Add files using upload-large-folder tool

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"definition","name":["IsAffineAddMonoid","mk","_flat_ctor"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M], (forall (a : M), IsAddLeftRegular.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3))) a) -> (forall (a : M), IsAddRightRegular.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3))) a) -> (AddSubmonoid.FG.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3) (Top.top.{u_1} (AddSubmonoid.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3))) (AddSubmonoid.instTop.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3))))) -> (forall {{n : Nat}}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Function.Injective.{succ u_1, succ u_1} M M (fun (a : M) => HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3))) n a))) -> (IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)","typeFull":"∀ {M : Type u_1} [inst : AddCommMonoid M],\n  (∀ (a : M), IsAddLeftRegular a) →\n    (∀ (a : M), IsAddRightRegular a) →\n      ⊤.FG → (∀ ⦃n : ℕ⦄, n ≠ 0 → Function.Injective fun a => n • a) → IsAffineAddMonoid M","typeReadable":"∀ {M : Type u_1} [inst : AddCommMonoid M],\n  (∀ (a : M), IsAddLeftRegular a) →\n    (∀ (a : M), IsAddRightRegular a) →\n      ⊤.FG → (∀ ⦃n : ℕ⦄, n ≠ 0 → Function.Injective fun a => n • a) → IsAffineAddMonoid M","typeReferences":[["IsAddRightRegular"],["AddSubmonoid","instTop"],["IsAffineAddMonoid"],["AddSubmonoid","FG"],["AddCommMonoid","toAddMonoid"],["OfNat","ofNat"],["AddCommMonoid"],["Nat"],["AddMonoid","toNSMul"],["AddCommMonoid","toAddCommSemigroup"],["instOfNatNat"],["AddSubmonoid"],["HSMul","hSMul"],["Top","top"],["instHSMul"],["Ne"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["IsAddLeftRegular"],["Function","Injective"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["IsAffineAddMonoid","mk"],["AddMonoid","FG","mk"],["AddCommMonoid","toAddCommSemigroup"],["IsAddTorsionFree","mk"],["IsCancelAdd","mk"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddCommMonoid","toAddMonoid"],["IsLeftCancelAdd","mk"],["IsRightCancelAdd","mk"]]},{"isProp":false,"kind":"recursor","name":["IsAffineAddMonoid","rec"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] {motive : (IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3) -> Sort.{u}}, (forall [toIsCancelAdd : IsCancelAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)))] [toFG : AddMonoid.FG.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)] [toIsAddTorsionFree : IsAddTorsionFree.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)], motive (IsAffineAddMonoid.mk.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 toIsCancelAdd toFG toIsAddTorsionFree)) -> (forall (t : IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3), motive t)","typeFull":"{M : Type u_1} →\n  [inst : AddCommMonoid M] →\n    {motive : IsAffineAddMonoid M → Sort u} →\n      ([toIsCancelAdd : IsCancelAdd M] →\n          [toFG : AddMonoid.FG M] → [toIsAddTorsionFree : IsAddTorsionFree M] → motive ⋯) →\n        (t : IsAffineAddMonoid M) → motive t","typeReadable":"{M : Type u_1} →\n  [inst : AddCommMonoid M] →\n    {motive : IsAffineAddMonoid M → Sort u} →\n      ([toIsCancelAdd : IsCancelAdd M] →\n          [toFG : AddMonoid.FG M] → [toIsAddTorsionFree : IsAddTorsionFree M] → motive ⋯) →\n        (t : IsAffineAddMonoid M) → motive t","typeReferences":[["IsAffineAddMonoid","mk"],["AddCommMonoid"],["IsAddTorsionFree"],["AddCommMonoid","toAddCommSemigroup"],["IsCancelAdd"],["AddMonoid","FG"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["IsAffineAddMonoid"],["AddCommMonoid","toAddMonoid"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["IsAffineMonoid","recOn"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M] {motive : (IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3) -> Sort.{u}} (t : IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3), (forall [toIsCancelMul : IsCancelMul.{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.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))))] [toFG : Monoid.FG.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)] [toIsMulTorsionFree : IsMulTorsionFree.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)], motive (IsAffineMonoid.mk.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 toIsCancelMul toFG toIsMulTorsionFree)) -> (motive t)","typeFull":"{M : Type u_1} →\n  [inst : CommMonoid M] →\n    {motive : IsAffineMonoid M → Sort u} →\n      (t : IsAffineMonoid M) →\n        ([toIsCancelMul : IsCancelMul M] →\n            [toFG : Monoid.FG M] → [toIsMulTorsionFree : IsMulTorsionFree M] → motive ⋯) →\n          motive t","typeReadable":"{M : Type u_1} →\n  [inst : CommMonoid M] →\n    {motive : IsAffineMonoid M → Sort u} →\n      (t : IsAffineMonoid M) →\n        ([toIsCancelMul : IsCancelMul M] →\n            [toFG : Monoid.FG M] → [toIsMulTorsionFree : IsMulTorsionFree M] → motive ⋯) →\n          motive t","typeReferences":[["IsMulTorsionFree"],["MulOneClass","toMulOne"],["Monoid","FG"],["MulOne","toMul"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["Monoid","toMulOneClass"],["IsCancelMul"],["IsAffineMonoid","mk"],["CommMonoid"]],"valueReferences":[["IsAffineMonoid","rec"]]},{"isProp":true,"kind":"theorem","name":["IsAffineMonoid","toIsMulTorsionFree"],"typeFallback":"forall {M : Type.{u_1}} {inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M} [self : IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3], IsMulTorsionFree.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)","typeFull":"∀ {M : Type u_1} {inst : CommMonoid M} [self : IsAffineMonoid M], IsMulTorsionFree M","typeReadable":"∀ {M : Type u_1} {inst : CommMonoid M} [self : IsAffineMonoid M], IsMulTorsionFree M","typeReferences":[["IsMulTorsionFree"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["CommMonoid"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["IsAffineAddMonoid","toIsAddTorsionFree"],"typeFallback":"forall {M : Type.{u_1}} {inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M} [self : IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3], IsAddTorsionFree.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)","typeFull":"∀ {M : Type u_1} {inst : AddCommMonoid M} [self : IsAffineAddMonoid M], IsAddTorsionFree M","typeReadable":"∀ {M : Type u_1} {inst : AddCommMonoid M} [self : IsAffineAddMonoid M], IsAddTorsionFree M","typeReferences":[["AddCommMonoid"],["IsAddTorsionFree"],["IsAffineAddMonoid"],["AddCommMonoid","toAddMonoid"]],"valueReferences":[]},{"isProp":false,"kind":"recursor","name":["IsAffineMonoid","rec"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M] {motive : (IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3) -> Sort.{u}}, (forall [toIsCancelMul : IsCancelMul.{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.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))))] [toFG : Monoid.FG.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)] [toIsMulTorsionFree : IsMulTorsionFree.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)], motive (IsAffineMonoid.mk.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 toIsCancelMul toFG toIsMulTorsionFree)) -> (forall (t : IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3), motive t)","typeFull":"{M : Type u_1} →\n  [inst : CommMonoid M] →\n    {motive : IsAffineMonoid M → Sort u} →\n      ([toIsCancelMul : IsCancelMul M] → [toFG : Monoid.FG M] → [toIsMulTorsionFree : IsMulTorsionFree M] → motive ⋯) →\n        (t : IsAffineMonoid M) → motive t","typeReadable":"{M : Type u_1} →\n  [inst : CommMonoid M] →\n    {motive : IsAffineMonoid M → Sort u} →\n      ([toIsCancelMul : IsCancelMul M] → [toFG : Monoid.FG M] → [toIsMulTorsionFree : IsMulTorsionFree M] → motive ⋯) →\n        (t : IsAffineMonoid M) → motive t","typeReferences":[["IsMulTorsionFree"],["MulOneClass","toMulOne"],["Monoid","FG"],["MulOne","toMul"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["Monoid","toMulOneClass"],["IsCancelMul"],["IsAffineMonoid","mk"],["CommMonoid"]],"valueReferences":null},{"isProp":true,"kind":"constructor","name":["IsAffineAddMonoid","mk"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] [toIsCancelAdd : IsCancelAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)))] [toFG : AddMonoid.FG.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)] [toIsAddTorsionFree : IsAddTorsionFree.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)], IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3","typeFull":"∀ {M : Type u_1} [inst : AddCommMonoid M] [toIsCancelAdd : IsCancelAdd M] [toFG : AddMonoid.FG M]\n  [toIsAddTorsionFree : IsAddTorsionFree M], IsAffineAddMonoid M","typeReadable":"∀ {M : Type u_1} [inst : AddCommMonoid M] [toIsCancelAdd : IsCancelAdd M] [toFG : AddMonoid.FG M]\n  [toIsAddTorsionFree : IsAddTorsionFree M], IsAffineAddMonoid M","typeReferences":[["AddCommMonoid"],["IsAddTorsionFree"],["AddCommMonoid","toAddCommSemigroup"],["IsCancelAdd"],["AddMonoid","FG"],["IsAffineAddMonoid"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["AddCommMonoid","toAddMonoid"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["IsAffineMonoid","casesOn"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M] {motive : (IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3) -> Sort.{u}} (t : IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3), (forall [toIsCancelMul : IsCancelMul.{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.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))))] [toFG : Monoid.FG.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)] [toIsMulTorsionFree : IsMulTorsionFree.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)], motive (IsAffineMonoid.mk.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 toIsCancelMul toFG toIsMulTorsionFree)) -> (motive t)","typeFull":"{M : Type u_1} →\n  [inst : CommMonoid M] →\n    {motive : IsAffineMonoid M → Sort u} →\n      (t : IsAffineMonoid M) →\n        ([toIsCancelMul : IsCancelMul M] →\n            [toFG : Monoid.FG M] → [toIsMulTorsionFree : IsMulTorsionFree M] → motive ⋯) →\n          motive t","typeReadable":"{M : Type u_1} →\n  [inst : CommMonoid M] →\n    {motive : IsAffineMonoid M → Sort u} →\n      (t : IsAffineMonoid M) →\n        ([toIsCancelMul : IsCancelMul M] →\n            [toFG : Monoid.FG M] → [toIsMulTorsionFree : IsMulTorsionFree M] → motive ⋯) →\n          motive t","typeReferences":[["IsMulTorsionFree"],["MulOneClass","toMulOne"],["Monoid","FG"],["MulOne","toMul"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["Monoid","toMulOneClass"],["IsCancelMul"],["IsAffineMonoid","mk"],["CommMonoid"]],"valueReferences":[["IsAffineMonoid","rec"]]},{"isProp":true,"kind":"theorem","name":["IsAffineAddMonoid","toIsCancelAdd"],"typeFallback":"forall {M : Type.{u_1}} {inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M} [self : IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3], IsCancelAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)))","typeFull":"∀ {M : Type u_1} {inst : AddCommMonoid M} [self : IsAffineAddMonoid M], IsCancelAdd M","typeReadable":"∀ {M : Type u_1} {inst : AddCommMonoid M} [self : IsAffineAddMonoid M], IsCancelAdd M","typeReferences":[["AddCommMonoid"],["AddCommMonoid","toAddCommSemigroup"],["IsCancelAdd"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["IsAffineAddMonoid"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["IsAffineMonoid","toIsCancelMul"],"typeFallback":"forall {M : Type.{u_1}} {inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M} [self : IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3], IsCancelMul.{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.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))))","typeFull":"∀ {M : Type u_1} {inst : CommMonoid M} [self : IsAffineMonoid M], IsCancelMul M","typeReadable":"∀ {M : Type u_1} {inst : CommMonoid M} [self : IsAffineMonoid M], IsCancelMul M","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["Monoid","toMulOneClass"],["IsCancelMul"],["CommMonoid"]],"valueReferences":[]},{"isProp":true,"kind":"theorem","name":["IsAffineMonoid","toFG"],"typeFallback":"forall {M : Type.{u_1}} {inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M} [self : IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3], Monoid.FG.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)","typeFull":"∀ {M : Type u_1} {inst : CommMonoid M} [self : IsAffineMonoid M], Monoid.FG M","typeReadable":"∀ {M : Type u_1} {inst : CommMonoid M} [self : IsAffineMonoid M], Monoid.FG M","typeReferences":[["Monoid","FG"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["CommMonoid"]],"valueReferences":[]},{"isProp":true,"kind":"constructor","name":["IsAffineMonoid","mk"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M] [toIsCancelMul : IsCancelMul.{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.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))))] [toFG : Monoid.FG.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)] [toIsMulTorsionFree : IsMulTorsionFree.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)], IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3","typeFull":"∀ {M : Type u_1} [inst : CommMonoid M] [toIsCancelMul : IsCancelMul M] [toFG : Monoid.FG M]\n  [toIsMulTorsionFree : IsMulTorsionFree M], IsAffineMonoid M","typeReadable":"∀ {M : Type u_1} [inst : CommMonoid M] [toIsCancelMul : IsCancelMul M] [toFG : Monoid.FG M]\n  [toIsMulTorsionFree : IsMulTorsionFree M], IsAffineMonoid M","typeReferences":[["IsMulTorsionFree"],["MulOneClass","toMulOne"],["Monoid","FG"],["MulOne","toMul"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["Monoid","toMulOneClass"],["IsCancelMul"],["CommMonoid"]],"valueReferences":null},{"isProp":false,"kind":"inductive","name":["IsAffineMonoid"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M], Prop","typeFull":"(M : Type u_1) → [CommMonoid M] → Prop","typeReadable":"(M : Type u_1) → [CommMonoid M] → Prop","typeReferences":[["CommMonoid"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["IsAffineAddMonoid","casesOn"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] {motive : (IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3) -> Sort.{u}} (t : IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3), (forall [toIsCancelAdd : IsCancelAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)))] [toFG : AddMonoid.FG.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)] [toIsAddTorsionFree : IsAddTorsionFree.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)], motive (IsAffineAddMonoid.mk.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 toIsCancelAdd toFG toIsAddTorsionFree)) -> (motive t)","typeFull":"{M : Type u_1} →\n  [inst : AddCommMonoid M] →\n    {motive : IsAffineAddMonoid M → Sort u} →\n      (t : IsAffineAddMonoid M) →\n        ([toIsCancelAdd : IsCancelAdd M] →\n            [toFG : AddMonoid.FG M] → [toIsAddTorsionFree : IsAddTorsionFree M] → motive ⋯) →\n          motive t","typeReadable":"{M : Type u_1} →\n  [inst : AddCommMonoid M] →\n    {motive : IsAffineAddMonoid M → Sort u} →\n      (t : IsAffineAddMonoid M) →\n        ([toIsCancelAdd : IsCancelAdd M] →\n            [toFG : AddMonoid.FG M] → [toIsAddTorsionFree : IsAddTorsionFree M] → motive ⋯) →\n          motive t","typeReferences":[["IsAffineAddMonoid","mk"],["AddCommMonoid"],["IsAddTorsionFree"],["AddCommMonoid","toAddCommSemigroup"],["IsCancelAdd"],["AddMonoid","FG"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["IsAffineAddMonoid"],["AddCommMonoid","toAddMonoid"]],"valueReferences":[["IsAffineAddMonoid","rec"]]},{"isProp":true,"kind":"theorem","name":["IsAffineAddMonoid","toFG"],"typeFallback":"forall {M : Type.