift.1989271912._hygCtx._hyg.3 q)","typeFull":"∀ {α : Type u} [inst : NNRatCast α] (q : ℚ≥0), (↑q).down = ↑q","typeReadable":"∀ {α : Type u} [inst : NNRatCast α] (q : ℚ≥0), (↑q).down = ↑q","typeReferences":[["ULift","down"],["NNRat"],["ULift"],["NNRatCast"],["ULift","instNNRatCast"],["NNRat","cast"],["Eq"]],"valueReferences":[["ULift","down"],["rfl"],["ULift"],["ULift","instNNRatCast"],["NNRat","cast"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_6"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_2} α] (a : ULift.{u_1, u_2} α), (Ne.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) a (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 0 (Zero.toOfNat0.{max u_2 u_1} (ULift.{u_1, u_2} α) (AddMonoid.toZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (AddCommMonoid.toAddMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toAddCommMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))))))) -> (Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) a)) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))))))))","typeFull":"∀ {α : Type u_2} [inst : Field α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReadable":"∀ {α : Type u_2} [inst : Field α] (a : ULift α), a ≠ 0 → a * a⁻¹ = 1","typeReferences":[["AddMonoid","toZero"],["Inv","inv"],["Semiring","toOne"],["NonUnitalNonAssocSemiring","toMul"],["ULift","divisionRing"],["Field"],["ULift"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["DivisionRing","toRing"],["HMul","hMul"],["Field","toDivisionRing"],["AddCommMonoid","toAddMonoid"],["DivisionRing","toInv"],["OfNat","ofNat"],["Ring","toSemiring"],["One","toOfNat1"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["Ne"],["Zero","toOfNat0"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["DivisionRing","mul_inv_cancel"],["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"]]},{"isProp":false,"kind":"definition","name":["ULift","semifield"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u} α], Semifield.{max u u_1} (ULift.{u_1, u} α)","typeFull":"{α : Type u} → [Semifield α] → Semifield (ULift α)","typeReadable":"{α : Type u} → [Semifield α] → Semifield (ULift α)","typeReferences":[["ULift"],["Semifield"]],"valueReferences":[["DivisionSemiring","toNNRatCast"],["ULift","semifield","_proof_7"],["ULift","semifield","_proof_8"],["ULift"],["ULift","semifield","_proof_1"],["DivisionSemiring","zpow"],["ULift","semifield","_proof_2"],["ULift","semifield","_proof_5"],["ULift","divisionSemiring"],["ULift","commGroupWithZero"],["DivisionSemiring","toSemiring"],["CommSemiring","mk"],["DivisionSemiring","toInv"],["DivisionSemiring","toDiv"],["DivisionSemiring","nnqsmul"],["ULift","semifield","_proof_3"],["ULift","semifield","_proof_6"],["ULift","semifield","_proof_10"],["Semifield","mk"],["ULift","semifield","_proof_4"],["Semifield","toDivisionSemiring"],["ULift","semifield","_proof_9"],["Semifield","toCommGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","up_nnratCast"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.1691088536._hygCtx._hyg.3 : NNRatCast.{u} α] (q : NNRat), Eq.{max (succ u) (succ u_1)} (ULift.{u_1, u} α) (ULift.up.{u_1, u} α (NNRat.cast.{u} α inst._@.Mathlib.Algebra.Field.ULift.1691088536._hygCtx._hyg.3 q)) (NNRat.cast.{max u u_1} (ULift.{u_1, u} α) (ULift.instNNRatCast.{u, u_1} α inst._@.Mathlib.Algebra.Field.ULift.1691088536._hygCtx._hyg.3) q)","typeFull":"∀ {α : Type u} [inst : NNRatCast α] (q : ℚ≥0), { down := ↑q } = ↑q","typeReadable":"∀ {α : Type u} [inst : NNRatCast α] (q : ℚ≥0), { down := ↑q } = ↑q","typeReferences":[["NNRat"],["ULift"],["NNRatCast"],["ULift","instNNRatCast"],["NNRat","cast"],["Eq"],["ULift","up"]],"valueReferences":[["rfl"],["ULift"],["NNRat","cast"],["ULift","up"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_10"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_1} α] (q : Rat), Eq.{succ (max u_1 u_2)} (ULift.{u_2, u_1} α) (Rat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toRatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) q) (HDiv.hDiv.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.{u_2, u_1} α) (ULift.{u_2, u_1} α) (instHDiv.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toDiv.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))) (Int.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toIntCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toRing.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))) (Rat.num q)) (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionRing.toRing.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionRing.{u_1, u_2} α (Field.toDivisionRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))))) (Rat.den q)))","typeFull":"∀ {α : Type u_1} [inst : Field α] (q : ℚ), ↑q = ↑q.num / ↑q.den","typeReadable":"∀ {α : Type u_1} [inst : Field α] (q : ℚ), ↑q = ↑q.num / ↑q.den","typeReferences":[["DivisionRing","toDiv"],["Nat","cast"],["ULift","divisionRing"],["Field"],["ULift"],["DivisionRing","toRing"],["Field","toDivisionRing"],["instHDiv"],["Int","cast"],["Rat","num"],["Ring","toSemiring"],["DivisionRing","toRatCast"],["HDiv","hDiv"],["Ring","toIntCast"],["Semiring","toNatCast"],["Rat","cast"],["Rat"],["Eq"],["Rat","den"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["DivisionRing","ratCast_def"],["Field","toDivisionRing"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_2"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_2} α] (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))) (OfNat.ofNat.{0} Int 0 (instOfNat 0)) a) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))))))))","typeFull":"∀ {α : Type u_2} [inst : DivisionRing α] (a : ULift α), GroupWithZero.zpow 0 a = 1","typeReadable":"∀ {α : Type u_2} [inst : DivisionRing α] (a : ULift α), GroupWithZero.zpow 0 a = 1","typeReferences":[["ULift"],["Monoid","toOne"],["DivisionSemiring","toGroupWithZero"],["OfNat","ofNat"],["Int"],["GroupWithZero","zpow"],["instOfNat"],["One","toOfNat1"],["DivisionRing","toDivisionSemiring"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["DivisionRing"],["Eq"],["ULift","groupWithZero"]],"valueReferences":[["DivisionRing","toDivisionSemiring"],["ULift"],["GroupWithZero","zpow_zero'"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"]]},{"isProp":false,"kind":"definition","name":["ULift","field"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u} α], Field.{max u u_1} (ULift.{u_1, u} α)","typeFull":"{α : Type u} → [Field α] → Field (ULift α)","typeReadable":"{α : Type u} → [Field α] → Field (ULift α)","typeReferences":[["ULift"],["Field"]],"valueReferences":[["DivisionRing","nnqsmul"],["ULift","field","_proof_5"],["DivisionRing","toDiv"],["Field","toCommRing"],["ULift","divisionRing"],["Field","mk"],["ULift"],["DivisionRing","qsmul"],["ULift","field","_proof_10"],["DivisionRing","toNNRatCast"],["Field","toDivisionRing"],["DivisionRing","toInv"],["ULift","field","_proof_4"],["ULift","field","_proof_7"],["DivisionRing","toRatCast"],["ULift","field","_proof_1"],["ULift","field","_proof_11"],["ULift","field","_proof_6"],["ULift","field","_proof_9"],["ULift","field","_proof_3"],["ULift","commRing"],["ULift","field","_proof_8"],["DivisionRing","zpow"],["ULift","field","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_9"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_2} α] (q : NNRat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionRing.nnqsmul.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)) q a) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))) (NNRat.cast.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toNNRatCast.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) q) a)","typeFull":"∀ {α : Type u_2} [inst : Field α] (q : ℚ≥0) (a : ULift α), DivisionRing.nnqsmul q a = ↑q * a","typeReadable":"∀ {α : Type u_2} [inst : Field α] (q : ℚ≥0) (a : ULift α), DivisionRing.nnqsmul q a = ↑q * a","typeReferences":[["DivisionRing","nnqsmul"],["ULift","divisionRing"],["NonUnitalNonAssocSemiring","toMul"],["Field"],["ULift"],["DivisionRing","toNNRatCast"],["DivisionRing","toRing"],["Field","toDivisionRing"],["HMul","hMul"],["Ring","toSemiring"],["NNRat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["NNRat","cast"],["instHMul"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"],["DivisionRing","nnqsmul_def"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_5"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_1} α], Nontrivial.{max u_1 u_2} (ULift.{u_2, u_1} α)","typeFull":"∀ {α : Type u_1} [DivisionSemiring α], Nontrivial (ULift α)","typeReadable":"∀ {α : Type u_1} [DivisionSemiring α], Nontrivial (ULift α)","typeReferences":[["ULift"],["Nontrivial"],["DivisionSemiring"]],"valueReferences":[["ULift"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"],["GroupWithZero","toNontrivial"]]},{"isProp":true,"kind":"theorem","name":["ULift","down_ratCast"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.2667121814._hygCtx._hyg.3 : RatCast.{u} α] (q : Rat), Eq.{succ u} α (ULift.down.{u_1, u} α (Rat.cast.{max u u_1} (ULift.{u_1, u} α) (ULift.instRatCast.{u, u_1} α inst._@.Mathlib.Algebra.Field.ULift.2667121814._hygCtx._hyg.3) q)) (Rat.cast.{u} α inst._@.Mathlib.Algebra.Field.ULift.2667121814._hygCtx._hyg.3 q)","typeFull":"∀ {α : Type u} [inst : RatCast α] (q : ℚ), (↑q).down = ↑q","typeReadable":"∀ {α : Type u} [inst : RatCast α] (q : ℚ), (↑q).down = ↑q","typeReferences":[["ULift","down"],["RatCast"],["ULift"],["Rat","cast"],["Rat"],["ULift","instRatCast"],["Eq"]],"valueReferences":[["ULift","down"],["rfl"],["ULift"],["Rat","cast"],["ULift","instRatCast"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_3"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_2} α] (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionSemiring.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)) (OfNat.ofNat.{0} Int 0 (instOfNat 0)) a) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))))))","typeFull":"∀ {α : Type u_2} [inst : Semifield α] (a : ULift α), DivisionSemiring.zpow 0 a = 1","typeReadable":"∀ {α : Type u_2} [inst : Semifield α] (a : ULift α), DivisionSemiring.zpow 0 a = 1","typeReferences":[["Semiring","toOne"],["One","toOfNat1"],["instOfNat"],["ULift"],["DivisionSemiring","zpow"],["Semifield"],["DivisionSemiring","toSemiring"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"],["Eq"],["OfNat","ofNat"],["Int"]],"valueReferences":[["ULift"],["DivisionSemiring","zpow_zero'"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":false,"kind":"definition","name":["ULift","divisionRing"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u} α], DivisionRing.{max u u_1} (ULift.{u_1, u} α)","typeFull":"{α : Type u} → [DivisionRing α] → DivisionRing (ULift α)","typeReadable":"{α : Type u} → [DivisionRing α] → DivisionRing (ULift α)","typeReferences":[["ULift"],["DivisionRing"]],"valueReferences":[["ULift","down"],["DivisionRing","qsmul"],["ULift","divisionRing","_proof_10"],["DivisionRing","toNNRatCast"],["ULift","instNNRatCast"],["GroupWithZero","toDiv"],["ULift","ring"],["ULift","divisionRing","_proof_2"],["ULift","up"],["GroupWithZero","zpow"],["DivisionRing","toDivisionSemiring"],["ULift","divisionRing","_proof_8"],["GroupWithZero","toInv"],["ULift","divisionRing","_proof_5"],["ULift","instRatCast"],["ULift","divisionRing","_proof_4"],["ULift","divisionRing","_proof_1"],["ULift"],["ULift","divisionRing","_proof_9"],["ULift","divisionRing","_proof_3"],["DivisionRing","toRing"],["ULift","divisionRing","_proof_6"],["DivisionRing","mk"],["DivisionSemiring","toGroupWithZero"],["DivisionSemiring","nnqsmul"],["ULift","divisionRing","_proof_11"],["DivisionRing","toRatCast"],["ULift","divisionRing","_proof_7"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_5"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionSemiring.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)) (Int.negSucc n) a) (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionSemiring.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))) (DivisionSemiring.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionSemiring.{u_2, u_1} α (Semifield.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a))","typeFull":"∀ {α : Type u_2} [inst : Semifield α] (n : ℕ) (a : ULift α),\n  DivisionSemiring.zpow (Int.negSucc n) a = (DivisionSemiring.zpow (↑n.succ) a)⁻¹","typeReadable":"∀ {α : Type u_2} [inst : Semifield α] (n : ℕ) (a : ULift α),\n  DivisionSemiring.zpow (Int.negSucc n) a = (DivisionSemiring.zpow (↑n.succ) a)⁻¹","typeReferences":[["Inv","inv"],["Nat","cast"],["ULift"],["DivisionSemiring","zpow"],["Int","negSucc"],["ULift","divisionSemiring"],["DivisionSemiring","toInv"],["Int"],["Nat"],["Nat","succ"],["Semifield"],["Semifield","toDivisionSemiring"],["Eq"],["instNatCastInt"]],"valueReferences":[["ULift"],["DivisionSemiring","zpow_neg'"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_9"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_1} α] (q : NNRat), Eq.{succ (max u_1 u_2)} (ULift.{u_2, u_1} α) (NNRat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionSemiring.toNNRatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionSemiring.{u_1, u_2} α (Semifield.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3))) q) (HDiv.hDiv.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.{u_2, u_1} α) (ULift.{u_2, u_1} α) (instHDiv.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionSemiring.toDiv.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionSemiring.{u_1, u_2} α (Semifield.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)))) (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionSemiring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionSemiring.{u_1, u_2} α (Semifield.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)))) (NNRat.num q)) (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (DivisionSemiring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.divisionSemiring.{u_1, u_2} α (Semifield.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)))) (NNRat.den q)))","typeFull":"∀ {α : Type u_1} [inst : Semifield α] (q : ℚ≥0), ↑q = ↑q.num / ↑q.den","typeReadable":"∀ {α : Type u_1} [inst : Semifield α] (q : ℚ≥0), ↑q = ↑q.num / ↑q.den","typeReferences":[["NNRat","den"],["DivisionSemiring","toNNRatCast"],["Nat","cast"],["ULift"],["DivisionSemiring","toSemiring"],["ULift","divisionSemiring"],["instHDiv"],["DivisionSemiring","toDiv"],["HDiv","hDiv"],["NNRat"],["Semiring","toNatCast"],["NNRat","num"],["Semifield"],["NNRat","cast"],["Semifield","toDivisionSemiring"],["Eq"]],"valueReferences":[["ULift"],["DivisionSemiring","nnratCast_def"],["Semifield","toDivisionSemiring"],["ULift","divisionSemiring"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_11"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_2} α] (a : Rat) (x : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionRing.qsmul.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)) a x) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalNonAssocSemiring.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (NonUnitalSemiring.toNonUnitalNonAssocSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toNonUnitalSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))) (Rat.cast.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRatCast.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) a) x)","typeFull":"∀ {α : Type u_2} [inst : Field α] (a : ℚ) (x : ULift α), DivisionRing.qsmul a x = ↑a * x","typeReadable":"∀ {α : Type u_2} [inst : Field α] (a : ℚ) (x : ULift α), DivisionRing.qsmul a x = ↑a * x","typeReferences":[["ULift","divisionRing"],["NonUnitalNonAssocSemiring","toMul"],["Field"],["ULift"],["DivisionRing","qsmul"],["DivisionRing","toRing"],["Field","toDivisionRing"],["HMul","hMul"],["Ring","toSemiring"],["DivisionRing","toRatCast"],["Rat","cast"],["Rat"],["NonUnitalSemiring","toNonUnitalNonAssocSemiring"],["instHMul"],["Eq"],["Semiring","toNonUnitalSemiring"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"],["DivisionRing","qsmul_def"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_4"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionRing.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)) (Int.negSucc n) a) (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3))) (DivisionRing.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a))","typeFull":"∀ {α : Type u_2} [inst : Field α] (n : ℕ) (a : ULift α),\n  DivisionRing.zpow (Int.negSucc n) a = (DivisionRing.zpow (↑n.succ) a)⁻¹","typeReadable":"∀ {α : Type u_2} [inst : Field α] (n : ℕ) (a : ULift α),\n  DivisionRing.zpow (Int.negSucc n) a = (DivisionRing.zpow (↑n.succ) a)⁻¹","typeReferences":[["Inv","inv"],["Nat","cast"],["ULift","divisionRing"],["Field"],["ULift"],["Int","negSucc"],["Field","toDivisionRing"],["DivisionRing","toInv"],["Int"],["Nat"],["Nat","succ"],["Eq"],["DivisionRing","zpow"],["instNatCastInt"]],"valueReferences":[["ULift","divisionRing"],["DivisionRing","zpow_neg'"],["ULift"],["Field","toDivisionRing"]]},{"isProp":true,"kind":"theorem","name":["ULift","field","_proof_2"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3 : Field.{u_2} α] (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (DivisionRing.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)) (OfNat.ofNat.{0} Int 0 (instOfNat 0)) a) (OfNat.ofNat.{max u_2 u_1} (ULift.{u_1, u_2} α) 1 (One.toOfNat1.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semiring.toOne.{max u_2 u_1} (ULift.{u_1, u_2} α) (Ring.toSemiring.{max u_2 u_1} (ULift.{u_1, u_2} α) (DivisionRing.toRing.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.divisionRing.{u_2, u_1} α (Field.toDivisionRing.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1296631047._hygCtx._hyg.3)))))))","typeFull":"∀ {α : Type u_2} [inst : Field α] (a : ULift α), DivisionRing.zpow 0 a = 1","typeReadable":"∀ {α : Type u_2} [inst : Field α] (a : ULift α), DivisionRing.zpow 0 a = 1","typeReferences":[["ULift","divisionRing"],["Semiring","toOne"],["Field"],["ULift"],["DivisionRing","toRing"],["Field","toDivisionRing"],["OfNat","ofNat"],["Int"],["Ring","toSemiring"],["instOfNat"],["One","toOfNat1"],["Eq"],["DivisionRing","zpow"]],"valueReferences":[["ULift","divisionRing"],["ULift"],["Field","toDivisionRing"],["DivisionRing","zpow_zero'"]]},{"isProp":false,"kind":"definition","name":["ULift","instNNRatCast"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Field.ULift.1126745635._hygCtx._hyg.3 : NNRatCast.{u} α], NNRatCast.{max u u_1} (ULift.{u_1, u} α)","typeFull":"{α : Type u} → [NNRatCast α] → NNRatCast (ULift α)","typeReadable":"{α : Type u} → [NNRatCast α] → NNRatCast (ULift α)","typeReferences":[["ULift"],["NNRatCast"]],"valueReferences":[["NNRatCast","mk"],["ULift"],["NNRat","cast"],["ULift","up"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionSemiring","_proof_1"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3 : DivisionSemiring.{u_2} α] (a : ULift.{u_1, u_2} α) (b : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HDiv.hDiv.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3)))) a b) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.3045832005._hygCtx._hyg.3))) b))","typeFull":"∀ {α : Type u_2} [inst : DivisionSemiring α] (a b : ULift α), a / b = a * b⁻¹","typeReadable":"∀ {α : Type u_2} [inst : DivisionSemiring α] (a b : ULift α), a / b = a * b⁻¹","typeReferences":[["Inv","inv"],["ULift"],["GroupWithZero","toDiv"],["HMul","hMul"],["DivisionSemiring","toGroupWithZero"],["instHDiv"],["DivisionSemiring"],["Semigroup","toMul"],["HDiv","hDiv"],["GroupWithZero","toInv"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["instHMul"],["Monoid","toSemigroup"],["Eq"],["ULift","groupWithZero"]],"valueReferences":[["ULift"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"],["GroupWithZero","div_eq_mul_inv"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_10"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_1} α] (x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.60 : Rat), Eq.{max (succ u_1) (succ u_2)} (ULift.{u_2, u_1} α) (ULift.up.{u_2, u_1} α (Rat.cast.{u_1} α (DivisionRing.toRatCast.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3) x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.60)) (ULift.up.{u_2, u_1} α (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 (DivisionSemiring.toGroupWithZero.{u_1} α (DivisionRing.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))))) (ULift.down.{u_2, u_1} α (Int.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toIntCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.ring.{u_1, u_2} α (DivisionRing.toRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))) (Rat.num x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.60))) (ULift.down.{u_2, u_1} α (Nat.cast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Semiring.toNatCast.{max u_1 u_2} (ULift.{u_2, u_1} α) (Ring.toSemiring.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.ring.{u_1, u_2} α (DivisionRing.toRing.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))) (Rat.den x._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.60)))))","typeFull":"∀ {α : Type u_1} [inst : DivisionRing α] (x : ℚ), { down := ↑x } = { down := (↑x.num).down / (↑x.den).down }","typeReadable":"∀ {α : Type u_1} [inst : DivisionRing α] (x : ℚ), { down := ↑x } = { down := (↑x.num).down / (↑x.den).down }","typeReferences":[["ULift","down"],["Nat","cast"],["ULift"],["GroupWithZero","toDivInvMonoid"],["DivisionRing","toRing"],["ULift","ring"],["DivisionSemiring","toGroupWithZero"],["Int","cast"],["ULift","up"],["instHDiv"],["Rat","num"],["DivInvMonoid","toDiv"],["Ring","toSemiring"],["DivisionRing","toRatCast"],["HDiv","hDiv"],["Ring","toIntCast"],["Semiring","toNatCast"],["DivisionRing","toDivisionSemiring"],["Rat","cast"],["Rat"],["DivisionRing"],["Eq"],["Rat","den"]],"valueReferences":[["ULift","down"],["Nat","cast"],["ULift"],["DivisionRing","ratCast_def"],["GroupWithZero","toDivInvMonoid"],["DivisionRing","toRing"],["ULift","ring"],["DivisionSemiring","toGroupWithZero"],["ULift","up"],["instHDiv"],["Int","cast"],["Rat","num"],["DivInvMonoid","toDiv"],["Ring","toSemiring"],["congrArg"],["DivisionRing","toRatCast"],["HDiv","hDiv"],["Ring","toIntCast"],["Semiring","toNatCast"],["DivisionRing","toDivisionSemiring"],["Rat","cast"],["Rat","den"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_4"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_2} α] (n : Nat) (a : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))) (Int.negSucc n) a) (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))) (GroupWithZero.zpow.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a))","typeFull":"∀ {α : Type u_2} [inst : DivisionRing α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (Int.negSucc n) a = (GroupWithZero.zpow (↑n.succ) a)⁻¹","typeReadable":"∀ {α : Type u_2} [inst : DivisionRing α] (n : ℕ) (a : ULift α),\n  GroupWithZero.zpow (Int.negSucc n) a = (GroupWithZero.zpow (↑n.succ) a)⁻¹","typeReferences":[["Inv","inv"],["Nat","cast"],["ULift"],["Int","negSucc"],["DivisionSemiring","toGroupWithZero"],["Int"],["Nat"],["GroupWithZero","zpow"],["Nat","succ"],["DivisionRing","toDivisionSemiring"],["GroupWithZero","toInv"],["DivisionRing"],["Eq"],["instNatCastInt"],["ULift","groupWithZero"]],"valueReferences":[["DivisionRing","toDivisionSemiring"],["ULift"],["DivisionSemiring","toGroupWithZero"],["GroupWithZero","zpow_neg'"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_7"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_1} α], Eq.{succ (max u_1 u_2)} (ULift.{u_2, u_1} α) (Inv.inv.{max u_1 u_2} (ULift.{u_2, u_1} α) (GroupWithZero.toInv.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.groupWithZero.{u_1, u_2} α (DivisionSemiring.toGroupWithZero.{u_1} α (DivisionRing.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (MonoidWithZero.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (GroupWithZero.toMonoidWithZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.groupWithZero.{u_1, u_2} α (DivisionSemiring.toGroupWithZero.{u_1} α (DivisionRing.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))))))) (OfNat.ofNat.{max u_1 u_2} (ULift.{u_2, u_1} α) 0 (Zero.toOfNat0.{max u_1 u_2} (ULift.{u_2, u_1} α) (MonoidWithZero.toZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (GroupWithZero.toMonoidWithZero.{max u_1 u_2} (ULift.{u_2, u_1} α) (ULift.groupWithZero.{u_1, u_2} α (DivisionSemiring.toGroupWithZero.{u_1} α (DivisionRing.toDivisionSemiring.{u_1} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))))))","typeFull":"∀ {α : Type u_1} [inst : DivisionRing α], 0⁻¹ = 0","typeReadable":"∀ {α : Type u_1} [inst : DivisionRing α], 0⁻¹ = 0","typeReferences":[["Inv","inv"],["DivisionRing","toDivisionSemiring"],["ULift"],["GroupWithZero","toInv"],["GroupWithZero","toMonoidWithZero"],["Zero","toOfNat0"],["DivisionSemiring","toGroupWithZero"],["MonoidWithZero","toZero"],["Eq"],["DivisionRing"],["OfNat","ofNat"],["ULift","groupWithZero"]],"valueReferences":[["DivisionRing","toDivisionSemiring"],["ULift"],["GroupWithZero","inv_zero"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"]]},{"isProp":true,"kind":"theorem","name":["ULift","divisionRing","_proof_1"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3 : DivisionRing.{u_2} α] (a : ULift.{u_1, u_2} α) (b : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HDiv.hDiv.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toDiv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3))))) a b) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (MonoidWithZero.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))))))) a (Inv.inv.{max u_2 u_1} (ULift.{u_1, u_2} α) (GroupWithZero.toInv.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.groupWithZero.{u_2, u_1} α (DivisionSemiring.toGroupWithZero.{u_2} α (DivisionRing.toDivisionSemiring.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.1533133584._hygCtx._hyg.3)))) b))","typeFull":"∀ {α : Type u_2} [inst : DivisionRing α] (a b : ULift α), a / b = a * b⁻¹","typeReadable":"∀ {α : Type u_2} [inst : DivisionRing α] (a b : ULift α), a / b = a * b⁻¹","typeReferences":[["Inv","inv"],["ULift"],["GroupWithZero","toDiv"],["HMul","hMul"],["DivisionSemiring","toGroupWithZero"],["instHDiv"],["Semigroup","toMul"],["HDiv","hDiv"],["DivisionRing","toDivisionSemiring"],["GroupWithZero","toInv"],["GroupWithZero","toMonoidWithZero"],["MonoidWithZero","toMonoid"],["instHMul"],["Monoid","toSemigroup"],["DivisionRing"],["Eq"],["ULift","groupWithZero"]],"valueReferences":[["DivisionRing","toDivisionSemiring"],["ULift"],["DivisionSemiring","toGroupWithZero"],["ULift","groupWithZero"],["GroupWithZero","div_eq_mul_inv"]]},{"isProp":true,"kind":"theorem","name":["ULift","semifield","_proof_1"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3 : Semifield.{u_2} α] (a : ULift.{u_1, u_2} α) (b : ULift.{u_1, u_2} α), Eq.{succ (max u_2 u_1)} (ULift.{u_1, u_2} α) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (CommMonoid.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (CommMonoidWithZero.toCommMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (CommGroupWithZero.toCommMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.commGroupWithZero.{u_2, u_1} α (Semifield.toCommGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)))))))) a b) (HMul.hMul.{max u_2 u_1, max u_2 u_1, max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (ULift.{u_1, u_2} α) (instHMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Semigroup.toMul.{max u_2 u_1} (ULift.{u_1, u_2} α) (Monoid.toSemigroup.{max u_2 u_1} (ULift.{u_1, u_2} α) (CommMonoid.toMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (CommMonoidWithZero.toCommMonoid.{max u_2 u_1} (ULift.{u_1, u_2} α) (CommGroupWithZero.toCommMonoidWithZero.{max u_2 u_1} (ULift.{u_1, u_2} α) (ULift.commGroupWithZero.{u_2, u_1} α (Semifield.toCommGroupWithZero.{u_2} α inst._@.Mathlib.Algebra.Field.ULift.4113141751._hygCtx._hyg.3)))))))) b a)","typeFull":"∀ {α : Type u_2} [inst : Semifield α] (a b : ULift α), a * b = b * a","typeReadable":"∀ {α : Type u_2} [inst : Semifield α] (a b : ULift α), a * b = b * a","typeReferences":[["CommMonoid","toMonoid"],["ULift"],["CommMonoidWithZero","toCommMonoid"],["instHMul"],["HMul","hMul"],["Semifield"],["ULift","commGroupWithZero"],["Monoid","toSemigroup"],["Eq"],["CommGroupWithZero","toCommMonoidWithZero"],["Semifield","toCommGroupWithZero"],["Semigroup","toMul"]],"valueReferences":[["ULift"],["CommMonoid","mul_comm"],["CommMonoidWithZero","toCommMonoid"],["ULift","commGroupWithZero"],["CommGroupWithZero","toCommMonoidWithZero"],["Semifield","toCommGroupWithZero"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Idempotent.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","coe_one"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.2552065102._hygCtx._hyg.5 : MulOneClass.{u_1} M], Eq.{succ u_1} M (Subtype.val.{succ u_1} M (fun (a : M) => IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2552065102._hygCtx._hyg.5)) a) (OfNat.ofNat.{u_1} (Subtype.{succ u_1} M (fun (a : M) => IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2552065102._hygCtx._hyg.5)) a)) 1 (One.toOfNat1.{u_1} (Subtype.{succ u_1} M (fun (a : M) => IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2552065102._hygCtx._hyg.5)) a)) (IsIdempotentElem.instOneSubtype.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2552065102._hygCtx._hyg.5)))) (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2552065102._hygCtx._hyg.5))))","typeFull":"∀ {M : Type u_1} [inst : MulOneClass M], ↑1 = 1","typeReadable":"∀ {M : Type u_1} [inst : MulOneClass M], ↑1 = 1","typeReferences":[["MulOneClass","toMulOne"],["IsIdempotentElem","instOneSubtype"],["MulOne","toMul"],["Subtype"],["MulOne","toOne"],["One","toOfNat1"],["IsIdempotentElem"],["MulOneClass"],["Eq"],["Subtype","val"],["OfNat","ofNat"]],"valueReferences":[["MulOneClass","toMulOne"],["rfl"],["IsIdempotentElem","instOneSubtype"],["MulOne","toMul"],["Subtype"],["One","toOfNat1"],["IsIdempotentElem"],["Subtype","val"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","mul"],"typeFallback":"forall {S : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Idempotent.2075127268._hygCtx._hyg.5 : CommSemigroup.{u_3} S] {a : S} {b : S}, (IsIdempotentElem.{u_3} S (CommMagma.toMul.{u_3} S (CommSemigroup.toCommMagma.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.2075127268._hygCtx._hyg.5)) a) -> (IsIdempotentElem.{u_3} S (CommMagma.toMul.{u_3} S (CommSemigroup.toCommMagma.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.2075127268._hygCtx._hyg.5)) b) -> (IsIdempotentElem.{u_3} S (CommMagma.toMul.{u_3} S (CommSemigroup.toCommMagma.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.2075127268._hygCtx._hyg.5)) (HMul.hMul.{u_3, u_3, u_3} S S S (instHMul.{u_3} S (CommMagma.toMul.{u_3} S (CommSemigroup.toCommMagma.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.2075127268._hygCtx._hyg.5))) a b))","typeFull":"∀ {S : Type u_3} [inst : CommSemigroup S] {a b : S}, IsIdempotentElem a → IsIdempotentElem b → IsIdempotentElem (a * b)","typeReadable":"∀ {S : Type u_3} [inst : CommSemigroup S] {a b : S}, IsIdempotentElem a → IsIdempotentElem b → IsIdempotentElem (a * b)","typeReferences":[["CommMagma","toMul"],["IsIdempotentElem"],["instHMul"],["HMul","hMul"],["CommSemigroup","toCommMagma"],["CommSemigroup"]],"valueReferences":[["IsIdempotentElem","mul_of_commute"],["Commute","all"],["CommSemigroup","toSemigroup"],["CommSemigroup","toCommMagma"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","pow_eq"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.2957677898._hygCtx._hyg.5 : Monoid.{u_1} M] {a : M}, (IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2957677898._hygCtx._hyg.5))) a) -> (forall {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Eq.{succ u_1} M (HPow.hPow.{u_1, 0, u_1} M Nat M (instHPow.{u_1, 0} M Nat (Monoid.toPow.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2957677898._hygCtx._hyg.5)) a n) a))","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {a : M}, IsIdempotentElem a → ∀ {n : ℕ}, n ≠ 0 → a ^ n = a","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {a : M}, IsIdempotentElem a → ∀ {n : ℕ}, n ≠ 0 → a ^ n = a","typeReferences":[["MulOneClass","toMulOne"],["instHPow"],["HPow","hPow"],["OfNat","ofNat"],["MulOne","toMul"],["Nat"],["Monoid","toPow"],["IsIdempotentElem"],["instOfNatNat"],["Monoid","toMulOneClass"],["Monoid"],["Ne"],["Eq"]],"valueReferences":[["instAddNat"],["instHPow"],["instHAdd"],["HPow","hPow"],["OfNat","ofNat"],["Exists","casesOn"],["HAdd","hAdd"],["Nat"],["Monoid","toPow"],["IsIdempotentElem","pow_succ_eq"],["instOfNatNat"],["Eq","symm"],["Ne"],["Eq"],["Eq","ndrec"],["Nat","exists_eq_add_one_of_ne_zero"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","one"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.603995082._hygCtx._hyg.5 : MulOneClass.{u_1} M], IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.603995082._hygCtx._hyg.5)) (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.603995082._hygCtx._hyg.5))))","typeFull":"∀ {M : Type u_1} [inst : MulOneClass M], IsIdempotentElem 1","typeReadable":"∀ {M : Type u_1} [inst : MulOneClass M], IsIdempotentElem 1","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["MulOne","toOne"],["One","toOfNat1"],["IsIdempotentElem"],["MulOneClass"],["OfNat","ofNat"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toOne"],["One","toOfNat1"],["mul_one"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","mul_of_commute"],"typeFallback":"forall {S : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Idempotent.96092743._hygCtx._hyg.5 : Semigroup.{u_3} S] {a : S} {b : S}, (Commute.{u_3} S (Semigroup.toMul.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.96092743._hygCtx._hyg.5) a b) -> (IsIdempotentElem.{u_3} S (Semigroup.toMul.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.96092743._hygCtx._hyg.5) a) -> (IsIdempotentElem.{u_3} S (Semigroup.toMul.