{u_1}} {inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M} [self : IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3], AddMonoid.FG.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)","typeFull":"∀ {M : Type u_1} {inst : AddCommMonoid M} [self : IsAffineAddMonoid M], AddMonoid.FG M","typeReadable":"∀ {M : Type u_1} {inst : AddCommMonoid M} [self : IsAffineAddMonoid M], AddMonoid.FG M","typeReferences":[["AddCommMonoid"],["AddMonoid","FG"],["IsAffineAddMonoid"],["AddCommMonoid","toAddMonoid"]],"valueReferences":[]},{"isProp":false,"kind":"inductive","name":["IsAffineAddMonoid"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M], Prop","typeFull":"(M : Type u_1) → [AddCommMonoid M] → Prop","typeReadable":"(M : Type u_1) → [AddCommMonoid M] → Prop","typeReferences":[["AddCommMonoid"]],"valueReferences":null},{"isProp":false,"kind":"definition","name":["IsAffineAddMonoid","recOn"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 : AddCommMonoid.{u_1} M] {motive : (IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3) -> Sort.{u}} (t : IsAffineAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3), (forall [toIsCancelAdd : IsCancelAdd.{u_1} M (AddCommMagma.toAdd.{u_1} M (AddCommSemigroup.toAddCommMagma.{u_1} M (AddCommMonoid.toAddCommSemigroup.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)))] [toFG : AddMonoid.FG.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)] [toIsAddTorsionFree : IsAddTorsionFree.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3)], motive (IsAffineAddMonoid.mk.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.3723108165._hygCtx._hyg.3 toIsCancelAdd toFG toIsAddTorsionFree)) -> (motive t)","typeFull":"{M : Type u_1} →\n  [inst : AddCommMonoid M] →\n    {motive : IsAffineAddMonoid M → Sort u} →\n      (t : IsAffineAddMonoid M) →\n        ([toIsCancelAdd : IsCancelAdd M] →\n            [toFG : AddMonoid.FG M] → [toIsAddTorsionFree : IsAddTorsionFree M] → motive ⋯) →\n          motive t","typeReadable":"{M : Type u_1} →\n  [inst : AddCommMonoid M] →\n    {motive : IsAffineAddMonoid M → Sort u} →\n      (t : IsAffineAddMonoid M) →\n        ([toIsCancelAdd : IsCancelAdd M] →\n            [toFG : AddMonoid.FG M] → [toIsAddTorsionFree : IsAddTorsionFree M] → motive ⋯) →\n          motive t","typeReferences":[["IsAffineAddMonoid","mk"],["AddCommMonoid"],["IsAddTorsionFree"],["AddCommMonoid","toAddCommSemigroup"],["IsCancelAdd"],["AddMonoid","FG"],["AddCommSemigroup","toAddCommMagma"],["AddCommMagma","toAdd"],["IsAffineAddMonoid"],["AddCommMonoid","toAddMonoid"]],"valueReferences":[["IsAffineAddMonoid","rec"]]},{"isProp":true,"kind":"definition","name":["IsAffineMonoid","mk","_flat_ctor"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3 : CommMonoid.{u_1} M], (forall (a : M), IsLeftRegular.{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.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)))) a) -> (forall (a : M), IsRightRegular.{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.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)))) a) -> (Submonoid.FG.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3) (Top.top.{u_1} (Submonoid.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))) (Submonoid.instTop.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))))) -> (forall {{n : Nat}}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Function.Injective.{succ u_1, succ u_1} M M (fun (a : M) => HPow.hPow.{u_1, 0, u_1} M Nat M (instHPow.{u_1, 0} M Nat (Monoid.toPow.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3))) a n))) -> (IsAffineMonoid.{u_1} M inst._@.Mathlib.Algebra.AffineMonoid.Basic.1256814968._hygCtx._hyg.3)","typeFull":"∀ {M : Type u_1} [inst : CommMonoid M],\n  (∀ (a : M), IsLeftRegular a) →\n    (∀ (a : M), IsRightRegular a) → ⊤.FG → (∀ ⦃n : ℕ⦄, n ≠ 0 → Function.Injective fun a => a ^ n) → IsAffineMonoid M","typeReadable":"∀ {M : Type u_1} [inst : CommMonoid M],\n  (∀ (a : M), IsLeftRegular a) →\n    (∀ (a : M), IsRightRegular a) → ⊤.FG → (∀ ⦃n : ℕ⦄, n ≠ 0 → Function.Injective fun a => a ^ n) → IsAffineMonoid M","typeReferences":[["instHPow"],["MulOneClass","toMulOne"],["CommMonoid","toMonoid"],["IsAffineMonoid"],["Submonoid","instTop"],["HPow","hPow"],["Submonoid","FG"],["OfNat","ofNat"],["IsRightRegular"],["Submonoid"],["MulOne","toMul"],["Nat"],["Monoid","toPow"],["instOfNatNat"],["Monoid","toMulOneClass"],["Top","top"],["IsLeftRegular"],["Ne"],["CommMonoid"],["Function","Injective"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["IsRightCancelMul","mk"],["IsLeftCancelMul","mk"],["CommMonoid","toMonoid"],["IsMulTorsionFree","mk"],["Monoid","toMulOneClass"],["Monoid","FG","mk"],["IsCancelMul","mk"],["IsAffineMonoid","mk"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["WithBot","bot_lt_sum_iff"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2091623205._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithBot.{u_2} M)} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2091623205._hygCtx._hyg.14 : LT.{u_2} M], Iff (LT.lt.{u_2} (WithBot.{u_2} M) (WithBot.instLT.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.2091623205._hygCtx._hyg.14) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M)) (Finset.sum.{u_1, u_2} ι (WithBot.{u_2} M) (WithBot.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.2091623205._hygCtx._hyg.5) s (fun (i : ι) => f i))) (forall (i : ι), (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) -> (LT.lt.{u_2} (WithBot.{u_2} M) (WithBot.instLT.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.2091623205._hygCtx._hyg.14) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M)) (f i)))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M} [inst_1 : LT M],\n  ⊥ < ∑ i ∈ s, f i ↔ ∀ i ∈ s, ⊥ < f i","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M} [inst_1 : LT M],\n  ⊥ < ∑ i ∈ s, f i ↔ ∀ i ∈ s, ⊥ < f i","typeReferences":[["Finset","instSetLike"],["Finset"],["SetLike","instMembership"],["Membership","mem"],["WithBot","addCommMonoid"],["Bot","bot"],["LT","lt"],["AddCommMonoid"],["WithBot","instLT"],["WithBot"],["Iff"],["Finset","sum"],["WithBot","bot"],["LT"]],"valueReferences":[["implies_congr"],["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_3"],["Finset","instSetLike"],["Finset"],["Eq","trans"],["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_2"],["Membership","mem"],["congrArg"],["WithBot","instLT"],["iff_self"],["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_4"],["WithBot"],["congr"],["forall_congr"],["Finset","sum"],["Eq"],["Not"],["SetLike","instMembership"],["Exists"],["True"],["WithBot","addCommMonoid"],["And"],["Bot","bot"],["LT","lt"],["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_1"],["of_eq_true"],["Eq","refl"],["Iff"],["Ne"],["WithBot","bot"]]},{"isProp":true,"kind":"theorem","name":["WithBot","coe_sum"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774651._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] (s : Finset.{u_1} ι) (f : ι -> M), Eq.{succ u_2} (WithBot.{u_2} M) (WithBot.some.{u_2} M (Finset.sum.{u_1, u_2} ι M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774651._hygCtx._hyg.5 s (fun (i : ι) => f i))) (Finset.sum.{u_1, u_2} ι (WithBot.{u_2} M) (WithBot.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774651._hygCtx._hyg.5) s (fun (i : ι) => WithBot.some.{u_2} M (f i)))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ↑(∑ i ∈ s, f i) = ∑ i ∈ s, ↑(f i)","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ↑(∑ i ∈ s, f i) = ∑ i ∈ s, ↑(f i)","typeReferences":[["AddCommMonoid"],["Finset"],["WithBot"],["WithBot","addCommMonoid"],["Finset","sum"],["Eq"],["WithBot","some"]],"valueReferences":[["WithBot","addZeroClass"],["WithBot"],["AddMonoidHom"],["WithBot","addCommMonoid"],["WithBot","addHom"],["AddMonoidHom","instFunLike"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["map_sum"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["WithTop","prod_ne_top"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_3} M₀)}, (forall (i : ι), (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) -> (Ne.{succ u_3} (WithTop.{u_3} M₀) (f i) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀)))) -> (Ne.{succ u_3} (WithTop.{u_3} M₀) (Finset.prod.{u_1, u_3} ι (WithTop.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithTop.{u_3} M₀) (WithTop.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.4212580315._hygCtx._hyg.11)) s (fun (i : ι) => f i)) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀)))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀}, (∀ i ∈ s, f i ≠ ⊤) → ∏ i ∈ s, f i ≠ ⊤","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀}, (∀ i ∈ s, f i ≠ ⊤) → ∏ i ∈ s, f i ≠ ⊤","typeReferences":[["Finset","instSetLike"],["CommMonoidWithZero"],["NoZeroDivisors"],["Finset"],["SetLike","instMembership"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["MulZeroClass","toMul"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Top","top"],["Ne"]],"valueReferences":[["MulOneClass","toMulOne"],["WithTop","coe_ne_top"],["MulOne","toOne"],["CommMonoidWithZero","toMonoidWithZero"],["WithTop"],["CommMonoidWithZero","toCommMonoid"],["MulZeroOneClass","toMulOneClass"],["Finset","prod_induction"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["WithTop","instCommMonoidWithZero"],["WithTop","mul_ne_top"],["One","toOfNat1"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["Top","top"],["Ne"]]},{"isProp":true,"kind":"theorem","name":["WithBot","sum_lt_bot"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.125132809._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithBot.{u_2} M)} [inst._@.Mathlib.Algebra.BigOperators.WithTop.125132809._hygCtx._hyg.14 : LT.{u_2} M], (forall (i : ι), (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) -> (Ne.{succ u_2} (WithBot.{u_2} M) (f i) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M)))) -> (LT.lt.{u_2} (WithBot.{u_2} M) (WithBot.instLT.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.125132809._hygCtx._hyg.14) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M)) (Finset.sum.{u_1, u_2} ι (WithBot.{u_2} M) (WithBot.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.125132809._hygCtx._hyg.5) s (fun (i : ι) => f i)))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M} [inst_1 : LT M],\n  (∀ i ∈ s, f i ≠ ⊥) → ⊥ < ∑ i ∈ s, f i","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M} [inst_1 : LT M],\n  (∀ i ∈ s, f i ≠ ⊥) → ⊥ < ∑ i ∈ s, f i","typeReferences":[["Finset","instSetLike"],["Finset"],["SetLike","instMembership"],["Membership","mem"],["WithBot","addCommMonoid"],["Bot","bot"],["LT","lt"],["AddCommMonoid"],["WithBot","instLT"],["WithBot"],["Finset","sum"],["Ne"],["WithBot","bot"],["LT"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["SetLike","instMembership"],["Membership","mem"],["WithBot","addCommMonoid"],["Bot","bot"],["LT","lt"],["WithBot","instLT"],["WithBot","bot_lt_sum_iff"],["WithBot"],["Iff","mpr"],["WithBot","bot_lt_iff_ne_bot"],["Finset","sum"],["Ne"],["WithBot","bot"]]},{"isProp":true,"kind":"theorem","name":["WithBot","sum_eq_bot_iff"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.53073213._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithBot.{u_2} M)}, Iff (Eq.{succ u_2} (WithBot.{u_2} M) (Finset.sum.{u_1, u_2} ι (WithBot.{u_2} M) (WithBot.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.53073213._hygCtx._hyg.5) s (fun (i : ι) => f i)) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M))) (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithBot.{u_2} M) (f i) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M)))))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M},\n  ∑ i ∈ s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M},\n  ∑ i ∈ s, f i = ⊥ ↔ ∃ i ∈ s, f i = ⊥","typeReferences":[["Finset","instSetLike"],["Finset"],["Exists"],["SetLike","instMembership"],["Membership","mem"],["WithBot","addCommMonoid"],["And"],["Bot","bot"],["AddCommMonoid"],["WithBot"],["Iff"],["Finset","sum"],["Eq"],["WithBot","bot"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["false_and"],["Eq","trans"],["WithBot","zero_ne_bot","_simp_1"],["Membership","mem"],["Finset","instEmptyCollection"],["AddCommMonoid","toAddMonoid"],["EmptyCollection","emptyCollection"],["Finset","cons"],["congrArg"],["Or"],["iff_self"],["exists_false","_simp_1"],["WithBot"],["congr"],["funext"],["Finset","sum"],["congrFun'"],["Finset","mem_cons","_simp_1"],["Eq"],["propext"],["AddSemigroup","toAdd"],["SetLike","instMembership"],["Exists"],["Finset","sum_cons"],["True"],["instHAdd"],["WithBot","add_eq_bot","_simp_1"],["Finset","notMem_empty","_simp_1"],["And"],["WithBot","addCommMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Bot","bot"],["HAdd","hAdd"],["of_eq_true"],["AddMonoid","toAddSemigroup"],["Iff"],["False"],["WithBot","bot"],["AddZero","toZero"],["exists_eq_or_imp","_simp_1"],["AddMonoid","toAddZeroClass"],["Finset","cons_induction"]]},{"isProp":true,"kind":"theorem","name":["WithTop","sum_lt_top","_simp_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_2} M)} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.14 : LT.{u_2} M], Eq.{1} Prop (LT.lt.{u_2} (WithTop.{u_2} M) (WithTop.instLT.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.14) (Finset.sum.{u_1, u_2} ι (WithTop.{u_2} M) (WithTop.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.5) s (fun (i : ι) => f i)) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))) (forall (i : ι), (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) -> (LT.lt.{u_2} (WithTop.{u_2} M) (WithTop.instLT.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.14) (f i) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M} [inst_1 : LT M],\n  (∑ i ∈ s, f i < ⊤) = ∀ i ∈ s, f i < ⊤","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M} [inst_1 : LT M],\n  (∑ i ∈ s, f i < ⊤) = ∀ i ∈ s, f i < ⊤","typeReferences":[["Finset","instSetLike"],["Finset"],["SetLike","instMembership"],["WithTop","addCommMonoid"],["Membership","mem"],["WithTop"],["WithTop","instLT"],["LT","lt"],["AddCommMonoid"],["WithTop","top"],["Top","top"],["Finset","sum"],["Eq"],["LT"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["SetLike","instMembership"],["WithTop","addCommMonoid"],["Membership","mem"],["WithTop"],["WithTop","instLT"],["LT","lt"],["WithTop","sum_lt_top"],["WithTop","top"],["Top","top"],["Finset","sum"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_3"],"typeFallback":"forall {α : Sort.{u_1}} {p : α -> Prop}, Eq.{1} Prop (Not (Exists.{u_1} α (fun (x : α) => p x))) (forall (x : α), Not (p x))","typeFull":"∀ {α : Sort u_1} {p : α → Prop}, (¬∃ x, p x) = ∀ (x : α), ¬p x","typeReadable":"∀ {α : Sort u_1} {p : α → Prop}, (¬∃ x, p x) = ∀ (x : α), ¬p x","typeReferences":[["Not"],["Exists"],["Eq"]],"valueReferences":[["Not"],["Exists"],["not_exists"],["propext"]]},{"isProp":true,"kind":"theorem","name":["WithTop","sum_lt_top"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_2} M)} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.14 : LT.{u_2} M], Iff (LT.lt.{u_2} (WithTop.{u_2} M) (WithTop.instLT.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.14) (Finset.sum.{u_1, u_2} ι (WithTop.{u_2} M) (WithTop.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.5) s (fun (i : ι) => f i)) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))) (forall (i : ι), (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) -> (LT.lt.{u_2} (WithTop.{u_2} M) (WithTop.instLT.