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.96092743._hygCtx._hyg.5) b) -> (IsIdempotentElem.{u_3} S (Semigroup.toMul.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.96092743._hygCtx._hyg.5) (HMul.hMul.{u_3, u_3, u_3} S S S (instHMul.{u_3} S (Semigroup.toMul.{u_3} S inst._@.Mathlib.Algebra.Group.Idempotent.96092743._hygCtx._hyg.5)) a b))","typeFull":"∀ {S : Type u_3} [inst : Semigroup S] {a b : S},\n  Commute a b → IsIdempotentElem a → IsIdempotentElem b → IsIdempotentElem (a * b)","typeReadable":"∀ {S : Type u_3} [inst : Semigroup S] {a b : S},\n  Commute a b → IsIdempotentElem a → IsIdempotentElem b → IsIdempotentElem (a * b)","typeReferences":[["IsIdempotentElem"],["Commute"],["instHMul"],["HMul","hMul"],["Semigroup"],["Semigroup","toMul"]],"valueReferences":[["Commute","mul_mul_mul_comm"],["HMul","hMul"],["Commute","symm"],["Semigroup","toMul"],["congrArg"],["IsIdempotentElem","eq_1"],["IsIdempotentElem"],["Eq","refl"],["IsIdempotentElem","eq"],["id"],["instHMul"],["Eq","mpr"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","mul_mul_self"],"typeFallback":"forall {M : Type.{u_4}} [inst._@.Mathlib.Algebra.Group.Idempotent.3510259847._hygCtx._hyg.6 : Semigroup.{u_4} M] {x : M}, (IsIdempotentElem.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.3510259847._hygCtx._hyg.6) x) -> (forall (y : M), Eq.{succ u_4} M (HMul.hMul.{u_4, u_4, u_4} M M M (instHMul.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.3510259847._hygCtx._hyg.6)) (HMul.hMul.{u_4, u_4, u_4} M M M (instHMul.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.3510259847._hygCtx._hyg.6)) y x) x) (HMul.hMul.{u_4, u_4, u_4} M M M (instHMul.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.3510259847._hygCtx._hyg.6)) y x))","typeFull":"∀ {M : Type u_4} [inst : Semigroup M] {x : M}, IsIdempotentElem x → ∀ (y : M), y * x * x = y * x","typeReadable":"∀ {M : Type u_4} [inst : Semigroup M] {x : M}, IsIdempotentElem x → ∀ (y : M), y * x * x = y * x","typeReferences":[["IsIdempotentElem"],["instHMul"],["HMul","hMul"],["Semigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["IsIdempotentElem","eq"],["Eq","symm"],["instHMul"],["HMul","hMul"],["mul_assoc"],["Eq"],["Eq","rec"],["congrArg"],["Semigroup","toMul"]]},{"isProp":true,"kind":"theorem","name":["isIdempotentElem_iff"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.583845154._hygCtx._hyg.5 : Mul.{u_1} M] {a : M}, Iff (IsIdempotentElem.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.583845154._hygCtx._hyg.5 a) (Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.583845154._hygCtx._hyg.5) a a) a)","typeFull":"∀ {M : Type u_1} [inst : Mul M] {a : M}, IsIdempotentElem a ↔ a * a = a","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {a : M}, IsIdempotentElem a ↔ a * a = a","typeReferences":[["IsIdempotentElem"],["Iff"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"]],"valueReferences":[["IsIdempotentElem"],["Iff","rfl"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","eq_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.1717780149._hygCtx._hyg.5 : Mul.{u_1} M] (a : M), Eq.{1} Prop (IsIdempotentElem.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1717780149._hygCtx._hyg.5 a) (Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1717780149._hygCtx._hyg.5) a a) a)","typeFull":"∀ {M : Type u_1} [inst : Mul M] (a : M), IsIdempotentElem a = (a * a = a)","typeReadable":"∀ {M : Type u_1} [inst : Mul M] (a : M), IsIdempotentElem a = (a * a = a)","typeReferences":[["IsIdempotentElem"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"]],"valueReferences":[["IsIdempotentElem"],["Eq","refl"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","eq"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.2921514041._hygCtx._hyg.5 : Mul.{u_1} M] {a : M}, (IsIdempotentElem.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2921514041._hygCtx._hyg.5 a) -> (Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.2921514041._hygCtx._hyg.5) a a) a)","typeFull":"∀ {M : Type u_1} [inst : Mul M] {a : M}, IsIdempotentElem a → a * a = a","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {a : M}, IsIdempotentElem a → a * a = a","typeReferences":[["IsIdempotentElem"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["IsIdempotentElem","instOneSubtype"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.4209232041._hygCtx._hyg.5 : MulOneClass.{u_1} M], One.{u_1} (Subtype.{succ u_1} M (fun (a : M) => IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.4209232041._hygCtx._hyg.5)) a))","typeFull":"{M : Type u_1} → [inst : MulOneClass M] → One { a // IsIdempotentElem a }","typeReadable":"{M : Type u_1} → [inst : MulOneClass M] → One { a // IsIdempotentElem a }","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Subtype"],["IsIdempotentElem"],["One"],["MulOneClass"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Subtype"],["MulOne","toOne"],["One","toOfNat1"],["IsIdempotentElem"],["One","mk"],["Subtype","mk"],["IsIdempotentElem","one"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","pow"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.427477602._hygCtx._hyg.5 : Monoid.{u_1} M] {a : M} (n : Nat), (IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.427477602._hygCtx._hyg.5))) a) -> (IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.427477602._hygCtx._hyg.5))) (HPow.hPow.{u_1, 0, u_1} M Nat M (instHPow.{u_1, 0} M Nat (Monoid.toPow.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.427477602._hygCtx._hyg.5)) a n))","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {a : M} (n : ℕ), IsIdempotentElem a → IsIdempotentElem (a ^ n)","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {a : M} (n : ℕ), IsIdempotentElem a → IsIdempotentElem (a ^ n)","typeReferences":[["instHPow"],["MulOneClass","toMulOne"],["MulOne","toMul"],["Nat"],["Monoid","toPow"],["IsIdempotentElem"],["Monoid","toMulOneClass"],["Monoid"],["HPow","hPow"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["HMul","hMul"],["congrArg"],["MulOne","toMul"],["Monoid","toPow"],["IsIdempotentElem"],["instOfNatNat"],["Monoid","toMulOneClass"],["instMulNat"],["Eq","symm"],["pow_mul"],["Eq","rec"],["Eq"],["instHPow"],["MulOne","toOne"],["HPow","hPow"],["OfNat","ofNat"],["Nat"],["One","toOfNat1"],["Nat","succ"],["Eq","refl"],["IsIdempotentElem","eq"],["sq"],["id"],["instHMul"],["pow_mul'"],["Eq","mpr"],["Nat","recOn"],["IsIdempotentElem","one"],["pow_zero"]]},{"isProp":false,"kind":"definition","name":["IsIdempotentElem"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.1717780149._hygCtx._hyg.5 : Mul.{u_1} M], M -> Prop","typeFull":"{M : Type u_1} → [Mul M] → M → Prop","typeReadable":"{M : Type u_1} → [Mul M] → M → Prop","typeReferences":[["Mul"]],"valueReferences":[["instHMul"],["HMul","hMul"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","iff_eq_one","_simp_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.3335447858._hygCtx._hyg.5 : CancelMonoid.{u_1} M] {a : M}, Eq.{1} Prop (IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (RightCancelMonoid.toMonoid.{u_1} M (CancelMonoid.toRightCancelMonoid.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.3335447858._hygCtx._hyg.5))))) a) (Eq.{succ u_1} M a (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (RightCancelMonoid.toMonoid.{u_1} M (CancelMonoid.toRightCancelMonoid.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.3335447858._hygCtx._hyg.5))))))))","typeFull":"∀ {M : Type u_1} [inst : CancelMonoid M] {a : M}, IsIdempotentElem a = (a = 1)","typeReadable":"∀ {M : Type u_1} [inst : CancelMonoid M] {a : M}, IsIdempotentElem a = (a = 1)","typeReferences":[["MulOneClass","toMulOne"],["RightCancelMonoid","toMonoid"],["MulOne","toMul"],["MulOne","toOne"],["One","toOfNat1"],["IsIdempotentElem"],["Monoid","toMulOneClass"],["Eq"],["OfNat","ofNat"],["CancelMonoid","toRightCancelMonoid"],["CancelMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["RightCancelMonoid","toMonoid"],["MulOne","toMul"],["IsIdempotentElem","iff_eq_one"],["MulOne","toOne"],["One","toOfNat1"],["IsIdempotentElem"],["Monoid","toMulOneClass"],["Eq"],["OfNat","ofNat"],["CancelMonoid","toRightCancelMonoid"],["propext"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","iff_eq_one_of_isUnit"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5 : Monoid.{u_1} M] {a : M}, (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5 a) -> (Iff (IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5))) a) (Eq.{succ u_1} M a (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5)))))))","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {a : M}, IsUnit a → (IsIdempotentElem a ↔ a = 1)","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {a : M}, IsUnit a → (IsIdempotentElem a ↔ a = 1)","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["MulOne","toOne"],["One","toOfNat1"],["IsIdempotentElem"],["Iff"],["Monoid","toMulOneClass"],["Monoid"],["IsUnit"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["MulOneClass","toMulOne"],["HMul","hMul"],["mul_assoc"],["Iff","intro"],["congrArg"],["Semigroup","toMul"],["MulOne","toMul"],["IsIdempotentElem"],["Monoid","toMulOneClass"],["Eq","symm"],["Monoid","toSemigroup"],["Eq"],["Eq","ndrec"],["MulOne","toOne"],["IsUnit"],["OfNat","ofNat"],["IsUnit","exists_left_inv"],["One","toOfNat1"],["Eq","refl"],["IsIdempotentElem","eq"],["id"],["instHMul"],["Eq","mpr"],["_private","Mathlib","Algebra","Group","Idempotent",0,"IsIdempotentElem","iff_eq_one_of_isUnit","match_1_1"],["IsIdempotentElem","one"],["one_mul"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","of_isIdempotent"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.5 : Mul.{u_1} M] [inst._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.9 : Std.IdempotentOp.{succ u_1} M (fun (x1._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.14 : M) (x2._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.14 : M) => HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.5) x1._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.14 x2._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.14)] (a : M), IsIdempotentElem.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.3861133547._hygCtx._hyg.5 a","typeFull":"∀ {M : Type u_1} [inst : Mul M] [Std.IdempotentOp fun x1 x2 => x1 * x2] (a : M), IsIdempotentElem a","typeReadable":"∀ {M : Type u_1} [inst : Mul M] [Std.IdempotentOp fun x1 x2 => x1 * x2] (a : M), IsIdempotentElem a","typeReferences":[["IsIdempotentElem"],["Mul"],["instHMul"],["HMul","hMul"],["Std","IdempotentOp"]],"valueReferences":[["instHMul"],["HMul","hMul"],["Std","IdempotentOp","idempotent"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","mul_self_mul"],"typeFallback":"forall {M : Type.{u_4}} [inst._@.Mathlib.Algebra.Group.Idempotent.1033445265._hygCtx._hyg.6 : Semigroup.{u_4} M] {x : M}, (IsIdempotentElem.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.1033445265._hygCtx._hyg.6) x) -> (forall (y : M), Eq.{succ u_4} M (HMul.hMul.{u_4, u_4, u_4} M M M (instHMul.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.1033445265._hygCtx._hyg.6)) x (HMul.hMul.{u_4, u_4, u_4} M M M (instHMul.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.1033445265._hygCtx._hyg.6)) x y)) (HMul.hMul.{u_4, u_4, u_4} M M M (instHMul.{u_4} M (Semigroup.toMul.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.1033445265._hygCtx._hyg.6)) x y))","typeFull":"∀ {M : Type u_4} [inst : Semigroup M] {x : M}, IsIdempotentElem x → ∀ (y : M), x * (x * y) = x * y","typeReadable":"∀ {M : Type u_4} [inst : Semigroup M] {x : M}, IsIdempotentElem x → ∀ (y : M), x * (x * y) = x * y","typeReferences":[["IsIdempotentElem"],["instHMul"],["HMul","hMul"],["Semigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["IsIdempotentElem","eq"],["instHMul"],["HMul","hMul"],["mul_assoc"],["Eq"],["Eq","rec"],["congrArg"],["Semigroup","toMul"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","iff_eq_one"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.3335447858._hygCtx._hyg.5 : CancelMonoid.{u_1} M] {a : M}, Iff (IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (RightCancelMonoid.toMonoid.{u_1} M (CancelMonoid.toRightCancelMonoid.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.3335447858._hygCtx._hyg.5))))) a) (Eq.{succ u_1} M a (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (RightCancelMonoid.toMonoid.{u_1} M (CancelMonoid.toRightCancelMonoid.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.3335447858._hygCtx._hyg.5))))))))","typeFull":"∀ {M : Type u_1} [inst : CancelMonoid M] {a : M}, IsIdempotentElem a ↔ a = 1","typeReadable":"∀ {M : Type u_1} [inst : CancelMonoid M] {a : M}, IsIdempotentElem a ↔ a = 1","typeReferences":[["MulOneClass","toMulOne"],["RightCancelMonoid","toMonoid"],["MulOne","toMul"],["MulOne","toOne"],["One","toOfNat1"],["IsIdempotentElem"],["Iff"],["Monoid","toMulOneClass"],["Eq"],["OfNat","ofNat"],["CancelMonoid","toRightCancelMonoid"],["CancelMonoid"]],"valueReferences":[["MulOneClass","toMulOne"],["RightCancelMonoid","toMonoid"],["LeftCancelMonoid","toLeftCancelSemigroup"],["True"],["MulOne","toOne"],["Eq","trans"],["CancelMonoid","toRightCancelMonoid"],["OfNat","ofNat"],["congrArg"],["LeftCancelSemigroup","toIsLeftCancelMul"],["MulOne","toMul"],["iff_self"],["of_eq_true"],["One","toOfNat1"],["IsIdempotentElem"],["Iff"],["Monoid","toMulOneClass"],["mul_eq_left","_simp_2"],["congrFun'"],["Eq"],["CancelMonoid","toLeftCancelMonoid"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","map"],"typeFallback":"forall {M : Type.{u_4}} {N : Type.{u_5}} {F : Type.{u_6}} [inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.8 : Mul.{u_4} M] [inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.11 : Mul.{u_5} N] [inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.14 : FunLike.{succ u_6, succ u_4, succ u_5} F M N] [inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.19 : MulHomClass.{u_6, u_4, u_5} F M N inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.14] {e : M}, (IsIdempotentElem.{u_4} M inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.8 e) -> (forall (f : F), IsIdempotentElem.{u_5} N inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.11 (DFunLike.coe.{succ u_6, succ u_4, succ u_5} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) inst._@.Mathlib.Algebra.Group.Idempotent.3880878766._hygCtx._hyg.14 f e))","typeFull":"∀ {M : Type u_4} {N : Type u_5} {F : Type u_6} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]\n  [MulHomClass F M N] {e : M}, IsIdempotentElem e → ∀ (f : F), IsIdempotentElem (f e)","typeReadable":"∀ {M : Type u_4} {N : Type u_5} {F : Type u_6} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]\n  [MulHomClass F M N] {e : M}, IsIdempotentElem e → ∀ (f : F), IsIdempotentElem (f e)","typeReferences":[["FunLike"],["IsIdempotentElem"],["Mul"],["DFunLike","coe"],["MulHomClass"]],"valueReferences":[["HMul","hMul"],["DFunLike","coe"],["congrArg"],["IsIdempotentElem","eq_1"],["IsIdempotentElem"],["Eq","refl"],["IsIdempotentElem","eq"],["id"],["Eq","symm"],["map_mul"],["instHMul"],["Eq","mpr"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["IsIdempotentElem","pow_succ_eq"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.391771498._hygCtx._hyg.5 : Monoid.{u_1} M] {a : M} (n : Nat), (IsIdempotentElem.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.391771498._hygCtx._hyg.5))) a) -> (Eq.{succ u_1} M (HPow.hPow.{u_1, 0, u_1} M Nat M (instHPow.{u_1, 0} M Nat (Monoid.toPow.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.391771498._hygCtx._hyg.5)) a (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) a)","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {a : M} (n : ℕ), IsIdempotentElem a → a ^ (n + 1) = a","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {a : M} (n : ℕ), IsIdempotentElem a → a ^ (n + 1) = a","typeReferences":[["MulOneClass","toMulOne"],["instHPow"],["instAddNat"],["instHAdd"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["MulOne","toMul"],["Monoid","toPow"],["instOfNatNat"],["IsIdempotentElem"],["Monoid","toMulOneClass"],["Monoid"],["Eq"]],"valueReferences":[["instAddNat"],["MulOneClass","toMulOne"],["HMul","hMul"],["congrArg"],["pow_one"],["MulOne","toMul"],["Monoid","toPow"],["pow_succ"],["instOfNatNat"],["Monoid","toMulOneClass"],["Eq","symm"],["Eq","rec"],["Eq"],["instHPow"],["instHAdd"],["Nat","zero_add"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["Nat","succ"],["Eq","refl"],["IsIdempotentElem","eq"],["id"],["instHMul"],["Eq","mpr"],["Nat","recOn"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Idempotent",0,"IsIdempotentElem","iff_eq_one_of_isUnit","match_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5 : Monoid.{u_1} M] {a : M} (motive : (Exists.{succ u_1} M (fun (b : M) => Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5)))) b a) (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5))))))) -> Prop) (x._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.47 : Exists.{succ u_1} M (fun (b : M) => Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5)))) b a) (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5))))))), (forall (q : M) (eq : Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5)))) q a) (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5)))))), motive (Exists.intro.{succ u_1} M (fun (b : M) => Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5)))) b a) (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (MulOne.toOne.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.5)))))) q eq)) -> (motive x._@.Mathlib.Algebra.Group.Idempotent.1945532467._hygCtx._hyg.47)","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {a : M} (motive : (∃ b, b * a = 1) → Prop) (x : ∃ b, b * a = 1),\n  (∀ (q : M) (eq : q * a = 1), motive ⋯) → motive x","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {a : M} (motive : (∃ b, b * a = 1) → Prop) (x : ∃ b, b * a = 1),\n  (∀ (q : M) (eq : q * a = 1), motive ⋯) → motive x","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Exists"],["MulOne","toOne"],["One","toOfNat1"],["Monoid","toMulOneClass"],["Monoid"],["Exists","intro"],["instHMul"],["HMul","hMul"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["MulOneClass","toMulOne"],["Exists","casesOn"],["MulOne","toMul"],["MulOne","toOne"],["One","toOfNat1"],["Monoid","toMulOneClass"],["instHMul"],["HMul","hMul"],["Eq"],["OfNat","ofNat"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Prod.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Prod","instModule","_proof_1"],"typeFallback":"forall {R : Type.{u_3}} {M : Type.{u_1}} {N : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11 : AddCommMonoid.{u_2} N] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.14 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.18 : Module.{u_3, u_2} R N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11] (x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.40 : R) (x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.42 : R) (x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.44 : Prod.{u_1, u_2} M N), Eq.{succ (max u_1 u_2)} (Prod.{u_1, u_2} M N) (HSMul.hSMul.{u_3, max u_1 u_2, max u_1 u_2} R (Prod.{u_1, u_2} M N) (Prod.{u_1, u_2} M N) (instHSMul.{u_3, max u_1 u_2} R (Prod.{u_1, u_2} M N) (SemigroupAction.toSMul.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5))) (MulAction.toSemigroupAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (Prod.instAddMonoid.{u_1, u_2} M N (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11)) (Prod.distribMulAction.{u_1, u_2, u_3} M N R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.14) (Module.toDistribMulAction.{u_3, u_2} R N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.18)))))) (HAdd.hAdd.{u_3, u_3, u_3} R R R (instHAdd.{u_3} R (Distrib.toAdd.{u_3} R (NonUnitalNonAssocSemiring.toDistrib.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5))))) x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.40 x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.42) x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.44) (HAdd.hAdd.{max u_1 u_2, max u_1 u_2, max u_1 u_2} (Prod.{u_1, u_2} M N) (Prod.{u_1, u_2} M N) (Prod.{u_1, u_2} M N) (instHAdd.{max u_1 u_2} (Prod.{u_1, u_2} M N) (AddCommMagma.toAdd.{max u_1 u_2} (Prod.{u_1, u_2} M N) (AddCommSemigroup.toAddCommMagma.{max u_1 u_2} (Prod.{u_1, u_2} M N) (AddCommMonoid.toAddCommSemigroup.{max u_1 u_2} (Prod.{u_1, u_2} M N) (Prod.instAddCommMonoid.{u_1, u_2} M N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11))))) (HSMul.hSMul.{u_3, max u_1 u_2, max u_1 u_2} R (Prod.{u_1, u_2} M N) (Prod.{u_1, u_2} M N) (instHSMul.{u_3, max u_1 u_2} R (Prod.{u_1, u_2} M N) (SemigroupAction.toSMul.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5))) (MulAction.toSemigroupAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (Prod.instAddMonoid.{u_1, u_2} M N (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11)) (Prod.distribMulAction.{u_1, u_2, u_3} M N R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.14) (Module.toDistribMulAction.{u_3, u_2} R N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.18)))))) x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.40 x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.44) (HSMul.hSMul.{u_3, max u_1 u_2, max u_1 u_2} R (Prod.{u_1, u_2} M N) (Prod.{u_1, u_2} M N) (instHSMul.{u_3, max u_1 u_2} R (Prod.{u_1, u_2} M N) (SemigroupAction.toSMul.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5))) (MulAction.toSemigroupAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (Prod.instAddMonoid.{u_1, u_2} M N (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11)) (Prod.distribMulAction.{u_1, u_2, u_3} M N R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.14) (Module.toDistribMulAction.{u_3, u_2} R N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.18)))))) x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.42 x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.44))","typeFull":"∀ {R : Type u_3} {M : Type u_1} {N : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N]\n  [inst_3 : Module R M] [inst_4 : Module R N] (x x_1 : R) (x_2 : M × N), (x + x_1) • x_2 = x • x_2 + x_1 • x_2","typeReadable":"∀ {R : Type u_3} {M : Type u_1} {N : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N]\n  [inst_3 : Module R M] [inst_4 : Module R N] (x x_1 : R) (x_2 : M × N), (x + x_1) • x_2 = x • x_2 + x_1 • x_2","typeReferences":[["Module"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["Prod","instAddCommMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Monoid","toSemigroup"],["Eq"],["Prod","distribMulAction"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Prod"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["Prod","instAddMonoid"],["MulAction","toSemigroupAction"],["Semiring"]],"valueReferences":[["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["Prod","instAddCommMonoid"],["Prod","fst"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Monoid","toSemigroup"],["Prod","distribMulAction"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Prod","ext"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Prod","snd"],["Prod"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommMonoid","toAddCommSemigroup"],["HSMul","hSMul"],["Prod","instAddMonoid"],["MulAction","toSemigroupAction"],["add_smul"]]},{"isProp":true,"kind":"theorem","name":["Prod","instModule","_proof_2"],"typeFallback":"forall {R : Type.{u_3}} {M : Type.{u_1}} {N : Type.{u_2}} [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 : Semiring.{u_3} R] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11 : AddCommMonoid.{u_2} N] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.14 : Module.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.18 : Module.{u_3, u_2} R N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11] (x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.49 : Prod.{u_1, u_2} M N), Eq.{succ (max u_1 u_2)} (Prod.{u_1, u_2} M N) (HSMul.hSMul.{u_3, max u_1 u_2, max u_1 u_2} R (Prod.{u_1, u_2} M N) (Prod.{u_1, u_2} M N) (instHSMul.{u_3, max u_1 u_2} R (Prod.{u_1, u_2} M N) (SemigroupAction.toSMul.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (Monoid.toSemigroup.{u_3} R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5))) (MulAction.toSemigroupAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (DistribMulAction.toMulAction.{u_3, max u_2 u_1} R (Prod.{u_1, u_2} M N) (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (Prod.instAddMonoid.{u_1, u_2} M N (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11)) (Prod.distribMulAction.{u_1, u_2, u_3} M N R (MonoidWithZero.toMonoid.{u_3} R (Semiring.toMonoidWithZero.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)) (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8) (AddCommMonoid.toAddMonoid.{u_2} N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11) (Module.toDistribMulAction.{u_3, u_1} R M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.14) (Module.toDistribMulAction.{u_3, u_2} R N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.18)))))) (OfNat.ofNat.{u_3} R 0 (Zero.toOfNat0.{u_3} R (MulZeroClass.toZero.{u_3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} R (Semiring.toNonAssocSemiring.{u_3} R inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5)))))) x._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.49) (OfNat.ofNat.{max u_1 u_2} (Prod.{u_1, u_2} M N) 0 (Zero.toOfNat0.{max u_1 u_2} (Prod.{u_1, u_2} M N) (AddZero.toZero.{max u_1 u_2} (Prod.{u_1, u_2} M N) (AddZeroClass.toAddZero.{max u_1 u_2} (Prod.{u_1, u_2} M N) (AddMonoid.toAddZeroClass.{max u_1 u_2} (Prod.{u_1, u_2} M N) (AddCommMonoid.toAddMonoid.{max u_1 u_2} (Prod.{u_1, u_2} M N) (Prod.instAddCommMonoid.{u_1, u_2} M N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11)))))))","typeFull":"∀ {R : Type u_3} {M : Type u_1} {N : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N]\n  [inst_3 : Module R M] [inst_4 : Module R N] (x : M × N), 0 • x = 0","typeReadable":"∀ {R : Type u_3} {M : Type u_1} {N : Type u_2} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : AddCommMonoid N]\n  [inst_3 : Module R M] [inst_4 : Module R N] (x : M × N), 0 • x = 0","typeReferences":[["Module"],["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["Prod","instAddCommMonoid"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["Eq"],["Prod","distribMulAction"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Prod"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Prod","instAddMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["SemigroupAction","toSMul"],["AddCommMonoid","toAddMonoid"],["Prod","instAddCommMonoid"],["Prod","fst"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["instHSMul"],["Zero","toOfNat0"],["Monoid","toSemigroup"],["MulActionWithZero","toSMulWithZero"],["Prod","distribMulAction"],["Prod","ext"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Module","toMulActionWithZero"],["Prod","snd"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["Prod"],["Module","toDistribMulAction"],["MulZeroClass","toZero"],["HSMul","hSMul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["zero_smul"],["Prod","instAddMonoid"],["MulAction","toSemigroupAction"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Prod","instModule"],"typeFallback":"forall {R : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 : AddCommMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11 : AddCommMonoid.{u_3} N] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.14 : Module.{u_1, u_2} R M inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.18 : Module.{u_1, u_3} R N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11], Module.{u_1, max u_3 u_2} R (Prod.{u_2, u_3} M N) inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.5 (Prod.instAddCommMonoid.{u_2, u_3} M N inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Prod.955123615._hygCtx._hyg.11)","typeFull":"{R : Type u_1} →\n  {M : Type u_2} →\n    {N : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] → [inst_2 : AddCommMonoid N] → [Module R M] → [Module R N] → Module R (M × N)","typeReadable":"{R : Type u_1} →\n  {M : Type u_2} →\n    {N : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid M] → [inst_2 : AddCommMonoid N] → [Module R M] → [Module R N] → Module R (M × N)","typeReferences":[["Prod"],["AddCommMonoid"],["Module"],["Prod","instAddCommMonoid"],["Semiring"]],"valueReferences":[["Prod"],["Module","toDistribMulAction"],["Prod","distribMulAction"],["Module","mk"],["Prod","instModule","_proof_1"],["Prod","instModule","_proof_2"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["Prod","instAddCommMonoid"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Notation.Pi.Basic.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Pi","single_op₂"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} {O : ι -> Type.{u_8}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.27 : forall (i : ι), Zero.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.36 : forall (i : ι), Zero.{u_8} (O i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (op : forall (i : ι), (M i) -> (N i) -> (O i)), (forall (i : ι), Eq.{succ u_8} (O i) (op i (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.18 i))) (OfNat.ofNat.{u_7} (N i) 0 (Zero.toOfNat0.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.27 i)))) (OfNat.ofNat.{u_8} (O i) 0 (Zero.toOfNat0.{u_8} (O i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.36 i)))) -> (forall (i : ι) (x : M i) (y : N i), Eq.{max (succ u_1) (succ u_8)} (forall (j : ι), O j) (Pi.single.{u_1, u_8} ι O inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 i (op i x y)) (fun (j : ι) => op j (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 i x j) (Pi.single.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 i y j)))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → Zero (M i)]\n  [inst_1 : (i : ι) → Zero (N i)] [inst_2 : (i : ι) → Zero (O i)] [inst_3 : DecidableEq ι]\n  (op : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), op i 0 0 = 0) →\n    ∀ (i : ι) (x : M i) (y : N i), Pi.single i (op i x y) = fun j => op j (Pi.single i x j) (Pi.single i y j)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → Zero (M i)]\n  [inst_1 : (i : ι) → Zero (N i)] [inst_2 : (i : ι) → Zero (O i)] [inst_3 : DecidableEq ι]\n  (op : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), op i 0 0 = 0) →\n    ∀ (i : ι) (x : M i) (y : N i), Pi.single i (op i x y) = fun j => op j (Pi.single i x j) (Pi.single i y j)","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["Pi","single"],["OfNat","ofNat"]],"valueReferences":[["Pi","apply_single₂"],["funext"],["Eq","symm"],["Pi","single"]]},{"isProp":true,"kind":"theorem","name":["Pi","single","congr_simp"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] {inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι} [inst_1._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) (x : M i) (x_1 : M i), (Eq.{succ u_6} (M i) x x_1) -> (forall (j : ι), Eq.{succ u_6} (M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 i x j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 inst_1._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 i x_1 j))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] {inst_1 : DecidableEq ι} [inst_2 : DecidableEq ι]\n  (i : ι) (x x_1 : M i), x = x_1 → ∀ (j : ι), Pi.single i x j = Pi.single i x_1 j","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] {inst_1 : DecidableEq ι} [inst_2 : DecidableEq ι]\n  (i : ι) (x x_1 : M i), x = x_1 → ∀ (j : ι), Pi.single i x j = Pi.single i x_1 j","typeReferences":[["DecidableEq"],["Zero"],["Eq"],["Pi","single"]],"valueReferences":[["Decidable"],["Subsingleton","elim"],["Eq","refl"],["DecidableEq"],["instSubsingletonDecidable"],["Eq","rec"],["Eq"],["Eq","ndrec"],["Pi","instSubsingleton"],["Pi","single"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_op₂"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} {O : ι -> Type.{u_8}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.27 : forall (i : ι), One.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.36 : forall (i : ι), One.{u_8} (O i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (op : forall (i : ι), (M i) -> (N i) -> (O i)), (forall (i : ι), Eq.{succ u_8} (O i) (op i (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.18 i))) (OfNat.ofNat.{u_7} (N i) 1 (One.toOfNat1.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.27 i)))) (OfNat.ofNat.{u_8} (O i) 1 (One.toOfNat1.{u_8} (O i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.36 i)))) -> (forall (i : ι) (x : M i) (y : N i), Eq.{max (succ u_1) (succ u_8)} (forall (j : ι), O j) (Pi.mulSingle.{u_1, u_8} ι O inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 i (op i x y)) (fun (j : ι) => op j (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 i x j) (Pi.mulSingle.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2315853854._hygCtx._hyg.45 i y j)))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → One (M i)]\n  [inst_1 : (i : ι) → One (N i)] [inst_2 : (i : ι) → One (O i)] [inst_3 : DecidableEq ι]\n  (op : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), op i 1 1 = 1) →\n    ∀ (i : ι) (x : M i) (y : N i), Pi.mulSingle i (op i x y) = fun j => op j (Pi.mulSingle i x j) (Pi.mulSingle i y j)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → One (M i)]\n  [inst_1 : (i : ι) → One (N i)] [inst_2 : (i : ι) → One (O i)] [inst_3 : DecidableEq ι]\n  (op : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), op i 1 1 = 1) →\n    ∀ (i : ι) (x : M i) (y : N i), Pi.mulSingle i (op i x y) = fun j => op j (Pi.mulSingle i x j) (Pi.mulSingle i y j)","typeReferences":[["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","apply_mulSingle₂"],["Pi","mulSingle"],["funext"],["Eq","symm"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_inj","_simp_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) {x : M i} {y : M i}, Eq.{1} Prop (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i x) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i y)) (Eq.{succ u_6} (M i) x y)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  (Pi.single i x = Pi.single i y) = (x = y)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  (Pi.single i x = Pi.single i y) = (x = y)","typeReferences":[["DecidableEq"],["Zero"],["Eq"],["Pi","single"]],"valueReferences":[["Pi","single_inj"],["Eq"],["Pi","single"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_eq_of_ne'"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {i' : ι}, (Ne.{succ u_1} ι i i') -> (forall (x : M i), Eq.{succ u_6} (M i') (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.45 i x i') (OfNat.ofNat.{u_6} (M i') 1 (One.toOfNat1.{u_6} (M i') (inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.18 i'))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i ≠ i' → ∀ (x : M i), Pi.mulSingle i x i' = 1","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i ≠ i' → ∀ (x : M i), Pi.mulSingle i x i' = 1","typeReferences":[["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Ne"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Ne","symm"],["Pi","mulSingle_eq_of_ne"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_one"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι), Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.45 i (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18 i)))) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 1 (One.toOfNat1.{max u_1 u_6} (forall (j : ι), M j) (Pi.instOne.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18)))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι), Pi.mulSingle i 1 = 1","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι), Pi.mulSingle i 1 = 1","typeReferences":[["Pi","instOne"],["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","instOne"],["Function","update_eq_self"],["One","toOfNat1"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_inj"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) {x : M i} {y : M i}, Iff (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i x) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i y)) (Eq.{succ u_6} (M i) x y)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  Pi.mulSingle i x = Pi.mulSingle i y ↔ x = y","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  Pi.mulSingle i x = Pi.mulSingle i y ↔ x = y","typeReferences":[["Pi","mulSingle"],["Iff"],["DecidableEq"],["One"],["Eq"]],"valueReferences":[["Pi","mulSingle_injective"],["Function","Injective","eq_iff"],["Pi","mulSingle"]]},{"isProp":true,"kind":"theorem","name":["Pi","single","eq_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) (x : M i), Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 i x) (Function.update.{succ u_1, succ u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 (OfNat.ofNat.{max u_1 u_6} (forall (a : ι), M a) 0 (Zero.toOfNat0.{max u_1 u_6} (forall (a : ι), M a) (Pi.instZero.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18))) i x)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.single i x = Function.update 0 i x","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.single i x = Function.update 0 i x","typeReferences":[["Function","update"],["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["OfNat","ofNat"],["Pi","single"],["Pi","instZero"]],"valueReferences":[["Eq","refl"],["Pi","single"]]},{"isProp":true,"kind":"theorem","name":["Pi","apply_single"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.27 : forall (i : ι), Zero.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (f' : forall (i : ι), (M i) -> (N i)), (forall (i : ι), Eq.{succ u_7} (N i) (f' i (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.18 i)))) (OfNat.ofNat.{u_7} (N i) 0 (Zero.toOfNat0.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.27 i)))) -> (forall (i : ι) (x : M i) (j : ι), Eq.{succ u_7} (N j) (f' j (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.45 i x j)) (Pi.single.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.45 i (f' i x) j))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → Zero (M i)] [inst_1 : (i : ι) → Zero (N i)]\n  [inst_2 : DecidableEq ι] (f' : (i : ι) → M i → N i),\n  (∀ (i : ι), f' i 0 = 0) → ∀ (i : ι) (x : M i) (j : ι), f' j (Pi.single i x j) = Pi.single i (f' i x) j","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → Zero (M i)] [inst_1 : (i : ι) → Zero (N i)]\n  [inst_2 : DecidableEq ι] (f' : (i : ι) → M i → N i),\n  (∀ (i : ι), f' i 0 = 0) → ∀ (i : ι) (x : M i) (j : ι), f' j (Pi.single i x j) = Pi.single i (f' i x) j","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["Pi","single"],["OfNat","ofNat"]],"valueReferences":[["Eq","mp"],["Function","update"],["Function","update","congr_simp"],["Pi","single"],["OfNat","ofNat"],["congrArg"],["Eq","refl"],["funext"],["id"],["Zero","toOfNat0"],["Function","apply_update"],["Eq"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_eq_same"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) (x : M i), Eq.{succ u_6} (M i) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.45 i x i) x","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.single i x i = x","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.single i x i = x","typeReferences":[["DecidableEq"],["Zero"],["Eq"],["Pi","single"]],"valueReferences":[["Function","update_self"],["Zero","toOfNat0"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","apply_mulSingle₂"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} {O : ι -> Type.{u_8}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.27 : forall (i : ι), One.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.36 : forall (i : ι), One.{u_8} (O i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (f' : forall (i : ι), (M i) -> (N i) -> (O i)), (forall (i : ι), Eq.{succ u_8} (O i) (f' i (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.18 i))) (OfNat.ofNat.{u_7} (N i) 1 (One.toOfNat1.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.27 i)))) (OfNat.ofNat.{u_8} (O i) 1 (One.toOfNat1.{u_8} (O i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.36 i)))) -> (forall (i : ι) (x : M i) (y : N i) (j : ι), Eq.{succ u_8} (O j) (f' j (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 i x j) (Pi.mulSingle.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 i y j)) (Pi.mulSingle.{u_1, u_8} ι O inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 i (f' i x y) j))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → One (M i)]\n  [inst_1 : (i : ι) → One (N i)] [inst_2 : (i : ι) → One (O i)] [inst_3 : DecidableEq ι]\n  (f' : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), f' i 1 1 = 1) →\n    ∀ (i : ι) (x : M i) (y : N i) (j : ι), f' j (Pi.mulSingle i x j) (Pi.mulSingle i y j) = Pi.mulSingle i (f' i x y) j","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → One (M i)]\n  [inst_1 : (i : ι) → One (N i)] [inst_2 : (i : ι) → One (O i)] [inst_3 : DecidableEq ι]\n  (f' : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), f' i 1 1 = 1) →\n    ∀ (i : ι) (x : M i) (y : N i) (j : ι), f' j (Pi.mulSingle i x j) (Pi.mulSingle i y j) = Pi.mulSingle i (f' i x y) j","typeReferences":[["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Eq","trans"],["True"],["Pi","mulSingle_eq_same"],["Pi","mulSingle"],["OfNat","ofNat"],["congrArg"],["eq_self"],["of_eq_true"],["One","toOfNat1"],["Pi","mulSingle_eq_of_ne"],["congr"],["dite"],["Eq"],["Eq","ndrec"]]},{"isProp":false,"kind":"definition","name":["Pi","single"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι), (M i) -> (forall (j : ι), M j)","typeFull":"{ι : Type u_1} → {M : ι → Type u_6} → [(i : ι) → Zero (M i)] → [DecidableEq ι] → (i : ι) → M i → (j : ι) → M j","typeReadable":"{ι : Type u_1} → {M : ι → Type u_6} → [(i : ι) → Zero (M i)] → [DecidableEq ι] → (i : ι) → M i → (j : ι) → M j","typeReferences":[["DecidableEq"],["Zero"]],"valueReferences":[["Function","update"],["Zero","toOfNat0"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_apply"],"typeFallback":"forall {ι : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.52 : One.{u_9} M] (i : ι) (x : M) (i' : ι), Eq.{succ u_9} M (Pi.mulSingle.{u_1, u_9} ι (fun (a._@._internal._hyg.0 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.45 i x i') (ite.{succ u_9} M (Eq.{succ u_1} ι i' i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.45 i' i) x (OfNat.ofNat.{u_9} M 1 (One.toOfNat1.{u_9} M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.52)))","typeFull":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] (i : ι) (x : M) (i' : ι),\n  Pi.mulSingle i x i' = if i' = i then x else 1","typeReadable":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] (i : ι) (x : M) (i' : ι),\n  Pi.mulSingle i x i' = if i' = i then x else 1","typeReferences":[["One","toOfNat1"],["ite"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","instOne"],["One","toOfNat1"],["Function","update_apply"],["OfNat","ofNat"]]},{"isProp":false,"kind":"definition","name":["Pi","mulSingle"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι), (M i) -> (forall (j : ι), M j)","typeFull":"{ι : Type u_1} → {M : ι → Type u_6} → [(i : ι) → One (M i)] → [DecidableEq ι] → (i : ι) → M i → (j : ι) → M j","typeReadable":"{ι : Type u_1} → {M : ι → Type u_6} → [(i : ι) → One (M i)] → [DecidableEq ι] → (i : ι) → M i → (j : ι) → M j","typeReferences":[["DecidableEq"],["One"]],"valueReferences":[["Pi","instOne"],["One","toOfNat1"],["Function","update"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_eq_of_ne'"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {i' : ι}, (Ne.{succ u_1} ι i i') -> (forall (x : M i), Eq.{succ u_6} (M i') (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.45 i x i') (OfNat.ofNat.{u_6} (M i') 0 (Zero.toOfNat0.{u_6} (M i') (inst._@.Mathlib.Algebra.Notation.Pi.Basic.594283409._hygCtx._hyg.18 i'))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i ≠ i' → ∀ (x : M i), Pi.single i x i' = 0","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i ≠ i' → ∀ (x : M i), Pi.single i x i' = 0","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Ne"],["Zero"],["Eq"],["OfNat","ofNat"],["Pi","single"]],"valueReferences":[["Pi","single_eq_of_ne"],["Ne","symm"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_inj","_simp_2"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) {x : M i} {y : M i}, Eq.{1} Prop (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i x) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i y)) (Eq.{succ u_6} (M i) x y)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  (Pi.mulSingle i x = Pi.mulSingle i y) = (x = y)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  (Pi.mulSingle i x = Pi.mulSingle i y) = (x = y)","typeReferences":[["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"]],"valueReferences":[["Pi","mulSingle"],["Pi","mulSingle_inj"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_op"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.27 : forall (i : ι), One.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (op : forall (i : ι), (M i) -> (N i)), (forall (i : ι), Eq.{succ u_7} (N i) (op i (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.18 i)))) (OfNat.ofNat.{u_7} (N i) 1 (One.toOfNat1.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.27 i)))) -> (forall (i : ι) (x : M i), Eq.{max (succ u_1) (succ u_7)} (forall (j : ι), N j) (Pi.mulSingle.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.45 i (op i x)) (fun (j : ι) => op j (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.45 i x j)))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → One (M i)] [inst_1 : (i : ι) → One (N i)]\n  [inst_2 : DecidableEq ι] (op : (i : ι) → M i → N i),\n  (∀ (i : ι), op i 1 = 1) → ∀ (i : ι) (x : M i), Pi.mulSingle i (op i x) = fun j => op j (Pi.mulSingle i x j)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → One (M i)] [inst_1 : (i : ι) → One (N i)]\n  [inst_2 : DecidableEq ι] (op : (i : ι) → M i → N i),\n  (∀ (i : ι), op i 1 = 1) → ∀ (i : ι) (x : M i), Pi.mulSingle i (op i x) = fun j => op j (Pi.mulSingle i x j)","typeReferences":[["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","mulSingle"],["funext"],["Eq","symm"],["Pi","apply_mulSingle"]]},{"isProp":true,"kind":"theorem","name":["Pi","curry_mulSingle"],"typeFallback":"forall {ι : Type.{u_1}} {ι' : Type.{u_2}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52 : One.{u_9} M] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.55 : DecidableEq.{succ u_2} ι'] (i : Prod.{u_1, u_2} ι ι') (b : M), Eq.{max (max (succ u_1) (succ u_2)) (succ u_9)} (ι -> ι' -> M) (Function.curry.{u_1, u_2, succ u_9} ι ι' M (Pi.mulSingle.{max u_2 u_1, u_9} (Prod.{u_1, u_2} ι ι') (fun (a._@._internal._hyg.0 : Prod.{u_1, u_2} ι ι') => M) (fun (i : Prod.{u_1, u_2} ι ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52) (fun (a : Prod.{u_1, u_2} ι ι') (b : Prod.{u_1, u_2} ι ι') => instDecidableEqProd.{u_1, u_2} ι ι' inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.55 a b) i b)) (Pi.mulSingle.{u_1, max u_2 u_9} ι (fun (j : ι) => ι' -> M) (fun (i : ι) => Pi.instOne.{u_2, u_9} ι' (fun (a._@._internal._hyg.0 : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52)) inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.45 (Prod.fst.{u_1, u_2} ι ι' i) (Pi.mulSingle.{u_2, u_9} ι' (fun (j : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.55 (Prod.snd.{u_1, u_2} ι ι' i) b))","typeFull":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] [inst_2 : DecidableEq ι']\n  (i : ι × ι') (b : M), Function.curry (Pi.mulSingle i b) = Pi.mulSingle i.fst (Pi.mulSingle i.snd b)","typeReadable":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] [inst_2 : DecidableEq ι']\n  (i : ι × ι') (b : M), Function.curry (Pi.mulSingle i b) = Pi.mulSingle i.fst (Pi.mulSingle i.snd b)","typeReferences":[["Pi","instOne"],["Prod"],["instDecidableEqProd"],["Pi","mulSingle"],["DecidableEq"],["One"],["Prod","snd"],["Function","curry"],["Eq"],["Prod","fst"]],"valueReferences":[["Pi","instOne"],["Prod"],["One","toOfNat1"],["Function","curry_update"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_eq_of_ne"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {i' : ι}, (Ne.{succ u_1} ι i' i) -> (forall (x : M i), Eq.{succ u_6} (M i') (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.45 i x i') (OfNat.ofNat.{u_6} (M i') 0 (Zero.toOfNat0.{u_6} (M i') (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.18 i'))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i' ≠ i → ∀ (x : M i), Pi.single i x i' = 0","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i' ≠ i → ∀ (x : M i), Pi.single i x i' = 0","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Ne"],["Zero"],["Eq"],["OfNat","ofNat"],["Pi","single"]],"valueReferences":[["Zero","toOfNat0"],["Function","update_of_ne"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_injective"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι), Function.Injective.{succ u_6, max (succ u_1) (succ u_6)} (M i) (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.45 i)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι),\n  Function.Injective (Pi.mulSingle i)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι),\n  Function.Injective (Pi.mulSingle i)","typeReferences":[["Pi","mulSingle"],["DecidableEq"],["One"],["Function","Injective"]],"valueReferences":[["Pi","instOne"],["One","toOfNat1"],["Function","update_injective"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_eq_one_iff"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {x : M i}, Iff (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 i x) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 1 (One.toOfNat1.{max u_1 u_6} (forall (j : ι), M j) (Pi.instOne.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18)))) (Eq.{succ u_6} (M i) x (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 i))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.mulSingle i x = 1 ↔ x = 1","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.mulSingle i x = 1 ↔ x = 1","typeReferences":[["Pi","instOne"],["One","toOfNat1"],["Pi","mulSingle"],["Iff"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","instOne"],["Pi","mulSingle_eq_same"],["Pi","mulSingle"],["Pi","mulSingle_one"],["OfNat","ofNat"],["Iff","intro"],["congrArg"],["Pi","one_apply"],["One","toOfNat1"],["Eq","refl"],["id"],["Eq","symm"],["Eq","mpr"],["Eq","rec"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_eq_of_ne"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {i' : ι}, (Ne.{succ u_1} ι i' i) -> (forall (x : M i), Eq.{succ u_6} (M i') (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.45 i x i') (OfNat.ofNat.{u_6} (M i') 1 (One.toOfNat1.{u_6} (M i') (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4112891430._hygCtx._hyg.18 i'))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i' ≠ i → ∀ (x : M i), Pi.mulSingle i x i' = 1","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i i' : ι},\n  i' ≠ i → ∀ (x : M i), Pi.mulSingle i x i' = 1","typeReferences":[["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Ne"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","instOne"],["One","toOfNat1"],["Function","update_of_ne"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_inj"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) {x : M i} {y : M i}, Iff (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i x) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3539862169._hygCtx._hyg.45 i y)) (Eq.{succ u_6} (M i) x y)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  Pi.single i x = Pi.single i y ↔ x = y","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι) {x y : M i},\n  Pi.single i x = Pi.single i y ↔ x = y","typeReferences":[["Iff"],["DecidableEq"],["Zero"],["Eq"],["Pi","single"]],"valueReferences":[["Pi","single_injective"],["Function","Injective","eq_iff"],["Pi","single"]]},{"isProp":true,"kind":"theorem","name":["Pi","uncurry_single_single"],"typeFallback":"forall {ι : Type.{u_1}} {ι' : Type.{u_2}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52 : Zero.{u_9} M] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.55 : DecidableEq.{succ u_2} ι'] (i : ι) (i' : ι') (b : M), Eq.{max (max (succ u_1) (succ u_2)) (succ u_9)} ((Prod.{u_1, u_2} ι ι') -> M) (Function.uncurry.{u_1, u_2, succ u_9} ι ι' M (Pi.single.{u_1, max u_2 u_9} ι (fun (a._@._internal._hyg.0 : ι) => ι' -> M) (fun (i : ι) => Pi.instZero.{u_2, u_9} ι' (fun (a._@._internal._hyg.0 : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52)) inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.45 i (Pi.single.{u_2, u_9} ι' (fun (a._@._internal._hyg.0 : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.55 i' b))) (Pi.single.{max u_2 u_1, u_9} (Prod.{u_1, u_2} ι ι') (fun (j : Prod.{u_1, u_2} ι ι') => M) (fun (i : Prod.{u_1, u_2} ι ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52) (fun (a : Prod.{u_1, u_2} ι ι') (b : Prod.{u_1, u_2} ι ι') => instDecidableEqProd.{u_1, u_2} ι ι' inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.55 a b) (Prod.mk.{u_1, u_2} ι ι' i i') b)","typeFull":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] [inst_2 : DecidableEq ι']\n  (i : ι) (i' : ι') (b : M), Function.uncurry (Pi.single i (Pi.single i' b)) = Pi.single (i, i') b","typeReadable":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] [inst_2 : DecidableEq ι']\n  (i : ι) (i' : ι') (b : M), Function.uncurry (Pi.single i (Pi.single i' b)) = Pi.single (i, i') b","typeReferences":[["Prod"],["Prod","mk"],["instDecidableEqProd"],["DecidableEq"],["Zero"],["Eq"],["Pi","single"],["Pi","instZero"],["Function","uncurry"]],"valueReferences":[["Function","uncurry_update_update"],["Zero","toOfNat0"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","uncurry_mulSingle_mulSingle"],"typeFallback":"forall {ι : Type.{u_1}} {ι' : Type.{u_2}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52 : One.{u_9} M] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.55 : DecidableEq.{succ u_2} ι'] (i : ι) (i' : ι') (b : M), Eq.{max (max (succ u_1) (succ u_2)) (succ u_9)} ((Prod.{u_1, u_2} ι ι') -> M) (Function.uncurry.{u_1, u_2, succ u_9} ι ι' M (Pi.mulSingle.{u_1, max u_2 u_9} ι (fun (a._@._internal._hyg.0 : ι) => ι' -> M) (fun (i : ι) => Pi.instOne.{u_2, u_9} ι' (fun (a._@._internal._hyg.0 : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52)) inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.45 i (Pi.mulSingle.{u_2, u_9} ι' (fun (a._@._internal._hyg.0 : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.55 i' b))) (Pi.mulSingle.{max u_2 u_1, u_9} (Prod.{u_1, u_2} ι ι') (fun (j : Prod.{u_1, u_2} ι ι') => M) (fun (i : Prod.{u_1, u_2} ι ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.52) (fun (a : Prod.{u_1, u_2} ι ι') (b : Prod.{u_1, u_2} ι ι') => instDecidableEqProd.{u_1, u_2} ι ι' inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4007770665._hygCtx._hyg.55 a b) (Prod.mk.{u_1, u_2} ι ι' i i') b)","typeFull":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] [inst_2 : DecidableEq ι']\n  (i : ι) (i' : ι') (b : M), Function.uncurry (Pi.mulSingle i (Pi.mulSingle i' b)) = Pi.mulSingle (i, i') b","typeReadable":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] [inst_2 : DecidableEq ι']\n  (i : ι) (i' : ι') (b : M), Function.uncurry (Pi.mulSingle i (Pi.mulSingle i' b)) = Pi.mulSingle (i, i') b","typeReferences":[["Pi","instOne"],["Prod"],["Prod","mk"],["instDecidableEqProd"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["Function","uncurry"]],"valueReferences":[["Pi","instOne"],["Function","uncurry_update_update"],["One","toOfNat1"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_op"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.27 : forall (i : ι), Zero.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (op : forall (i : ι), (M i) -> (N i)), (forall (i : ι), Eq.{succ u_7} (N i) (op i (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.18 i)))) (OfNat.ofNat.{u_7} (N i) 0 (Zero.toOfNat0.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.27 i)))) -> (forall (i : ι) (x : M i), Eq.{max (succ u_1) (succ u_7)} (forall (j : ι), N j) (Pi.single.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.45 i (op i x)) (fun (j : ι) => op j (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.519996119._hygCtx._hyg.45 i x j)))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → Zero (M i)] [inst_1 : (i : ι) → Zero (N i)]\n  [inst_2 : DecidableEq ι] (op : (i : ι) → M i → N i),\n  (∀ (i : ι), op i 0 = 0) → ∀ (i : ι) (x : M i), Pi.single i (op i x) = fun j => op j (Pi.single i x j)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → Zero (M i)] [inst_1 : (i : ι) → Zero (N i)]\n  [inst_2 : DecidableEq ι] (op : (i : ι) → M i → N i),\n  (∀ (i : ι), op i 0 = 0) → ∀ (i : ι) (x : M i), Pi.single i (op i x) = fun j => op j (Pi.single i x j)","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["Pi","single"],["OfNat","ofNat"]],"valueReferences":[["funext"],["Eq","symm"],["Pi","apply_single"],["Pi","single"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_eq_same"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) (x : M i), Eq.{succ u_6} (M i) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3780085894._hygCtx._hyg.45 i x i) x","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.mulSingle i x i = x","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.mulSingle i x i = x","typeReferences":[["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"]],"valueReferences":[["Pi","instOne"],["One","toOfNat1"],["Function","update_self"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_comm"],"typeFallback":"forall {ι : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.52 : Zero.{u_9} M] (i : ι) (x : M) (j : ι), Eq.{succ u_9} M (Pi.single.{u_1, u_9} ι (fun (a._@._internal._hyg.0 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.45 i x j) (Pi.single.{u_1, u_9} ι (fun (a._@._internal._hyg.0 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.45 j x i)","typeFull":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] (i : ι) (x : M) (j : ι),\n  Pi.single i x j = Pi.single j x i","typeReadable":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] (i : ι) (x : M) (j : ι),\n  Pi.single i x j = Pi.single j x i","typeReferences":[["DecidableEq"],["Zero"],["Eq"],["Pi","single"]],"valueReferences":[["Eq","trans"],["True"],["ite"],["eq_comm"],["Pi","single_apply"],["Pi","single"],["OfNat","ofNat"],["congrArg"],["ite_congr"],["eq_self"],["of_eq_true"],["Eq","refl"],["congr"],["Zero","toOfNat0"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Notation","Pi","Basic",0,"Pi","mulSingle_comm","_simp_1_1"],"typeFallback":"forall {α : Sort.{u_1}} {a : α} {b : α}, Eq.{1} Prop (Eq.{u_1} α a b) (Eq.{u_1} α b a)","typeFull":"∀ {α : Sort u_1} {a b : α}, (a = b) = (b = a)","typeReadable":"∀ {α : Sort u_1} {a b : α}, (a = b) = (b = a)","typeReferences":[["Eq"]],"valueReferences":[["eq_comm"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Pi","apply_mulSingle"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.27 : forall (i : ι), One.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (f' : forall (i : ι), (M i) -> (N i)), (forall (i : ι), Eq.{succ u_7} (N i) (f' i (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.18 i)))) (OfNat.ofNat.{u_7} (N i) 1 (One.toOfNat1.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.27 i)))) -> (forall (i : ι) (x : M i) (j : ι), Eq.{succ u_7} (N j) (f' j (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.45 i x j)) (Pi.mulSingle.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.3978888526._hygCtx._hyg.45 i (f' i x) j))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → One (M i)] [inst_1 : (i : ι) → One (N i)]\n  [inst_2 : DecidableEq ι] (f' : (i : ι) → M i → N i),\n  (∀ (i : ι), f' i 1 = 1) → ∀ (i : ι) (x : M i) (j : ι), f' j (Pi.mulSingle i x j) = Pi.mulSingle i (f' i x) j","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} [inst : (i : ι) → One (M i)] [inst_1 : (i : ι) → One (N i)]\n  [inst_2 : DecidableEq ι] (f' : (i : ι) → M i → N i),\n  (∀ (i : ι), f' i 1 = 1) → ∀ (i : ι) (x : M i) (j : ι), f' j (Pi.mulSingle i x j) = Pi.mulSingle i (f' i x) j","typeReferences":[["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","instOne"],["Pi","mulSingle"],["Eq","mp"],["Function","update"],["Function","update","congr_simp"],["OfNat","ofNat"],["congrArg"],["One","toOfNat1"],["Eq","refl"],["funext"],["id"],["Function","apply_update"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_eq_zero_iff"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {x : M i}, Iff (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 i x) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 0 (Zero.toOfNat0.{max u_1 u_6} (forall (j : ι), M j) (Pi.instZero.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18)))) (Eq.{succ u_6} (M i) x (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 i))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.single i x = 0 ↔ x = 0","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.single i x = 0 ↔ x = 0","typeReferences":[["Iff"],["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["OfNat","ofNat"],["Pi","single"],["Pi","instZero"]],"valueReferences":[["Pi","single"],["OfNat","ofNat"],["Iff","intro"],["congrArg"],["Pi","single_eq_same"],["Pi","single_zero"],["Eq","refl"],["id"],["Eq","symm"],["Zero","toOfNat0"],["Eq","mpr"],["Eq","rec"],["Eq"],["Pi","zero_apply"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle","congr_simp"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] {inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι} [inst_1._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) (x : M i) (x_1 : M i), (Eq.{succ u_6} (M i) x x_1) -> (forall (j : ι), Eq.{succ u_6} (M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 i x j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 inst_1._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 i x_1 j))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] {inst_1 : DecidableEq ι} [inst_2 : DecidableEq ι]\n  (i : ι) (x x_1 : M i), x = x_1 → ∀ (j : ι), Pi.mulSingle i x j = Pi.mulSingle i x_1 j","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] {inst_1 : DecidableEq ι} [inst_2 : DecidableEq ι]\n  (i : ι) (x x_1 : M i), x = x_1 → ∀ (j : ι), Pi.mulSingle i x j = Pi.mulSingle i x_1 j","typeReferences":[["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"]],"valueReferences":[["Decidable"],["Pi","mulSingle"],["Subsingleton","elim"],["Eq","refl"],["DecidableEq"],["instSubsingletonDecidable"],["Eq","rec"],["Eq"],["Eq","ndrec"],["Pi","instSubsingleton"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_zero"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι), Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.45 i (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18 i)))) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 0 (Zero.toOfNat0.{max u_1 u_6} (forall (j : ι), M j) (Pi.instZero.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.2797108460._hygCtx._hyg.18)))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι), Pi.single i 0 = 0","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι), Pi.single i 0 = 0","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["OfNat","ofNat"],["Pi","single"],["Pi","instZero"]],"valueReferences":[["Function","update_eq_self"],["Zero","toOfNat0"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_comm"],"typeFallback":"forall {ι : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.52 : One.{u_9} M] (i : ι) (x : M) (j : ι), Eq.{succ u_9} M (Pi.mulSingle.{u_1, u_9} ι (fun (a._@._internal._hyg.0 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.45 i x j) (Pi.mulSingle.{u_1, u_9} ι (fun (a._@._internal._hyg.0 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.608237876._hygCtx._hyg.45 j x i)","typeFull":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] (i : ι) (x : M) (j : ι),\n  Pi.mulSingle i x j = Pi.mulSingle j x i","typeReadable":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : One M] (i : ι) (x : M) (j : ι),\n  Pi.mulSingle i x j = Pi.mulSingle j x i","typeReferences":[["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"]],"valueReferences":[["Eq","trans"],["True"],["Pi","mulSingle"],["ite"],["_private","Mathlib","Algebra","Notation","Pi","Basic",0,"Pi","mulSingle_comm","_simp_1_1"],["OfNat","ofNat"],["congrArg"],["ite_congr"],["eq_self"],["of_eq_true"],["One","toOfNat1"],["Eq","refl"],["congr"],["Pi","mulSingle_apply"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_eq_one_iff","_simp_2"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {x : M i}, Eq.{1} Prop (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 i x) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 1 (One.toOfNat1.{max u_1 u_6} (forall (j : ι), M j) (Pi.instOne.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18)))) (Eq.{succ u_6} (M i) x (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 i))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  (Pi.mulSingle i x = 1) = (x = 1)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  (Pi.mulSingle i x = 1) = (x = 1)","typeReferences":[["Pi","instOne"],["One","toOfNat1"],["Pi","mulSingle"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","instOne"],["Pi","mulSingle_eq_one_iff"],["One","toOfNat1"],["Pi","mulSingle"],["Eq"],["OfNat","ofNat"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle","eq_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι) (x : M i), Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 i x) (Function.update.{succ u_1, succ u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.45 (OfNat.ofNat.{max u_1 u_6} (forall (a : ι), M a) 1 (One.toOfNat1.{max u_1 u_6} (forall (a : ι), M a) (Pi.instOne.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1883711519._hygCtx._hyg.18))) i x)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.mulSingle i x = Function.update 1 i x","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] (i : ι) (x : M i),\n  Pi.mulSingle i x = Function.update 1 i x","typeReferences":[["Pi","instOne"],["One","toOfNat1"],["Pi","mulSingle"],["Function","update"],["DecidableEq"],["One"],["Eq"],["OfNat","ofNat"]],"valueReferences":[["Pi","mulSingle"],["Eq","refl"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_injective"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (i : ι), Function.Injective.{succ u_6, max (succ u_1) (succ u_6)} (M i) (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.1709218144._hygCtx._hyg.45 i)","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι),\n  Function.Injective (Pi.single i)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] (i : ι),\n  Function.Injective (Pi.single i)","typeReferences":[["DecidableEq"],["Zero"],["Pi","single"],["Function","Injective"]],"valueReferences":[["Function","update_injective"],["Zero","toOfNat0"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","apply_single₂"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} {N : ι -> Type.{u_7}} {O : ι -> Type.{u_8}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.27 : forall (i : ι), Zero.{u_7} (N i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.36 : forall (i : ι), Zero.{u_8} (O i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] (f' : forall (i : ι), (M i) -> (N i) -> (O i)), (forall (i : ι), Eq.{succ u_8} (O i) (f' i (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.18 i))) (OfNat.ofNat.{u_7} (N i) 0 (Zero.toOfNat0.{u_7} (N i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.27 i)))) (OfNat.ofNat.{u_8} (O i) 0 (Zero.toOfNat0.{u_8} (O i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.36 i)))) -> (forall (i : ι) (x : M i) (y : N i) (j : ι), Eq.{succ u_8} (O j) (f' j (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 i x j) (Pi.single.