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1759458191._hygCtx._hyg.14) (f i) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M} [inst_1 : LT M],\n  ∑ i ∈ s, f i < ⊤ ↔ ∀ i ∈ s, f i < ⊤","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M} [inst_1 : LT M],\n  ∑ i ∈ s, f i < ⊤ ↔ ∀ i ∈ s, f i < ⊤","typeReferences":[["Finset","instSetLike"],["Finset"],["SetLike","instMembership"],["WithTop","addCommMonoid"],["Membership","mem"],["WithTop"],["WithTop","instLT"],["LT","lt"],["AddCommMonoid"],["WithTop","top"],["Iff"],["Top","top"],["Finset","sum"],["LT"]],"valueReferences":[["implies_congr"],["Finset","instSetLike"],["Finset"],["Eq","trans"],["Membership","mem"],["WithTop"],["not_exists","_simp_1"],["WithTop","instLT"],["congrArg"],["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithTop","sum_lt_top","_simp_1_1"],["iff_self"],["WithTop","top"],["congr"],["forall_congr"],["Finset","sum"],["Eq"],["not_and","_simp_1"],["Not"],["WithTop","sum_eq_top","_simp_1"],["SetLike","instMembership"],["Exists"],["True"],["WithTop","addCommMonoid"],["And"],["LT","lt"],["of_eq_true"],["Eq","refl"],["Iff"],["Top","top"],["Ne"]]},{"isProp":true,"kind":"theorem","name":["WithTop","prod_eq_top_ex_top"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_3} M₀)}, (Eq.{succ u_3} (WithTop.{u_3} M₀) (Finset.prod.{u_1, u_3} ι (WithTop.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithTop.{u_3} M₀) (WithTop.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.3188381333._hygCtx._hyg.11)) s (fun (j : ι) => f j)) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀))) -> (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_3} (WithTop.{u_3} M₀) (f i) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀)))))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀}, ∏ j ∈ s, f j = ⊤ → ∃ i ∈ s, f i = ⊤","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀}, ∏ j ∈ s, f j = ⊤ → ∃ i ∈ s, f i = ⊤","typeReferences":[["Finset","instSetLike"],["CommMonoidWithZero"],["NoZeroDivisors"],["SetLike","instMembership"],["Exists"],["Finset"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["Membership","mem"],["MulZeroClass","toMul"],["And"],["CommMonoidWithZero","toCommMonoid"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Top","top"],["Eq"]],"valueReferences":[["implies_congr"],["Finset","instSetLike"],["Finset"],["Eq","trans"],["Membership","mem"],["WithTop"],["Mathlib","Tactic","Push","not_exists","_simp_1"],["WithTop","prod_ne_top"],["WithTop","top"],["forall_congr"],["Eq"],["Not"],["binderNameHint"],["SetLike","instMembership"],["Exists"],["And"],["CommMonoidWithZero","toCommMonoid"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["Eq","refl"],["Top","top"],["id"],["Eq","mpr"],["Ne"],["Mathlib","Tactic","Push","not_and_eq"],["Mathlib","Tactic","Contrapose","contrapose₁"]]},{"isProp":true,"kind":"theorem","name":["WithBot","prod_ne_bot"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithBot.{u_3} M₀)}, (forall (i : ι), (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) -> (Ne.{succ u_3} (WithBot.{u_3} M₀) (f i) (Bot.bot.{u_3} (WithBot.{u_3} M₀) (WithBot.bot.{u_3} M₀)))) -> (Ne.{succ u_3} (WithBot.{u_3} M₀) (Finset.prod.{u_1, u_3} ι (WithBot.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithBot.{u_3} M₀) (WithBot.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.2913267311._hygCtx._hyg.11)) s (fun (i : ι) => f i)) (Bot.bot.{u_3} (WithBot.{u_3} M₀) (WithBot.bot.{u_3} M₀)))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithBot M₀}, (∀ i ∈ s, f i ≠ ⊥) → ∏ i ∈ s, f i ≠ ⊥","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithBot M₀}, (∀ i ∈ s, f i ≠ ⊥) → ∏ i ∈ s, f i ≠ ⊥","typeReferences":[["Finset","instSetLike"],["CommMonoidWithZero"],["NoZeroDivisors"],["Finset"],["SetLike","instMembership"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["MulZeroClass","toMul"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Bot","bot"],["Finset","prod"],["WithBot","instCommMonoidWithZero"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["WithBot"],["Nontrivial"],["Ne"],["WithBot","bot"]],"valueReferences":[["MulOneClass","toMulOne"],["SemigroupWithZero","toMulZeroClass"],["WithBot","coe_ne_bot"],["MulOne","toOne"],["CommMonoidWithZero","toMonoidWithZero"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Finset","prod_induction"],["Bot","bot"],["MonoidWithZero","toSemigroupWithZero"],["OfNat","ofNat"],["WithBot","mul_ne_bot"],["WithBot","instCommMonoidWithZero"],["One","toOfNat1"],["WithBot"],["Ne"],["WithBot","bot"]]},{"isProp":true,"kind":"theorem","name":["WithTop","sum_eq_top"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1414735597._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_2} M)}, Iff (Eq.{succ u_2} (WithTop.{u_2} M) (Finset.sum.{u_1, u_2} ι (WithTop.{u_2} M) (WithTop.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1414735597._hygCtx._hyg.5) s (fun (i : ι) => f i)) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))) (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithTop.{u_2} M) (f i) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M)))))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M},\n  ∑ i ∈ s, f i = ⊤ ↔ ∃ i ∈ s, f i = ⊤","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M},\n  ∑ i ∈ s, f i = ⊤ ↔ ∃ i ∈ s, f i = ⊤","typeReferences":[["Finset","instSetLike"],["Finset"],["Exists"],["SetLike","instMembership"],["WithTop","addCommMonoid"],["Membership","mem"],["And"],["WithTop"],["AddCommMonoid"],["WithTop","top"],["Iff"],["Top","top"],["Finset","sum"],["Eq"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["false_and"],["Eq","trans"],["Membership","mem"],["Finset","instEmptyCollection"],["WithTop"],["AddCommMonoid","toAddMonoid"],["EmptyCollection","emptyCollection"],["Finset","cons"],["congrArg"],["WithTop","add_eq_top","_simp_1"],["Or"],["iff_self"],["WithTop","top"],["exists_false","_simp_1"],["congr"],["funext"],["Finset","sum"],["congrFun'"],["Finset","mem_cons","_simp_1"],["Eq"],["propext"],["AddSemigroup","toAdd"],["SetLike","instMembership"],["Exists"],["Finset","sum_cons"],["True"],["instHAdd"],["Finset","notMem_empty","_simp_1"],["WithTop","addCommMonoid"],["And"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["WithTop","zero_ne_top","_simp_1"],["HAdd","hAdd"],["of_eq_true"],["AddMonoid","toAddSemigroup"],["Iff"],["Top","top"],["False"],["AddZero","toZero"],["exists_eq_or_imp","_simp_1"],["AddMonoid","toAddZeroClass"],["Finset","cons_induction"]]},{"isProp":true,"kind":"theorem","name":["WithBot","coe_sum","_simp_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774651._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] (s : Finset.{u_1} ι) (f : ι -> M), Eq.{succ u_2} (WithBot.{u_2} M) (Finset.sum.{u_1, u_2} ι (WithBot.{u_2} M) (WithBot.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774651._hygCtx._hyg.5) s (fun (i : ι) => WithBot.some.{u_2} M (f i))) (WithBot.some.{u_2} M (Finset.sum.{u_1, u_2} ι M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774651._hygCtx._hyg.5 s (fun (i : ι) => f i)))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ∑ i ∈ s, ↑(f i) = ↑(∑ i ∈ s, f i)","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ∑ i ∈ s, ↑(f i) = ↑(∑ i ∈ s, f i)","typeReferences":[["AddCommMonoid"],["Finset"],["WithBot"],["WithBot","addCommMonoid"],["Finset","sum"],["Eq"],["WithBot","some"]],"valueReferences":[["WithBot","coe_sum"],["WithBot"],["WithBot","addCommMonoid"],["Eq","symm"],["Finset","sum"],["WithBot","some"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithTop","prod_eq_top","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Data.Finset.Erase.3951670936._hygCtx._hyg.5 : DecidableEq.{succ u_1} α] {a : α} {b : α} {s : Finset.{u_1} α}, Eq.{1} Prop (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) (Finset.erase.{u_1} α inst._@.Mathlib.Data.Finset.Erase.3951670936._hygCtx._hyg.5 s b) a) (And (Ne.{succ u_1} α a b) (Membership.mem.{u_1, u_1} α (Finset.{u_1} α) (SetLike.instMembership.{u_1, u_1} (Finset.{u_1} α) α (Finset.instSetLike.{u_1} α)) s a))","typeFull":"∀ {α : Type u_1} [inst : DecidableEq α] {a b : α} {s : Finset α}, (a ∈ s.erase b) = (a ≠ b ∧ a ∈ s)","typeReadable":"∀ {α : Type u_1} [inst : DecidableEq α] {a b : α} {s : Finset α}, (a ∈ s.erase b) = (a ≠ b ∧ a ∈ s)","typeReferences":[["Finset","instSetLike"],["SetLike","instMembership"],["Finset"],["DecidableEq"],["Membership","mem"],["And"],["Finset","erase"],["Ne"],["Eq"]],"valueReferences":[["Finset","instSetLike"],["SetLike","instMembership"],["Finset"],["Membership","mem"],["And"],["Finset","erase"],["Ne"],["Finset","mem_erase"],["propext"]]},{"isProp":true,"kind":"theorem","name":["WithBot","bot_lt_prod"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithBot.{u_3} M₀)} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.23 : LT.{u_3} M₀], (forall (i : ι), (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) -> (LT.lt.{u_3} (WithBot.{u_3} M₀) (WithBot.instLT.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.23) (Bot.bot.{u_3} (WithBot.{u_3} M₀) (WithBot.bot.{u_3} M₀)) (f i))) -> (LT.lt.{u_3} (WithBot.{u_3} M₀) (WithBot.instLT.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.23) (Bot.bot.{u_3} (WithBot.{u_3} M₀) (WithBot.bot.{u_3} M₀)) (Finset.prod.{u_1, u_3} ι (WithBot.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithBot.{u_3} M₀) (WithBot.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.1899828770._hygCtx._hyg.11)) s (fun (i : ι) => f i)))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithBot M₀} [inst_4 : LT M₀], (∀ i ∈ s, ⊥ < f i) → ⊥ < ∏ i ∈ s, f i","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithBot M₀} [inst_4 : LT M₀], (∀ i ∈ s, ⊥ < f i) → ⊥ < ∏ i ∈ s, f i","typeReferences":[["Finset","instSetLike"],["CommMonoidWithZero"],["NoZeroDivisors"],["SetLike","instMembership"],["Finset"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["MulZeroClass","toMul"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["Bot","bot"],["LT","lt"],["Finset","prod"],["WithBot","instCommMonoidWithZero"],["WithBot","instLT"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["WithBot"],["Nontrivial"],["WithBot","bot"],["LT"]],"valueReferences":[["MulOneClass","toMulOne"],["WithBot","bot_lt_mul"],["SemigroupWithZero","toMulZeroClass"],["MulOne","toOne"],["CommMonoidWithZero","toMonoidWithZero"],["CommMonoidWithZero","toCommMonoid"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["Finset","prod_induction"],["WithBot","bot_lt_coe"],["Bot","bot"],["MonoidWithZero","toSemigroupWithZero"],["OfNat","ofNat"],["LT","lt"],["WithBot","instCommMonoidWithZero"],["WithBot","instLT"],["One","toOfNat1"],["WithBot"],["WithBot","bot"]]},{"isProp":true,"kind":"theorem","name":["WithTop","prod_eq_top_iff"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_3} M₀)}, Iff (Eq.{succ u_3} (WithTop.{u_3} M₀) (Finset.prod.{u_1, u_3} ι (WithTop.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithTop.{u_3} M₀) (WithTop.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.11)) s (fun (j : ι) => f j)) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀))) (And (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_3} (WithTop.{u_3} M₀) (f i) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀))))) (forall (i : ι), (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) -> (Ne.{succ u_3} (WithTop.{u_3} M₀) (f i) (OfNat.ofNat.{u_3} (WithTop.{u_3} M₀) 0 (Zero.toOfNat0.{u_3} (WithTop.{u_3} M₀) (WithTop.zero.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5))))))))))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀},\n  ∏ j ∈ s, f j = ⊤ ↔ (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀},\n  ∏ j ∈ s, f j = ⊤ ↔ (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0","typeReferences":[["Finset","instSetLike"],["Finset"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["WithTop","zero"],["MulZeroClass","toMul"],["Membership","mem"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["WithTop","top"],["Zero","toOfNat0"],["Eq"],["NoZeroDivisors"],["CommMonoidWithZero"],["Exists"],["SetLike","instMembership"],["And"],["CommMonoidWithZero","toCommMonoid"],["OfNat","ofNat"],["Finset","prod"],["WithTop","instCommMonoidWithZero"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["Nontrivial"],["Top","top"],["Ne"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["WithTop","zero"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["Iff","intro"],["And","intro"],["WithTop","top"],["WithTop","prod_eq_top_ex_top"],["Zero","toOfNat0"],["Eq"],["And","left"],["WithTop","prod_eq_top"],["SetLike","instMembership"],["Exists"],["Exists","choose_spec"],["And","right"],["And"],["CommMonoidWithZero","toCommMonoid"],["Exists","choose"],["OfNat","ofNat"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["WithTop","prod_eq_top_ne_zero"],["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithTop","prod_eq_top_iff","match_1_1"],["Top","top"],["Ne"]]},{"isProp":true,"kind":"theorem","name":["WithTop","prod_eq_top"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_3} M₀)} {i : ι}, (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) -> (Eq.{succ u_3} (WithTop.{u_3} M₀) (f i) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀))) -> (forall (j : ι), (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 j) -> (Ne.{succ u_3} (WithTop.{u_3} M₀) (f j) (OfNat.ofNat.{u_3} (WithTop.{u_3} M₀) 0 (Zero.toOfNat0.{u_3} (WithTop.{u_3} M₀) (WithTop.zero.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.5))))))))) -> (Eq.{succ u_3} (WithTop.{u_3} M₀) (Finset.prod.{u_1, u_3} ι (WithTop.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithTop.{u_3} M₀) (WithTop.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.2886805248._hygCtx._hyg.11)) s (fun (j : ι) => f j)) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀)))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀} {i : ι},\n  i ∈ s → f i = ⊤ → (∀ j ∈ s, f j ≠ 0) → ∏ j ∈ s, f j = ⊤","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀} {i : ι},\n  i ∈ s → f i = ⊤ → (∀ j ∈ s, f j ≠ 0) → ∏ j ∈ s, f j = ⊤","typeReferences":[["Finset","instSetLike"],["CommMonoidWithZero"],["NoZeroDivisors"],["Finset"],["SetLike","instMembership"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["WithTop","zero"],["MulZeroClass","toMul"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Top","top"],["Zero","toOfNat0"],["Ne"],["Eq"]],"valueReferences":[["Finset"],["CommMonoidWithZero","toMonoidWithZero"],["Eq","trans"],["MulZeroClass","toMul"],["Membership","mem"],["Classical","propDecidable"],["WithTop"],["HMul","hMul"],["MonoidWithZero","toMulZeroOneClass"],["WithTop","mul_eq_top_iff"],["And","intro"],["not_false_eq_true"],["Or"],["Eq","symm"],["WithTop","instNoZeroDivisors"],["SetLike","instMembership"],["And"],["CommMonoidWithZero","toCommMonoid"],["WithTop","instCommMonoidWithZero"],["MulZeroOneClass","toMulZeroClass"],["eq_false"],["Iff","mpr"],["id"],["Top","top"],["WithTop","instMulZeroClass"],["instHMul"],["Eq","mpr"],["WithTop","nontrivial"],["MulOneClass","toMulOne"],["Finset","instSetLike"],["CommMonoid","toMonoid"],["Eq","mp"],["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithTop","prod_eq_top","_simp_1_1"],["WithTop","zero"],["Finset","prod_ne_zero_iff"],["congrArg"],["MulOne","toMul"],["WithTop","top"],["Or","inl"],["Monoid","toMulOneClass"],["Finset","erase"],["Zero","toOfNat0"],["Eq"],["Finset","prod_erase_mul"],["Not"],["True"],["Nontrivial","to_nonempty"],["OfNat","ofNat"],["Finset","prod"],["of_eq_true"],["MulZeroClass","toZero"],["False"],["Ne"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithTop","prod_eq_top_iff","match_1_1"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5 : CommMonoidWithZero.{u_2} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_2} M₀)} (motive : (And (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithTop.{u_2} M₀) (f i) (Top.top.{u_2} (WithTop.{u_2} M₀) (WithTop.top.{u_2} M₀))))) (forall (i : ι), (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) -> (Ne.{succ u_2} (WithTop.{u_2} M₀) (f i) (OfNat.ofNat.{u_2} (WithTop.{u_2} M₀) 0 (Zero.toOfNat0.{u_2} (WithTop.{u_2} M₀) (WithTop.zero.{u_2} M₀ (MulZeroClass.toZero.