{u_1, u_7} ι N inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.27 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 i y j)) (Pi.single.{u_1, u_8} ι O inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.36 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4075143953._hygCtx._hyg.45 i (f' i x y) j))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → Zero (M i)]\n  [inst_1 : (i : ι) → Zero (N i)] [inst_2 : (i : ι) → Zero (O i)] [inst_3 : DecidableEq ι]\n  (f' : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), f' i 0 0 = 0) →\n    ∀ (i : ι) (x : M i) (y : N i) (j : ι), f' j (Pi.single i x j) (Pi.single i y j) = Pi.single i (f' i x y) j","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} {N : ι → Type u_7} {O : ι → Type u_8} [inst : (i : ι) → Zero (M i)]\n  [inst_1 : (i : ι) → Zero (N i)] [inst_2 : (i : ι) → Zero (O i)] [inst_3 : DecidableEq ι]\n  (f' : (i : ι) → M i → N i → O i),\n  (∀ (i : ι), f' i 0 0 = 0) →\n    ∀ (i : ι) (x : M i) (y : N i) (j : ι), f' j (Pi.single i x j) (Pi.single i y j) = Pi.single i (f' i x y) j","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["Pi","single"],["OfNat","ofNat"]],"valueReferences":[["Eq","trans"],["True"],["Pi","single"],["OfNat","ofNat"],["congrArg"],["Pi","single_eq_same"],["eq_self"],["Pi","single_eq_of_ne"],["of_eq_true"],["congr"],["Zero","toOfNat0"],["dite"],["Eq"],["Eq","ndrec"]]},{"isProp":true,"kind":"theorem","name":["Pi","mulSingle_ne_one_iff"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18 : forall (i : ι), One.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {x : M i}, Iff (Ne.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.mulSingle.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.45 i x) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 1 (One.toOfNat1.{max u_1 u_6} (forall (j : ι), M j) (Pi.instOne.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18)))) (Ne.{succ u_6} (M i) x (OfNat.ofNat.{u_6} (M i) 1 (One.toOfNat1.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18 i))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.mulSingle i x ≠ 1 ↔ x ≠ 1","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → One (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.mulSingle i x ≠ 1 ↔ x ≠ 1","typeReferences":[["Pi","instOne"],["One","toOfNat1"],["Pi","mulSingle"],["Iff"],["DecidableEq"],["One"],["Ne"],["OfNat","ofNat"]],"valueReferences":[["Pi","instOne"],["Pi","mulSingle_eq_one_iff"],["One","toOfNat1"],["Pi","mulSingle"],["Iff","ne"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_eq_zero_iff","_simp_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {x : M i}, Eq.{1} Prop (Eq.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.45 i x) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 0 (Zero.toOfNat0.{max u_1 u_6} (forall (j : ι), M j) (Pi.instZero.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18)))) (Eq.{succ u_6} (M i) x (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.4063372067._hygCtx._hyg.18 i))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  (Pi.single i x = 0) = (x = 0)","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  (Pi.single i x = 0) = (x = 0)","typeReferences":[["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["OfNat","ofNat"],["Pi","single"],["Pi","instZero"]],"valueReferences":[["Pi","single_eq_zero_iff"],["Zero","toOfNat0"],["Eq"],["OfNat","ofNat"],["Pi","single"],["propext"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","curry_single"],"typeFallback":"forall {ι : Type.{u_1}} {ι' : Type.{u_2}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52 : Zero.{u_9} M] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.55 : DecidableEq.{succ u_2} ι'] (i : Prod.{u_1, u_2} ι ι') (b : M), Eq.{max (max (succ u_1) (succ u_2)) (succ u_9)} (ι -> ι' -> M) (Function.curry.{u_1, u_2, succ u_9} ι ι' M (Pi.single.{max u_2 u_1, u_9} (Prod.{u_1, u_2} ι ι') (fun (a._@._internal._hyg.0 : Prod.{u_1, u_2} ι ι') => M) (fun (i : Prod.{u_1, u_2} ι ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52) (fun (a : Prod.{u_1, u_2} ι ι') (b : Prod.{u_1, u_2} ι ι') => instDecidableEqProd.{u_1, u_2} ι ι' inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.45 inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.55 a b) i b)) (Pi.single.{u_1, max u_2 u_9} ι (fun (j : ι) => ι' -> M) (fun (i : ι) => Pi.instZero.{u_2, u_9} ι' (fun (a._@._internal._hyg.0 : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52)) inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.45 (Prod.fst.{u_1, u_2} ι ι' i) (Pi.single.{u_2, u_9} ι' (fun (j : ι') => M) (fun (i : ι') => inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.691379539._hygCtx._hyg.55 (Prod.snd.{u_1, u_2} ι ι' i) b))","typeFull":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] [inst_2 : DecidableEq ι']\n  (i : ι × ι') (b : M), Function.curry (Pi.single i b) = Pi.single i.fst (Pi.single i.snd b)","typeReadable":"∀ {ι : Type u_1} {ι' : Type u_2} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] [inst_2 : DecidableEq ι']\n  (i : ι × ι') (b : M), Function.curry (Pi.single i b) = Pi.single i.fst (Pi.single i.snd b)","typeReferences":[["Prod"],["instDecidableEqProd"],["DecidableEq"],["Prod","snd"],["Function","curry"],["Zero"],["Eq"],["Pi","single"],["Prod","fst"],["Pi","instZero"]],"valueReferences":[["Prod"],["Function","curry_update"],["Zero","toOfNat0"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_apply"],"typeFallback":"forall {ι : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {M : Type.{u_9}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.52 : Zero.{u_9} M] (i : ι) (x : M) (i' : ι), Eq.{succ u_9} M (Pi.single.{u_1, u_9} ι (fun (a._@._internal._hyg.0 : ι) => M) (fun (i : ι) => inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.52) inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.45 i x i') (ite.{succ u_9} M (Eq.{succ u_1} ι i' i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.45 i' i) x (OfNat.ofNat.{u_9} M 0 (Zero.toOfNat0.{u_9} M inst._@.Mathlib.Algebra.Notation.Pi.Basic.3521610820._hygCtx._hyg.52)))","typeFull":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] (i : ι) (x : M) (i' : ι),\n  Pi.single i x i' = if i' = i then x else 0","typeReadable":"∀ {ι : Type u_1} [inst : DecidableEq ι] {M : Type u_9} [inst_1 : Zero M] (i : ι) (x : M) (i' : ι),\n  Pi.single i x i' = if i' = i then x else 0","typeReferences":[["ite"],["DecidableEq"],["Zero","toOfNat0"],["Zero"],["Eq"],["OfNat","ofNat"],["Pi","single"]],"valueReferences":[["Zero","toOfNat0"],["Function","update_apply"],["OfNat","ofNat"],["Pi","instZero"]]},{"isProp":true,"kind":"theorem","name":["Pi","single_ne_zero_iff"],"typeFallback":"forall {ι : Type.{u_1}} {M : ι -> Type.{u_6}} [inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18 : forall (i : ι), Zero.{u_6} (M i)] [inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.45 : DecidableEq.{succ u_1} ι] {i : ι} {x : M i}, Iff (Ne.{max (succ u_1) (succ u_6)} (forall (j : ι), M j) (Pi.single.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.45 i x) (OfNat.ofNat.{max u_1 u_6} (forall (j : ι), M j) 0 (Zero.toOfNat0.{max u_1 u_6} (forall (j : ι), M j) (Pi.instZero.{u_1, u_6} ι M inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18)))) (Ne.{succ u_6} (M i) x (OfNat.ofNat.{u_6} (M i) 0 (Zero.toOfNat0.{u_6} (M i) (inst._@.Mathlib.Algebra.Notation.Pi.Basic.893117752._hygCtx._hyg.18 i))))","typeFull":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.single i x ≠ 0 ↔ x ≠ 0","typeReadable":"∀ {ι : Type u_1} {M : ι → Type u_6} [inst : (i : ι) → Zero (M i)] [inst_1 : DecidableEq ι] {i : ι} {x : M i},\n  Pi.single i x ≠ 0 ↔ x ≠ 0","typeReferences":[["Iff"],["DecidableEq"],["Zero","toOfNat0"],["Ne"],["Zero"],["OfNat","ofNat"],["Pi","single"],["Pi","instZero"]],"valueReferences":[["Pi","single_eq_zero_iff"],["Zero","toOfNat0"],["Iff","ne"],["OfNat","ofNat"],["Pi","single"],["Pi","instZero"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Order","Group","Ideal",0,"SemigroupIdeal","fg_of_wellQuasiOrderedLE","_simp_1_2"],"typeFallback":"forall {A : Type.{u_1}} {B : Type.{u_2}} [i : SetLike.{u_1, u_2} A B] {p : A} {x : B}, Eq.{1} Prop (Membership.mem.{u_2, u_2} B (Set.{u_2} B) (Set.instMembership.{u_2} B) (SetLike.coe.{u_1, u_2} A B i p) x) (Membership.mem.{u_2, u_1} B A (SetLike.instMembership.{u_1, u_2} A B i) p x)","typeFull":"∀ {A : Type u_1} {B : Type u_2} [i : SetLike A B] {p : A} {x : B}, (x ∈ ↑p) = (x ∈ p)","typeReadable":"∀ {A : Type u_1} {B : Type u_2} [i : SetLike A B] {p : A} {x : B}, (x ∈ ↑p) = (x ∈ p)","typeReferences":[["SetLike","instMembership"],["SetLike","coe"],["Set"],["Membership","mem"],["SetLike"],["Eq"],["Set","instMembership"]],"valueReferences":[["SetLike","mem_coe"],["SetLike","instMembership"],["SetLike","coe"],["Set"],["Membership","mem"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Order","Group","Ideal",0,"SemigroupIdeal","fg_of_wellQuasiOrderedLE","_simp_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3 : Monoid.{u_1} M] {s : Set.{u_1} M} {x : M}, Eq.{1} Prop (Membership.mem.{u_1, u_1} M (SemigroupIdeal.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))) (SetLike.instMembership.{u_1, u_1} (SemigroupIdeal.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))))) (SemigroupIdeal.closure.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3))) s) x) (Exists.{succ u_1} M (fun (y : M) => Exists.{succ u_1} M (fun (z : M) => And (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s z) (Eq.{succ u_1} M (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))) y z) x))))","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {s : Set M} {x : M}, (x ∈ SemigroupIdeal.closure s) = ∃ y, ∃ z ∈ s, y * z = x","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {s : Set M} {x : M}, (x ∈ SemigroupIdeal.closure s) = ∃ y, ∃ z ∈ s, y * z = x","typeReferences":[["MulOneClass","toMulOne"],["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["SetLike","instMembership"],["Exists"],["Set"],["Membership","mem"],["And"],["HMul","hMul"],["instSMulOfMul"],["Set","instMembership"],["SemigroupIdeal"],["MulOne","toMul"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["Eq"]],"valueReferences":[["MulOneClass","toMulOne"],["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["SetLike","instMembership"],["Exists"],["Set"],["Membership","mem"],["And"],["HMul","hMul"],["instSMulOfMul"],["Set","instMembership"],["SemigroupIdeal","mem_closure''"],["SemigroupIdeal"],["MulOne","toMul"],["Monoid","toMulOneClass"],["instHMul"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","instWellFoundedGT"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.6 : PartialOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.9 : WellQuasiOrderedLE.{u_1} M (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.12 : CanonicallyOrderedAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3)))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.6))], WellFoundedGT.{u_1} (AddSemigroupIdeal.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))) (Preorder.toLT.{u_1} (AddSemigroupIdeal.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))) (PartialOrder.toPreorder.{u_1} (AddSemigroupIdeal.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))) (SubAddAction.instPartialOrder.{u_1, u_1} M M (Add.toVAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))))))","typeFull":"∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedAdd M],\n  WellFoundedGT (AddSemigroupIdeal M)","typeReadable":"∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedAdd M],\n  WellFoundedGT (AddSemigroupIdeal M)","typeReferences":[["WellQuasiOrderedLE"],["PartialOrder","toPreorder"],["Preorder","toLT"],["CanonicallyOrderedAdd"],["AddCommMonoid","toAddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["AddCommMonoid"],["PartialOrder"],["SubAddAction","instPartialOrder"],["WellFoundedGT"],["Add","toVAdd"],["AddSemigroupIdeal"],["Preorder","toLE"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Classical","choose_spec"],["Lattice","toSemilatticeSup"],["SubAddAction"],["PartialOrder","toPreorder"],["Finset"],["Eq","trans"],["Membership","mem"],["Preorder","toLT"],["Iff","mp"],["SubAddAction","mem_iSup"],["Exists","intro"],["SubAddAction","instSupSet"],["Classical","choose"],["iSup"],["LE","le","antisymm"],["Eq","symm"],["Nat","instOrderBot"],["Eq","ndrec"],["wellFoundedGT_iff_monotone_chain_condition"],["AddSemigroupIdeal","fg_of_wellQuasiOrderedLE"],["AddSemigroupIdeal","closure"],["Exists"],["SetLike","instMembership"],["And"],["SubAddAction","instCompleteLattice"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Exists","casesOn"],["Nat"],["SetLike","coe"],["instNonemptyOfInhabited"],["Nat","instLattice"],["AddSemigroupIdeal","subset_closure"],["id"],["instInhabitedNat"],["Eq","mpr"],["SubAddAction","instSetLike"],["AddSemigroupIdeal","add_mem"],["AddMonoid","toAddZeroClass"],["OrderHom","instFunLike"],["Finset","instSetLike"],["Function","sometimes_spec"],["Finset","le_sup"],["Eq","mp"],["OrderHom"],["CompleteSemilatticeSup","toSupSet"],["AddCommMonoid","toAddMonoid"],["CompleteLattice","toCompleteSemilatticeSup"],["AddSemigroupIdeal","FG"],["DFunLike","coe"],["LE","le","trans"],["congrArg"],["Nat","instPreorder"],["OrderHom","mono"],["SetLike","mem_coe"],["Finset","sup"],["WellFoundedGT"],["SubAddAction","instPartialOrder"],["Function","sometimes"],["Add","toVAdd"],["congrFun'"],["Eq"],["propext"],["AddSemigroupIdeal","fg_iff"],["instHAdd"],["Set"],["AddZero","toAdd"],["le_iSup"],["AddSemigroupIdeal","mem_closure''"],["HAdd","hAdd"],["LE","le"],["AddSemigroupIdeal"],["instLENat"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","fg_of_wellQuasiOrderedLE"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3 : CommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.6 : PartialOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.9 : WellQuasiOrderedLE.{u_1} M (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.12 : CanonicallyOrderedMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3)))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.6))] (I : SemigroupIdeal.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3))))), SemigroupIdeal.FG.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3)))) I","typeFull":"∀ {M : Type u_1} [inst : CommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedMul M]\n  (I : SemigroupIdeal M), I.FG","typeReadable":"∀ {M : Type u_1} [inst : CommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedMul M]\n  (I : SemigroupIdeal M), I.FG","typeReferences":[["MulOneClass","toMulOne"],["SemigroupIdeal"],["MulOne","toMul"],["WellQuasiOrderedLE"],["PartialOrder","toPreorder"],["SemigroupIdeal","FG"],["CommMonoid","toMonoid"],["PartialOrder"],["Monoid","toMulOneClass"],["CanonicallyOrderedMul"],["CommMonoid"],["Preorder","toLE"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["_private","Mathlib","Algebra","Order","Group","Ideal",0,"SemigroupIdeal","fg_of_wellQuasiOrderedLE","_simp_1_2"],["Membership","mem"],["Exists","intro"],["HMul","hMul"],["SubMulAction"],["Set","IsPWO","exists_le_minimal"],["And","intro"],["Minimal"],["SubMulAction","ext"],["funext"],["Eq","symm"],["SubMulAction","smul_mem"],["Eq","ndrec"],["And","left"],["rfl"],["_private","Mathlib","Algebra","Order","Group","Ideal",0,"SemigroupIdeal","fg_of_wellQuasiOrderedLE","_simp_1_1"],["SetLike","instMembership"],["Exists"],["CommMagma","toMul"],["And"],["Set","instMembership"],["Exists","casesOn"],["SetLike","coe"],["Iff"],["id"],["IsAntichain","finite_of_partiallyWellOrderedOn"],["instHMul"],["Eq","mpr"],["Set","Finite"],["Set","IsPWO","mono"],["setOf"],["SemigroupIdeal","closure"],["MulOneClass","toMulOne"],["setOf_minimal_subset"],["CommMonoid","toMonoid"],["Eq","mp"],["congrArg"],["Iff","intro"],["MulOne","toMul"],["le_iff_exists_mul'"],["Monoid","toMulOneClass"],["setOf_minimal_antichain"],["congrFun'"],["Preorder","toLE"],["Eq"],["propext"],["CommSemigroup","toCommMagma"],["Set","isPWO_of_wellQuasiOrderedLE"],["SubMulAction","instSetLike"],["Set"],["instSMulOfMul"],["SemigroupIdeal"],["CommMonoid","toCommSemigroup"],["LE","le"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","instWellFoundedGT"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3 : CommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.6 : PartialOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.9 : WellQuasiOrderedLE.{u_1} M (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.12 : CanonicallyOrderedMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3)))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.6))], WellFoundedGT.{u_1} (SemigroupIdeal.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))) (Preorder.toLT.{u_1} (SemigroupIdeal.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))) (PartialOrder.toPreorder.{u_1} (SemigroupIdeal.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))) (SubMulAction.instPartialOrder.{u_1, u_1} M M (instSMulOfMul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.3669512330._hygCtx._hyg.3))))))))","typeFull":"∀ {M : Type u_1} [inst : CommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedMul M],\n  WellFoundedGT (SemigroupIdeal M)","typeReadable":"∀ {M : Type u_1} [inst : CommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedMul M],\n  WellFoundedGT (SemigroupIdeal M)","typeReferences":[["MulOneClass","toMulOne"],["WellQuasiOrderedLE"],["PartialOrder","toPreorder"],["CommMonoid","toMonoid"],["Preorder","toLT"],["instSMulOfMul"],["SubMulAction","instPartialOrder"],["MulOne","toMul"],["SemigroupIdeal"],["PartialOrder"],["Monoid","toMulOneClass"],["CanonicallyOrderedMul"],["WellFoundedGT"],["CommMonoid"],["Preorder","toLE"]],"valueReferences":[["Classical","choose_spec"],["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Finset"],["Eq","trans"],["Membership","mem"],["Preorder","toLT"],["Iff","mp"],["HMul","hMul"],["Exists","intro"],["SubMulAction"],["Classical","choose"],["iSup"],["LE","le","antisymm"],["Eq","symm"],["Nat","instOrderBot"],["Eq","ndrec"],["wellFoundedGT_iff_monotone_chain_condition"],["SubMulAction","instCompleteLattice"],["Exists"],["SetLike","instMembership"],["And"],["Set","instMembership"],["SemigroupIdeal","mem_closure''"],["Exists","casesOn"],["Nat"],["SetLike","coe"],["SemigroupIdeal","mul_mem"],["instNonemptyOfInhabited"],["Nat","instLattice"],["id"],["instHMul"],["instInhabitedNat"],["Eq","mpr"],["SemigroupIdeal","fg_of_wellQuasiOrderedLE"],["OrderHom","instFunLike"],["MulOneClass","toMulOne"],["SemigroupIdeal","closure"],["SubMulAction","instSupSet"],["Finset","instSetLike"],["Function","sometimes_spec"],["CommMonoid","toMonoid"],["Finset","le_sup"],["Eq","mp"],["OrderHom"],["CompleteSemilatticeSup","toSupSet"],["CompleteLattice","toCompleteSemilatticeSup"],["DFunLike","coe"],["LE","le","trans"],["congrArg"],["Nat","instPreorder"],["OrderHom","mono"],["MulOne","toMul"],["Monoid","toMulOneClass"],["Finset","sup"],["WellFoundedGT"],["Function","sometimes"],["congrFun'"],["Eq"],["propext"],["SubMulAction","instSetLike"],["Set"],["instSMulOfMul"],["SubMulAction","mem_iSup","_simp_2"],["le_iSup"],["SubMulAction","instPartialOrder"],["SemigroupIdeal","fg_iff"],["SemigroupIdeal"],["SemigroupIdeal","FG"],["SetLike","mem_coe","_simp_1"],["SemigroupIdeal","subset_closure"],["LE","le"],["instLENat"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","fg_of_wellQuasiOrderedLE"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.6 : PartialOrder.{u_1} M] [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.9 : WellQuasiOrderedLE.{u_1} M (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.12 : CanonicallyOrderedAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3)))) (Preorder.toLE.{u_1} M (PartialOrder.toPreorder.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.6))] (I : AddSemigroupIdeal.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3))))), AddSemigroupIdeal.FG.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.Order.Group.Ideal.2271001167._hygCtx._hyg.3)))) I","typeFull":"∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedAdd M]\n  (I : AddSemigroupIdeal M), I.FG","typeReadable":"∀ {M : Type u_1} [inst : AddCommMonoid M] [inst_1 : PartialOrder M] [WellQuasiOrderedLE M] [CanonicallyOrderedAdd M]\n  (I : AddSemigroupIdeal M), I.FG","typeReferences":[["WellQuasiOrderedLE"],["AddCommMonoid"],["PartialOrder","toPreorder"],["PartialOrder"],["CanonicallyOrderedAdd"],["AddSemigroupIdeal"],["AddCommMonoid","toAddMonoid"],["AddSemigroupIdeal","FG"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Preorder","toLE"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SubAddAction"],["PartialOrder","toPreorder"],["Eq","trans"],["Membership","mem"],["Exists","intro"],["Set","IsPWO","exists_le_minimal"],["And","intro"],["Minimal"],["funext"],["Eq","symm"],["Eq","ndrec"],["And","left"],["rfl"],["AddSemigroupIdeal","closure"],["SetLike","instMembership"],["Exists"],["And"],["AddZeroClass","toAddZero"],["Set","instMembership"],["Exists","casesOn"],["SetLike","coe"],["le_iff_exists_add'"],["Iff"],["id"],["IsAntichain","finite_of_partiallyWellOrderedOn"],["Eq","mpr"],["SubAddAction","instSetLike"],["Set","Finite"],["Set","IsPWO","mono"],["setOf"],["AddMonoid","toAddZeroClass"],["setOf_minimal_subset"],["Eq","mp"],["AddCommMonoid","toAddMonoid"],["congrArg"],["Iff","intro"],["SetLike","mem_coe"],["setOf_minimal_antichain"],["Add","toVAdd"],["congrFun'"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Preorder","toLE"],["Eq"],["propext"],["Set","isPWO_of_wellQuasiOrderedLE"],["Set"],["instHAdd"],["AddZero","toAdd"],["AddSemigroupIdeal","mem_closure''"],["HAdd","hAdd"],["SubAddAction","ext"],["SubAddAction","vadd_mem"],["AddCommMonoid","toAddCommSemigroup"],["LE","le"],["AddSemigroupIdeal"],["And","casesOn"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Polynomial.Eval.Subring.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Polynomial","mem_map_rangeS"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29 : Semiring.{v} S] (f : RingHom.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29)) {p : Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29}, Iff (Membership.mem.{v, v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Subsemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29))) (SetLike.instMembership.{v, v} (Subsemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29))) (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Subsemiring.instSetLike.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29)))) (RingHom.rangeS.{u, v} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.14) (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.14) (Polynomial.semiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.14)) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29)) (Polynomial.mapRingHom.{u, v} R S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29 f)) p) (forall (n : Nat), Membership.mem.{v, v} S (Subsemiring.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29)) (SetLike.instMembership.{v, v} (Subsemiring.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29)) S (Subsemiring.instSetLike.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29))) (RingHom.rangeS.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29) f) (Polynomial.coeff.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1340172270._hygCtx._hyg.29 p n))","typeFull":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (f : R →+* S) {p : Polynomial S},\n  p ∈ (Polynomial.mapRingHom f).rangeS ↔ ∀ (n : ℕ), p.coeff n ∈ f.rangeS","typeReadable":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (f : R →+* S) {p : Polynomial S},\n  p ∈ (Polynomial.mapRingHom f).rangeS ↔ ∀ (n : ℕ), p.coeff n ∈ f.rangeS","typeReferences":[["RingHom"],["Polynomial","mapRingHom"],["Polynomial","coeff"],["SetLike","instMembership"],["Membership","mem"],["Subsemiring","instSetLike"],["Polynomial","semiring"],["Polynomial"],["Nat"],["Semiring","toNonAssocSemiring"],["RingHom","rangeS"],["Iff"],["Subsemiring"],["Semiring"]],"valueReferences":[["instAddNat"],["RingHom"],["Polynomial","mapRingHom"],["Polynomial","map_mul"],["Polynomial","as_sum_range_C_mul_X_pow"],["Membership","mem"],["Subsemiring","instSetLike"],["HMul","hMul"],["Exists","intro"],["Polynomial","coe_mapRingHom"],["Semiring","toNonAssocSemiring"],["RingHom","rangeS"],["Monoid","toPow"],["Polynomial","instMul"],["Finset","sum"],["Polynomial","map_pow"],["Eq","ndrec"],["Set","mem_range_self"],["Polynomial","coeff_map"],["instHPow"],["Subsemiring","sum_mem"],["SetLike","instMembership"],["Polynomial","coeff"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Exists","casesOn"],["Polynomial"],["Polynomial","map_X"],["Nat"],["Eq","refl"],["id"],["instHMul"],["Eq","mpr"],["Polynomial","map"],["Polynomial","natDegree"],["RingHom","instFunLike"],["DFunLike","coe"],["congrArg"],["Iff","intro"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Eq"],["Finset","range"],["Polynomial","X"],["Polynomial","C"],["instHAdd"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["Polynomial","semiring"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["Polynomial","map_C"],["Subsemiring"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","mem_map_range"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_2}} [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.39 : Ring.{u_1} R] [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42 : Ring.{u_2} S] (f : RingHom.{u_1, u_2} R S (Semiring.toNonAssocSemiring.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.39)) (Semiring.toNonAssocSemiring.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42))) {p : Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)}, Iff (Membership.mem.{u_2, u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Subring.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42))) (SetLike.instMembership.{u_2, u_2} (Subring.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42))) (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Subring.instSetLike.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)))) (RingHom.range.{u_1, u_2} (Polynomial.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.39)) (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_1} (Polynomial.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.39)) (Polynomial.ring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.39)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (Polynomial.mapRingHom.{u_1, u_2} R S (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.39) (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42) f)) p) (forall (n : Nat), Membership.mem.{u_2, u_2} S (Subring.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) (SetLike.instMembership.{u_2, u_2} (Subring.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42)) S (Subring.instSetLike.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42))) (RingHom.range.{u_1, u_2} R S (Ring.toNonAssocRing.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.39) (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42) f) (Polynomial.coeff.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.1070345515._hygCtx._hyg.42) p n))","typeFull":"∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] (f : R →+* S) {p : Polynomial S},\n  p ∈ (Polynomial.mapRingHom f).range ↔ ∀ (n : ℕ), p.coeff n ∈ f.range","typeReadable":"∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] (f : R →+* S) {p : Polynomial S},\n  p ∈ (Polynomial.mapRingHom f).range ↔ ∀ (n : ℕ), p.coeff n ∈ f.range","typeReferences":[["RingHom"],["Polynomial","mapRingHom"],["Polynomial","ring"],["Polynomial","coeff"],["Ring","toNonAssocRing"],["SetLike","instMembership"],["Subring"],["Membership","mem"],["Ring","toSemiring"],["Polynomial"],["Nat"],["Semiring","toNonAssocSemiring"],["Iff"],["RingHom","range"],["Subring","instSetLike"],["Ring"]],"valueReferences":[["Polynomial","mem_map_rangeS"],["Ring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","notMem_map_range"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_2}} [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.39 : Ring.{u_1} R] [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42 : Ring.{u_2} S] (f : RingHom.{u_1, u_2} R S (Semiring.toNonAssocSemiring.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.39)) (Semiring.toNonAssocSemiring.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42))) {p : Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)}, Iff (Not (Membership.mem.{u_2, u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Subring.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42))) (SetLike.instMembership.{u_2, u_2} (Subring.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42))) (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Subring.instSetLike.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)))) (RingHom.range.{u_1, u_2} (Polynomial.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.39)) (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Ring.toNonAssocRing.{u_1} (Polynomial.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.39)) (Polynomial.ring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.39)) (Ring.toNonAssocRing.{u_2} (Polynomial.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Polynomial.ring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (Polynomial.mapRingHom.{u_1, u_2} R S (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.39) (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42) f)) p)) (Exists.{1} Nat (fun (n : Nat) => Not (Membership.mem.{u_2, u_2} S (Subring.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) (SetLike.instMembership.{u_2, u_2} (Subring.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42)) S (Subring.instSetLike.{u_2} S (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42))) (RingHom.range.{u_1, u_2} R S (Ring.toNonAssocRing.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.39) (Ring.toNonAssocRing.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42) f) (Polynomial.coeff.{u_2} S (Ring.toSemiring.{u_2} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.6055197._hygCtx._hyg.42) p n))))","typeFull":"∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] (f : R →+* S) {p : Polynomial S},\n  p ∉ (Polynomial.mapRingHom f).range ↔ ∃ n, p.coeff n ∉ f.range","typeReadable":"∀ {R : Type u_1} {S : Type u_2} [inst : Ring R] [inst_1 : Ring S] (f : R →+* S) {p : Polynomial S},\n  p ∉ (Polynomial.mapRingHom f).range ↔ ∃ n, p.coeff n ∉ f.range","typeReferences":[["Not"],["RingHom"],["Polynomial","mapRingHom"],["Polynomial","ring"],["Polynomial","coeff"],["Exists"],["Ring","toNonAssocRing"],["SetLike","instMembership"],["Subring"],["Membership","mem"],["Ring","toSemiring"],["Polynomial"],["Nat"],["Semiring","toNonAssocSemiring"],["Iff"],["RingHom","range"],["Subring","instSetLike"],["Ring"]],"valueReferences":[["Polynomial","notMem_map_rangeS"],["Ring","toSemiring"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","notMem_map_rangeS"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29 : Semiring.{v} S] (f : RingHom.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29)) {p : Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29}, Iff (Not (Membership.mem.{v, v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Subsemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29))) (SetLike.instMembership.{v, v} (Subsemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29))) (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Subsemiring.instSetLike.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29)))) (RingHom.rangeS.{u, v} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.14) (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.14) (Polynomial.semiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.14)) (Semiring.toNonAssocSemiring.{v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) (Polynomial.semiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29)) (Polynomial.mapRingHom.{u, v} R S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29 f)) p)) (Exists.{1} Nat (fun (n : Nat) => Not (Membership.mem.{v, v} S (Subsemiring.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29)) (SetLike.instMembership.{v, v} (Subsemiring.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29)) S (Subsemiring.instSetLike.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29))) (RingHom.rangeS.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29) f) (Polynomial.coeff.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.Subring.4147948050._hygCtx._hyg.29 p n))))","typeFull":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (f : R →+* S) {p : Polynomial S},\n  p ∉ (Polynomial.mapRingHom f).rangeS ↔ ∃ n, p.coeff n ∉ f.rangeS","typeReadable":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (f : R →+* S) {p : Polynomial S},\n  p ∉ (Polynomial.mapRingHom f).rangeS ↔ ∃ n, p.coeff n ∉ f.rangeS","typeReferences":[["Not"],["RingHom"],["Polynomial","mapRingHom"],["Polynomial","coeff"],["Exists"],["SetLike","instMembership"],["Membership","mem"],["Subsemiring","instSetLike"],["Polynomial","semiring"],["Polynomial"],["Nat"],["Semiring","toNonAssocSemiring"],["RingHom","rangeS"],["Iff"],["Subsemiring"],["Semiring"]],"valueReferences":[["Iff","trans"],["Not"],["Polynomial","mapRingHom"],["Exists"],["SetLike","instMembership"],["Polynomial","coeff"],["Membership","mem"],["Subsemiring","instSetLike"],["Polynomial","semiring"],["Classical","not_forall"],["Polynomial"],["Nat"],["Semiring","toNonAssocSemiring"],["RingHom","rangeS"],["Iff","not"],["Subsemiring"],["Polynomial","mem_map_rangeS"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["IsAddRegular","right"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R] {c : R}, (IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> (IsAddRightRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c)","typeFull":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddRegular c → IsAddRightRegular c","typeReadable":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddRegular c → IsAddRightRegular c","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"]],"valueReferences":[]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Regular","Defs",0,"isAddRegular_iff","match_1_3"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 : Add.{u_1} R] {c : R} (motive : (And (IsAddLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsAddRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c)) -> Prop) (x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.47.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.54 : And (IsAddLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsAddRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c)), (forall (h1 : IsAddLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (h2 : IsAddRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c), motive (And.intro (IsAddLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsAddRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) h1 h2)) -> (motive x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.47.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.54)","typeFull":"∀ {R : Type u_1} [inst : Add R] {c : R} (motive : IsAddLeftRegular c ∧ IsAddRightRegular c → Prop)\n  (x : IsAddLeftRegular c ∧ IsAddRightRegular c),\n  (∀ (h1 : IsAddLeftRegular c) (h2 : IsAddRightRegular c), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_1} [inst : Add R] {c : R} (motive : IsAddLeftRegular c ∧ IsAddRightRegular c → Prop)\n  (x : IsAddLeftRegular c ∧ IsAddRightRegular c),\n  (∀ (h1 : IsAddLeftRegular c) (h2 : IsAddRightRegular c), motive ⋯) → motive x","typeReferences":[["And","intro"],["IsAddRightRegular"],["Add"],["And"],["IsAddLeftRegular"]],"valueReferences":[["IsAddRightRegular"],["IsAddLeftRegular"],["And","casesOn"]]},{"isProp":false,"kind":"definition","name":["IsAddRegular","recOn"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R] {c : R} {motive : (IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> Sort.{u}} (t : IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c), (forall (left : IsAddLeftRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) (right : IsAddRightRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c), motive (IsAddRegular.mk.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c left right)) -> (motive t)","typeFull":"{R : Type u_2} →\n  [inst : Add R] →\n    {c : R} →\n      {motive : IsAddRegular c → Sort u} →\n        (t : IsAddRegular c) → ((left : IsAddLeftRegular c) → (right : IsAddRightRegular c) → motive ⋯) → motive t","typeReadable":"{R : Type u_2} →\n  [inst : Add R] →\n    {c : R} →\n      {motive : IsAddRegular c → Sort u} →\n        (t : IsAddRegular c) → ((left : IsAddLeftRegular c) → (right : IsAddRightRegular c) → motive ⋯) → motive t","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"],["IsAddLeftRegular"],["IsAddRegular","mk"]],"valueReferences":[["IsAddRegular","rec"]]},{"isProp":true,"kind":"theorem","name":["isAddRegular_iff"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 : Add.{u_1} R] {c : R}, Iff (IsAddRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (And (IsAddLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsAddRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c))","typeFull":"∀ {R : Type u_1} [inst : Add R] {c : R}, IsAddRegular c ↔ IsAddLeftRegular c ∧ IsAddRightRegular c","typeReadable":"∀ {R : Type u_1} [inst : Add R] {c : R}, IsAddRegular c ↔ IsAddLeftRegular c ∧ IsAddRightRegular c","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"],["Iff"],["And"],["IsAddLeftRegular"]],"valueReferences":[["_private","Mathlib","Algebra","Regular","Defs",0,"isAddRegular_iff","match_1_3"],["And","intro"],["IsAddRightRegular"],["IsAddRegular"],["And"],["_private","Mathlib","Algebra","Regular","Defs",0,"isAddRegular_iff","match_1_1"],["IsAddLeftRegular"],["IsAddRegular","mk"],["Iff","intro"]]},{"isProp":true,"kind":"definition","name":["IsAddRegular","mk","_flat_ctor"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R] {c : R}, (IsAddLeftRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> (IsAddRightRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> (IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c)","typeFull":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddLeftRegular c → IsAddRightRegular c → IsAddRegular c","typeReadable":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddLeftRegular c → IsAddRightRegular c → IsAddRegular c","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"],["IsAddLeftRegular"]],"valueReferences":[["IsAddRegular","mk"]]},{"isProp":false,"kind":"definition","name":["IsRightRegular"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3 : Mul.{u_1} R], R -> Prop","typeFull":"{R : Type u_1} → [Mul R] → R → Prop","typeReadable":"{R : Type u_1} → [Mul R] → R → Prop","typeReferences":[["Mul"]],"valueReferences":[["instHMul"],["HMul","hMul"],["Function","Injective"]]},{"isProp":true,"kind":"theorem","name":["IsLeftRegular","eq_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3 : Mul.{u_1} R] (c : R), Eq.{1} Prop (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3 c) (Function.Injective.{succ u_1, succ u_1} R R (fun (x._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.11 : R) => HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3) c x._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.11))","typeFull":"∀ {R : Type u_1} [inst : Mul R] (c : R), IsLeftRegular c = Function.Injective fun x => c * x","typeReadable":"∀ {R : Type u_1} [inst : Mul R] (c : R), IsLeftRegular c = Function.Injective fun x => c * x","typeReferences":[["Mul"],["instHMul"],["HMul","hMul"],["IsLeftRegular"],["Eq"],["Function","Injective"]],"valueReferences":[["Eq","refl"],["IsLeftRegular"]]},{"isProp":true,"kind":"theorem","name":["IsAddLeftRegular","eq_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3 : Add.{u_1} R] (c : R), Eq.{1} Prop (IsAddLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3 c) (Function.Injective.{succ u_1, succ u_1} R R (fun (x._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.11 : R) => HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3) c x._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.11))","typeFull":"∀ {R : Type u_1} [inst : Add R] (c : R), IsAddLeftRegular c = Function.Injective fun x => c + x","typeReadable":"∀ {R : Type u_1} [inst : Add R] (c : R), IsAddLeftRegular c = Function.Injective fun x => c + x","typeReferences":[["HAdd","hAdd"],["instHAdd"],["Add"],["IsAddLeftRegular"],["Eq"],["Function","Injective"]],"valueReferences":[["Eq","refl"],["IsAddLeftRegular"]]},{"isProp":false,"kind":"definition","name":["IsAddRegular","casesOn"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R] {c : R} {motive : (IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> Sort.{u}} (t : IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c), (forall (left : IsAddLeftRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) (right : IsAddRightRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c), motive (IsAddRegular.mk.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c left right)) -> (motive t)","typeFull":"{R : Type u_2} →\n  [inst : Add R] →\n    {c : R} →\n      {motive : IsAddRegular c → Sort u} →\n        (t : IsAddRegular c) → ((left : IsAddLeftRegular c) → (right : IsAddRightRegular c) → motive ⋯) → motive t","typeReadable":"{R : Type u_2} →\n  [inst : Add R] →\n    {c : R} →\n      {motive : IsAddRegular c → Sort u} →\n        (t : IsAddRegular c) → ((left : IsAddLeftRegular c) → (right : IsAddRightRegular c) → motive ⋯) → motive t","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"],["IsAddLeftRegular"],["IsAddRegular","mk"]],"valueReferences":[["IsAddRegular","rec"]]},{"isProp":false,"kind":"recursor","name":["IsRegular","rec"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R} {motive : (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> Sort.{u}}, (forall (left : IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) (right : IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c), motive (IsRegular.mk.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c left right)) -> (forall (t : IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c), motive t)","typeFull":"{R : Type u_1} →\n  [inst : Mul R] →\n    {c : R} →\n      {motive : IsRegular c → Sort u} →\n        ((left : IsLeftRegular c) → (right : IsRightRegular c) → motive ⋯) → (t : IsRegular c) → motive t","typeReadable":"{R : Type u_1} →\n  [inst : Mul R] →\n    {c : R} →\n      {motive : IsRegular c → Sort u} →\n        ((left : IsLeftRegular c) → (right : IsRightRegular c) → motive ⋯) → (t : IsRegular c) → motive t","typeReferences":[["Mul"],["IsRegular","mk"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["IsRegular","left","_simp_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R}, (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (Eq.{1} Prop (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) True)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsLeftRegular c = True","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsLeftRegular c = True","typeReferences":[["True"],["Mul"],["IsLeftRegular"],["Eq"],["IsRegular"]],"valueReferences":[["IsRegular","left"],["eq_true"],["IsLeftRegular"]]},{"isProp":true,"kind":"theorem","name":["IsRegular","right"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R}, (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsRightRegular c","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsRightRegular c","typeReferences":[["Mul"],["IsRegular"],["IsRightRegular"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["IsRegular","left"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R}, (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsLeftRegular c","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsLeftRegular c","typeReferences":[["Mul"],["IsLeftRegular"],["IsRegular"]],"valueReferences":[]},{"isProp":false,"kind":"recursor","name":["IsAddRegular","rec"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R] {c : R} {motive : (IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> Sort.{u}}, (forall (left : IsAddLeftRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) (right : IsAddRightRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c), motive (IsAddRegular.mk.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c left right)) -> (forall (t : IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c), motive t)","typeFull":"{R : Type u_2} →\n  [inst : Add R] →\n    {c : R} →\n      {motive : IsAddRegular c → Sort u} →\n        ((left : IsAddLeftRegular c) → (right : IsAddRightRegular c) → motive ⋯) → (t : IsAddRegular c) → motive t","typeReadable":"{R : Type u_2} →\n  [inst : Add R] →\n    {c : R} →\n      {motive : IsAddRegular c → Sort u} →\n        ((left : IsAddLeftRegular c) → (right : IsAddRightRegular c) → motive ⋯) → (t : IsAddRegular c) → motive t","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"],["IsAddLeftRegular"],["IsAddRegular","mk"]],"valueReferences":null},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Regular","Defs",0,"isRegular_iff","match_1_3"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 : Mul.{u_1} R] {c : R} (motive : (And (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c)) -> Prop) (x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.47.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.54 : And (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c)), (forall (h1 : IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (h2 : IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c), motive (And.intro (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) h1 h2)) -> (motive x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.47.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.54)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R} (motive : IsLeftRegular c ∧ IsRightRegular c → Prop)\n  (x : IsLeftRegular c ∧ IsRightRegular c), (∀ (h1 : IsLeftRegular c) (h2 : IsRightRegular c), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R} (motive : IsLeftRegular c ∧ IsRightRegular c → Prop)\n  (x : IsLeftRegular c ∧ IsRightRegular c), (∀ (h1 : IsLeftRegular c) (h2 : IsRightRegular c), motive ⋯) → motive x","typeReferences":[["And","intro"],["Mul"],["And"],["IsLeftRegular"],["IsRightRegular"]],"valueReferences":[["IsLeftRegular"],["IsRightRegular"],["And","casesOn"]]},{"isProp":true,"kind":"definition","name":["IsRegular","mk","_flat_ctor"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R}, (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsLeftRegular c → IsRightRegular c → IsRegular c","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsLeftRegular c → IsRightRegular c → IsRegular c","typeReferences":[["Mul"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"]],"valueReferences":[["IsRegular","mk"]]},{"isProp":true,"kind":"theorem","name":["IsAddRegular","left"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R] {c : R}, (IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> (IsAddLeftRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c)","typeFull":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddRegular c → IsAddLeftRegular c","typeReadable":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddRegular c → IsAddLeftRegular c","typeReferences":[["IsAddRegular"],["Add"],["IsAddLeftRegular"]],"valueReferences":[]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Regular","Defs",0,"isAddRegular_iff","match_1_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 : Add.{u_1} R] {c : R} (motive : (IsAddRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) -> Prop) (x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.23.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.30 : IsAddRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c), (forall (h1 : IsAddLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (h2 : IsAddRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c), motive (IsAddRegular.mk.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c h1 h2)) -> (motive x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.23.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.30)","typeFull":"∀ {R : Type u_1} [inst : Add R] {c : R} (motive : IsAddRegular c → Prop) (x : IsAddRegular c),\n  (∀ (h1 : IsAddLeftRegular c) (h2 : IsAddRightRegular c), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_1} [inst : Add R] {c : R} (motive : IsAddRegular c → Prop) (x : IsAddRegular c),\n  (∀ (h1 : IsAddLeftRegular c) (h2 : IsAddRightRegular c), motive ⋯) → motive x","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"],["IsAddLeftRegular"],["IsAddRegular","mk"]],"valueReferences":[["IsAddRegular","casesOn"]]},{"isProp":false,"kind":"definition","name":["IsLeftRegular"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3 : Mul.{u_1} R], R -> Prop","typeFull":"{R : Type u_1} → [Mul R] → R → Prop","typeReadable":"{R : Type u_1} → [Mul R] → R → Prop","typeReferences":[["Mul"]],"valueReferences":[["instHMul"],["HMul","hMul"],["Function","Injective"]]},{"isProp":false,"kind":"definition","name":["IsAddLeftRegular"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.4110891431._hygCtx._hyg.3 : Add.{u_1} R], R -> Prop","typeFull":"{R : Type u_1} → [Add R] → R → Prop","typeReadable":"{R : Type u_1} → [Add R] → R → Prop","typeReferences":[["Add"]],"valueReferences":[["HAdd","hAdd"],["instHAdd"],["Function","Injective"]]},{"isProp":false,"kind":"inductive","name":["IsRegular"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R], R -> Prop","typeFull":"{R : Type u_1} → [Mul R] → R → Prop","typeReadable":"{R : Type u_1} → [Mul R] → R → Prop","typeReferences":[["Mul"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["IsAddRightRegular"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3 : Add.{u_1} R], R -> Prop","typeFull":"{R : Type u_1} → [Add R] → R → Prop","typeReadable":"{R : Type u_1} → [Add R] → R → Prop","typeReferences":[["Add"]],"valueReferences":[["HAdd","hAdd"],["instHAdd"],["Function","Injective"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Regular","Defs",0,"isRegular_iff","match_1_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 : Mul.{u_1} R] {c : R} (motive : (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) -> Prop) (x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.23.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.30 : IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c), (forall (h1 : IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (h2 : IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c), motive (IsRegular.mk.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c h1 h2)) -> (motive x._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx.23.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.30)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R} (motive : IsRegular c → Prop) (x : IsRegular c),\n  (∀ (h1 : IsLeftRegular c) (h2 : IsRightRegular c), motive ⋯) → motive x","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R} (motive : IsRegular c → Prop) (x : IsRegular c),\n  (∀ (h1 : IsLeftRegular c) (h2 : IsRightRegular c), motive ⋯) → motive x","typeReferences":[["Mul"],["IsRegular","mk"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"]],"valueReferences":[["IsRegular","casesOn"]]},{"isProp":true,"kind":"theorem","name":["IsRightRegular","eq_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3 : Mul.{u_1} R] (c : R), Eq.{1} Prop (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3 c) (Function.Injective.{succ u_1, succ u_1} R R (fun (x._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.11 : R) => HMul.hMul.{u_1, u_1, u_1} R R R (instHMul.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3) x._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.11 c))","typeFull":"∀ {R : Type u_1} [inst : Mul R] (c : R), IsRightRegular c = Function.Injective fun x => x * c","typeReadable":"∀ {R : Type u_1} [inst : Mul R] (c : R), IsRightRegular c = Function.Injective fun x => x * c","typeReferences":[["Mul"],["instHMul"],["HMul","hMul"],["Eq"],["IsRightRegular"],["Function","Injective"]],"valueReferences":[["Eq","refl"],["IsRightRegular"]]},{"isProp":false,"kind":"definition","name":["IsRegular","casesOn"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R} {motive : (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> Sort.{u}} (t : IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c), (forall (left : IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) (right : IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c), motive (IsRegular.mk.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c left right)) -> (motive t)","typeFull":"{R : Type u_1} →\n  [inst : Mul R] →\n    {c : R} →\n      {motive : IsRegular c → Sort u} →\n        (t : IsRegular c) → ((left : IsLeftRegular c) → (right : IsRightRegular c) → motive ⋯) → motive t","typeReadable":"{R : Type u_1} →\n  [inst : Mul R] →\n    {c : R} →\n      {motive : IsRegular c → Sort u} →\n        (t : IsRegular c) → ((left : IsLeftRegular c) → (right : IsRightRegular c) → motive ⋯) → motive t","typeReferences":[["Mul"],["IsRegular","mk"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"]],"valueReferences":[["IsRegular","rec"]]},{"isProp":true,"kind":"theorem","name":["isRegular_iff"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 : Mul.{u_1} R] {c : R}, Iff (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (And (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c) (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.281246656._hygCtx._hyg.3 c))","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c ↔ IsLeftRegular c ∧ IsRightRegular c","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c ↔ IsLeftRegular c ∧ IsRightRegular c","typeReferences":[["Iff"],["Mul"],["And"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"]],"valueReferences":[["_private","Mathlib","Algebra","Regular","Defs",0,"isRegular_iff","match_1_3"],["And","intro"],["_private","Mathlib","Algebra","Regular","Defs",0,"isRegular_iff","match_1_1"],["And"],["IsRegular","mk"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"],["Iff","intro"]]},{"isProp":true,"kind":"constructor","name":["IsAddRegular","mk"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R] {c : R}, (IsAddLeftRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> (IsAddRightRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c) -> (IsAddRegular.{u_2} R inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 c)","typeFull":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddLeftRegular c → IsAddRightRegular c → IsAddRegular c","typeReadable":"∀ {R : Type u_2} [inst : Add R] {c : R}, IsAddLeftRegular c → IsAddRightRegular c → IsAddRegular c","typeReferences":[["IsAddRightRegular"],["IsAddRegular"],["Add"],["IsAddLeftRegular"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["IsRegular","right","_simp_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R}, (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (Eq.{1} Prop (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) True)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsRightRegular c = True","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsRegular c → IsRightRegular c = True","typeReferences":[["True"],["Mul"],["Eq"],["IsRegular"],["IsRightRegular"]],"valueReferences":[["IsRegular","right"],["eq_true"],["IsRightRegular"]]},{"isProp":true,"kind":"theorem","name":["IsAddRightRegular","eq_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3 : Add.{u_1} R] (c : R), Eq.{1} Prop (IsAddRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3 c) (Function.Injective.{succ u_1, succ u_1} R R (fun (x._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.11 : R) => HAdd.hAdd.{u_1, u_1, u_1} R R R (instHAdd.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.3) x._@.Mathlib.Algebra.Regular.Defs.1696579642._hygCtx._hyg.11 c))","typeFull":"∀ {R : Type u_1} [inst : Add R] (c : R), IsAddRightRegular c = Function.Injective fun x => x + c","typeReadable":"∀ {R : Type u_1} [inst : Add R] (c : R), IsAddRightRegular c = Function.Injective fun x => x + c","typeReferences":[["HAdd","hAdd"],["IsAddRightRegular"],["instHAdd"],["Add"],["Eq"],["Function","Injective"]],"valueReferences":[["IsAddRightRegular"],["Eq","refl"]]},{"isProp":false,"kind":"inductive","name":["IsAddRegular"],"typeFallback":"forall {R : Type.{u_2}} [inst._@.Mathlib.Algebra.Regular.Defs.3794537213._hygCtx._hyg.11 : Add.{u_2} R], R -> Prop","typeFull":"{R : Type u_2} → [Add R] → R → Prop","typeReadable":"{R : Type u_2} → [Add R] → R → Prop","typeReferences":[["Add"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["IsRegular","recOn"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R} {motive : (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> Sort.{u}} (t : IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c), (forall (left : IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) (right : IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c), motive (IsRegular.mk.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c left right)) -> (motive t)","typeFull":"{R : Type u_1} →\n  [inst : Mul R] →\n    {c : R} →\n      {motive : IsRegular c → Sort u} →\n        (t : IsRegular c) → ((left : IsLeftRegular c) → (right : IsRightRegular c) → motive ⋯) → motive t","typeReadable":"{R : Type u_1} →\n  [inst : Mul R] →\n    {c : R} →\n      {motive : IsRegular c → Sort u} →\n        (t : IsRegular c) → ((left : IsLeftRegular c) → (right : IsRightRegular c) → motive ⋯) → motive t","typeReferences":[["Mul"],["IsRegular","mk"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"]],"valueReferences":[["IsRegular","rec"]]},{"isProp":true,"kind":"constructor","name":["IsRegular","mk"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 : Mul.{u_1} R] {c : R}, (IsLeftRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (IsRightRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c) -> (IsRegular.{u_1} R inst._@.Mathlib.Algebra.Regular.Defs.1527490745._hygCtx._hyg.7 c)","typeFull":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsLeftRegular c → IsRightRegular c → IsRegular c","typeReadable":"∀ {R : Type u_1} [inst : Mul R] {c : R}, IsLeftRegular c → IsRightRegular c → IsRegular c","typeReferences":[["Mul"],["IsLeftRegular"],["IsRegular"],["IsRightRegular"]],"valueReferences":null}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Ring.AddAut.sym.json

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["AddAut","mulRight"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.AddAut.1264609845._hygCtx._hyg.3 : Semiring.{u_1} R], (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.1264609845._hygCtx._hyg.3))) -> (AddAut.{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.Ring.AddAut.1264609845._hygCtx._hyg.3)))))","typeFull":"{R : Type u_1} → [inst : Semiring R] → Rˣ → AddAut R","typeReadable":"{R : Type u_1} → [inst : Semiring R] → Rˣ → AddAut R","typeReferences":[["AddAut"],["Distrib","toAdd"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["Units"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Units","instMul"],["Units","instGroup"],["MonoidHom"],["MonoidHom","instFunLike"],["MulEquiv","symm"],["Units","opEquiv"],["MulEquiv","instEquivLike"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["MulOpposite"],["Semiring","toNonAssocSemiring"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Group","toDivInvMonoid"],["Units"],["MulOpposite","instSemiring"],["AddSemigroup","toAdd"],["MulEquiv"],["MulOpposite","instMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Semiring","toMonoidWithZero"],["MulOpposite","instMonoid"],["AddAut"],["Module","toDistribMulAction"],["DivInvMonoid","toMonoid"],["Units","instDistribMulAction"],["MulOpposite","op"],["AddMonoid","toAddSemigroup"],["AddAut","instGroup"],["Semiring","toOppositeModule"],["DistribMulAction","toAddAut"]]},{"isProp":false,"kind":"definition","name":["AddAut","mulLeft"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3 : Semiring.{u_1} R], MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Monoid.toMulOneClass.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (DivInvMonoid.toMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Group.toDivInvMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (AddAut.instGroup.{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.Ring.AddAut.3455109452._hygCtx._hyg.3)))))))))","typeFull":"{R : Type u_1} → [inst : Semiring R] → Rˣ →* AddAut R","typeReadable":"{R : Type u_1} → [inst : Semiring R] → Rˣ →* AddAut R","typeReferences":[["MulOneClass","toMulOne"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["MonoidHom"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["AddAut"],["Semiring","toNonAssocSemiring"],["DivInvMonoid","toMonoid"],["AddAut","instGroup"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["Group","toDivInvMonoid"],["Units"],["Semiring"]],"valueReferences":[["Units","instGroup"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["AddMonoidWithOne","toAddMonoid"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["Units","instDistribMulAction"],["MonoidWithZero","toMonoid"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribMulAction","toAddAut"],["Units"]]},{"isProp":true,"kind":"theorem","name":["AddAut","mulLeft_apply_symm_apply"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3 : Semiring.{u_1} R] (x : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (a._@._internal._hyg.0 : R), Eq.{succ u_1} R (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (EquivLike.toFunLike.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) R R (AddEquiv.instEquivLike.{u_1, u_1} R R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))))) (AddEquiv.symm.{u_1, u_1} R R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Monoid.toMulOneClass.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (DivInvMonoid.toMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Group.toDivInvMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (AddAut.instGroup.{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.Ring.AddAut.3455109452._hygCtx._hyg.3)))))))))) (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) => AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (MonoidHom.instFunLike.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Monoid.toMulOneClass.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (DivInvMonoid.toMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Group.toDivInvMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (AddAut.instGroup.{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.Ring.AddAut.3455109452._hygCtx._hyg.3)))))))))) (AddAut.mulLeft.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3) x)) a._@._internal._hyg.0) (HSMul.hSMul.{u_1, u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R R (instHSMul.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (SemigroupAction.toSMul.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (Monoid.toSemigroup.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (DivInvMonoid.toMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Group.toDivInvMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (MulAction.toSemigroupAction.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (DivInvMonoid.toMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Group.toDivInvMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (DistribMulAction.toMulAction.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (DivInvMonoid.toMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Group.toDivInvMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (Units.instDistribMulAction.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (Module.toDistribMulAction.{u_1, u_1} R R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) (Inv.inv.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (InvOneClass.toInv.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (DivInvOneMonoid.toInvOneClass.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (DivisionMonoid.toDivInvOneMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Group.toDivisionMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) x) a._@._internal._hyg.0)","typeFull":"∀ {R : Type u_1} [inst : Semiring R] (x : Rˣ) (a : R), (AddEquiv.symm (AddAut.mulLeft x)) a = x⁻¹ • a","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] (x : Rˣ) (a : R), (AddEquiv.symm (AddAut.mulLeft x)) a = x⁻¹ • a","typeReferences":[["MulOneClass","toMulOne"],["Units","instGroup"],["MonoidHom"],["MonoidHom","instFunLike"],["SemigroupAction","toSMul"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["AddEquiv","instEquivLike"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["Monoid","toMulOneClass"],["DistribMulAction","toMulAction"],["instHSMul"],["Monoid","toSemigroup"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Semiring","toModule"],["Eq"],["Group","toDivInvMonoid"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Units"],["AddSemigroup","toAdd"],["AddEquiv","symm"],["Group","toDivisionMonoid"],["Inv","inv"],["Distrib","toAdd"],["InvOneClass","toInv"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["AddEquiv"],["AddAut"],["Module","toDistribMulAction"],["DivInvOneMonoid","toInvOneClass"],["Units","instDistribMulAction"],["DivInvMonoid","toMonoid"],["AddAut","instGroup"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["AddAut","mulLeft"],["MulAction","toSemigroupAction"],["DivisionMonoid","toDivInvOneMonoid"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["MonoidHom"],["MonoidHom","instFunLike"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["AddEquiv","instEquivLike"],["Semiring","toNonAssocSemiring"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Group","toDivInvMonoid"],["Units"],["AddSemigroup","toAdd"],["AddEquiv","symm"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["AddEquiv"],["AddAut"],["DivInvMonoid","toMonoid"],["AddAut","instGroup"],["AddMonoid","toAddSemigroup"],["Eq","refl"],["AddAut","mulLeft"]]},{"isProp":true,"kind":"theorem","name":["AddAut","mulRight_apply"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3 : Semiring.{u_1} R] (u : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3))) (x : R), Eq.{succ u_1} R (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3)))) (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3))))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (EquivLike.toFunLike.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3)))) (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3))))) R R (AddEquiv.instEquivLike.{u_1, u_1} R R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3)))) (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3)))))) (AddAut.mulRight.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3 u) x) (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.Ring.AddAut.2104618878._hygCtx._hyg.3))))) x (Units.val.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2104618878._hygCtx._hyg.3)) u))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] (u : Rˣ) (x : R), (AddAut.mulRight u) x = x * ↑u","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] (u : Rˣ) (x : R), (AddAut.mulRight u) x = x * ↑u","typeReferences":[["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["HMul","hMul"],["Semiring","toMonoidWithZero"],["DFunLike","coe"],["AddEquiv"],["AddEquiv","instEquivLike"],["Semiring","toNonAssocSemiring"],["Units","val"],["AddAut","mulRight"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["instHMul"],["Eq"],["Units"],["Semiring"]],"valueReferences":[["rfl"],["Distrib","toAdd"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocSemiring","toDistrib"],["AddAut","mulRight"],["EquivLike","toFunLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddEquiv"],["DFunLike","coe"],["AddEquiv","instEquivLike"]]},{"isProp":true,"kind":"theorem","name":["AddAut","mulLeft_apply_apply"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3 : Semiring.