{u_2} M₀ (MulZeroOneClass.toMulZeroClass.{u_2} M₀ (MonoidWithZero.toMulZeroOneClass.{u_2} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_2} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5)))))))))) -> Prop) (x._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx.133.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.141 : And (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithTop.{u_2} M₀) (f i) (Top.top.{u_2} (WithTop.{u_2} M₀) (WithTop.top.{u_2} M₀))))) (forall (i : ι), (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) -> (Ne.{succ u_2} (WithTop.{u_2} M₀) (f i) (OfNat.ofNat.{u_2} (WithTop.{u_2} M₀) 0 (Zero.toOfNat0.{u_2} (WithTop.{u_2} M₀) (WithTop.zero.{u_2} M₀ (MulZeroClass.toZero.{u_2} M₀ (MulZeroOneClass.toMulZeroClass.{u_2} M₀ (MonoidWithZero.toMulZeroOneClass.{u_2} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_2} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5)))))))))), (forall (h : Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithTop.{u_2} M₀) (f i) (Top.top.{u_2} (WithTop.{u_2} M₀) (WithTop.top.{u_2} M₀))))) (h' : forall (i : ι), (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) -> (Ne.{succ u_2} (WithTop.{u_2} M₀) (f i) (OfNat.ofNat.{u_2} (WithTop.{u_2} M₀) 0 (Zero.toOfNat0.{u_2} (WithTop.{u_2} M₀) (WithTop.zero.{u_2} M₀ (MulZeroClass.toZero.{u_2} M₀ (MulZeroOneClass.toMulZeroClass.{u_2} M₀ (MonoidWithZero.toMulZeroOneClass.{u_2} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_2} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5))))))))), motive (And.intro (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithTop.{u_2} M₀) (f i) (Top.top.{u_2} (WithTop.{u_2} M₀) (WithTop.top.{u_2} M₀))))) (forall (i : ι), (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) -> (Ne.{succ u_2} (WithTop.{u_2} M₀) (f i) (OfNat.ofNat.{u_2} (WithTop.{u_2} M₀) 0 (Zero.toOfNat0.{u_2} (WithTop.{u_2} M₀) (WithTop.zero.{u_2} M₀ (MulZeroClass.toZero.{u_2} M₀ (MulZeroOneClass.toMulZeroClass.{u_2} M₀ (MonoidWithZero.toMulZeroOneClass.{u_2} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_2} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.5))))))))) h h')) -> (motive x._@.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx.133.Mathlib.Algebra.BigOperators.WithTop.2495973660._hygCtx._hyg.141)","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_2} [inst : CommMonoidWithZero M₀] {s : Finset ι} {f : ι → WithTop M₀}\n  (motive : ((∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0) → Prop) (x : (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0),\n  (∀ (h : ∃ i ∈ s, f i = ⊤) (h' : ∀ i ∈ s, f i ≠ 0), motive ⋯) → motive x","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_2} [inst : CommMonoidWithZero M₀] {s : Finset ι} {f : ι → WithTop M₀}\n  (motive : ((∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0) → Prop) (x : (∃ i ∈ s, f i = ⊤) ∧ ∀ i ∈ s, f i ≠ 0),\n  (∀ (h : ∃ i ∈ s, f i = ⊤) (h' : ∀ i ∈ s, f i ≠ 0), motive ⋯) → motive x","typeReferences":[["CommMonoidWithZero"],["Finset","instSetLike"],["Finset"],["Exists"],["SetLike","instMembership"],["CommMonoidWithZero","toMonoidWithZero"],["WithTop","zero"],["Membership","mem"],["And"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["And","intro"],["MulZeroOneClass","toMulZeroClass"],["WithTop","top"],["MulZeroClass","toZero"],["Top","top"],["Zero","toOfNat0"],["Ne"],["Eq"]],"valueReferences":[["Finset","instSetLike"],["Exists"],["Finset"],["SetLike","instMembership"],["CommMonoidWithZero","toMonoidWithZero"],["WithTop","zero"],["Membership","mem"],["And"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["WithTop","top"],["MulZeroClass","toZero"],["Top","top"],["Zero","toOfNat0"],["Ne"],["Eq"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_4"],"typeFallback":"forall {a : Prop} {b : Prop}, Eq.{1} Prop (Not (And a b)) (a -> (Not b))","typeFull":"∀ {a b : Prop}, (¬(a ∧ b)) = (a → ¬b)","typeReadable":"∀ {a b : Prop}, (¬(a ∧ b)) = (a → ¬b)","typeReferences":[["Not"],["And"],["Eq"]],"valueReferences":[["Not"],["not_and"],["And"],["propext"]]},{"isProp":true,"kind":"theorem","name":["WithTop","prod_eq_top_ne_zero"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_3} M₀)} {i : ι}, (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) -> (Eq.{succ u_3} (WithTop.{u_3} M₀) (Finset.prod.{u_1, u_3} ι (WithTop.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithTop.{u_3} M₀) (WithTop.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.11)) s (fun (j : ι) => f j)) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀))) -> (Ne.{succ u_3} (WithTop.{u_3} M₀) (f i) (OfNat.ofNat.{u_3} (WithTop.{u_3} M₀) 0 (Zero.toOfNat0.{u_3} (WithTop.{u_3} M₀) (WithTop.zero.{u_3} M₀ (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.2710061930._hygCtx._hyg.5))))))))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀} {i : ι}, i ∈ s → ∏ j ∈ s, f j = ⊤ → f i ≠ 0","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀} {i : ι}, i ∈ s → ∏ j ∈ s, f j = ⊤ → f i ≠ 0","typeReferences":[["Finset","instSetLike"],["CommMonoidWithZero"],["NoZeroDivisors"],["Finset"],["SetLike","instMembership"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["WithTop","zero"],["MulZeroClass","toMul"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Top","top"],["Zero","toOfNat0"],["Ne"],["Eq"]],"valueReferences":[["instTransEq"],["WithTop","top_ne_zero"],["Trans","trans"],["CommMonoidWithZero","toMonoidWithZero"],["WithTop","zero"],["CommMonoidWithZero","toCommMonoid"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["Finset","prod_eq_zero"],["OfNat","ofNat"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Eq","symm"],["Top","top"],["id"],["Ne"],["Zero","toOfNat0"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithTop","sum_lt_top","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.WithBot.3003386878._hygCtx._hyg.8 : LT.{u_1} α] {x : WithTop.{u_1} α}, Eq.{1} Prop (LT.lt.{u_1} (WithTop.{u_1} α) (WithTop.instLT.{u_1} α inst._@.Mathlib.Order.WithBot.3003386878._hygCtx._hyg.8) x (Top.top.{u_1} (WithTop.{u_1} α) (WithTop.top.{u_1} α))) (Ne.{succ u_1} (WithTop.{u_1} α) x (Top.top.{u_1} (WithTop.{u_1} α) (WithTop.top.{u_1} α)))","typeFull":"∀ {α : Type u_1} [inst : LT α] {x : WithTop α}, (x < ⊤) = (x ≠ ⊤)","typeReadable":"∀ {α : Type u_1} [inst : LT α] {x : WithTop α}, (x < ⊤) = (x ≠ ⊤)","typeReferences":[["LT","lt"],["WithTop","top"],["Top","top"],["WithTop"],["Ne"],["Eq"],["LT"],["WithTop","instLT"]],"valueReferences":[["LT","lt"],["WithTop","top"],["Top","top"],["WithTop"],["WithTop","lt_top_iff_ne_top"],["Ne"],["WithTop","instLT"],["propext"]]},{"isProp":true,"kind":"theorem","name":["WithTop","prod_lt_top"],"typeFallback":"forall {ι : Type.{u_1}} {M₀ : Type.{u_3}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.5 : CommMonoidWithZero.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.8 : NoZeroDivisors.{u_3} M₀ (MulZeroClass.toMul.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.5)))) (MulZeroClass.toZero.{u_3} M₀ (MulZeroOneClass.toMulZeroClass.{u_3} M₀ (MonoidWithZero.toMulZeroOneClass.{u_3} M₀ (CommMonoidWithZero.toMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.5))))] [inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.11 : Nontrivial.{u_3} M₀] [inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.14 : DecidableEq.{succ u_3} M₀] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_3} M₀)} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.24 : LT.{u_3} M₀], (forall (i : ι), (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) -> (LT.lt.{u_3} (WithTop.{u_3} M₀) (WithTop.instLT.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.24) (f i) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀)))) -> (LT.lt.{u_3} (WithTop.{u_3} M₀) (WithTop.instLT.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.24) (Finset.prod.{u_1, u_3} ι (WithTop.{u_3} M₀) (CommMonoidWithZero.toCommMonoid.{u_3} (WithTop.{u_3} M₀) (WithTop.instCommMonoidWithZero.{u_3} M₀ inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.14 inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.5 inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.8 inst._@.Mathlib.Algebra.BigOperators.WithTop.1286552746._hygCtx._hyg.11)) s (fun (i : ι) => f i)) (Top.top.{u_3} (WithTop.{u_3} M₀) (WithTop.top.{u_3} M₀)))","typeFull":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀} [inst_4 : LT M₀], (∀ i ∈ s, f i < ⊤) → ∏ i ∈ s, f i < ⊤","typeReadable":"∀ {ι : Type u_1} {M₀ : Type u_3} [inst : CommMonoidWithZero M₀] [inst_1 : NoZeroDivisors M₀] [inst_2 : Nontrivial M₀]\n  [inst_3 : DecidableEq M₀] {s : Finset ι} {f : ι → WithTop M₀} [inst_4 : LT M₀], (∀ i ∈ s, f i < ⊤) → ∏ i ∈ s, f i < ⊤","typeReferences":[["Finset","instSetLike"],["CommMonoidWithZero"],["NoZeroDivisors"],["SetLike","instMembership"],["Finset"],["CommMonoidWithZero","toMonoidWithZero"],["DecidableEq"],["MulZeroClass","toMul"],["Membership","mem"],["CommMonoidWithZero","toCommMonoid"],["WithTop"],["MonoidWithZero","toMulZeroOneClass"],["WithTop","instLT"],["LT","lt"],["WithTop","instCommMonoidWithZero"],["Finset","prod"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Top","top"],["LT"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toOne"],["CommMonoidWithZero","toMonoidWithZero"],["WithTop"],["CommMonoidWithZero","toCommMonoid"],["MulZeroOneClass","toMulOneClass"],["Finset","prod_induction"],["MonoidWithZero","toMulZeroOneClass"],["OfNat","ofNat"],["WithTop","instLT"],["WithTop","coe_lt_top"],["LT","lt"],["WithTop","instCommMonoidWithZero"],["WithTop","mul_lt_top"],["One","toOfNat1"],["WithTop","top"],["MulZeroOneClass","toMulZeroClass"],["Top","top"]]},{"isProp":true,"kind":"theorem","name":["WithTop","coe_sum","_simp_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774650._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] (s : Finset.{u_1} ι) (f : ι -> M), Eq.{succ u_2} (WithTop.{u_2} M) (Finset.sum.{u_1, u_2} ι (WithTop.{u_2} M) (WithTop.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774650._hygCtx._hyg.5) s (fun (i : ι) => WithTop.some.{u_2} M (f i))) (WithTop.some.{u_2} M (Finset.sum.{u_1, u_2} ι M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774650._hygCtx._hyg.5 s (fun (i : ι) => f i)))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ∑ i ∈ s, ↑(f i) = ↑(∑ i ∈ s, f i)","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ∑ i ∈ s, ↑(f i) = ↑(∑ i ∈ s, f i)","typeReferences":[["AddCommMonoid"],["Finset"],["WithTop","addCommMonoid"],["WithTop","some"],["Finset","sum"],["WithTop"],["Eq"]],"valueReferences":[["WithTop","coe_sum"],["WithTop","addCommMonoid"],["Eq","symm"],["Finset","sum"],["WithTop","some"],["WithTop"]]},{"isProp":true,"kind":"theorem","name":["WithTop","coe_sum"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774650._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] (s : Finset.{u_1} ι) (f : ι -> M), Eq.{succ u_2} (WithTop.{u_2} M) (WithTop.some.{u_2} M (Finset.sum.{u_1, u_2} ι M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774650._hygCtx._hyg.5 s (fun (i : ι) => f i))) (Finset.sum.{u_1, u_2} ι (WithTop.{u_2} M) (WithTop.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1082774650._hygCtx._hyg.5) s (fun (i : ι) => WithTop.some.{u_2} M (f i)))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ↑(∑ i ∈ s, f i) = ∑ i ∈ s, ↑(f i)","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] (s : Finset ι) (f : ι → M), ↑(∑ i ∈ s, f i) = ∑ i ∈ s, ↑(f i)","typeReferences":[["AddCommMonoid"],["Finset"],["WithTop","addCommMonoid"],["Finset","sum"],["WithTop","some"],["WithTop"],["Eq"]],"valueReferences":[["WithTop","addHom"],["AddMonoidHom"],["WithTop","addCommMonoid"],["WithTop","addZeroClass"],["AddMonoidHom","instFunLike"],["WithTop"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["AddZeroClass","toAddZero"],["map_sum"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["WithTop","sum_eq_top","_simp_1"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.1414735597._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_2} M)}, Eq.{1} Prop (Eq.{succ u_2} (WithTop.{u_2} M) (Finset.sum.{u_1, u_2} ι (WithTop.{u_2} M) (WithTop.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.1414735597._hygCtx._hyg.5) s (fun (i : ι) => f i)) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))) (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithTop.{u_2} M) (f i) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M)))))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M},\n  (∑ i ∈ s, f i = ⊤) = ∃ i ∈ s, f i = ⊤","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M},\n  (∑ i ∈ s, f i = ⊤) = ∃ i ∈ s, f i = ⊤","typeReferences":[["Finset","instSetLike"],["Finset"],["Exists"],["SetLike","instMembership"],["WithTop","addCommMonoid"],["Membership","mem"],["And"],["WithTop"],["AddCommMonoid"],["WithTop","top"],["Top","top"],["Finset","sum"],["Eq"]],"valueReferences":[["Finset","instSetLike"],["Exists"],["Finset"],["SetLike","instMembership"],["WithTop","addCommMonoid"],["Membership","mem"],["And"],["WithTop"],["WithTop","top"],["Top","top"],["Finset","sum"],["WithTop","sum_eq_top"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_2"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.53073213._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithBot.{u_2} M)}, Eq.{1} Prop (Eq.{succ u_2} (WithBot.{u_2} M) (Finset.sum.{u_1, u_2} ι (WithBot.{u_2} M) (WithBot.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.53073213._hygCtx._hyg.5) s (fun (i : ι) => f i)) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M))) (Exists.{succ u_1} ι (fun (i : ι) => And (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) (Eq.{succ u_2} (WithBot.{u_2} M) (f i) (Bot.bot.{u_2} (WithBot.{u_2} M) (WithBot.bot.{u_2} M)))))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M},\n  (∑ i ∈ s, f i = ⊥) = ∃ i ∈ s, f i = ⊥","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithBot M},\n  (∑ i ∈ s, f i = ⊥) = ∃ i ∈ s, f i = ⊥","typeReferences":[["Finset","instSetLike"],["Finset"],["Exists"],["SetLike","instMembership"],["Membership","mem"],["WithBot","addCommMonoid"],["And"],["Bot","bot"],["AddCommMonoid"],["WithBot"],["Finset","sum"],["Eq"],["WithBot","bot"]],"valueReferences":[["Finset","instSetLike"],["Exists"],["Finset"],["SetLike","instMembership"],["Membership","mem"],["WithBot","addCommMonoid"],["And"],["Bot","bot"],["WithBot","sum_eq_bot_iff"],["WithBot"],["Finset","sum"],["Eq"],["WithBot","bot"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","BigOperators","WithTop",0,"WithBot","bot_lt_sum_iff","_simp_1_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Order.WithBot.3003386878._hygCtx._hyg.8 : LT.{u_1} α] {x : WithBot.{u_1} α}, Eq.{1} Prop (LT.lt.{u_1} (WithBot.{u_1} α) (WithBot.instLT.{u_1} α inst._@.Mathlib.Order.WithBot.3003386878._hygCtx._hyg.8) (Bot.bot.{u_1} (WithBot.{u_1} α) (WithBot.bot.{u_1} α)) x) (Ne.{succ u_1} (WithBot.{u_1} α) x (Bot.bot.{u_1} (WithBot.{u_1} α) (WithBot.bot.{u_1} α)))","typeFull":"∀ {α : Type u_1} [inst : LT α] {x : WithBot α}, (⊥ < x) = (x ≠ ⊥)","typeReadable":"∀ {α : Type u_1} [inst : LT α] {x : WithBot α}, (⊥ < x) = (x ≠ ⊥)","typeReferences":[["LT","lt"],["WithBot","instLT"],["WithBot"],["Ne"],["WithBot","bot"],["Eq"],["Bot","bot"],["LT"]],"valueReferences":[["LT","lt"],["WithBot","instLT"],["WithBot"],["WithBot","bot_lt_iff_ne_bot"],["Ne"],["WithBot","bot"],["Bot","bot"],["propext"]]},{"isProp":true,"kind":"theorem","name":["WithTop","sum_ne_top"],"typeFallback":"forall {ι : Type.{u_1}} {M : Type.{u_2}} [inst._@.Mathlib.Algebra.BigOperators.WithTop.2151015906._hygCtx._hyg.5 : AddCommMonoid.{u_2} M] {s : Finset.{u_1} ι} {f : ι -> (WithTop.{u_2} M)}, Iff (Ne.{succ u_2} (WithTop.{u_2} M) (Finset.sum.{u_1, u_2} ι (WithTop.{u_2} M) (WithTop.addCommMonoid.{u_2} M inst._@.Mathlib.Algebra.BigOperators.WithTop.2151015906._hygCtx._hyg.5) s (fun (i : ι) => f i)) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))) (forall (i : ι), (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) -> (Ne.{succ u_2} (WithTop.{u_2} M) (f i) (Top.top.{u_2} (WithTop.{u_2} M) (WithTop.top.{u_2} M))))","typeFull":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M},\n  ∑ i ∈ s, f i ≠ ⊤ ↔ ∀ i ∈ s, f i ≠ ⊤","typeReadable":"∀ {ι : Type u_1} {M : Type u_2} [inst : AddCommMonoid M] {s : Finset ι} {f : ι → WithTop M},\n  ∑ i ∈ s, f i ≠ ⊤ ↔ ∀ i ∈ s, f i ≠ ⊤","typeReferences":[["Finset","instSetLike"],["AddCommMonoid"],["SetLike","instMembership"],["Finset"],["WithTop","top"],["WithTop","addCommMonoid"],["Iff"],["Membership","mem"],["Top","top"],["Finset","sum"],["WithTop"],["Ne"]],"valueReferences":[["Finset","instSetLike"],["Finset"],["Eq","trans"],["Membership","mem"],["WithTop"],["not_exists","_simp_1"],["congrArg"],["WithTop","top"],["iff_self"],["forall_congr"],["Finset","sum"],["congrFun'"],["Eq"],["Not"],["not_and","_simp_1"],["WithTop","sum_eq_top","_simp_1"],["SetLike","instMembership"],["Exists"],["True"],["WithTop","addCommMonoid"],["And"],["of_eq_true"],["Iff"],["Top","top"],["Ne"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.Grp.EnoughInjectives.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["AddCommGrpCat","enoughInjectives"],"typeFallback":"CategoryTheory.EnoughInjectives.{u, succ u} AddCommGrpCat.{u} AddCommGrpCat.instCategory.{u}","typeFull":"CategoryTheory.EnoughInjectives AddCommGrpCat","typeReadable":"CategoryTheory.EnoughInjectives AddCommGrpCat","typeReferences":[["AddCommGrpCat"],["AddCommGrpCat","instCategory"],["CategoryTheory","EnoughInjectives"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["RingHom"],["AddCommGroup","toAddGroup"],["AddGroupWithOne","toAddMonoidWithOne"],["CategoryTheory","Mono"],["AddGroup","toSubtractionMonoid"],["AddCommGrpCat","ofHom"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddCommGrpCat","of"],["RingHom","id"],["CategoryTheory","EnoughInjectives","mk"],["CategoryTheory","InjectivePresentation"],["ZeroHom","mk"],["congr_fun"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["congr_arg"],["AddCommGroup","toIntModule"],["map_add"],["Eq","refl"],["AddMonoidWithOne","toOne"],["Rat"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["Rat","instPartialOrder"],["Rat","instZeroLEOneClass"],["Rat","instCharZero"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddMonoidHom","instAddMonoidHomClass"],["AddCommMonoid","toAddMonoid"],["Rat","instIsStrictOrderedRing"],["ZeroLEOneClass","factZeroLtOne"],["AddCommMagma","toAdd"],["Eq"],["map_zero"],["CategoryTheory","InjectivePresentation","mk"],["Rat","instField"],["DivisionRing","toRing"],["ZeroHom","toFun"],["ULift","addCommGroup"],["AddZero","toAdd"],["OfNat","ofNat"],["Int"],["AddCommGrpCat"],["HAdd","hAdd"],["Module","toDistribMulAction"],["AddCommGrpCat","str"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["CharacterModule","eq_zero_of_character_apply"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddSubgroup","zmultiples"],["Pi","addCommGroup"],["Function","Injective"],["AddCommGrpCat","mono_iff_injective"],["Int","instCommRing"],["Pi","addZeroClass"],["ULift","addZeroClass"],["ULift","subNegAddMonoid"],["DistribMulActionSemiHomClass","toAddMonoidHomClass"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["AddMonoidHom"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["funext"],["injective_iff_map_eq_zero"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["Rat","linearOrder"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["QuotientAddGroup","Quotient","addCommGroup"],["ULift"],["Rat","addCommGroup"],["AddZeroClass","toAddZero"],["Iff","mpr"],["NegZeroClass","toZero"],["id"],["NeZero","charZero_one"],["SubNegMonoid","toZSMul"],["Rat","instFloorRing"],["AddZero","toZero"],["Rat","instDivisionRing"],["ULift","down"],["CharacterModule","instLinearMapClassIntAddCircleRatOfNat"],["RingHom","instFunLike"],["CommRing","toNonUnitalCommRing"],["AddCircle"],["AddCircle","instDivisibleByInt"],["ULift","up"],["DFunLike","coe"],["SubNegZeroMonoid","toNegZeroClass"],["Rat","instOfNat"],["congrArg"],["CategoryTheory","ConcreteCategory","hom"],["AddSubgroup","normal_of_comm"],["Nonempty","intro"],["MonoidWithZero","toMonoid"],["QuotientAddGroup","Quotient","addGroup"],["CharacterModule","instFunLikeAddCircleRatOfNat"],["AddCommSemigroup","toAddCommMagma"],["Zero","toOfNat0"],["congrFun'"],["AddCommGrpCat","carrier"],["ULift","instDivisibleBy"],["AddCommGrpCat","instCategory"],["Field","toCommRing"],["instHAdd"],["AddMonoidHomClass","toZeroHomClass"],["AddCommGrpCat","injective_of_divisible"],["Semiring","toMonoidWithZero"],["SemilinearMapClass","toAddHomClass"],["CharacterModule"],["AddMonoidHom","mk"],["AddCommMonoid","toAddCommSemigroup"],["SubNegMonoid","toAddMonoid"],["Int","instSemiring"],["Pi","divisibleBy"],["SemilinearMapClass","distribMulActionSemiHomClass"]]},{"isProp":true,"kind":"theorem","name":["CommGrpCat","enoughInjectives"],"typeFallback":"CategoryTheory.EnoughInjectives.{u, succ u} CommGrpCat.{u} CommGrpCat.instCategory.{u}","typeFull":"CategoryTheory.EnoughInjectives CommGrpCat","typeReadable":"CategoryTheory.EnoughInjectives CommGrpCat","typeReferences":[["CommGrpCat"],["CommGrpCat","instCategory"],["CategoryTheory","EnoughInjectives"]],"valueReferences":[["AddCommGrpCat"],["AddCommGrpCat","instCategory"],["CommGrpCat"],["CommGrpCat","instCategory"],["CategoryTheory","Equivalence","isEquivalence_functor"],["commGroupAddCommGroupEquivalence"],["CategoryTheory","Equivalence","functor"],["CategoryTheory","EnoughInjectives","of_equivalence"],["AddCommGrpCat","enoughInjectives"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.ModuleCat.AB.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["instHasSeparatorModuleCatOfSmall"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.1678865837._hygCtx._hyg.3 : Ring.{u} R] [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.1678865837._hygCtx._hyg.6 : Small.{v, u} R], CategoryTheory.HasSeparator.{v, max (succ v) u} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.1678865837._hygCtx._hyg.3) (ModuleCat.moduleCategory.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.1678865837._hygCtx._hyg.3)","typeFull":"∀ (R : Type u) [inst : Ring R] [Small.{v, u} R], CategoryTheory.HasSeparator (ModuleCat R)","typeReadable":"∀ (R : Type u) [inst : Ring R] [Small.{v, u} R], CategoryTheory.HasSeparator (ModuleCat R)","typeReferences":[["Small"],["CategoryTheory","HasSeparator"],["ModuleCat","moduleCategory"],["ModuleCat"],["Ring"]],"valueReferences":[["Ring","toNonAssocRing"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CategoryTheory","HasSeparator","mk"],["ModuleCat","moduleCategory"],["Exists","intro"],["ModuleCat","of"],["Shrink"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["Shrink","instModule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Shrink","instAddCommGroup"],["CategoryTheory","IsSeparator"],["Semiring","toModule"],["ModuleCat","isSeparator"],["NonAssocRing","toNonUnitalNonAssocRing"],["ModuleCat"]]},{"isProp":true,"kind":"theorem","name":["instAB5ModuleCat"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.160834823._hygCtx._hyg.3 : Ring.{u} R], CategoryTheory.AB5.{u, succ u} (ModuleCat.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.160834823._hygCtx._hyg.3) (ModuleCat.moduleCategory.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.160834823._hygCtx._hyg.3) (CategoryTheory.Limits.hasFilteredColimitsOfSize_of_hasColimitsOfSize.{u, u, u, succ u} (ModuleCat.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.160834823._hygCtx._hyg.3) (ModuleCat.moduleCategory.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.160834823._hygCtx._hyg.3) (ModuleCat.hasColimitsOfSize.{u, u, u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.160834823._hygCtx._hyg.3 (AddCommGrpCat.hasColimitsOfSize.{u, u, u} UnivLE.self.{u})))","typeFull":"∀ (R : Type u) [inst : Ring R], CategoryTheory.AB5 (ModuleCat R)","typeReadable":"∀ (R : Type u) [inst : Ring R], CategoryTheory.AB5 (ModuleCat R)","typeReferences":[["UnivLE","self"],["CategoryTheory","AB5"],["AddCommGrpCat","hasColimitsOfSize"],["ModuleCat","hasColimitsOfSize"],["ModuleCat","moduleCategory"],["CategoryTheory","Limits","hasFilteredColimitsOfSize_of_hasColimitsOfSize"],["ModuleCat"],["Ring"]],"valueReferences":[["instAB5AddCommGrpCat"],["ModuleCat","isModule"],["LinearMap","instFunLike"],["CategoryTheory","forget₂"],["AddCommGroup","toAddGroup"],["UnivLE","small"],["CategoryTheory","Limits","hasColimitsOfShape_of_has_filtered_colimits"],["ModuleCat","moduleCategory"],["ModuleCat","forget₂PreservesColimitsOfShape"],["AddCommGrpCat","hasColimitsOfShape"],["CategoryTheory","Limits","PreservesLimits","preservesFiniteLimits"],["CategoryTheory","AB5OfSize","ofShape"],["Semiring","toNonAssocSemiring"],["CategoryTheory","AB5OfSize","mk"],["RingHom","id"],["AddMonoidHom"],["ModuleCat","instConcreteCategoryLinearMapIdCarrier"],["instHasFilteredColimitsAddCommGrpCat"],["CategoryTheory","Limits","hasCountableLimits_of_hasLimits"],["ModuleCat","hasColimitsOfSize"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["CategoryTheory","Limits","hasFilteredColimitsOfSize_of_hasColimitsOfSize"],["AddCommGrpCat","carrier"],["CategoryTheory","Limits","hasFiniteLimits_of_hasCountableLimits"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["ModuleCat"],["ModuleCat","carrier"],["AddCommGrpCat","instCategory"],["ModuleCat","isAddCommGroup"],["CategoryTheory","HasExactColimitsOfShape","domain_of_functor"],["ModuleCat","forget₂AddCommGroup_preservesLimits"],["ModuleCat","hasLimits'"],["AddZeroClass","toAddZero"],["LinearMap"],["ModuleCat","instReflectsIsomorphismsAddCommGrpCatForget₂LinearMapIdCarrierAddMonoidHomCarrier"],["Ring","toSemiring"],["AddCommGrpCat"],["AddCommGrpCat","str"],["UnivLE","self"],["SubNegMonoid","toAddMonoid"],["CategoryTheory","Limits","reflectsFiniteLimits_of_reflectsIsomorphisms"],["AddCommGrpCat","hasColimitsOfSize"],["ModuleCat","hasForgetToAddCommGroup"],["AddCommGroup","toAddCommMonoid"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["instIsGrothendieckAbelianModuleCat"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.4251028603._hygCtx._hyg.3 : Ring.{u} R], CategoryTheory.IsGrothendieckAbelian.{u, u, succ u} (ModuleCat.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.4251028603._hygCtx._hyg.3) (ModuleCat.moduleCategory.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.4251028603._hygCtx._hyg.3) (ModuleCat.abelian.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.4251028603._hygCtx._hyg.3)","typeFull":"∀ (R : Type u) [inst : Ring R], CategoryTheory.IsGrothendieckAbelian.{u, u, u + 1} (ModuleCat R)","typeReadable":"∀ (R : Type u) [inst : Ring R], CategoryTheory.IsGrothendieckAbelian.{u, u, u + 1} (ModuleCat R)","typeReferences":[["CategoryTheory","IsGrothendieckAbelian"],["ModuleCat","moduleCategory"],["ModuleCat","abelian"],["ModuleCat"],["Ring"]],"valueReferences":[["UnivLE","small"],["CategoryTheory","HasSeparator"],["ModuleCat","moduleCategory"],["CategoryTheory","Limits","HasFilteredColimitsOfSize"],["ModuleCat","abelian"],["instHasSeparatorModuleCatOfSmall"],["instAB5ModuleCat"],["UnivLE","self"],["CategoryTheory","IsGrothendieckAbelian","mk"],["AddCommGrpCat","hasColimitsOfSize"],["CategoryTheory","AB5OfSize"],["CategoryTheory","LocallySmall"],["ModuleCat","hasColimitsOfSize"],["inferInstance"],["CategoryTheory","Limits","hasFilteredColimitsOfSize_of_hasColimitsOfSize"],["ModuleCat"],["CategoryTheory","locallySmall_of_univLE"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_4"],"typeFallback":"forall {α : Sort.{u_1}} {p : α -> Prop} {a' : α}, Eq.{1} Prop (forall (a : α), (Eq.{u_1} α a' a) -> (p a)) (p a')","typeFull":"∀ {α : Sort u_1} {p : α → Prop} {a' : α}, (∀ (a : α), a' = a → p a) = p a'","typeReadable":"∀ {α : Sort u_1} {p : α → Prop} {a' : α}, (∀ (a : α), a' = a → p a) = p a'","typeReferences":[["Eq"]],"valueReferences":[["forall_eq'"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["instAB4StarModuleCat"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3 : Ring.{u} R], CategoryTheory.AB4Star.{u, succ u} (ModuleCat.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3) (ModuleCat.moduleCategory.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3) (fun (J : Type.{u}) => CategoryTheory.Limits.hasProductsOfShape_of_hasProducts.{u, u, succ u} (ModuleCat.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3) (ModuleCat.moduleCategory.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3) (fun (J : Type.{u}) => CategoryTheory.Limits.hasLimitsOfShapeOfHasLimits.{u, u, u, succ u} (ModuleCat.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3) (ModuleCat.moduleCategory.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3) (CategoryTheory.Discrete.{u} J) (CategoryTheory.discreteCategory.{u} J) (ModuleCat.hasLimits'.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3275254160._hygCtx._hyg.3)) J)","typeFull":"∀ (R : Type u) [inst : Ring R], CategoryTheory.AB4Star (ModuleCat R)","typeReadable":"∀ (R : Type u) [inst : Ring R], CategoryTheory.AB4Star (ModuleCat R)","typeReferences":[["CategoryTheory","discreteCategory"],["CategoryTheory","Limits","hasProductsOfShape_of_hasProducts"],["CategoryTheory","Discrete"],["ModuleCat","moduleCategory"],["CategoryTheory","AB4Star"],["CategoryTheory","Limits","hasLimitsOfShapeOfHasLimits"],["ModuleCat","hasLimits'"],["ModuleCat"],["Ring"]],"valueReferences":[["ModuleCat","isModule"],["LinearMap","instFunLike"],["CategoryTheory","forget₂"],["AddCommGroup","toAddGroup"],["UnivLE","small"],["ModuleCat","instPreservesColimitsOfSizeAddCommGrpCatForget₂LinearMapIdCarrierAddMonoidHomCarrierOfHasColimitsOfSizeAddCommGrpMax"],["ModuleCat","moduleCategory"],["CategoryTheory","Limits","PreservesColimits","preservesFiniteColimits"],["instHasExactLimitsOfShapeDiscreteAddCommGrpCat"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["univLE_of_max"],["AddMonoidHom"],["ModuleCat","instConcreteCategoryLinearMapIdCarrier"],["CategoryTheory","Limits","reflectsFiniteColimitsOfReflectsIsomorphisms"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["CategoryTheory","Limits","hasLimitsOfShapeOfHasLimits"],["ModuleCat","instHasFiniteColimits"],["AddCommGrpCat","carrier"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["CategoryTheory","instSmallDiscrete"],["ModuleCat"],["ModuleCat","carrier"],["AddCommGrpCat","instCategory"],["ModuleCat","isAddCommGroup"],["CategoryTheory","Discrete"],["ModuleCat","forget₂AddCommGroup_preservesLimits"],["AddZeroClass","toAddZero"],["ModuleCat","hasLimits'"],["LinearMap"],["ModuleCat","instReflectsIsomorphismsAddCommGrpCatForget₂LinearMapIdCarrierAddMonoidHomCarrier"],["Ring","toSemiring"],["AddCommGrpCat"],["CategoryTheory","discreteCategory"],["AddCommGrpCat","str"],["CategoryTheory","Limits","hasProductsOfShape_of_hasProducts"],["UnivLE","self"],["SubNegMonoid","toAddMonoid"],["CategoryTheory","HasExactLimitsOfShape","domain_of_functor"],["AddCommGrpCat","hasLimitsOfShape"],["AddCommGrpCat","hasColimitsOfSize"],["CategoryTheory","AB4StarOfSize","mk"],["ModuleCat","hasForgetToAddCommGroup"],["AddCommGroup","toAddCommMonoid"],["CategoryTheory","Limits","PreservesLimitsOfSize","preservesLimitsOfShape"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["instAB4ModuleCat"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3416217441._hygCtx._hyg.3 : Ring.{u} R], CategoryTheory.AB4.{u, succ u} (ModuleCat.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3416217441._hygCtx._hyg.3) (ModuleCat.moduleCategory.{u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3416217441._hygCtx._hyg.3) (fun (J : Type.