{u_1} R] (x : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (a._@._internal._hyg.0 : R), Eq.{succ u_1} R (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (EquivLike.toFunLike.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) R R (AddEquiv.instEquivLike.{u_1, u_1} R R (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))) (AddSemigroup.toAdd.{u_1} R (AddMonoid.toAddSemigroup.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))))))) (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (MonoidHom.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Monoid.toMulOneClass.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (DivInvMonoid.toMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Group.toDivInvMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (AddAut.instGroup.{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.Ring.AddAut.3455109452._hygCtx._hyg.3)))))))))) (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) => AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (MonoidHom.instFunLike.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (MulOneClass.toMulOne.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instMulOneClass.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (MulOneClass.toMulOne.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Monoid.toMulOneClass.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (DivInvMonoid.toMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (Group.toDivInvMonoid.{u_1} (AddAut.{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.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (AddAut.instGroup.{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.Ring.AddAut.3455109452._hygCtx._hyg.3)))))))))) (AddAut.mulLeft.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3) x) a._@._internal._hyg.0) (HSMul.hSMul.{u_1, u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R R (instHSMul.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (SMulZeroClass.toSMul.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (AddZero.toZero.{u_1} R (AddZeroClass.toAddZero.{u_1} R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) (DistribSMul.toSMulZeroClass.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (AddMonoid.toAddZeroClass.{u_1} R (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (DistribMulAction.toDistribSMul.{u_1, u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) R (DivInvMonoid.toMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Group.toDivInvMonoid.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Units.instGroup.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (Units.instDistribMulAction.{u_1, u_1} R R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)) (AddMonoidWithOne.toAddMonoid.{u_1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3)))) (Module.toDistribMulAction.{u_1, u_1} R R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))) (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.3455109452._hygCtx._hyg.3))))))) x a._@._internal._hyg.0)","typeFull":"∀ {R : Type u_1} [inst : Semiring R] (x : Rˣ) (a : R), (AddAut.mulLeft x) a = x • a","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] (x : Rˣ) (a : R), (AddAut.mulLeft x) a = x • a","typeReferences":[["MulOneClass","toMulOne"],["Units","instGroup"],["MonoidHom"],["MonoidHom","instFunLike"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["AddEquiv","instEquivLike"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["Monoid","toMulOneClass"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Semiring","toModule"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Group","toDivInvMonoid"],["Units"],["DistribSMul","toSMulZeroClass"],["AddSemigroup","toAdd"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["AddZeroClass","toAddZero"],["AddEquiv"],["AddAut"],["Module","toDistribMulAction"],["Units","instDistribMulAction"],["DivInvMonoid","toMonoid"],["AddAut","instGroup"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["AddAut","mulLeft"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["MonoidHom"],["MonoidHom","instFunLike"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["AddEquiv","instEquivLike"],["Semiring","toNonAssocSemiring"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Group","toDivInvMonoid"],["Units"],["AddSemigroup","toAdd"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["AddEquiv"],["AddAut"],["DivInvMonoid","toMonoid"],["AddMonoid","toAddSemigroup"],["Eq","refl"],["AddAut","instGroup"],["AddAut","mulLeft"]]},{"isProp":true,"kind":"theorem","name":["AddAut","mulRight_symm_apply"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3 : Semiring.{u_1} R] (u : Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3))) (x : R), Eq.{succ u_1} R (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3)))) (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3))))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => R) (EquivLike.toFunLike.{succ u_1, succ u_1, succ u_1} (AddEquiv.{u_1, u_1} R R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3)))) (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3))))) R R (AddEquiv.instEquivLike.{u_1, u_1} R R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3)))) (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3)))))) (AddEquiv.symm.{u_1, u_1} R R (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3)))) (Distrib.toAdd.{u_1} R (NonUnitalNonAssocSemiring.toDistrib.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3)))) (AddAut.mulRight.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3 u)) x) (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.Ring.AddAut.2641806681._hygCtx._hyg.3))))) x (Units.val.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3)) (Inv.inv.{u_1} (Units.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3))) (Units.instInv.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R inst._@.Mathlib.Algebra.Ring.AddAut.2641806681._hygCtx._hyg.3))) u)))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] (u : Rˣ) (x : R), (AddEquiv.symm (AddAut.mulRight u)) x = x * ↑u⁻¹","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] (u : Rˣ) (x : R), (AddEquiv.symm (AddAut.mulRight u)) x = x * ↑u⁻¹","typeReferences":[["AddEquiv","symm"],["Inv","inv"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Units","instInv"],["HMul","hMul"],["Semiring","toMonoidWithZero"],["DFunLike","coe"],["AddEquiv"],["AddEquiv","instEquivLike"],["Semiring","toNonAssocSemiring"],["Units","val"],["AddAut","mulRight"],["MonoidWithZero","toMonoid"],["EquivLike","toFunLike"],["instHMul"],["Eq"],["Units"],["Semiring"]],"valueReferences":[["rfl"],["AddEquiv","symm"],["Distrib","toAdd"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocSemiring","toDistrib"],["AddAut","mulRight"],["EquivLike","toFunLike"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddEquiv"],["DFunLike","coe"],["AddEquiv","instEquivLike"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["IsUnit","neg"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5)))] {a : α}, (IsUnit.{u} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 a) -> (IsUnit.{u} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.8)) a))","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] {a : α}, IsUnit a → IsUnit (-a)","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] {a : α}, IsUnit a → IsUnit (-a)","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["Neg","neg"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["IsUnit"],["InvolutiveNeg","toNeg"]],"valueReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["Neg","neg"],["IsUnit"],["_private","Mathlib","Algebra","Ring","Units",0,"IsUnit","neg","match_1_1"],["MulOne","toMul"],["Units","isUnit"],["HasDistribNeg","toInvolutiveNeg"],["Units","val"],["Monoid","toMulOneClass"],["InvolutiveNeg","toNeg"],["Eq","rec"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","divp_sub_divp_same"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.2623939083._hygCtx._hyg.5 : Ring.{u} α] (a : α) (b : α) (u : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2623939083._hygCtx._hyg.5)))), Eq.{succ u} α (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.2623939083._hygCtx._hyg.5))))) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2623939083._hygCtx._hyg.5))) a u) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2623939083._hygCtx._hyg.5))) b u)) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2623939083._hygCtx._hyg.5))) (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.2623939083._hygCtx._hyg.5))))) a b) u)","typeFull":"∀ {α : Type u} [inst : Ring α] (a b : α) (u : αˣ), a /ₚ u - b /ₚ u = (a - b) /ₚ u","typeReadable":"∀ {α : Type u} [inst : Ring α] (a b : α) (u : αˣ), a /ₚ u - b /ₚ u = (a - b) /ₚ u","typeReferences":[["divp"],["Semiring","toMonoidWithZero"],["Ring","toSemiring"],["AddGroupWithOne","toAddGroup"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["Units"],["Ring"]],"valueReferences":[["Units","divp_add_divp_same"],["MulOneClass","toMulOne"],["Ring","toNonAssocRing"],["divp"],["congrArg"],["sub_eq_add_neg"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["NonUnitalNonAssocRing","toHasDistribNeg"],["AddGroup","toSubNegMonoid"],["InvolutiveNeg","toNeg"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["Units","neg_divp"],["instHAdd"],["Neg","neg"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["SubNegMonoid","toNeg"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Ring","toSemiring"],["HAdd","hAdd"],["AddGroupWithOne","toAddGroup"],["SubNegMonoid","toAddMonoid"],["HasDistribNeg","toInvolutiveNeg"],["Eq","refl"],["id"],["Eq","mpr"],["instHSub"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Units","coe_neg_one"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5)))], Eq.{succ u} α (Units.val.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5 (Neg.neg.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5) (Units.instNeg.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.8) (OfNat.ofNat.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5) 1 (One.toOfNat1.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5) (Units.instOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5))))) (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.8)) (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3998613915._hygCtx._hyg.5))))))","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α], ↑(-1) = -1","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α], ↑(-1) = -1","typeReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["MulOne","toOne"],["Neg","neg"],["OfNat","ofNat"],["Units","instOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["One","toOfNat1"],["Units","val"],["Monoid","toMulOneClass"],["Monoid"],["HasDistribNeg"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["rfl"],["Units","instNeg"],["One","toOfNat1"],["Neg","neg"],["Units","val"],["OfNat","ofNat"],["Units"],["Units","instOne"]]},{"isProp":false,"kind":"definition","name":["Units","instNeg"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)))], Neg.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)","typeFull":"{α : Type u} → [inst : Monoid α] → [HasDistribNeg α] → Neg αˣ","typeReadable":"{α : Type u} → [inst : Monoid α] → [HasDistribNeg α] → Neg αˣ","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["Neg"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["Inv","inv"],["Neg","neg"],["Units","instInv"],["Units","mk"],["Units","instNeg","_proof_2"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["Units","val"],["Monoid","toMulOneClass"],["InvolutiveNeg","toNeg"],["Neg","mk"],["Units","instNeg","_proof_1"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","divp_sub"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5 : Ring.{u} α] (a : α) (b : α) (u : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5)))), Eq.{succ u} α (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5))))) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5))) a u) b) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5))) (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5))))) a (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} α (NonAssocRing.toNonUnitalNonAssocRing.{u} α (Ring.toNonAssocRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5)))))) b (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2980190669._hygCtx._hyg.5))) u))) u)","typeFull":"∀ {α : Type u} [inst : Ring α] (a b : α) (u : αˣ), a /ₚ u - b = (a - b * ↑u) /ₚ u","typeReadable":"∀ {α : Type u} [inst : Ring α] (a b : α) (u : αˣ), a /ₚ u - b = (a - b * ↑u) /ₚ u","typeReferences":[["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toDistrib"],["divp"],["Distrib","toMul"],["HMul","hMul"],["Semiring","toMonoidWithZero"],["Ring","toSemiring"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddGroupWithOne","toAddGroup"],["Ring","toAddGroupWithOne"],["Units","val"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["instHMul"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["NonAssocRing","toNonUnitalNonAssocRing"],["Units"],["Ring"]],"valueReferences":[["NonUnitalNonAssocRing","toAddCommGroup"],["MulOneClass","toMulOne"],["Ring","toNonAssocRing"],["Eq","trans"],["divp"],["AddCommGroup","toAddGroup"],["Units","instInv"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["mul_assoc"],["Semigroup","toMul"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["Units"],["sub_mul"],["Inv","inv"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["Units","mul_inv"],["Distrib","toMul"],["_private","Mathlib","Algebra","Ring","Units",0,"Units","divp_sub","_simp_1_1"],["Semiring","toMonoidWithZero"],["mul_one"],["OfNat","ofNat"],["Ring","toSemiring"],["AddGroupWithOne","toAddGroup"],["One","toOfNat1"],["Units","val"],["Eq","refl"],["id"],["instHMul"],["Eq","mpr"],["SemigroupWithZero","toSemigroup"],["instHSub"]]},{"isProp":true,"kind":"theorem","name":["Units","divp_add"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5 : Semiring.{u} α] (a : α) (b : α) (u : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5))), Eq.{succ u} α (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5))))) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5)) a u) b) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5)) (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5))))) a (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5))))) b (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.3247562239._hygCtx._hyg.5)) u))) u)","typeFull":"∀ {α : Type u} [inst : Semiring α] (a b : α) (u : αˣ), a /ₚ u + b = (a + b * ↑u) /ₚ u","typeReadable":"∀ {α : Type u} [inst : Semiring α] (a b : α) (u : αˣ), a /ₚ u + b = (a + b * ↑u) /ₚ u","typeReferences":[["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["divp"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["HMul","hMul"],["HAdd","hAdd"],["Semiring","toNonAssocSemiring"],["Units","val"],["MonoidWithZero","toMonoid"],["instHMul"],["Eq"],["Units"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["divp"],["Units","instInv"],["HMul","hMul"],["congrArg"],["add_mul"],["Distrib","rightDistribClass"],["MulOne","toMul"],["Units","mul_inv_cancel_right"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Eq"],["Units"],["Distrib","toAdd"],["Inv","inv"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["HAdd","hAdd"],["eq_self"],["of_eq_true"],["Units","val"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["Units","instNeg","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.8 : HasDistribNeg.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)))] (u : Units.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5), Eq.{succ u_1} α (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)))) (Neg.neg.{u_1} α (InvolutiveNeg.toNeg.{u_1} α (HasDistribNeg.toInvolutiveNeg.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.8)) (Units.val.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5 u)) (Neg.neg.{u_1} α (InvolutiveNeg.toNeg.{u_1} α (HasDistribNeg.toInvolutiveNeg.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.8)) (Units.val.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5 (Inv.inv.{u_1} (Units.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5) (Units.instInv.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5) u)))) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)))))","typeFull":"∀ {α : Type u_1} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), -↑u * -↑u⁻¹ = 1","typeReadable":"∀ {α : Type u_1} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), -↑u * -↑u⁻¹ = 1","typeReferences":[["MulOneClass","toMulOne"],["Inv","inv"],["MulOne","toOne"],["Neg","neg"],["Units","instInv"],["HMul","hMul"],["OfNat","ofNat"],["MulOne","toMul"],["One","toOfNat1"],["HasDistribNeg","toInvolutiveNeg"],["Units","val"],["Monoid","toMulOneClass"],["Monoid"],["HasDistribNeg"],["instHMul"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["Units","instInv"],["HMul","hMul"],["neg_neg"],["congrArg"],["MulOne","toMul"],["mul_neg"],["neg_mul"],["Monoid","toMulOneClass"],["congrFun'"],["InvolutiveNeg","toNeg"],["Eq"],["Units"],["Inv","inv"],["MulOne","toOne"],["True"],["Neg","neg"],["Units","mul_inv"],["OfNat","ofNat"],["eq_self"],["HasDistribNeg","toInvolutiveNeg"],["of_eq_true"],["One","toOfNat1"],["Units","val"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["Units","divp_add_divp_same"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.1001560643._hygCtx._hyg.5 : Semiring.{u} α] (a : α) (b : α) (u : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.1001560643._hygCtx._hyg.5))), Eq.{succ u} α (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.1001560643._hygCtx._hyg.5))))) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.1001560643._hygCtx._hyg.5)) a u) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.1001560643._hygCtx._hyg.5)) b u)) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.1001560643._hygCtx._hyg.5)) (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.1001560643._hygCtx._hyg.5))))) a b) u)","typeFull":"∀ {α : Type u} [inst : Semiring α] (a b : α) (u : αˣ), a /ₚ u + b /ₚ u = (a + b) /ₚ u","typeReadable":"∀ {α : Type u} [inst : Semiring α] (a b : α) (u : αˣ), a /ₚ u + b /ₚ u = (a + b) /ₚ u","typeReferences":[["HAdd","hAdd"],["Distrib","toAdd"],["Semiring","toNonAssocSemiring"],["NonUnitalNonAssocSemiring","toDistrib"],["divp"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["Eq"],["Units"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["divp"],["Units","instInv"],["HMul","hMul"],["congrArg"],["add_mul"],["Distrib","rightDistribClass"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Eq"],["Units"],["Distrib","toAdd"],["Inv","inv"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["HAdd","hAdd"],["eq_self"],["of_eq_true"],["Units","val"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["Units","divp_sub_divp"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5 : CommRing.{u} α] (a : α) (b : α) (u₁ : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))) (u₂ : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))), Eq.{succ u} α (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α (CommRing.toRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))))) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))) a u₁) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))) b u₂)) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))) (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α (CommRing.toRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))))) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} α (CommRing.toNonUnitalCommRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))))) a (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))) u₂)) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} α (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} α (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} α (CommRing.toNonUnitalCommRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))))) (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))) u₁) b)) (HMul.hMul.{u, u, u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))) (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))) (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))) (instHMul.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5))))) (Units.instMul.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α (CommRing.toCommSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2007046379._hygCtx._hyg.5)))))) u₁ u₂))","typeFull":"∀ {α : Type u} [inst : CommRing α] (a b : α) (u₁ u₂ : αˣ), a /ₚ u₁ - b /ₚ u₂ = (a * ↑u₂ - ↑u₁ * b) /ₚ (u₁ * u₂)","typeReadable":"∀ {α : Type u} [inst : CommRing α] (a b : α) (u₁ u₂ : αˣ), a /ₚ u₁ - b /ₚ u₂ = (a * ↑u₂ - ↑u₁ * b) /ₚ (u₁ * u₂)","typeReferences":[["Units","instMul"],["divp"],["HMul","hMul"],["CommRing","toNonUnitalCommRing"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["Units"],["CommRing","toCommSemiring"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CommSemiring","toSemiring"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["CommRing"],["CommRing","toRing"],["AddGroupWithOne","toAddGroup"],["Units","val"],["instHMul"],["instHSub"]],"valueReferences":[["Units","instMul"],["Eq","trans"],["HMul","hMul"],["sub_eq_add_neg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["mul_neg"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["NonUnitalNonAssocRing","toHasDistribNeg"],["AddGroup","toSubNegMonoid"],["InvolutiveNeg","toNeg"],["Units"],["Units","divp_add_divp"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddZeroClass","toAddZero"],["Units","val"],["instHMul"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["divp"],["CommRing","toNonUnitalCommRing"],["congrArg"],["MulOne","toMul"],["congr"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["congrFun'"],["Eq"],["CommRing","toCommSemiring"],["Distrib","toAdd"],["True"],["Units","neg_divp"],["instHAdd"],["Distrib","toMul"],["CommSemiring","toSemiring"],["SubNegMonoid","toNeg"],["Semiring","toMonoidWithZero"],["AddZero","toAdd"],["HAdd","hAdd"],["eq_self"],["CommRing","toRing"],["AddGroupWithOne","toAddGroup"],["of_eq_true"],["HasDistribNeg","toInvolutiveNeg"],["SubNegMonoid","toAddMonoid"],["instHSub"]]},{"isProp":true,"kind":"theorem","name":["IsUnit","sub_iff"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.3774554476._hygCtx._hyg.5 : Ring.{u} α] {x : α} {y : α}, Iff (IsUnit.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3774554476._hygCtx._hyg.5))) (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.3774554476._hygCtx._hyg.5))))) x y)) (IsUnit.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3774554476._hygCtx._hyg.5))) (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.3774554476._hygCtx._hyg.5))))) y x))","typeFull":"∀ {α : Type u} [inst : Ring α] {x y : α}, IsUnit (x - y) ↔ IsUnit (y - x)","typeReadable":"∀ {α : Type u} [inst : Ring α] {x y : α}, IsUnit (x - y) ↔ IsUnit (y - x)","typeReferences":[["AddGroupWithOne","toAddGroup"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["Iff"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["Semiring","toMonoidWithZero"],["IsUnit"],["AddGroup","toSubNegMonoid"],["instHSub"],["Ring","toSemiring"],["Ring"]],"valueReferences":[["MulOneClass","toMulOne"],["Ring","toNonAssocRing"],["SubtractionCommMonoid","toSubtractionMonoid"],["IsUnit","neg_iff"],["MulOne","toMul"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["HSub","hSub"],["NonUnitalNonAssocRing","toHasDistribNeg"],["InvolutiveNeg","toNeg"],["AddGroup","toSubNegMonoid"],["Eq","rec"],["NonAssocRing","toNonUnitalNonAssocRing"],["Iff","trans"],["Neg","neg"],["SubNegMonoid","toNeg"],["neg_sub"],["Semiring","toMonoidWithZero"],["IsUnit"],["Iff","rfl"],["Iff","symm"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["AddGroupWithOne","toAddGroup"],["AddCommGroup","toDivisionAddCommMonoid"],["HasDistribNeg","toInvolutiveNeg"],["SubtractionMonoid","toSubNegMonoid"],["Iff"],["instHSub"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Ring","Units",0,"IsUnit","neg","match_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 : Monoid.{u_1} α] {a : α} (motive : (IsUnit.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 a) -> Prop) (x._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx.22.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.34 : IsUnit.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 a), (forall (x : Units.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5) (hx : Eq.{succ u_1} α (Units.val.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 x) a), motive (Exists.intro.{succ u_1} (Units.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5) (fun (u : Units.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5) => Eq.{succ u_1} α (Units.val.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.5 u) a) x hx)) -> (motive x._@.Mathlib.Algebra.Ring.Units.1717405508._hygCtx.22.Mathlib.Algebra.Ring.Units.1717405508._hygCtx._hyg.34)","typeFull":"∀ {α : Type u_1} [inst : Monoid α] {a : α} (motive : IsUnit a → Prop) (x : IsUnit a),\n  (∀ (x : αˣ) (hx : ↑x = a), motive ⋯) → motive x","typeReadable":"∀ {α : Type u_1} [inst : Monoid α] {a : α} (motive : IsUnit a → Prop) (x : IsUnit a),\n  (∀ (x : αˣ) (hx : ↑x = a), motive ⋯) → motive x","typeReferences":[["Units","val"],["Monoid"],["Exists","intro"],["IsUnit"],["Eq"],["Units"]],"valueReferences":[["Exists","casesOn"],["Units","val"],["Eq"],["Units"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Ring","Units",0,"Units","divp_sub","_simp_1_1"],"typeFallback":"forall {G : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Basic.3905289566._hygCtx._hyg.6 : AddGroup.{u_3} G] {a : G} {b : G} {c : G}, Eq.{1} Prop (Eq.{succ u_3} G (HSub.hSub.{u_3, u_3, u_3} G G G (instHSub.{u_3} G (SubNegMonoid.toSub.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3905289566._hygCtx._hyg.6))) a b) (HSub.hSub.{u_3, u_3, u_3} G G G (instHSub.{u_3} G (SubNegMonoid.toSub.{u_3} G (AddGroup.toSubNegMonoid.{u_3} G inst._@.Mathlib.Algebra.Group.Basic.3905289566._hygCtx._hyg.6))) a c)) (Eq.{succ u_3} G b c)","typeFull":"∀ {G : Type u_3} [inst : AddGroup G] {a b c : G}, (a - b = a - c) = (b = c)","typeReadable":"∀ {G : Type u_3} [inst : AddGroup G] {a b c : G}, (a - b = a - c) = (b = c)","typeReferences":[["SubNegMonoid","toSub"],["HSub","hSub"],["AddGroup"],["AddGroup","toSubNegMonoid"],["instHSub"],["Eq"]],"valueReferences":[["SubNegMonoid","toSub"],["HSub","hSub"],["sub_right_inj"],["AddGroup","toSubNegMonoid"],["instHSub"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["IsUnit","neg_iff"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5)))] (a : α), Iff (IsUnit.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5 (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.8)) a)) (IsUnit.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5 a)","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (a : α), IsUnit (-a) ↔ IsUnit a","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (a : α), IsUnit (-a) ↔ IsUnit a","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["Neg","neg"],["Iff"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["IsUnit"],["InvolutiveNeg","toNeg"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["IsUnit","neg"],["Neg","neg"],["Monoid","toMulOneClass"],["neg_neg"],["IsUnit"],["InvolutiveNeg","toNeg"],["Eq","rec"],["Iff","intro"]]},{"isProp":true,"kind":"theorem","name":["RingHom","isUnit_map"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} [inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.5 : Semiring.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.8 : Semiring.{v} β] (f : RingHom.{u, v} α β (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.5) (Semiring.toNonAssocSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.8)) {a : α}, (IsUnit.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.5)) a) -> (IsUnit.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.8)) (DFunLike.coe.{max (succ u) (succ v), succ u, succ v} (RingHom.{u, v} α β (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.5) (Semiring.toNonAssocSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.8)) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (RingHom.instFunLike.{u, v} α β (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.5) (Semiring.toNonAssocSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.633642759._hygCtx._hyg.8)) f a))","typeFull":"∀ {α : Type u} {β : Type v} [inst : Semiring α] [inst_1 : Semiring β] (f : α →+* β) {a : α}, IsUnit a → IsUnit (f a)","typeReadable":"∀ {α : Type u} {β : Type v} [inst : Semiring α] [inst_1 : Semiring β] (f : α →+* β) {a : α}, IsUnit a → IsUnit (f a)","typeReferences":[["RingHom"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["RingHom","instFunLike"],["Semiring","toMonoidWithZero"],["IsUnit"],["DFunLike","coe"],["Semiring"]],"valueReferences":[["RingHom"],["IsUnit","map"],["Semiring","toNonAssocSemiring"],["RingHomClass","toMonoidWithZeroHomClass"],["MonoidWithZero","toMonoid"],["Semiring","toMonoidWithZero"],["RingHom","instFunLike"],["MonoidWithZeroHomClass","toMonoidHomClass"],["RingHom","instRingHomClass"],["NonAssocSemiring","toMulZeroOneClass"]]},{"isProp":true,"kind":"theorem","name":["Units","val_eq_neg_one","_simp_1"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5)))] {a : Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5}, Eq.{1} Prop (Eq.{succ u} α (Units.val.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5 a) (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.8)) (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5))))))) (Eq.{succ u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) a (Neg.neg.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) (Units.instNeg.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.8) (OfNat.ofNat.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) 1 (One.toOfNat1.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) (Units.instOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5)))))","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] {a : αˣ}, (↑a = -1) = (a = -1)","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] {a : αˣ}, (↑a = -1) = (a = -1)","typeReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["MulOne","toOne"],["Neg","neg"],["OfNat","ofNat"],["Units","instOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["One","toOfNat1"],["Units","val"],["Monoid","toMulOneClass"],["Monoid"],["HasDistribNeg"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["MulOne","toOne"],["Neg","neg"],["OfNat","ofNat"],["Units","instOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["One","toOfNat1"],["Units","val"],["Monoid","toMulOneClass"],["Units","val_eq_neg_one"],["InvolutiveNeg","toNeg"],["Eq"],["propext"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","divp_add_divp"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5 : CommSemiring.{u} α] (a : α) (b : α) (u₁ : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))) (u₂ : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))), Eq.{succ u} α (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))))) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5))) a u₁) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5))) b u₂)) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5))) (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))))) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))))) a (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5))) u₂)) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))))) (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5))) u₁) b)) (HMul.hMul.{u, u, u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))) (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))) (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))) (instHMul.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5)))) (Units.instMul.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.3601599087._hygCtx._hyg.5))))) u₁ u₂))","typeFull":"∀ {α : Type u} [inst : CommSemiring α] (a b : α) (u₁ u₂ : αˣ), a /ₚ u₁ + b /ₚ u₂ = (a * ↑u₂ + ↑u₁ * b) /ₚ (u₁ * u₂)","typeReadable":"∀ {α : Type u} [inst : CommSemiring α] (a b : α) (u₁ u₂ : αˣ), a /ₚ u₁ + b /ₚ u₂ = (a * ↑u₂ + ↑u₁ * b) /ₚ (u₁ * u₂)","typeReferences":[["Units","instMul"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["divp"],["Distrib","toMul"],["CommSemiring"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["HMul","hMul"],["Semiring","toMonoidWithZero"],["HAdd","hAdd"],["Semiring","toNonAssocSemiring"],["Units","val"],["MonoidWithZero","toMonoid"],["instHMul"],["Eq"],["Units"]],"valueReferences":[["DivInvMonoid","toInv"],["Units","instMul"],["Units","instCommGroupUnits"],["Eq","trans"],["HMul","hMul"],["NonUnitalSemiring","toSemigroupWithZero"],["CommGroup","toDivisionCommMonoid"],["Distrib","rightDistribClass"],["DivisionMonoid","toDivInvMonoid"],["NonUnitalCommSemiring","toNonUnitalNonAssocCommSemiring"],["Semiring","toNonAssocSemiring"],["mul_comm"],["Eq","symm"],["NonAssocSemiring","toMulZeroOneClass"],["Units"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["CommMagma","toMul"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["Units","inv_mul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommSemiring","toCommMonoid"],["DivInvMonoid","toMonoid"],["Units","val"],["Eq","refl"],["id"],["instHMul"],["Eq","mpr"],["MulOneClass","toMulOne"],["divp"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["Units","instInv"],["MulZeroOneClass","toMulOneClass"],["mul_assoc"],["congrArg"],["add_mul"],["Semigroup","toMul"],["MulOne","toMul"],["CommSemiring","toNonUnitalCommSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["mul_inv_rev"],["Inv","inv"],["Distrib","toAdd"],["instHAdd"],["Units","mul_inv"],["Distrib","toMul"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["mul_one"],["OfNat","ofNat"],["HAdd","hAdd"],["One","toOfNat1"],["SemigroupWithZero","toSemigroup"]]},{"isProp":true,"kind":"theorem","name":["Units","instNeg","_proof_2"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5 : Monoid.{u_1} α] [inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.8 : HasDistribNeg.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)))] (u : Units.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5), Eq.{succ u_1} α (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)))) (Neg.neg.{u_1} α (InvolutiveNeg.toNeg.{u_1} α (HasDistribNeg.toInvolutiveNeg.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.8)) (Units.val.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5 (Inv.inv.{u_1} (Units.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5) (Units.instInv.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5) u))) (Neg.neg.{u_1} α (InvolutiveNeg.toNeg.{u_1} α (HasDistribNeg.toInvolutiveNeg.{u_1} α (MulOne.toMul.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.8)) (Units.val.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5 u))) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (Monoid.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Ring.Units.1827621731._hygCtx._hyg.5)))))","typeFull":"∀ {α : Type u_1} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), -↑u⁻¹ * -↑u = 1","typeReadable":"∀ {α : Type u_1} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), -↑u⁻¹ * -↑u = 1","typeReferences":[["MulOneClass","toMulOne"],["Inv","inv"],["MulOne","toOne"],["Neg","neg"],["Units","instInv"],["HMul","hMul"],["OfNat","ofNat"],["MulOne","toMul"],["One","toOfNat1"],["HasDistribNeg","toInvolutiveNeg"],["Units","val"],["Monoid","toMulOneClass"],["Monoid"],["HasDistribNeg"],["instHMul"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["Units","instInv"],["HMul","hMul"],["neg_neg"],["congrArg"],["MulOne","toMul"],["mul_neg"],["neg_mul"],["Monoid","toMulOneClass"],["congrFun'"],["InvolutiveNeg","toNeg"],["Eq"],["Units"],["Inv","inv"],["MulOne","toOne"],["True"],["Neg","neg"],["Units","inv_mul"],["OfNat","ofNat"],["eq_self"],["HasDistribNeg","toInvolutiveNeg"],["of_eq_true"],["One","toOfNat1"],["Units","val"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["Units","map_neg_one"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} [inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5 : Ring.{u} α] {F : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9 : Ring.{v} β] [inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.12 : FunLike.{succ u_1, succ u, succ v} F α β] [inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.17 : RingHomClass.{u_1, u, v} F α β (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)) (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.12] (f : F), Eq.{succ v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (DFunLike.coe.{max (succ u) (succ v), succ u, succ v} (MonoidHom.{u, v} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (MulOneClass.toMulOne.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (Units.instMulOneClass.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (Units.instMulOneClass.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))))) (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) => Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (MonoidHom.instFunLike.{u, v} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (MulOneClass.toMulOne.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (Units.instMulOneClass.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (Units.instMulOneClass.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))))) (Units.map.{u, v} α β (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5))) (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9))) (MonoidHomClass.toMonoidHom.{u, v, u_1} α β F (MulOneClass.toMulOne.{u} α (MulZeroOneClass.toMulOneClass.{u} α (NonAssocSemiring.toMulZeroOneClass.{u} α (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} β (MulZeroOneClass.toMulOneClass.{v} β (NonAssocSemiring.toMulZeroOneClass.{v} β (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9))))) inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.12 (MonoidWithZeroHomClass.toMonoidHomClass.{u_1, u, v} F α β (NonAssocSemiring.toMulZeroOneClass.{u} α (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5))) (NonAssocSemiring.toMulZeroOneClass.{v} β (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9))) inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.12 (RingHomClass.toMonoidWithZeroHomClass.{u_1, u, v} F α β (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)) (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.17)) f)) (Neg.neg.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (Units.instNeg.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5))) (NonUnitalNonAssocRing.toHasDistribNeg.{u} α (NonAssocRing.toNonUnitalNonAssocRing.{u} α (Ring.toNonAssocRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (OfNat.ofNat.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) 1 (One.toOfNat1.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))) (Units.instOne.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.5)))))))) (Neg.neg.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (Units.instNeg.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9))) (NonUnitalNonAssocRing.toHasDistribNeg.{v} β (NonAssocRing.toNonUnitalNonAssocRing.{v} β (Ring.toNonAssocRing.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (OfNat.ofNat.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) 1 (One.toOfNat1.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))) (Units.instOne.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.2651577279._hygCtx._hyg.9)))))))","typeFull":"∀ {α : Type u} {β : Type v} [inst : Ring α] {F : Type u_1} [inst_1 : Ring β] [inst_2 : FunLike F α β]\n  [inst_3 : RingHomClass F α β] (f : F), (Units.map ↑f) (-1) = -1","typeReadable":"∀ {α : Type u} {β : Type v} [inst : Ring α] {F : Type u_1} [inst_1 : Ring β] [inst_2 : FunLike F α β]\n  [inst_3 : RingHomClass F α β] (f : F), (Units.map ↑f) (-1) = -1","typeReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["Ring","toNonAssocRing"],["MonoidHom"],["MonoidHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZeroHomClass","toMonoidHomClass"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["RingHomClass","toMonoidWithZeroHomClass"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocRing","toHasDistribNeg"],["Eq"],["Units"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["FunLike"],["Neg","neg"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["OfNat","ofNat"],["Units","instOne"],["Ring","toSemiring"],["One","toOfNat1"],["Units","map"],["MonoidHomClass","toMonoidHom"],["RingHomClass"],["Ring"]],"valueReferences":[["MonoidHomClass","toOneHomClass"],["MulOneClass","toMulOne"],["Units","instNeg"],["Ring","toNonAssocRing"],["Eq","trans"],["MonoidHom"],["MonoidHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZeroHomClass","toMonoidHomClass"],["DFunLike","coe"],["congrArg"],["Semiring","toNonAssocSemiring"],["RingHomClass","toMonoidWithZeroHomClass"],["MonoidWithZero","toMonoid"],["congrFun'"],["NonUnitalNonAssocRing","toHasDistribNeg"],["map_one"],["Eq"],["Units"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["True"],["Neg","neg"],["Semiring","toMonoidWithZero"],["Units","map_neg"],["Units","instMulOneClass"],["MonoidHom","instMonoidHomClass"],["OfNat","ofNat"],["Units","instOne"],["Ring","toSemiring"],["eq_self"],["of_eq_true"],["One","toOfNat1"],["Units","map"],["MonoidHomClass","toMonoidHom"]]},{"isProp":true,"kind":"theorem","name":["Units","val_eq_neg_one"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5)))] {a : Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5}, Iff (Eq.{succ u} α (Units.val.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5 a) (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.8)) (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5))))))) (Eq.{succ u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) a (Neg.neg.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) (Units.instNeg.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.8) (OfNat.ofNat.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) 1 (One.toOfNat1.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5) (Units.instOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.2384965532._hygCtx._hyg.5)))))","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] {a : αˣ}, ↑a = -1 ↔ a = -1","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] {a : αˣ}, ↑a = -1 ↔ a = -1","typeReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["MulOne","toOne"],["Neg","neg"],["OfNat","ofNat"],["Units","instOne"],["MulOne","toMul"],["One","toOfNat1"],["HasDistribNeg","toInvolutiveNeg"],["Units","val"],["Iff"],["Monoid","toMulOneClass"],["Monoid"],["HasDistribNeg"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["Units","val_inj"],["Units","instNeg"],["Units","coe_neg_one"],["MulOne","toOne"],["Neg","neg"],["Iff","rfl"],["OfNat","ofNat"],["congrArg"],["Units","instOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["One","toOfNat1"],["Units","val"],["Iff"],["Monoid","toMulOneClass"],["Eq","symm"],["id"],["Eq","mpr"],["InvolutiveNeg","toNeg"],["Eq"],["propext"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","add_divp"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5 : Semiring.{u} α] (a : α) (b : α) (u : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5))), Eq.{succ u} α (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5))))) a (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5)) b u)) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5)) (HAdd.hAdd.{u, u, u} α α α (instHAdd.{u} α (Distrib.toAdd.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5))))) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5))))) a (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α inst._@.Mathlib.Algebra.Ring.Units.892516224._hygCtx._hyg.5)) u)) b) u)","typeFull":"∀ {α : Type u} [inst : Semiring α] (a b : α) (u : αˣ), a + b /ₚ u = (a * ↑u + b) /ₚ u","typeReadable":"∀ {α : Type u} [inst : Semiring α] (a b : α) (u : αˣ), a + b /ₚ u = (a * ↑u + b) /ₚ u","typeReferences":[["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["divp"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["HMul","hMul"],["HAdd","hAdd"],["Semiring","toNonAssocSemiring"],["Units","val"],["MonoidWithZero","toMonoid"],["instHMul"],["Eq"],["Units"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["divp"],["Units","instInv"],["HMul","hMul"],["congrArg"],["add_mul"],["Distrib","rightDistribClass"],["MulOne","toMul"],["Units","mul_inv_cancel_right"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["congrFun'"],["Eq"],["Units"],["Distrib","toAdd"],["Inv","inv"],["NonUnitalNonAssocSemiring","toDistrib"],["True"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["HAdd","hAdd"],["eq_self"],["of_eq_true"],["Units","val"],["instHMul"]]},{"isProp":true,"kind":"theorem","name":["Units","val_neg"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5)))] (u : Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5), Eq.{succ u} α (Units.val.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 (Neg.neg.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5) (Units.instNeg.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.8) u)) (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.8)) (Units.val.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 u))","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), ↑(-u) = -↑u","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), ↑(-u) = -↑u","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","instNeg"],["HasDistribNeg","toInvolutiveNeg"],["Neg","neg"],["Units","val"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["rfl"],["Units","instNeg"],["Neg","neg"],["Units","val"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","val_neg","_simp_1"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5)))] (u : Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5), Eq.{succ u} α (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.8)) (Units.val.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 u)) (Units.val.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 (Neg.neg.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5) (Units.instNeg.{u} α inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Ring.Units.1783226093._hygCtx._hyg.8) u))","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), -↑u = ↑(-u)","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (u : αˣ), -↑u = ↑(-u)","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","instNeg"],["HasDistribNeg","toInvolutiveNeg"],["Units","val"],["Neg","neg"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["Units","val_neg"],["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","instNeg"],["HasDistribNeg","toInvolutiveNeg"],["Neg","neg"],["Units","val"],["Monoid","toMulOneClass"],["Eq","symm"],["InvolutiveNeg","toNeg"],["Units"]]},{"isProp":true,"kind":"theorem","name":["isUnit_neg_one"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.3023479982._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.3023479982._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3023479982._hygCtx._hyg.5)))], IsUnit.{u} α inst._@.Mathlib.Algebra.Ring.Units.3023479982._hygCtx._hyg.5 (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3023479982._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.3023479982._hygCtx._hyg.8)) (OfNat.ofNat.{u} α 1 (One.toOfNat1.{u} α (MulOne.toOne.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3023479982._hygCtx._hyg.5))))))","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α], IsUnit (-1)","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α], IsUnit (-1)","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["MulOne","toOne"],["One","toOfNat1"],["HasDistribNeg","toInvolutiveNeg"],["Neg","neg"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["IsUnit"],["InvolutiveNeg","toNeg"],["OfNat","ofNat"]],"valueReferences":[["MulOneClass","toMulOne"],["isUnit_one"],["MulOne","toOne"],["One","toOfNat1"],["IsUnit","neg"],["Monoid","toMulOneClass"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Units","sub_divp"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5 : Ring.{u} α] (a : α) (b : α) (u : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5)))), Eq.{succ u} α (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5))))) a (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5))) b u)) (divp.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5))) (HSub.hSub.{u, u, u} α α α (instHSub.{u} α (SubNegMonoid.toSub.{u} α (AddGroup.toSubNegMonoid.{u} α (AddGroupWithOne.toAddGroup.{u} α (Ring.toAddGroupWithOne.{u} α inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5))))) (HMul.hMul.{u, u, u} α α α (instHMul.{u} α (Distrib.toMul.{u} α (NonUnitalNonAssocSemiring.toDistrib.{u} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} α (NonAssocRing.toNonUnitalNonAssocRing.{u} α (Ring.toNonAssocRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5)))))) a (Units.val.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.1512667377._hygCtx._hyg.5))) u)) b) u)","typeFull":"∀ {α : Type u} [inst : Ring α] (a b : α) (u : αˣ), a - b /ₚ u = (a * ↑u - b) /ₚ u","typeReadable":"∀ {α : Type u} [inst : Ring α] (a b : α) (u : αˣ), a - b /ₚ u = (a * ↑u - b) /ₚ u","typeReferences":[["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toDistrib"],["divp"],["Distrib","toMul"],["HMul","hMul"],["Semiring","toMonoidWithZero"],["Ring","toSemiring"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddGroupWithOne","toAddGroup"],["Ring","toAddGroupWithOne"],["Units","val"],["SubNegMonoid","toSub"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["instHMul"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["NonAssocRing","toNonUnitalNonAssocRing"],["Units"],["Ring"]],"valueReferences":[["NonUnitalNonAssocRing","toAddCommGroup"],["MulOneClass","toMulOne"],["Ring","toNonAssocRing"],["Eq","trans"],["divp"],["AddCommGroup","toAddGroup"],["Units","instInv"],["HMul","hMul"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["MulOne","toMul"],["Units","mul_inv_cancel_right"],["Ring","toAddGroupWithOne"],["SubNegMonoid","toSub"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["HSub","hSub"],["congrFun'"],["AddGroup","toSubNegMonoid"],["Eq"],["Units"],["NonAssocRing","toNonUnitalNonAssocRing"],["sub_mul"],["Inv","inv"],["True"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["Ring","toSemiring"],["eq_self"],["AddGroupWithOne","toAddGroup"],["of_eq_true"],["Units","val"],["instHMul"],["instHSub"]]},{"isProp":true,"kind":"theorem","name":["Units","neg_divp"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.5)))] (a : α) (u : Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.5), Eq.{succ u} α (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.8)) (divp.{u} α inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.5 a u)) (divp.{u} α inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.5 (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.2521660621._hygCtx._hyg.8)) a) u)","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (a : α) (u : αˣ), -(a /ₚ u) = -a /ₚ u","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (a : α) (u : αˣ), -(a /ₚ u) = -a /ₚ u","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["divp"],["Neg","neg"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["Inv","inv"],["True"],["Eq","trans"],["divp"],["Neg","neg"],["Units","instInv"],["HMul","hMul"],["congrArg"],["eq_self"],["MulOne","toMul"],["of_eq_true"],["HasDistribNeg","toInvolutiveNeg"],["Units","val"],["neg_mul"],["Monoid","toMulOneClass"],["instHMul"],["InvolutiveNeg","toNeg"],["Eq"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Units","map_neg"],"typeFallback":"forall {α : Type.{u}} {β : Type.{v}} [inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5 : Ring.{u} α] {F : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9 : Ring.{v} β] [inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12 : FunLike.{succ u_1, succ u, succ v} F α β] [inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.17 : RingHomClass.{u_1, u, v} F α β (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)) (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12] (f : F) (u : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))), Eq.{succ v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (DFunLike.coe.{max (succ u) (succ v), succ u, succ v} (MonoidHom.{u, v} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (MulOneClass.toMulOne.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.instMulOneClass.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (Units.instMulOneClass.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))))) (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) => Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (MonoidHom.instFunLike.{u, v} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (MulOneClass.toMulOne.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.instMulOneClass.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (Units.instMulOneClass.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))))) (Units.map.{u, v} α β (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))) (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9))) (MonoidHomClass.toMonoidHom.{u, v, u_1} α β F (MulOneClass.toMulOne.{u} α (MulZeroOneClass.toMulOneClass.{u} α (NonAssocSemiring.toMulZeroOneClass.{u} α (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} β (MulZeroOneClass.toMulOneClass.{v} β (NonAssocSemiring.toMulZeroOneClass.{v} β (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9))))) inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12 (MonoidWithZeroHomClass.toMonoidHomClass.{u_1, u, v} F α β (NonAssocSemiring.toMulZeroOneClass.{u} α (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))) (NonAssocSemiring.toMulZeroOneClass.{v} β (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9))) inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12 (RingHomClass.toMonoidWithZeroHomClass.{u_1, u, v} F α β (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)) (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.17)) f)) (Neg.neg.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.instNeg.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))) (NonUnitalNonAssocRing.toHasDistribNeg.{u} α (NonAssocRing.toNonUnitalNonAssocRing.{u} α (Ring.toNonAssocRing.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) u)) (Neg.neg.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (Units.instNeg.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9))) (NonUnitalNonAssocRing.toHasDistribNeg.{v} β (NonAssocRing.toNonUnitalNonAssocRing.{v} β (Ring.toNonAssocRing.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (DFunLike.coe.{max (succ u) (succ v), succ u, succ v} (MonoidHom.{u, v} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (MulOneClass.toMulOne.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.instMulOneClass.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (Units.instMulOneClass.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))))) (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) => Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (MonoidHom.instFunLike.{u, v} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (MulOneClass.toMulOne.{u} (Units.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)))) (Units.instMulOneClass.{u} α (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} (Units.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))) (Units.instMulOneClass.{v} β (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)))))) (Units.map.{u, v} α β (MonoidWithZero.toMonoid.{u} α (Semiring.toMonoidWithZero.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))) (MonoidWithZero.toMonoid.{v} β (Semiring.toMonoidWithZero.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9))) (MonoidHomClass.toMonoidHom.{u, v, u_1} α β F (MulOneClass.toMulOne.{u} α (MulZeroOneClass.toMulOneClass.{u} α (NonAssocSemiring.toMulZeroOneClass.{u} α (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))))) (MulOneClass.toMulOne.{v} β (MulZeroOneClass.toMulOneClass.{v} β (NonAssocSemiring.toMulZeroOneClass.{v} β (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9))))) inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12 (MonoidWithZeroHomClass.toMonoidHomClass.{u_1, u, v} F α β (NonAssocSemiring.toMulZeroOneClass.{u} α (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5))) (NonAssocSemiring.toMulZeroOneClass.{v} β (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9))) inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12 (RingHomClass.toMonoidWithZeroHomClass.{u_1, u, v} F α β (Semiring.toNonAssocSemiring.{u} α (Ring.toSemiring.{u} α inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.5)) (Semiring.toNonAssocSemiring.{v} β (Ring.toSemiring.{v} β inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.9)) inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Ring.Units.580316088._hygCtx._hyg.17)) f)) u))","typeFull":"∀ {α : Type u} {β : Type v} [inst : Ring α] {F : Type u_1} [inst_1 : Ring β] [inst_2 : FunLike F α β]\n  [inst_3 : RingHomClass F α β] (f : F) (u : αˣ), (Units.map ↑f) (-u) = -(Units.map ↑f) u","typeReadable":"∀ {α : Type u} {β : Type v} [inst : Ring α] {F : Type u_1} [inst_1 : Ring β] [inst_2 : FunLike F α β]\n  [inst_3 : RingHomClass F α β] (f : F) (u : αˣ), (Units.map ↑f) (-u) = -(Units.map ↑f) u","typeReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["Ring","toNonAssocRing"],["MonoidHom"],["MonoidHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZeroHomClass","toMonoidHomClass"],["DFunLike","coe"],["Semiring","toNonAssocSemiring"],["RingHomClass","toMonoidWithZeroHomClass"],["MonoidWithZero","toMonoid"],["NonUnitalNonAssocRing","toHasDistribNeg"],["Eq"],["Units"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["FunLike"],["Neg","neg"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["Ring","toSemiring"],["Units","map"],["MonoidHomClass","toMonoidHom"],["RingHomClass"],["Ring"]],"valueReferences":[["MulOneClass","toMulOne"],["Units","instNeg"],["Ring","toNonAssocRing"],["Eq","trans"],["map_neg"],["MonoidHom"],["MonoidHom","instFunLike"],["SubtractionCommMonoid","toSubtractionMonoid"],["MulZeroOneClass","toMulOneClass"],["MonoidWithZeroHomClass","toMonoidHomClass"],["DFunLike","coe"],["congrArg"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["RingHomClass","toMonoidWithZeroHomClass"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["congrFun'"],["NonUnitalNonAssocRing","toHasDistribNeg"],["InvolutiveNeg","toNeg"],["Eq"],["RingHomClass","toAddMonoidHomClass"],["Units"],["NonAssocSemiring","toMulZeroOneClass"],["NonAssocRing","toNonUnitalNonAssocRing"],["True"],["Neg","neg"],["SubNegMonoid","toNeg"],["Semiring","toMonoidWithZero"],["Units","instMulOneClass"],["Ring","toSemiring"],["Units","ext"],["Ring","toAddCommGroup"],["eq_self"],["AddGroupWithOne","toAddGroup"],["AddCommGroup","toDivisionAddCommMonoid"],["HasDistribNeg","toInvolutiveNeg"],["of_eq_true"],["SubtractionMonoid","toSubNegMonoid"],["Units","map"],["Units","val"],["MonoidHomClass","toMonoidHom"]]},{"isProp":true,"kind":"theorem","name":["Units","add_eq_mul_one_add_div"],"typeFallback":"forall {R : Type.{x}} [inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5 : Semiring.{x} R] {a : Units.{x} R (MonoidWithZero.toMonoid.{x} R (Semiring.toMonoidWithZero.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5))} {b : R}, Eq.{succ x} R (HAdd.hAdd.{x, x, x} R R R (instHAdd.{x} R (Distrib.toAdd.{x} R (NonUnitalNonAssocSemiring.toDistrib.{x} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{x} R (Semiring.toNonAssocSemiring.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5))))) (Units.val.{x} R (MonoidWithZero.toMonoid.{x} R (Semiring.toMonoidWithZero.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5)) a) b) (HMul.hMul.{x, x, x} R R R (instHMul.{x} R (Distrib.toMul.{x} R (NonUnitalNonAssocSemiring.toDistrib.{x} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{x} R (Semiring.toNonAssocSemiring.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5))))) (Units.val.{x} R (MonoidWithZero.toMonoid.{x} R (Semiring.toMonoidWithZero.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5)) a) (HAdd.hAdd.{x, x, x} R R R (instHAdd.{x} R (Distrib.toAdd.{x} R (NonUnitalNonAssocSemiring.toDistrib.{x} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{x} R (Semiring.toNonAssocSemiring.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5))))) (OfNat.ofNat.{x} R 1 (One.toOfNat1.{x} R (AddMonoidWithOne.toOne.{x} R (AddCommMonoidWithOne.toAddMonoidWithOne.{x} R (NonAssocSemiring.toAddCommMonoidWithOne.{x} R (Semiring.toNonAssocSemiring.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5)))))) (HMul.hMul.{x, x, x} R R R (instHMul.{x} R (Distrib.toMul.{x} R (NonUnitalNonAssocSemiring.toDistrib.{x} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{x} R (Semiring.toNonAssocSemiring.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5))))) (Units.val.{x} R (MonoidWithZero.toMonoid.{x} R (Semiring.toMonoidWithZero.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5)) (Inv.inv.{x} (Units.{x} R (MonoidWithZero.toMonoid.{x} R (Semiring.toMonoidWithZero.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5))) (Units.instInv.{x} R (MonoidWithZero.toMonoid.{x} R (Semiring.toMonoidWithZero.{x} R inst._@.Mathlib.Algebra.Ring.Units.2149472103._hygCtx._hyg.5))) a)) b)))","typeFull":"∀ {R : Type x} [inst : Semiring R] {a : Rˣ} {b : R}, ↑a + b = ↑a * (1 + ↑a⁻¹ * b)","typeReadable":"∀ {R : Type x} [inst : Semiring R] {a : Rˣ} {b : R}, ↑a + b = ↑a * (1 + ↑a⁻¹ * b)","typeReferences":[["Inv","inv"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Units","instInv"],["Semiring","toMonoidWithZero"],["HMul","hMul"],["OfNat","ofNat"],["HAdd","hAdd"],["Semiring","toNonAssocSemiring"],["One","toOfNat1"],["Units","val"],["AddMonoidWithOne","toOne"],["MonoidWithZero","toMonoid"],["instHMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Units"],["Semiring"]],"valueReferences":[["MulOneClass","toMulOne"],["Units","instInv"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["mul_assoc"],["MulZeroOneClass","toMulOneClass"],["Semigroup","toMul"],["congrArg"],["MulOne","toMul"],["Semiring","toNonAssocSemiring"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["Eq","symm"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["Units"],["Distrib","toAdd"],["Inv","inv"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["instHAdd"],["Units","mul_inv"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Distrib","toMul"],["mul_add"],["Semiring","toMonoidWithZero"],["mul_one"],["OfNat","ofNat"],["HAdd","hAdd"],["One","toOfNat1"],["Units","val"],["Eq","refl"],["AddMonoidWithOne","toOne"],["id"],["instHMul"],["Eq","mpr"],["SemigroupWithZero","toSemigroup"],["one_mul"],["Distrib","leftDistribClass"]]},{"isProp":false,"kind":"definition","name":["Units","instHasDistribNeg"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.662237809._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.662237809._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.662237809._hygCtx._hyg.5)))], HasDistribNeg.{u} (Units.{u} α inst._@.Mathlib.Algebra.Ring.Units.662237809._hygCtx._hyg.5) (Units.instMul.{u} α inst._@.Mathlib.Algebra.Ring.Units.662237809._hygCtx._hyg.5)","typeFull":"{α : Type u} → [inst : Monoid α] → [HasDistribNeg α] → HasDistribNeg αˣ","typeReadable":"{α : Type u} → [inst : Monoid α] → [HasDistribNeg α] → HasDistribNeg αˣ","typeReferences":[["MulOneClass","toMulOne"],["Units","instMul"],["MulOne","toMul"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["Units"]],"valueReferences":[["Units","val_neg"],["MulOneClass","toMulOne"],["Units","instMul"],["MulOne","toMul"],["Units","instNeg"],["Units","val"],["Monoid","toMulOneClass"],["Units","val_mul"],["Units","val_injective"],["Units"],["Function","Injective","hasDistribNeg"]]},{"isProp":true,"kind":"theorem","name":["IsUnit","neg_iff","_simp_1"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5 : Monoid.{u} α] [inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.8 : HasDistribNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5)))] (a : α), Eq.{1} Prop (IsUnit.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5 (Neg.neg.{u} α (InvolutiveNeg.toNeg.{u} α (HasDistribNeg.toInvolutiveNeg.{u} α (MulOne.toMul.{u} α (MulOneClass.toMulOne.{u} α (Monoid.toMulOneClass.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5))) inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.8)) a)) (IsUnit.{u} α inst._@.Mathlib.Algebra.Ring.Units.3427928639._hygCtx._hyg.5 a)","typeFull":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (a : α), IsUnit (-a) = IsUnit a","typeReadable":"∀ {α : Type u} [inst : Monoid α] [inst_1 : HasDistribNeg α] (a : α), IsUnit (-a) = IsUnit a","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["Neg","neg"],["Monoid","toMulOneClass"],["HasDistribNeg"],["Monoid"],["IsUnit"],["InvolutiveNeg","toNeg"],["Eq"]],"valueReferences":[["IsUnit","neg_iff"],["MulOneClass","toMulOne"],["MulOne","toMul"],["HasDistribNeg","toInvolutiveNeg"],["Neg","neg"],["Monoid","toMulOneClass"],["IsUnit"],["InvolutiveNeg","toNeg"],["propext"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.AlgebraicGeometry.Morphisms.LocalIso.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","instIsMultiplicativeScheme"],"typeFallback":"CategoryTheory.MorphismProperty.IsMultiplicative.{u_1, succ u_1} AlgebraicGeometry.Scheme.{u_1} AlgebraicGeometry.Scheme.instCategory.{u_1} AlgebraicGeometry.IsLocalIso.{u_1}","typeFull":"CategoryTheory.MorphismProperty.IsMultiplicative @AlgebraicGeometry.IsLocalIso","typeReadable":"CategoryTheory.MorphismProperty.IsMultiplicative @AlgebraicGeometry.IsLocalIso","typeReferences":[["CategoryTheory","MorphismProperty","IsMultiplicative"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"]],"valueReferences":[["CategoryTheory","MorphismProperty","IsMultiplicative"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","sourceLocalClosure","instIsMultiplicativeSchemeOfIsStableUnderBaseChange"],["AlgebraicGeometry","IsLocalIso","eq_sourceLocalClosure_isOpenImmersion"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","isOpenImmersion_stableUnderBaseChange"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","sourceLocalClosure"],["congrArg"],["AlgebraicGeometry","Scheme","instIsJointlySurjectivePreservingIsOpenImmersion"],["CategoryTheory","CategoryStruct","toQuiver"],["Quiver","Hom"],["id"],["inferInstance"],["AlgebraicGeometry","isOpenImmersion_isMultiplicative"],["Eq","mpr"],["AlgebraicGeometry","IsOpenImmersion"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","isLocalIso_iff"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}} (f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y), Iff (AlgebraicGeometry.IsLocalIso.{u} X Y f) (forall (x : TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))), Exists.{succ u} (AlgebraicGeometry.Scheme.Opens.{u} X) (fun (U : AlgebraicGeometry.Scheme.Opens.{u} X) => And (Membership.mem.{u, u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.Scheme.Opens.{u} X) (SetLike.instMembership.{u, u} (AlgebraicGeometry.Scheme.Opens.{u} X) (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (TopologicalSpace.Opens.instSetLike.