{u}) => ModuleCat.hasColimitsOfShape.{u, u, u, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.3416217441._hygCtx._hyg.3 (CategoryTheory.Discrete.{u} J) (CategoryTheory.discreteCategory.{u} J) (AddCommGrpCat.hasColimitsOfShape.{u, u, u} (CategoryTheory.Discrete.{u} J) (CategoryTheory.discreteCategory.{u} J) (CategoryTheory.instSmallDiscrete.{u, u} J (UnivLE.small.{u, u} UnivLE.self.{u} J))))","typeFull":"∀ (R : Type u) [inst : Ring R], CategoryTheory.AB4 (ModuleCat R)","typeReadable":"∀ (R : Type u) [inst : Ring R], CategoryTheory.AB4 (ModuleCat R)","typeReferences":[["CategoryTheory","discreteCategory"],["UnivLE","self"],["ModuleCat","hasColimitsOfShape"],["UnivLE","small"],["CategoryTheory","Discrete"],["ModuleCat","moduleCategory"],["CategoryTheory","AB4"],["AddCommGrpCat","hasColimitsOfShape"],["CategoryTheory","instSmallDiscrete"],["ModuleCat"],["Ring"]],"valueReferences":[["ModuleCat","instPreadditive"],["ModuleCat","moduleCategory"],["CategoryTheory","Abelian","hasFiniteBiproducts"],["ModuleCat","abelian"],["ModuleCat","hasLimits'"],["instAB5ModuleCat"],["UnivLE","self"],["CategoryTheory","AB4","of_AB5"],["AddCommGrpCat","hasColimitsOfSize"],["CategoryTheory","Limits","hasCountableLimits_of_hasLimits"],["ModuleCat","hasColimitsOfSize"],["CategoryTheory","Preadditive","preadditiveHasZeroMorphisms"],["CategoryTheory","Limits","hasFilteredColimitsOfSize_of_hasColimitsOfSize"],["CategoryTheory","Limits","hasFiniteLimits_of_hasCountableLimits"],["ModuleCat"]]},{"isProp":true,"kind":"theorem","name":["ModuleCat","isSeparator"],"typeFallback":"forall (R : Type.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3 : Ring.{u} R] [inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.6 : Small.{v, u} R], CategoryTheory.IsSeparator.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3) (ModuleCat.moduleCategory.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3) (ModuleCat.of.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3 (Shrink.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.6) (Shrink.instAddCommGroup.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.6 (Ring.toAddCommGroup.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3)) (Shrink.instModule.{v, u, u} R R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.6 (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonAssocRing.toNonUnitalNonAssocRing.{u} R (Ring.toNonAssocRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3)))) (Semiring.toModule.{u} R (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.AB.712464633._hygCtx._hyg.3))))","typeFull":"∀ (R : Type u) [inst : Ring R] [inst_1 : Small.{v, u} R], CategoryTheory.IsSeparator (ModuleCat.of R (Shrink.{v, u} R))","typeReadable":"∀ (R : Type u) [inst : Ring R] [inst_1 : Small.{v, u} R], CategoryTheory.IsSeparator (ModuleCat.of R (Shrink.{v, u} R))","typeReferences":[["Ring","toNonAssocRing"],["Small"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["ModuleCat","moduleCategory"],["ModuleCat","of"],["Shrink"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["Shrink","instModule"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Shrink","instAddCommGroup"],["CategoryTheory","IsSeparator"],["Semiring","toModule"],["NonAssocRing","toNonUnitalNonAssocRing"],["Ring"],["ModuleCat"]],"valueReferences":[["implies_congr"],["Ring","toNonAssocRing"],["ModuleCat","ofHom"],["Eq","trans"],["RingHomCompTriple","ids"],["AddGroupWithOne","toAddMonoidWithOne"],["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_1"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["RingHom","id"],["LinearMap","ext"],["forall_congr"],["DistribMulAction","toMulAction"],["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_4"],["Semiring","toModule"],["NonAssocRing","toNonUnitalNonAssocRing"],["Shrink","linearEquiv"],["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_2"],["ModuleCat","Hom","hom"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["LinearMap"],["CategoryTheory","Category","toCategoryStruct"],["Ring","toSemiring"],["Shrink","instAddCommGroup"],["CategoryTheory","CategoryStruct","comp"],["AddMonoidWithOne","toOne"],["one_smul"],["AddCommGroup","toAddCommMonoid"],["RingHomInvPair","ids"],["ModuleCat","isModule"],["LinearMap","toSpanSingleton"],["LinearMap","instFunLike"],["Eq","mp"],["ModuleCat","moduleCategory"],["LinearMap","comp"],["CategoryTheory","ObjectProperty","singleton"],["AddCommMonoid","toAddMonoid"],["LinearEquiv"],["Shrink"],["DFunLike","coe"],["congrArg"],["Shrink","instModule"],["Quiver","Hom"],["congr"],["EquivLike","toFunLike"],["MonoidWithZero","toMonoid"],["Eq"],["LinearEquiv","instEquivLike"],["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_3"],["Shrink","instAddCommMonoid"],["ModuleCat"],["ModuleCat","carrier"],["ModuleCat","isAddCommGroup"],["ModuleCat","hom_ext"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["equivShrink_symm_one"],["Semiring","toMonoidWithZero"],["ModuleCat","of"],["LinearEquiv","toLinearMap"],["OfNat","ofNat"],["Ring","toAddCommGroup"],["Module","toDistribMulAction"],["CategoryTheory","CategoryStruct","toQuiver"],["One","toOfNat1"],["Shrink","instOne"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_2"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 : Ring.{u} R] {M : ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3} {N : ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3} {f : Quiver.Hom.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (ModuleCat.moduleCategory.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3))) M N} {g : Quiver.Hom.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (ModuleCat.moduleCategory.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3))) M N}, Eq.{1} Prop (Eq.{succ v} (Quiver.Hom.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (CategoryTheory.CategoryStruct.toQuiver.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (CategoryTheory.Category.toCategoryStruct.{v, max u (succ v)} (ModuleCat.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (ModuleCat.moduleCategory.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3))) M N) f g) (Eq.{succ v} (LinearMap.{u, u, v, v} R R (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R (Ring.toSemiring.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3))) (ModuleCat.carrier.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 M) (ModuleCat.carrier.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 N) (AddCommGroup.toAddCommMonoid.{v} (ModuleCat.carrier.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 M) (ModuleCat.isAddCommGroup.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 M)) (AddCommGroup.toAddCommMonoid.{v} (ModuleCat.carrier.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 N) (ModuleCat.isAddCommGroup.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 N)) (ModuleCat.isModule.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 M) (ModuleCat.isModule.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 N)) (ModuleCat.Hom.hom.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 M N f) (ModuleCat.Hom.hom.{v, u} R inst._@.Mathlib.Algebra.Category.ModuleCat.Basic.1595704765._hygCtx._hyg.3 M N g))","typeFull":"∀ {R : Type u} [inst : Ring R] {M N : ModuleCat R} {f g : M ⟶ N}, (f = g) = (ModuleCat.Hom.hom f = ModuleCat.Hom.hom g)","typeReadable":"∀ {R : Type u} [inst : Ring R] {M N : ModuleCat R} {f g : M ⟶ N}, (f = g) = (ModuleCat.Hom.hom f = ModuleCat.Hom.hom g)","typeReferences":[["ModuleCat","isModule"],["ModuleCat","isAddCommGroup"],["ModuleCat","Hom","hom"],["ModuleCat","moduleCategory"],["LinearMap"],["CategoryTheory","Category","toCategoryStruct"],["Ring","toSemiring"],["CategoryTheory","CategoryStruct","toQuiver"],["Semiring","toNonAssocSemiring"],["Quiver","Hom"],["RingHom","id"],["AddCommGroup","toAddCommMonoid"],["Eq"],["Ring"],["ModuleCat"],["ModuleCat","carrier"]],"valueReferences":[["ModuleCat","isModule"],["ModuleCat","isAddCommGroup"],["ModuleCat","Hom","hom"],["ModuleCat","moduleCategory"],["ModuleCat","hom_ext_iff"],["LinearMap"],["CategoryTheory","Category","toCategoryStruct"],["Ring","toSemiring"],["CategoryTheory","CategoryStruct","toQuiver"],["Semiring","toNonAssocSemiring"],["Quiver","Hom"],["RingHom","id"],["AddCommGroup","toAddCommMonoid"],["Eq"],["propext"],["ModuleCat"],["ModuleCat","carrier"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_3"],"typeFallback":"forall {R : Type.{u_1}} {S : Type.{u_5}} {M : Type.{u_8}} {M₃ : Type.{u_11}} [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 : Semiring.{u_5} S] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 : AddCommMonoid.{u_8} M] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 : AddCommMonoid.{u_11} M₃] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 : Module.{u_1, u_8} R M inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21] [inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41 : Module.{u_5, u_11} S M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30] {σ : RingHom.{u_1, u_5} R S (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15) (Semiring.toNonAssocSemiring.{u_5} S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18)} {f : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41} {g : LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41}, Eq.{1} Prop (Eq.{max (succ u_11) (succ u_8)} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41) f g) (forall (x : M), Eq.{succ u_11} M₃ (DFunLike.coe.{max (succ u_11) (succ u_8), succ u_8, succ u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41 σ) f x) (DFunLike.coe.{max (succ u_11) (succ u_8), succ u_8, succ u_11} (LinearMap.{u_1, u_5, u_8, u_11} R S inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 σ M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => M₃) (LinearMap.instFunLike.{u_1, u_5, u_8, u_11} R S M M₃ inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.18 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.30 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.33 inst._@.Mathlib.Algebra.Module.LinearMap.Defs.722126854._hygCtx._hyg.41 σ) g x))","typeFull":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  {f g : M →ₛₗ[σ] M₃}, (f = g) = ∀ (x : M), f x = g x","typeReadable":"∀ {R : Type u_1} {S : Type u_5} {M : Type u_8} {M₃ : Type u_11} [inst : Semiring R] [inst_1 : Semiring S]\n  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₃] [inst_4 : Module R M] [inst_5 : Module S M₃] {σ : R →+* S}\n  {f g : M →ₛₗ[σ] M₃}, (f = g) = ∀ (x : M), f x = g x","typeReferences":[["RingHom"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["LinearMap","instFunLike"],["Module"],["Eq"],["LinearMap"],["DFunLike","coe"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["LinearMap","ext_iff"],["LinearMap"],["Eq"],["DFunLike","coe"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Category","ModuleCat","AB",0,"ModuleCat","isSeparator","_simp_1_1"],"typeFallback":"forall {C : Type.{u}} [inst._@.Mathlib.CategoryTheory.ObjectProperty.Basic.2740281896._hygCtx._hyg.4 : CategoryTheory.CategoryStruct.{v, u} C] (X : C) (Y : C), Eq.{1} Prop (CategoryTheory.ObjectProperty.singleton.{v, u} C inst._@.Mathlib.CategoryTheory.ObjectProperty.Basic.2740281896._hygCtx._hyg.4 X Y) (Eq.{succ u} C X Y)","typeFull":"∀ {C : Type u} [inst : CategoryTheory.CategoryStruct.{v, u} C] (X Y : C),\n  CategoryTheory.ObjectProperty.singleton X Y = (X = Y)","typeReadable":"∀ {C : Type u} [inst : CategoryTheory.CategoryStruct.{v, u} C] (X Y : C),\n  CategoryTheory.ObjectProperty.singleton X Y = (X = Y)","typeReferences":[["CategoryTheory","CategoryStruct"],["CategoryTheory","ObjectProperty","singleton"],["Eq"]],"valueReferences":[["CategoryTheory","ObjectProperty","singleton_iff"],["CategoryTheory","ObjectProperty","singleton"],["Eq"],["propext"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["SheafOfModules","instHasColimitsOfSizeOfPresheafOfModulesObjFunctorOppositeRingCatIsSheaf"],"typeFallback":"forall {C : Type.{u'}} [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3} (R : CategoryTheory.Sheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 J RingCat.{u} RingCat.instCategory.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.11 : CategoryTheory.HasWeakSheafify.{v', v, u', succ v} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 J AddCommGrpCat.{v} AddCommGrpCat.instCategory.{v}] [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.15 : CategoryTheory.GrothendieckTopology.WEqualsLocallyBijective.{v, v, v', succ v, u', v} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 J AddCommGrpCat.{v} AddCommGrpCat.instCategory.{v} (fun (x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6 : AddCommGrpCat.{v}) (x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6 : AddCommGrpCat.{v}) => AddMonoidHom.{v, v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGrpCat.str.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6)))))) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGrpCat.str.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6))))))) AddCommGrpCat.carrier.{v} (fun (X : AddCommGrpCat.{v}) (Y : AddCommGrpCat.{v}) => AddMonoidHom.instFunLike.{v, v} (AddCommGrpCat.carrier.{v} X) (AddCommGrpCat.carrier.{v} Y) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} X) ((fun (X : AddCommGrpCat.{v}) => AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} X) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} X) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} X) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} X) (AddCommGrpCat.str.{v} X))))) X)) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} Y) (AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} Y) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} Y) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} Y) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} Y) (AddCommGrpCat.str.{v} Y))))))) AddCommGrpCat.instConcreteCategoryAddMonoidHomCarrier.{v}] [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.22 : CategoryTheory.Limits.HasColimitsOfSize.{w', w, max u' v, max (max (max (succ v) u) u') v'} (PresheafOfModules.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R)) (PresheafOfModules.instCategory.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R))], CategoryTheory.Limits.HasColimitsOfSize.{w', w, max u' v, max (max (max (succ v) u) u') v'} (SheafOfModules.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 J R) (SheafOfModules.instCategory.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.485427707._hygCtx._hyg.3 J R)","typeFull":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  (R : CategoryTheory.Sheaf J RingCat) [CategoryTheory.HasWeakSheafify J AddCommGrpCat]\n  [J.WEqualsLocallyBijective AddCommGrpCat]\n  [CategoryTheory.Limits.HasColimitsOfSize.{w', w, max u' v, max (max (max (v + 1) u) u') v'}\n      (PresheafOfModules R.obj)],\n  CategoryTheory.Limits.HasColimitsOfSize.{w', w, max u' v, max (max (max (v + 1) u) u') v'} (SheafOfModules R)","typeReadable":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  (R : CategoryTheory.Sheaf J RingCat) [CategoryTheory.HasWeakSheafify J AddCommGrpCat]\n  [J.WEqualsLocallyBijective AddCommGrpCat]\n  [CategoryTheory.Limits.HasColimitsOfSize.{w', w, max u' v, max (max (max (v + 1) u) u') v'}\n      (PresheafOfModules R.obj)],\n  CategoryTheory.Limits.HasColimitsOfSize.