{u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) U x) (AlgebraicGeometry.IsOpenImmersion.{u} (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) Y (CategoryTheory.CategoryStruct.comp.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}) (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) X Y (AlgebraicGeometry.Scheme.Opens.ι.{u} X U) f))))","typeFull":"∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y),\n  AlgebraicGeometry.IsLocalIso f ↔\n    ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)","typeReadable":"∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y),\n  AlgebraicGeometry.IsLocalIso f ↔\n    ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)","typeReferences":[["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"],["AlgebraicGeometry","PresheafedSpace","carrier"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["CommRingCat"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["AlgebraicGeometry","Scheme","Opens","ι"],["AlgebraicGeometry","IsOpenImmersion"],["Exists"],["SetLike","instMembership"],["And"],["TopologicalSpace","Opens","instSetLike"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","instCategory"],["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","comp"],["Iff"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"]],"valueReferences":[["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","IsLocalIso","casesOn"],["AlgebraicGeometry","Scheme"],["AlgebraicGeometry","PresheafedSpace","carrier"],["Iff","intro"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["AlgebraicGeometry","IsLocalIso","mk"],["CommRingCat"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["AlgebraicGeometry","Scheme","Opens","ι"],["AlgebraicGeometry","IsOpenImmersion"],["Exists"],["SetLike","instMembership"],["And"],["TopologicalSpace","Opens","instSetLike"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","instCategory"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","comp"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"]]},{"isProp":false,"kind":"inductive","name":["AlgebraicGeometry","IsLocalIso"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}}, (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y) -> Prop","typeFull":"{X Y : AlgebraicGeometry.Scheme} → (X ⟶ Y) → Prop","typeReadable":"{X Y : AlgebraicGeometry.Scheme} → (X ⟶ Y) → Prop","typeReferences":[["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","instCategory"],["Quiver","Hom"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["AlgebraicGeometry","IsLocalIso","recOn"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}} {f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y} {motive : (AlgebraicGeometry.IsLocalIso.{u} X Y f) -> Sort.{u_1}} (t : AlgebraicGeometry.IsLocalIso.{u} X Y f), (forall (exists_isOpenImmersion : forall (x : TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))), Exists.{succ u} (AlgebraicGeometry.Scheme.Opens.{u} X) (fun (U : AlgebraicGeometry.Scheme.Opens.{u} X) => And (Membership.mem.{u, u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.Scheme.Opens.{u} X) (SetLike.instMembership.{u, u} (AlgebraicGeometry.Scheme.Opens.{u} X) (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (TopologicalSpace.Opens.instSetLike.{u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) U x) (AlgebraicGeometry.IsOpenImmersion.{u} (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) Y (CategoryTheory.CategoryStruct.comp.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}) (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) X Y (AlgebraicGeometry.Scheme.Opens.ι.{u} X U) f)))), motive (AlgebraicGeometry.IsLocalIso.mk.{u} X Y f exists_isOpenImmersion)) -> (motive t)","typeFull":"{X Y : AlgebraicGeometry.Scheme} →\n  {f : X ⟶ Y} →\n    {motive : AlgebraicGeometry.IsLocalIso f → Sort u_1} →\n      (t : AlgebraicGeometry.IsLocalIso f) →\n        ((exists_isOpenImmersion :\n              ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n            motive ⋯) →\n          motive t","typeReadable":"{X Y : AlgebraicGeometry.Scheme} →\n  {f : X ⟶ Y} →\n    {motive : AlgebraicGeometry.IsLocalIso f → Sort u_1} →\n      (t : AlgebraicGeometry.IsLocalIso f) →\n        ((exists_isOpenImmersion :\n              ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n            motive ⋯) →\n          motive t","typeReferences":[["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"],["AlgebraicGeometry","PresheafedSpace","carrier"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["AlgebraicGeometry","IsLocalIso","mk"],["CommRingCat"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["AlgebraicGeometry","Scheme","Opens","ι"],["AlgebraicGeometry","IsOpenImmersion"],["Exists"],["SetLike","instMembership"],["And"],["TopologicalSpace","Opens","instSetLike"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","instCategory"],["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","comp"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"]],"valueReferences":[["AlgebraicGeometry","IsLocalIso","rec"]]},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","le_of_isZariskiLocalAtSource"],"typeFallback":"forall (P : CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) [inst._@.Mathlib.AlgebraicGeometry.Morphisms.LocalIso.3544533415._hygCtx._hyg.11 : CategoryTheory.MorphismProperty.ContainsIdentities.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u} P] [inst._@.Mathlib.AlgebraicGeometry.Morphisms.LocalIso.3544533415._hygCtx._hyg.13 : AlgebraicGeometry.IsZariskiLocalAtSource.{u} P], LE.le.{succ u} (forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}}, (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y) -> Prop) (Pi.hasLe.{succ u, succ u} AlgebraicGeometry.Scheme.{u} (fun (X : AlgebraicGeometry.Scheme.{u}) => forall {Y : AlgebraicGeometry.Scheme.{u}}, (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y) -> Prop) (fun (i : AlgebraicGeometry.Scheme.{u}) => Pi.hasLe.{succ u, u} AlgebraicGeometry.Scheme.{u} (fun (Y : AlgebraicGeometry.Scheme.{u}) => (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i Y) -> Prop) (fun (i_1 : AlgebraicGeometry.Scheme.{u}) => Pi.hasLe.{u, 0} (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i i_1) (fun (f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i i_1) => Prop) (fun (i : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i i_1) => Prop.le)))) AlgebraicGeometry.IsLocalIso.{u} P","typeFull":"∀ (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) [P.ContainsIdentities]\n  [AlgebraicGeometry.IsZariskiLocalAtSource P], @AlgebraicGeometry.IsLocalIso ≤ P","typeReadable":"∀ (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) [P.ContainsIdentities]\n  [AlgebraicGeometry.IsZariskiLocalAtSource P], @AlgebraicGeometry.IsLocalIso ≤ P","typeReferences":[["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","instCategory"],["Quiver","Hom"],["CategoryTheory","MorphismProperty","ContainsIdentities"],["Pi","hasLe"],["LE","le"],["Prop","le"],["CategoryTheory","MorphismProperty"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","IsZariskiLocalAtSource"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"]],"valueReferences":[["CategoryTheory","PreZeroHypercover","f"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","IsLocalIso","eq_sourceLocalClosure_isOpenImmersion"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"],["AlgebraicGeometry","IsZariskiLocalAtSource","of_isOpenImmersion"],["congrArg"],["AlgebraicGeometry","IsZariskiLocalAtSource","iff_of_openCover"],["Quiver","Hom"],["Eq","rec"],["AlgebraicGeometry","IsOpenImmersion"],["Eq"],["propext"],["CategoryTheory","PreZeroHypercover","I₀"],["CategoryTheory","PreZeroHypercover","X"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","sourceLocalClosure"],["Exists","casesOn"],["AlgebraicGeometry","Scheme","precoverage"],["CategoryTheory","CategoryStruct","toQuiver"],["CategoryTheory","CategoryStruct","comp"],["CategoryTheory","Precoverage","ZeroHypercover","toPreZeroHypercover"],["id"],["Eq","mpr"],["AlgebraicGeometry","Scheme","Cover"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","AlgebraicGeometry","Morphisms","LocalIso",0,"AlgebraicGeometry","IsLocalIso","eq_iInf","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} {ι : Sort.{u_4}} [inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13 : CompleteLattice.{u_1} α] {f : ι -> α} {a : α}, Eq.{1} Prop (LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeSup.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeSup.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13)))) a (iInf.{u_1, u_4} α ι (CompleteSemilatticeInf.toInfSet.{u_1} α (CompleteLattice.toCompleteSemilatticeInf.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13)) f)) (forall (i : ι), LE.le.{u_1} α (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α (CompleteSemilatticeSup.toPartialOrder.{u_1} α (CompleteLattice.toCompleteSemilatticeSup.{u_1} α inst._@.Mathlib.Order.CompleteLattice.Basic.10424780._hygCtx._hyg.13)))) a (f i))","typeFull":"∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLattice α] {f : ι → α} {a : α}, (a ≤ iInf f) = ∀ (i : ι), a ≤ f i","typeReadable":"∀ {α : Type u_1} {ι : Sort u_4} [inst : CompleteLattice α] {f : ι → α} {a : α}, (a ≤ iInf f) = ∀ (i : ι), a ≤ f i","typeReferences":[["CompleteSemilatticeInf","toInfSet"],["PartialOrder","toPreorder"],["LE","le"],["CompleteSemilatticeSup","toPartialOrder"],["iInf"],["CompleteLattice","toCompleteSemilatticeInf"],["CompleteLattice","toCompleteSemilatticeSup"],["Preorder","toLE"],["Eq"],["CompleteLattice"]],"valueReferences":[["CompleteSemilatticeInf","toInfSet"],["le_iInf_iff"],["PartialOrder","toPreorder"],["LE","le"],["CompleteSemilatticeSup","toPartialOrder"],["iInf"],["CompleteLattice","toCompleteSemilatticeInf"],["CompleteLattice","toCompleteSemilatticeSup"],["Preorder","toLE"],["propext"]]},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","instIsStableUnderBaseChangeScheme"],"typeFallback":"CategoryTheory.MorphismProperty.IsStableUnderBaseChange.{u_1, succ u_1} AlgebraicGeometry.Scheme.{u_1} AlgebraicGeometry.Scheme.instCategory.{u_1} AlgebraicGeometry.IsLocalIso.{u_1}","typeFull":"CategoryTheory.MorphismProperty.IsStableUnderBaseChange @AlgebraicGeometry.IsLocalIso","typeReadable":"CategoryTheory.MorphismProperty.IsStableUnderBaseChange @AlgebraicGeometry.IsLocalIso","typeReferences":[["CategoryTheory","MorphismProperty","IsStableUnderBaseChange"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"]],"valueReferences":[["CategoryTheory","MorphismProperty","IsStableUnderBaseChange"],["AlgebraicGeometry","sourceLocalClosure","instIsStableUnderBaseChangeScheme"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","IsLocalIso","eq_sourceLocalClosure_isOpenImmersion"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","isOpenImmersion_stableUnderBaseChange"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","sourceLocalClosure"],["congrArg"],["AlgebraicGeometry","Scheme","instIsJointlySurjectivePreservingIsOpenImmersion"],["CategoryTheory","CategoryStruct","toQuiver"],["Quiver","Hom"],["id"],["inferInstance"],["Eq","mpr"],["AlgebraicGeometry","IsOpenImmersion"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","le_of_isLocalAtSource"],"typeFallback":"forall (P : CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) [inst._@.Mathlib.AlgebraicGeometry.Morphisms.LocalIso.3544533415._hygCtx._hyg.11 : CategoryTheory.MorphismProperty.ContainsIdentities.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u} P] [inst._@.Mathlib.AlgebraicGeometry.Morphisms.LocalIso.3544533415._hygCtx._hyg.13 : AlgebraicGeometry.IsZariskiLocalAtSource.{u} P], LE.le.{succ u} (forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}}, (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y) -> Prop) (Pi.hasLe.{succ u, succ u} AlgebraicGeometry.Scheme.{u} (fun (X : AlgebraicGeometry.Scheme.{u}) => forall {Y : AlgebraicGeometry.Scheme.{u}}, (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y) -> Prop) (fun (i : AlgebraicGeometry.Scheme.{u}) => Pi.hasLe.{succ u, u} AlgebraicGeometry.Scheme.{u} (fun (Y : AlgebraicGeometry.Scheme.{u}) => (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i Y) -> Prop) (fun (i_1 : AlgebraicGeometry.Scheme.{u}) => Pi.hasLe.{u, 0} (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i i_1) (fun (f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i i_1) => Prop) (fun (i : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) i i_1) => Prop.le)))) AlgebraicGeometry.IsLocalIso.{u} P","typeFull":"∀ (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) [P.ContainsIdentities]\n  [AlgebraicGeometry.IsZariskiLocalAtSource P], @AlgebraicGeometry.IsLocalIso ≤ P","typeReadable":"∀ (P : CategoryTheory.MorphismProperty AlgebraicGeometry.Scheme) [P.ContainsIdentities]\n  [AlgebraicGeometry.IsZariskiLocalAtSource P], @AlgebraicGeometry.IsLocalIso ≤ P","typeReferences":[["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","instCategory"],["Quiver","Hom"],["CategoryTheory","MorphismProperty","ContainsIdentities"],["Pi","hasLe"],["LE","le"],["Prop","le"],["CategoryTheory","MorphismProperty"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","IsZariskiLocalAtSource"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"]],"valueReferences":[["AlgebraicGeometry","IsLocalIso","le_of_isZariskiLocalAtSource"]]},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","exists_isOpenImmersion"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}} {f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y} [self : AlgebraicGeometry.IsLocalIso.{u} X Y f] (x : TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))), Exists.{succ u} (AlgebraicGeometry.Scheme.Opens.{u} X) (fun (U : AlgebraicGeometry.Scheme.Opens.{u} X) => And (Membership.mem.{u, u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.Scheme.Opens.{u} X) (SetLike.instMembership.{u, u} (AlgebraicGeometry.Scheme.Opens.{u} X) (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (TopologicalSpace.Opens.instSetLike.{u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) U x) (AlgebraicGeometry.IsOpenImmersion.{u} (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) Y (CategoryTheory.CategoryStruct.comp.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}) (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) X Y (AlgebraicGeometry.Scheme.Opens.ι.{u} X U) f)))","typeFull":"∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [self : AlgebraicGeometry.IsLocalIso f] (x : ↥X),\n  ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)","typeReadable":"∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y} [self : AlgebraicGeometry.IsLocalIso f] (x : ↥X),\n  ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)","typeReferences":[["SetLike","instMembership"],["Exists"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["And"],["AlgebraicGeometry","IsLocalIso"],["TopologicalSpace","Opens","instSetLike"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","PresheafedSpace","carrier"],["CommRingCat","instCategory"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","toQuiver"],["CommRingCat"],["CategoryTheory","CategoryStruct","comp"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","ι"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"],["AlgebraicGeometry","IsOpenImmersion"]],"valueReferences":[]},{"isProp":false,"kind":"definition","name":["AlgebraicGeometry","IsLocalIso","casesOn"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}} {f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y} {motive : (AlgebraicGeometry.IsLocalIso.{u} X Y f) -> Sort.{u_1}} (t : AlgebraicGeometry.IsLocalIso.{u} X Y f), (forall (exists_isOpenImmersion : forall (x : TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))), Exists.{succ u} (AlgebraicGeometry.Scheme.Opens.{u} X) (fun (U : AlgebraicGeometry.Scheme.Opens.{u} X) => And (Membership.mem.{u, u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.Scheme.Opens.{u} X) (SetLike.instMembership.{u, u} (AlgebraicGeometry.Scheme.Opens.{u} X) (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (TopologicalSpace.Opens.instSetLike.{u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) U x) (AlgebraicGeometry.IsOpenImmersion.{u} (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) Y (CategoryTheory.CategoryStruct.comp.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}) (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) X Y (AlgebraicGeometry.Scheme.Opens.ι.{u} X U) f)))), motive (AlgebraicGeometry.IsLocalIso.mk.{u} X Y f exists_isOpenImmersion)) -> (motive t)","typeFull":"{X Y : AlgebraicGeometry.Scheme} →\n  {f : X ⟶ Y} →\n    {motive : AlgebraicGeometry.IsLocalIso f → Sort u_1} →\n      (t : AlgebraicGeometry.IsLocalIso f) →\n        ((exists_isOpenImmersion :\n              ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n            motive ⋯) →\n          motive t","typeReadable":"{X Y : AlgebraicGeometry.Scheme} →\n  {f : X ⟶ Y} →\n    {motive : AlgebraicGeometry.IsLocalIso f → Sort u_1} →\n      (t : AlgebraicGeometry.IsLocalIso f) →\n        ((exists_isOpenImmersion :\n              ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n            motive ⋯) →\n          motive t","typeReferences":[["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"],["AlgebraicGeometry","PresheafedSpace","carrier"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["AlgebraicGeometry","IsLocalIso","mk"],["CommRingCat"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["AlgebraicGeometry","Scheme","Opens","ι"],["AlgebraicGeometry","IsOpenImmersion"],["Exists"],["SetLike","instMembership"],["And"],["TopologicalSpace","Opens","instSetLike"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","instCategory"],["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","comp"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"]],"valueReferences":[["AlgebraicGeometry","IsLocalIso","rec"]]},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","eq_iInf"],"typeFallback":"Eq.{succ (succ u)} (forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}}, (Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y) -> Prop) AlgebraicGeometry.IsLocalIso.{u} (iInf.{succ u, succ (succ u)} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (ConditionallyCompletePartialOrderInf.toInfSet.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CompleteLattice.toConditionallyCompleteLattice.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CompleteBooleanAlgebra.toCompleteLattice.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}))))))) (fun (P : CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) => iInf.{succ u, 0} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CategoryTheory.MorphismProperty.ContainsIdentities.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u} P) (ConditionallyCompletePartialOrderInf.toInfSet.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CompleteLattice.toConditionallyCompleteLattice.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CompleteBooleanAlgebra.toCompleteLattice.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}))))))) (fun (x._@.Mathlib.AlgebraicGeometry.Morphisms.LocalIso.1724003552._hygCtx._hyg.31 : CategoryTheory.MorphismProperty.ContainsIdentities.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u} P) => iInf.{succ u, 0} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (AlgebraicGeometry.IsZariskiLocalAtSource.{u} P) (ConditionallyCompletePartialOrderInf.toInfSet.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (ConditionallyCompleteLattice.toConditionallyCompletePartialOrder.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CompleteLattice.toConditionallyCompleteLattice.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CompleteBooleanAlgebra.toCompleteLattice.{succ u} (CategoryTheory.MorphismProperty.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) (CategoryTheory.MorphismProperty.instCompleteBooleanAlgebra.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}))))))) (fun (x._@.Mathlib.AlgebraicGeometry.Morphisms.LocalIso.1724003552._hygCtx._hyg.40 : AlgebraicGeometry.IsZariskiLocalAtSource.{u} P) => P))))","typeFull":"@AlgebraicGeometry.IsLocalIso = ⨅ P, ⨅ (_ : P.ContainsIdentities), ⨅ (_ : AlgebraicGeometry.IsZariskiLocalAtSource P), P","typeReadable":"@AlgebraicGeometry.IsLocalIso = ⨅ P, ⨅ (_ : P.ContainsIdentities), ⨅ (_ : AlgebraicGeometry.IsZariskiLocalAtSource P), P","typeReferences":[["CategoryTheory","MorphismProperty","instCompleteBooleanAlgebra"],["AlgebraicGeometry","Scheme","instCategory"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["CompleteBooleanAlgebra","toCompleteLattice"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","IsZariskiLocalAtSource"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","CategoryStruct","toQuiver"],["Quiver","Hom"],["CategoryTheory","MorphismProperty","ContainsIdentities"],["CategoryTheory","MorphismProperty"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["Eq"],["CompleteLattice","toConditionallyCompleteLattice"]],"valueReferences":[["implies_congr"],["Prop","instCompleteLattice"],["PartialOrder","toPreorder"],["_private","Mathlib","AlgebraicGeometry","Morphisms","LocalIso",0,"AlgebraicGeometry","IsLocalIso","eq_iInf","_simp_1_1"],["Eq","trans"],["AlgebraicGeometry","Scheme","instCategory"],["iInf_le_of_le"],["ConditionallyCompleteLattice","toConditionallyCompletePartialOrder"],["AlgebraicGeometry","IsLocalIso"],["CompleteBooleanAlgebra","toCompleteLattice"],["AlgebraicGeometry","Scheme"],["AlgebraicGeometry","IsZariskiLocalAtSource"],["CompleteLattice","toCompleteSemilatticeSup"],["AlgebraicGeometry","IsLocalIso","instIsMultiplicativeScheme"],["Quiver","Hom"],["CategoryTheory","MorphismProperty","ContainsIdentities"],["forall_congr"],["iInf_le"],["iInf"],["ConditionallyCompletePartialOrder","toConditionallyCompletePartialOrderInf"],["ConditionallyCompletePartialOrderInf","toInfSet"],["Prop","partialOrder"],["CategoryTheory","MorphismProperty","IsMultiplicative","toContainsIdentities"],["Eq"],["Preorder","toLE"],["le_antisymm"],["CompleteLattice","toConditionallyCompleteLattice"],["CategoryTheory","MorphismProperty","instCompleteBooleanAlgebra"],["Pi","instCompleteLattice"],["AlgebraicGeometry","IsLocalIso","instIsZariskiLocalAtSource"],["AlgebraicGeometry","IsLocalIso","le_of_isZariskiLocalAtSource"],["CategoryTheory","Category","toCategoryStruct"],["CompleteSemilatticeInf","toInfSet"],["CategoryTheory","CategoryStruct","toQuiver"],["Eq","refl"],["LE","le"],["id"],["CategoryTheory","MorphismProperty"],["CompleteSemilatticeSup","toPartialOrder"],["inferInstance"],["Eq","mpr"],["CompleteLattice","toCompleteSemilatticeInf"],["Pi","partialOrder"]]},{"isProp":false,"kind":"recursor","name":["AlgebraicGeometry","IsLocalIso","rec"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}} {f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y} {motive : (AlgebraicGeometry.IsLocalIso.{u} X Y f) -> Sort.{u_1}}, (forall (exists_isOpenImmersion : forall (x : TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))), Exists.{succ u} (AlgebraicGeometry.Scheme.Opens.{u} X) (fun (U : AlgebraicGeometry.Scheme.Opens.{u} X) => And (Membership.mem.{u, u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.Scheme.Opens.{u} X) (SetLike.instMembership.{u, u} (AlgebraicGeometry.Scheme.Opens.{u} X) (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (TopologicalSpace.Opens.instSetLike.{u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) U x) (AlgebraicGeometry.IsOpenImmersion.{u} (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) Y (CategoryTheory.CategoryStruct.comp.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}) (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) X Y (AlgebraicGeometry.Scheme.Opens.ι.{u} X U) f)))), motive (AlgebraicGeometry.IsLocalIso.mk.{u} X Y f exists_isOpenImmersion)) -> (forall (t : AlgebraicGeometry.IsLocalIso.{u} X Y f), motive t)","typeFull":"{X Y : AlgebraicGeometry.Scheme} →\n  {f : X ⟶ Y} →\n    {motive : AlgebraicGeometry.IsLocalIso f → Sort u_1} →\n      ((exists_isOpenImmersion :\n            ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n          motive ⋯) →\n        (t : AlgebraicGeometry.IsLocalIso f) → motive t","typeReadable":"{X Y : AlgebraicGeometry.Scheme} →\n  {f : X ⟶ Y} →\n    {motive : AlgebraicGeometry.IsLocalIso f → Sort u_1} →\n      ((exists_isOpenImmersion :\n            ∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n          motive ⋯) →\n        (t : AlgebraicGeometry.IsLocalIso f) → motive t","typeReferences":[["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"],["AlgebraicGeometry","PresheafedSpace","carrier"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["AlgebraicGeometry","IsLocalIso","mk"],["CommRingCat"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["AlgebraicGeometry","Scheme","Opens","ι"],["AlgebraicGeometry","IsOpenImmersion"],["Exists"],["SetLike","instMembership"],["And"],["TopologicalSpace","Opens","instSetLike"],["CategoryTheory","Category","toCategoryStruct"],["CommRingCat","instCategory"],["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","comp"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"]],"valueReferences":null},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","instIsZariskiLocalAtSource"],"typeFallback":"AlgebraicGeometry.IsZariskiLocalAtSource.{u_1} AlgebraicGeometry.IsLocalIso.{u_1}","typeFull":"AlgebraicGeometry.IsZariskiLocalAtSource @AlgebraicGeometry.IsLocalIso","typeReadable":"AlgebraicGeometry.IsZariskiLocalAtSource @AlgebraicGeometry.IsLocalIso","typeReferences":[["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","IsZariskiLocalAtSource"]],"valueReferences":[["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","IsLocalIso","eq_sourceLocalClosure_isOpenImmersion"],["CategoryTheory","MorphismProperty","instRespectsOfIsStableUnderComposition"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","IsZariskiLocalAtSource"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","sourceLocalClosure"],["congrArg"],["CategoryTheory","CategoryStruct","toQuiver"],["Quiver","Hom"],["AlgebraicGeometry","sourceLocalClosure","instIsZariskiLocalAtSourceIsOpenImmersionOfRespectsIsoOfRespectsLeftScheme"],["CategoryTheory","MorphismProperty","Respects","toRespectsLeft"],["id"],["inferInstance"],["Eq","mpr"],["AlgebraicGeometry","isOpenImmersion_isStableUnderComposition"],["AlgebraicGeometry","isOpenImmersion_respectsIso"],["AlgebraicGeometry","IsOpenImmersion"],["Eq"]]},{"isProp":true,"kind":"definition","name":["AlgebraicGeometry","IsLocalIso","mk","_flat_ctor"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}} {f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y}, (forall (x : TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))), Exists.{succ u} (AlgebraicGeometry.Scheme.Opens.{u} X) (fun (U : AlgebraicGeometry.Scheme.Opens.{u} X) => And (Membership.mem.{u, u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.Scheme.Opens.{u} X) (SetLike.instMembership.{u, u} (AlgebraicGeometry.Scheme.Opens.{u} X) (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (TopologicalSpace.Opens.instSetLike.{u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) U x) (AlgebraicGeometry.IsOpenImmersion.{u} (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) Y (CategoryTheory.CategoryStruct.comp.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}) (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) X Y (AlgebraicGeometry.Scheme.Opens.ι.{u} X U) f)))) -> (AlgebraicGeometry.IsLocalIso.{u} X Y f)","typeFull":"∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},\n  (∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n    AlgebraicGeometry.IsLocalIso f","typeReadable":"∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},\n  (∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n    AlgebraicGeometry.IsLocalIso f","typeReferences":[["SetLike","instMembership"],["Exists"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["And"],["AlgebraicGeometry","IsLocalIso"],["TopologicalSpace","Opens","instSetLike"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","PresheafedSpace","carrier"],["CommRingCat","instCategory"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","toQuiver"],["CommRingCat"],["CategoryTheory","CategoryStruct","comp"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","ι"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"],["AlgebraicGeometry","IsOpenImmersion"]],"valueReferences":[["AlgebraicGeometry","IsLocalIso","mk"]]},{"isProp":true,"kind":"theorem","name":["AlgebraicGeometry","IsLocalIso","eq_sourceLocalClosure_isOpenImmersion"],"typeFallback":"Eq.{succ (succ u_1)} (forall {X : AlgebraicGeometry.Scheme.{u_1}} {Y : AlgebraicGeometry.Scheme.{u_1}}, (Quiver.Hom.{u_1, succ u_1} AlgebraicGeometry.Scheme.{u_1} (CategoryTheory.CategoryStruct.toQuiver.{u_1, succ u_1} AlgebraicGeometry.Scheme.{u_1} (CategoryTheory.Category.toCategoryStruct.{u_1, succ u_1} AlgebraicGeometry.Scheme.{u_1} AlgebraicGeometry.Scheme.instCategory.{u_1})) X Y) -> Prop) AlgebraicGeometry.IsLocalIso.{u_1} (AlgebraicGeometry.sourceLocalClosure.{u_1} AlgebraicGeometry.IsOpenImmersion.{u_1} AlgebraicGeometry.IsOpenImmersion.{u_1})","typeFull":"@AlgebraicGeometry.IsLocalIso =\n  AlgebraicGeometry.sourceLocalClosure AlgebraicGeometry.IsOpenImmersion AlgebraicGeometry.IsOpenImmersion","typeReadable":"@AlgebraicGeometry.IsLocalIso =\n  AlgebraicGeometry.sourceLocalClosure AlgebraicGeometry.IsOpenImmersion AlgebraicGeometry.IsOpenImmersion","typeReferences":[["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","instCategory"],["Quiver","Hom"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","IsOpenImmersion"],["AlgebraicGeometry","Scheme"],["Eq"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","sourceLocalClosure"]],"valueReferences":[["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["AlgebraicGeometry","isLocalIso_iff"],["AlgebraicGeometry","IsLocalIso"],["AlgebraicGeometry","Scheme"],["congrArg"],["AlgebraicGeometry","PresheafedSpace","carrier"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["CommRingCat"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","ι"],["funext"],["Eq"],["AlgebraicGeometry","IsOpenImmersion"],["propext"],["Exists"],["SetLike","instMembership"],["And"],["Iff","rfl"],["TopologicalSpace","Opens","instSetLike"],["AlgebraicGeometry","sourceLocalClosure","iff_forall_exists"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","sourceLocalClosure"],["CommRingCat","instCategory"],["CategoryTheory","CategoryStruct","toQuiver"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","comp"],["Iff"],["TopCat","carrier"],["id"],["Eq","mpr"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"],["AlgebraicGeometry","isOpenImmersion_respectsIso"]]},{"isProp":true,"kind":"constructor","name":["AlgebraicGeometry","IsLocalIso","mk"],"typeFallback":"forall {X : AlgebraicGeometry.Scheme.{u}} {Y : AlgebraicGeometry.Scheme.{u}} {f : Quiver.Hom.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.CategoryStruct.toQuiver.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u})) X Y}, (forall (x : TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))), Exists.{succ u} (AlgebraicGeometry.Scheme.Opens.{u} X) (fun (U : AlgebraicGeometry.Scheme.Opens.{u} X) => And (Membership.mem.{u, u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.Scheme.Opens.{u} X) (SetLike.instMembership.{u, u} (AlgebraicGeometry.Scheme.Opens.{u} X) (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (TopologicalSpace.Opens.instSetLike.{u} (TopCat.carrier.{u} (AlgebraicGeometry.PresheafedSpace.carrier.{u, succ u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.SheafedSpace.toPresheafedSpace.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) (AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier.{succ u, u, u} CommRingCat.{u} CommRingCat.instCategory.{u} (AlgebraicGeometry.LocallyRingedSpace.toSheafedSpace.{u} (AlgebraicGeometry.Scheme.toLocallyRingedSpace.{u} X))))) U x) (AlgebraicGeometry.IsOpenImmersion.{u} (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) Y (CategoryTheory.CategoryStruct.comp.{u, succ u} AlgebraicGeometry.Scheme.{u} (CategoryTheory.Category.toCategoryStruct.{u, succ u} AlgebraicGeometry.Scheme.{u} AlgebraicGeometry.Scheme.instCategory.{u}) (AlgebraicGeometry.Scheme.Opens.toScheme.{u} X U) X Y (AlgebraicGeometry.Scheme.Opens.ι.{u} X U) f)))) -> (AlgebraicGeometry.IsLocalIso.{u} X Y f)","typeFull":"∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},\n  (∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n    AlgebraicGeometry.IsLocalIso f","typeReadable":"∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},\n  (∀ (x : ↥X), ∃ U, x ∈ U ∧ AlgebraicGeometry.IsOpenImmersion (CategoryTheory.CategoryStruct.comp U.ι f)) →\n    AlgebraicGeometry.IsLocalIso f","typeReferences":[["SetLike","instMembership"],["Exists"],["AlgebraicGeometry","Scheme","instCategory"],["AlgebraicGeometry","Scheme","toLocallyRingedSpace"],["AlgebraicGeometry","SheafedSpace","instTopologicalSpaceCarrierCarrier"],["Membership","mem"],["And"],["AlgebraicGeometry","IsLocalIso"],["TopologicalSpace","Opens","instSetLike"],["AlgebraicGeometry","Scheme"],["CategoryTheory","Category","toCategoryStruct"],["AlgebraicGeometry","PresheafedSpace","carrier"],["CommRingCat","instCategory"],["AlgebraicGeometry","SheafedSpace","toPresheafedSpace"],["AlgebraicGeometry","Scheme","Opens"],["CategoryTheory","CategoryStruct","toQuiver"],["CommRingCat"],["CategoryTheory","CategoryStruct","comp"],["AlgebraicGeometry","Scheme","Opens","toScheme"],["Quiver","Hom"],["AlgebraicGeometry","Scheme","Opens","ι"],["TopCat","carrier"],["AlgebraicGeometry","LocallyRingedSpace","toSheafedSpace"],["AlgebraicGeometry","IsOpenImmersion"]],"valueReferences":null}]