{w', w, max u' v, max (max (max (v + 1) u) u') v'} (SheafOfModules R)","typeReferences":[["AddCommGroup","toAddGroup"],["Opposite"],["CategoryTheory","Category"],["CategoryTheory","Limits","HasColimitsOfSize"],["PresheafOfModules"],["PresheafOfModules","instCategory"],["AddMonoidHom"],["CategoryTheory","Presheaf","IsSheaf"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["AddCommGrpCat","carrier"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["SheafOfModules","instCategory"],["SheafOfModules"],["AddCommGrpCat","instCategory"],["CategoryTheory","Functor"],["CategoryTheory","Category","opposite"],["CategoryTheory","HasWeakSheafify"],["CategoryTheory","GrothendieckTopology"],["CategoryTheory","GrothendieckTopology","WEqualsLocallyBijective"],["AddZeroClass","toAddZero"],["CategoryTheory","ObjectProperty","FullSubcategory","obj"],["AddCommGrpCat"],["RingCat"],["AddCommGrpCat","str"],["SubNegMonoid","toAddMonoid"],["RingCat","instCategory"],["CategoryTheory","Functor","category"],["CategoryTheory","Sheaf"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["SheafOfModules","instCategory"],["SheafOfModules"],["CategoryTheory","Functor"],["CategoryTheory","Category","opposite"],["Opposite"],["CategoryTheory","Category"],["CategoryTheory","Limits","HasColimitsOfShape"],["CategoryTheory","Limits","hasColimitsOfShapeOfHasColimitsOfSize"],["PresheafOfModules"],["CategoryTheory","ObjectProperty","FullSubcategory","obj"],["PresheafOfModules","instCategory"],["SheafOfModules","instHasColimitsOfShapeOfPresheafOfModulesObjFunctorOppositeRingCatIsSheaf"],["CategoryTheory","Limits","HasColimitsOfSize","mk"],["RingCat"],["CategoryTheory","Presheaf","IsSheaf"],["inferInstance"],["RingCat","instCategory"],["CategoryTheory","Functor","category"]]},{"isProp":true,"kind":"theorem","name":["SheafOfModules","instHasColimitsOfShapeOfPresheafOfModulesObjFunctorOppositeRingCatIsSheaf"],"typeFallback":"forall {C : Type.{u'}} [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3} (R : CategoryTheory.Sheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 J RingCat.{u} RingCat.instCategory.{u}) [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.11 : CategoryTheory.HasWeakSheafify.{v', v, u', succ v} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 J AddCommGrpCat.{v} AddCommGrpCat.instCategory.{v}] [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.15 : CategoryTheory.GrothendieckTopology.WEqualsLocallyBijective.{v, v, v', succ v, u', v} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 J AddCommGrpCat.{v} AddCommGrpCat.instCategory.{v} (fun (x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6 : AddCommGrpCat.{v}) (x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6 : AddCommGrpCat.{v}) => AddMonoidHom.{v, v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGrpCat.str.{v} x1._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6)))))) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6) (AddCommGrpCat.str.{v} x2._@.Mathlib.Algebra.Category.Grp.Basic.4010222601._hygCtx._hyg.6))))))) AddCommGrpCat.carrier.{v} (fun (X : AddCommGrpCat.{v}) (Y : AddCommGrpCat.{v}) => AddMonoidHom.instFunLike.{v, v} (AddCommGrpCat.carrier.{v} X) (AddCommGrpCat.carrier.{v} Y) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} X) ((fun (X : AddCommGrpCat.{v}) => AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} X) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} X) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} X) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} X) (AddCommGrpCat.str.{v} X))))) X)) (AddZeroClass.toAddZero.{v} (AddCommGrpCat.carrier.{v} Y) (AddMonoid.toAddZeroClass.{v} (AddCommGrpCat.carrier.{v} Y) (SubNegMonoid.toAddMonoid.{v} (AddCommGrpCat.carrier.{v} Y) (AddGroup.toSubNegMonoid.{v} (AddCommGrpCat.carrier.{v} Y) (AddCommGroup.toAddGroup.{v} (AddCommGrpCat.carrier.{v} Y) (AddCommGrpCat.str.{v} Y))))))) AddCommGrpCat.instConcreteCategoryAddMonoidHomCarrier.{v}] (K : Type.{w}) [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.19 : CategoryTheory.Category.{w', w} K] [inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.22 : CategoryTheory.Limits.HasColimitsOfShape.{w', w, max u' v, max (max (max (succ v) u) u') v'} K inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.19 (PresheafOfModules.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R)) (PresheafOfModules.instCategory.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 (CategoryTheory.ObjectProperty.FullSubcategory.obj.{max u u', max (max (succ u) u') v'} (CategoryTheory.Functor.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Functor.category.{v', u, u', succ u} (Opposite.{succ u'} C) (CategoryTheory.Category.opposite.{v', u'} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3) RingCat.{u} RingCat.instCategory.{u}) (CategoryTheory.Presheaf.IsSheaf.{v', u, u', succ u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 RingCat.{u} RingCat.instCategory.{u} J) R))], CategoryTheory.Limits.HasColimitsOfShape.{w', w, max u' v, max (max (max (succ v) u) u') v'} K inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.19 (SheafOfModules.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 J R) (SheafOfModules.instCategory.{v, v', u', u} C inst._@.Mathlib.Algebra.Category.ModuleCat.Sheaf.Colimits.4116382175._hygCtx._hyg.3 J R)","typeFull":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  (R : CategoryTheory.Sheaf J RingCat) [CategoryTheory.HasWeakSheafify J AddCommGrpCat]\n  [J.WEqualsLocallyBijective AddCommGrpCat] (K : Type w) [inst_3 : CategoryTheory.Category.{w', w} K]\n  [CategoryTheory.Limits.HasColimitsOfShape K (PresheafOfModules R.obj)],\n  CategoryTheory.Limits.HasColimitsOfShape K (SheafOfModules R)","typeReadable":"∀ {C : Type u'} [inst : CategoryTheory.Category.{v', u'} C] {J : CategoryTheory.GrothendieckTopology C}\n  (R : CategoryTheory.Sheaf J RingCat) [CategoryTheory.HasWeakSheafify J AddCommGrpCat]\n  [J.WEqualsLocallyBijective AddCommGrpCat] (K : Type w) [inst_3 : CategoryTheory.Category.{w', w} K]\n  [CategoryTheory.Limits.HasColimitsOfShape K (PresheafOfModules R.obj)],\n  CategoryTheory.Limits.HasColimitsOfShape K (SheafOfModules R)","typeReferences":[["AddCommGroup","toAddGroup"],["Opposite"],["CategoryTheory","Category"],["CategoryTheory","Limits","HasColimitsOfShape"],["PresheafOfModules"],["PresheafOfModules","instCategory"],["AddMonoidHom"],["CategoryTheory","Presheaf","IsSheaf"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["AddCommGrpCat","carrier"],["AddCommGrpCat","instConcreteCategoryAddMonoidHomCarrier"],["SheafOfModules","instCategory"],["SheafOfModules"],["AddCommGrpCat","instCategory"],["CategoryTheory","Functor"],["CategoryTheory","Category","opposite"],["CategoryTheory","HasWeakSheafify"],["CategoryTheory","GrothendieckTopology"],["CategoryTheory","GrothendieckTopology","WEqualsLocallyBijective"],["AddZeroClass","toAddZero"],["CategoryTheory","ObjectProperty","FullSubcategory","obj"],["AddCommGrpCat"],["RingCat"],["AddCommGrpCat","str"],["SubNegMonoid","toAddMonoid"],["RingCat","instCategory"],["CategoryTheory","Functor","category"],["CategoryTheory","Sheaf"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["CategoryTheory","Adjunction","counit"],["RingHom"],["CategoryTheory","Functor","id"],["PresheafOfModules","instIsLeftAdjointSheafOfModulesSheafification"],["Opposite"],["RingHom","instFunLike"],["CategoryTheory","Presheaf","isLocallySurjective_of_iso"],["CategoryTheory","Functor","comp"],["PresheafOfModules"],["CategoryTheory","asIso"],["PresheafOfModules","instCategory"],["PresheafOfModules","restrictScalars"],["CategoryTheory","CategoryStruct","id"],["RingCat","carrier"],["CategoryTheory","IsIso","id"],["Semiring","toNonAssocSemiring"],["CategoryTheory","Presheaf","IsSheaf"],["CategoryTheory","Limits","PreservesColimitsOfShape","preservesColimit"],["SheafOfModules","instCategory"],["CategoryTheory","Category","opposite"],["CategoryTheory","Functor"],["SheafOfModules"],["CategoryTheory","Iso","symm"],["PresheafOfModules","instIsIsoFunctorSheafOfModulesCounitSheafificationAdjunction"],["CategoryTheory","Functor","instPreservesColimitsOfShapeOfIsLeftAdjoint"],["CategoryTheory","ObjectProperty","FullSubcategory","obj"],["CategoryTheory","Functor","isoWhiskerLeft"],["CategoryTheory","Category","toCategoryStruct"],["PresheafOfModules","sheafification"],["Ring","toSemiring"],["CategoryTheory","Limits","HasColimitsOfShape","mk"],["CategoryTheory","Limits","hasColimit_of_iso"],["RingCat"],["PresheafOfModules","sheafificationAdjunction"],["SheafOfModules","forget"],["CategoryTheory","Presheaf","instIsLocallyInjectiveOfIsIsoFunctorOpposite"],["CategoryTheory","Limits","instHasColimitCompOfPreservesColimit"],["CategoryTheory","Limits","hasColimitOfHasColimitsOfShape"],["RingCat","instCategory"],["RingCat","instConcreteCategoryRingHomCarrier"],["RingCat","ring"],["CategoryTheory","Functor","category"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.CharP.LocalRing.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["charP_zero_or_prime_power"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.CharP.LocalRing.2419030685._hygCtx._hyg.3 : CommRing.{u_1} R] [inst._@.Mathlib.Algebra.CharP.LocalRing.2419030685._hygCtx._hyg.6 : IsLocalRing.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.LocalRing.2419030685._hygCtx._hyg.3))] (q : Nat) [char_R_q : CharP.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.LocalRing.2419030685._hygCtx._hyg.3))) q], Or (Eq.{1} Nat q (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (IsPrimePow.{0} Nat Nat.instCommMonoidWithZero q)","typeFull":"∀ (R : Type u_1) [inst : CommRing R] [IsLocalRing R] (q : ℕ) [char_R_q : CharP R q], q = 0 ∨ IsPrimePow q","typeReadable":"∀ (R : Type u_1) [inst : CommRing R] [IsLocalRing R] (q : ℕ) [char_R_q : CharP R q], q = 0 ∨ IsPrimePow q","typeReferences":[["CommRing","toCommSemiring"],["Nat","instCommMonoidWithZero"],["CharP"],["CommSemiring","toSemiring"],["AddGroupWithOne","toAddMonoidWithOne"],["IsLocalRing"],["CommRing"],["OfNat","ofNat"],["CommRing","toRing"],["Nat"],["Ring","toAddGroupWithOne"],["Or"],["instOfNatNat"],["IsPrimePow"],["Eq"]],"valueReferences":[["RingHom"],["Nat","instCommMonoidWithZero"],["Eq","trans"],["MulZeroClass","toMul"],["Exists","intro"],["AddGroupWithOne","toAddMonoidWithOne"],["IsLocalRing","maximalIdeal"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["NonUnitalCommSemiring","toNonUnitalNonAssocCommSemiring"],["mul_comm"],["Eq","symm"],["Nat","ordProj_dvd"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Units"],["Nat","mul_div_cancel'"],["pos_iff_ne_zero"],["instLTNat"],["Finsupp","instFunLike"],["Exists"],["Nat","dvd_antisymm"],["Dvd","dvd"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["RingHom","charZero"],["Nat","instCommSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["ringChar","charP"],["MulZeroClass","mul_zero"],["DivisionSemiring","toSemiring"],["Ideal","instHasQuotient"],["Ring","toSemiring"],["Set","instMembership"],["Units","val"],["Eq","refl"],["Classical","byContradiction"],["Eq","mpr"],["pow_zero"],["one_mul"],["AddMonoid","toAddZeroClass"],["IsLocalRing","residue"],["MulOneClass","toMulOne"],["IsDomain","to_noZeroDivisors"],["HasQuotient","Quotient"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["Nat","instDiv"],["Submodule","Quotient","instZeroQuotient"],["CharP","char_ne_one"],["MulZeroOneClass","toMulOneClass"],["instHDiv"],["Ideal","Quotient","ring"],["Semigroup","toMul"],["MulOne","toMul"],["instOfNatNat"],["Ideal","Quotient","mk"],["Eq"],["propext"],["Nat","instAddMonoid"],["instIsDomain"],["Set"],["DivisionRing","toRing"],["CharP","charP_to_charZero"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Field","toDivisionRing"],["IsLocalRing","toNontrivial"],["CharP","eq"],["map_natCast"],["CharP","ofCharZero"],["CharP","char_is_prime_or_zero"],["HPow","hPow"],["OfNat","ofNat"],["CommRing","toRing"],["MulZeroClass","toZero"],["mem_nonunits_iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Nat","not_dvd_ordCompl"],["Ne"],["IsUnit","unit"],["PartialOrder","toPreorder"],["CommMonoidWithZero","toMonoidWithZero"],["Membership","mem"],["ringChar","spec"],["Preorder","toLT"],["Iff","mp"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["CharP","cast_eq_zero_iff"],["AddMonoidWithOne","toAddMonoid"],["RingHom","instRingHomClass"],["HDiv","hDiv"],["And","intro"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["Or"],["Ring","toAddGroupWithOne"],["ringChar"],["Ideal","instIsTwoSided_1"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Nat","cast_mul"],["Semifield","toDivisionSemiring"],["Nat","instMulZeroClass"],["Semiring","toModule"],["Nat","Prime"],["NonAssocSemiring","toMulZeroOneClass"],["instHPow"],["Ideal","Quotient","semiring"],["SetLike","instMembership"],["CommMagma","toMul"],["NonUnitalNonAssocSemiring","toDistrib"],["MulOne","toOne"],["Nat","instPartialOrder"],["outParam"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["NonUnitalCommRing","toNonUnitalCommSemiring"],["AddZeroClass","toAddZero"],["Nat","instCanonicallyOrderedAdd"],["absurd"],["Classical","or_iff_not_imp_left"],["Nat"],["Nat","instMonoid"],["AddMonoidWithOne","toNatCast"],["Iff","mpr"],["id"],["instHMul"],["AddZero","toZero"],["Field","instIsLocalRing"],["Nat","instDvd"],["Nat","cast"],["Prime"],["Eq","mp"],["Units","instInv"],["nonunits"],["RingHom","instFunLike"],["mul_assoc"],["Ideal","Quotient","eq_zero_iff_mem"],["CommRing","toNonUnitalCommRing"],["Nat","factorization"],["DFunLike","coe"],["congrArg"],["CommSemiring","toNonUnitalCommSemiring"],["Ideal"],["Monoid","toMulOneClass"],["MonoidWithZero","toMonoid"],["instMulNat"],["IsPrimePow"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Nat","Prime","prime"],["Not"],["Inv","inv"],["CommRing","toCommSemiring"],["Distrib","toMul"],["CommSemiring","toSemiring"],["IsLocalRing","ResidueField"],["Semiring","toMonoidWithZero"],["IsUnit"],["LT","lt"],["Or","casesOn"],["Ring","toAddCommGroup"],["One","toOfNat1"],["Submodule","setLike"],["CharP","cast_eq_zero"],["IsLocalRing","mem_maximalIdeal"],["Field","toSemifield"],["False"],["SemigroupWithZero","toSemigroup"],["IsUnit","val_inv_mul"],["IsLocalRing","ResidueField","field"],["Finsupp"],["ringChar","of_eq"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.CharP.Quotient.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["CharP","quotient"],"typeFallback":"forall (R : Type.{u_1}) [inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3 : CommRing.{u_1} R] (p : Nat) [hp1 : Fact (Nat.Prime p)], (Membership.mem.{u_1, u_1} R (Set.{u_1} R) (Set.instMembership.{u_1} R) (nonunits.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3))))) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)))) p)) -> (CharP.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3) (Ideal.span.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)) (Singleton.singleton.{u_1, u_1} R (Set.{u_1} R) (Set.instSingletonSet.{u_1} R) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)))) p)))) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3) (Ideal.span.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)) (Singleton.singleton.{u_1, u_1} R (Set.{u_1} R) (Set.instSingletonSet.{u_1} R) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)))) p)))) (Ring.toAddGroupWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3) (Ideal.span.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)) (Singleton.singleton.{u_1, u_1} R (Set.{u_1} R) (Set.instSingletonSet.{u_1} R) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)))) p)))) (Ideal.Quotient.instRingQuotient.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3 (Ideal.span.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)) (Singleton.singleton.{u_1, u_1} R (Set.{u_1} R) (Set.instSingletonSet.{u_1} R) (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2403509443._hygCtx._hyg.3)))) p)))))) p)","typeFull":"∀ (R : Type u_1) [inst : CommRing R] (p : ℕ) [hp1 : Fact (Nat.Prime p)], ↑p ∈ nonunits R → CharP (R ⧸ Ideal.span {↑p}) p","typeReadable":"∀ (R : Type u_1) [inst : CommRing R] (p : ℕ) [hp1 : Fact (Nat.Prime p)], ↑p ∈ nonunits R → CharP (R ⧸ Ideal.span {↑p}) p","typeReferences":[["Ideal","span"],["Nat","cast"],["HasQuotient","Quotient"],["Singleton","singleton"],["Membership","mem"],["nonunits"],["AddGroupWithOne","toAddMonoidWithOne"],["Ideal","instHasQuotient_1"],["Ring","toAddGroupWithOne"],["Ideal"],["MonoidWithZero","toMonoid"],["Ideal","Quotient","instRingQuotient"],["Nat","Prime"],["CommRing","toCommSemiring"],["Fact"],["Set"],["CharP"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["CommRing"],["Set","instSingletonSet"],["Set","instMembership"],["CommRing","toRing"],["Nat"],["AddMonoidWithOne","toNatCast"]],"valueReferences":[["RingHom"],["semigroupDvd"],["Subsingleton","elim"],["Singleton","singleton"],["Membership","mem"],["Iff","mp"],["NonUnitalSemiring","toSemigroupWithZero"],["AddGroupWithOne","toAddMonoidWithOne"],["Ideal","subset_span"],["RingHom","instRingHomClass"],["Ideal","instHasQuotient_1"],["Semiring","toNonAssocSemiring"],["Or"],["Ring","toAddGroupWithOne"],["Ideal","instIsTwoSided_1"],["ringChar"],["CharP","CharOne","subsingleton"],["Monoid","toSemigroup"],["Eq","rec"],["Semiring","toModule"],["Nat","Prime"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Ideal","Quotient","semiring"],["SetLike","instMembership"],["MulOne","toOne"],["Dvd","dvd"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Or","resolve_left"],["Ideal","instHasQuotient"],["Ring","toSemiring"],["isUnit_iff_dvd_one"],["Nat","dvd_prime"],["Nat"],["AddMonoidWithOne","toNatCast"],["Iff","mpr"],["Ideal","mem_span_singleton"],["Nat","instDvd"],["MulOneClass","toMulOne"],["Ideal","span"],["Nat","cast"],["CommMonoid","toMonoid"],["HasQuotient","Quotient"],["Submodule","Quotient","instZeroQuotient"],["RingHom","instFunLike"],["Ideal","Quotient","eq_zero_iff_mem"],["Ideal","Quotient","ring"],["DFunLike","coe"],["ringChar","dvd"],["CommSemiring","toNonUnitalCommSemiring"],["Ideal"],["Ideal","Quotient","mk"],["instOfNatNat"],["Monoid","toMulOneClass"],["Fact","out"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Set","mem_singleton"],["CommRing","toCommSemiring"],["Set"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["IsUnit"],["map_natCast"],["Set","instSingletonSet"],["OfNat","ofNat"],["CommRing","toCommMonoid"],["Ring","toAddCommGroup"],["CommRing","toRing"],["One","toOfNat1"],["Submodule","setLike"],["SemigroupWithZero","toSemigroup"],["ringChar","of_eq"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","CharP","Quotient",0,"CharP","ker_intAlgebraMap_eq_span","_simp_1_1"],"typeFallback":"forall {α : Type.{u}} [inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3 : CommSemiring.{u} α] {x : α} {y : α}, Eq.{1} Prop (Membership.mem.{u, u} α (Ideal.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3)) (SetLike.instMembership.{u, u} (Ideal.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3)) α (Submodule.setLike.{u, u} α α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} α (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} α (Semiring.toNonAssocSemiring.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3)))) (Semiring.toModule.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3)))) (Ideal.span.{u} α (CommSemiring.toSemiring.{u} α inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3) (Singleton.singleton.{u, u} α (Set.{u} α) (Set.instSingletonSet.{u} α) y)) x) (Dvd.dvd.{u} α (semigroupDvd.{u} α (SemigroupWithZero.toSemigroup.{u} α (NonUnitalSemiring.toSemigroupWithZero.{u} α (NonUnitalCommSemiring.toNonUnitalSemiring.{u} α (CommSemiring.toNonUnitalCommSemiring.{u} α inst._@.Mathlib.RingTheory.Ideal.Span.2364764113._hygCtx._hyg.3))))) y x)","typeFull":"∀ {α : Type u} [inst : CommSemiring α] {x y : α}, (x ∈ Ideal.span {y}) = (y ∣ x)","typeReadable":"∀ {α : Type u} [inst : CommSemiring α] {x y : α}, (x ∈ Ideal.span {y}) = (y ∣ x)","typeReferences":[["semigroupDvd"],["Ideal","span"],["SetLike","instMembership"],["Dvd","dvd"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["Set"],["Singleton","singleton"],["CommSemiring"],["Membership","mem"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalSemiring","toSemigroupWithZero"],["Set","instSingletonSet"],["Semiring","toNonAssocSemiring"],["CommSemiring","toNonUnitalCommSemiring"],["Ideal"],["Submodule","setLike"],["SemigroupWithZero","toSemigroup"],["Eq"],["Semiring","toModule"]],"valueReferences":[["semigroupDvd"],["Ideal","span"],["SetLike","instMembership"],["Dvd","dvd"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["Set"],["Singleton","singleton"],["Membership","mem"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalSemiring","toSemigroupWithZero"],["Set","instSingletonSet"],["Semiring","toNonAssocSemiring"],["CommSemiring","toNonUnitalCommSemiring"],["Ideal"],["Submodule","setLike"],["SemigroupWithZero","toSemigroup"],["Semiring","toModule"],["Ideal","mem_span_singleton"],["propext"]]},{"isProp":true,"kind":"theorem","name":["CharP","quotient_iff_le_ker_natCast"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3 : CommRing.{u_1} R] (n : Nat) [inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.9 : CharP.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3))) n] (I : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3))), Iff (CharP.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3) I) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3) I) (Ring.toAddGroupWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3) I) (Ideal.Quotient.instRingQuotient.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3 I))) n) (LE.le.{0} (Ideal.{0} Nat Nat.instSemiring) (Preorder.toLE.{0} (Ideal.{0} Nat Nat.instSemiring) (PartialOrder.toPreorder.{0} (Ideal.{0} Nat Nat.instSemiring) (Submodule.instPartialOrder.{0, 0} Nat Nat Nat.instSemiring (NonUnitalNonAssocSemiring.toAddCommMonoid.{0} Nat (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Nat (Semiring.toNonAssocSemiring.{0} Nat Nat.instSemiring))) (Semiring.toModule.{0} Nat Nat.instSemiring)))) (Ideal.comap.{0, u_1, u_1} Nat R (RingHom.{0, u_1} Nat R Nat.instNonAssocSemiring (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)))) Nat.instSemiring (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)) (RingHom.instFunLike.{0, u_1} Nat R Nat.instNonAssocSemiring (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)))) (Nat.castRingHom.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)))) (RingHom.instRingHomClass.{0, u_1} Nat R Nat.instNonAssocSemiring (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)))) I) (RingHom.ker.{0, u_1, u_1} Nat R (RingHom.{0, u_1} Nat R Nat.instNonAssocSemiring (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)))) Nat.instSemiring (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)) (RingHom.instFunLike.{0, u_1} Nat R Nat.instNonAssocSemiring (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)))) (RingHom.instRingHomClass.{0, u_1} Nat R Nat.instNonAssocSemiring (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3)))) (Nat.castRingHom.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2840209580._hygCtx._hyg.3))))))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (n : ℕ) [CharP R n] (I : Ideal R),\n  CharP (R ⧸ I) n ↔ Ideal.comap (Nat.castRingHom R) I ≤ RingHom.ker (Nat.castRingHom R)","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (n : ℕ) [CharP R n] (I : Ideal R),\n  CharP (R ⧸ I) n ↔ Ideal.comap (Nat.castRingHom R) I ≤ RingHom.ker (Nat.castRingHom R)","typeReferences":[["RingHom"],["PartialOrder","toPreorder"],["HasQuotient","Quotient"],["RingHom","instFunLike"],["AddGroupWithOne","toAddMonoidWithOne"],["RingHom","instRingHomClass"],["Ideal","instHasQuotient_1"],["RingHom","ker"],["Nat","castRingHom"],["Semiring","toNonAssocSemiring"],["Ideal"],["Ring","toAddGroupWithOne"],["Nat","instSemiring"],["Nat","instNonAssocSemiring"],["Ideal","comap"],["Ideal","Quotient","instRingQuotient"],["Preorder","toLE"],["Semiring","toModule"],["CommRing","toCommSemiring"],["CharP"],["CommSemiring","toSemiring"],["Submodule","instPartialOrder"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommRing"],["CommRing","toRing"],["Nat"],["Iff"],["LE","le"]],"valueReferences":[["RingHom"],["PartialOrder","toPreorder"],["Membership","mem"],["AddGroupWithOne","toAddMonoidWithOne"],["RingHom","instRingHomClass"],["Ideal","instHasQuotient_1"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Nat","castRingHom"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["Ideal","comap"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Semiring","toModule"],["SetLike","instMembership"],["RingHom","ker_eq_comap_bot"],["CharP"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Bot","bot"],["CharP","quotient_iff"],["Nat"],["AddMonoidWithOne","toNatCast"],["Iff"],["Submodule","instBot"],["id"],["Eq","mpr"],["Nat","cast"],["HasQuotient","Quotient"],["RingHom","instFunLike"],["CommRing","toNonUnitalCommRing"],["RingHom","ker"],["congrArg"],["Nat","instNonAssocSemiring"],["Nat","instSemiring"],["Ideal"],["Ideal","Quotient","instRingQuotient"],["Zero","toOfNat0"],["Preorder","toLE"],["Eq"],["propext"],["CommRing","toCommSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Submodule","instPartialOrder"],["CommSemiring","toSemiring"],["Iff","rfl"],["OfNat","ofNat"],["CommRing","toRing"],["Submodule","setLike"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["CharP","quotient_iff"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3 : CommRing.{u_1} R] (n : Nat) [inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.9 : CharP.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))) n] (I : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))), Iff (CharP.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3) I) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3) I) (Ring.toAddGroupWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3) I) (Ideal.Quotient.instRingQuotient.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3 I))) n) (forall (x : Nat), (Membership.mem.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))) (SetLike.instMembership.{u_1, u_1} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))) R (Submodule.setLike.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))))) I (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3)))) x)) -> (Eq.{succ u_1} R (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3)))) x) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R (CommRing.toNonUnitalCommRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.1139478891._hygCtx._hyg.3))))))))))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (n : ℕ) [CharP R n] (I : Ideal R), CharP (R ⧸ I) n ↔ ∀ (x : ℕ), ↑x ∈ I → ↑x = 0","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (n : ℕ) [CharP R n] (I : Ideal R), CharP (R ⧸ I) n ↔ ∀ (x : ℕ), ↑x ∈ I → ↑x = 0","typeReferences":[["Nat","cast"],["HasQuotient","Quotient"],["Membership","mem"],["AddGroupWithOne","toAddMonoidWithOne"],["CommRing","toNonUnitalCommRing"],["Ideal","instHasQuotient_1"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ideal"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Ideal","Quotient","instRingQuotient"],["Zero","toOfNat0"],["Semiring","toModule"],["Eq"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CharP"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommRing"],["OfNat","ofNat"],["CommRing","toRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["MulZeroClass","toZero"],["Submodule","setLike"],["Iff"],["NonUnitalNonAssocSemiring","toMulZeroClass"]],"valueReferences":[["Submodule","Quotient","mk"],["Membership","mem"],["AddGroupWithOne","toAddMonoidWithOne"],["CharP","cast_eq_zero_iff"],["AddMonoidWithOne","toAddMonoid"],["Ideal","instHasQuotient_1"],["CharP","quotient'"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Eq","symm"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SetLike","instMembership"],["Dvd","dvd"],["CharP"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["outParam"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddZeroClass","toAddZero"],["Submodule","Quotient","mk_eq_zero"],["Ring","toSemiring"],["Nat"],["AddMonoidWithOne","toNatCast"],["Iff","mpr"],["id"],["AddCommGroup","toAddCommMonoid"],["Eq","mpr"],["AddZero","toZero"],["Nat","instDvd"],["AddMonoid","toAddZeroClass"],["Submodule","hasQuotient"],["Nat","cast"],["HasQuotient","Quotient"],["Submodule","Quotient","instZeroQuotient"],["CommRing","toNonUnitalCommRing"],["Iff","intro"],["congrArg"],["Submodule"],["Ideal"],["Zero","toOfNat0"],["Ideal","Quotient","instRingQuotient"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["propext"],["CommRing","toCommSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["OfNat","ofNat"],["Ring","toAddCommGroup"],["CommRing","toRing"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"]]},{"isProp":true,"kind":"theorem","name":["CharP","quotient'"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3 : CommRing.{u_1} R] (p : Nat) [inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.9 : CharP.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))) p] (I : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))), (forall (x : Nat), (Membership.mem.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))) (SetLike.instMembership.{u_1, u_1} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))) R (Submodule.setLike.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))))) I (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3)))) x)) -> (Eq.{succ u_1} R (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3)))) x) (OfNat.ofNat.{u_1} R 0 (Zero.toOfNat0.{u_1} R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u_1} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u_1} R (CommRing.toNonUnitalCommRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3)))))))))) -> (CharP.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3) I) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3) I) (Ring.toAddGroupWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3) I) (Ideal.Quotient.instRingQuotient.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2304705457._hygCtx._hyg.3 I))) p)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (p : ℕ) [CharP R p] (I : Ideal R), (∀ (x : ℕ), ↑x ∈ I → ↑x = 0) → CharP (R ⧸ I) p","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (p : ℕ) [CharP R p] (I : Ideal R), (∀ (x : ℕ), ↑x ∈ I → ↑x = 0) → CharP (R ⧸ I) p","typeReferences":[["Nat","cast"],["HasQuotient","Quotient"],["Membership","mem"],["AddGroupWithOne","toAddMonoidWithOne"],["CommRing","toNonUnitalCommRing"],["Ideal","instHasQuotient_1"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ideal"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Ideal","Quotient","instRingQuotient"],["Zero","toOfNat0"],["Semiring","toModule"],["Eq"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["CharP"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["CommRing"],["OfNat","ofNat"],["CommRing","toRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["Submodule","setLike"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"]],"valueReferences":[["RingHom"],["SubtractionMonoid","toSubNegZeroMonoid"],["CharP","mk"],["Membership","mem"],["AddGroupWithOne","toAddMonoidWithOne"],["CharP","cast_eq_zero_iff"],["sub_zero"],["AddMonoidWithOne","toAddMonoid"],["RingHom","instRingHomClass"],["Ideal","instHasQuotient_1"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["Ring","toAddGroupWithOne"],["Ideal","instIsTwoSided_1"],["SubNegMonoid","toSub"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["Eq","symm"],["AddGroup","toSubNegMonoid"],["Eq","rec"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["SetLike","instMembership"],["Dvd","dvd"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["outParam"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddZeroClass","toAddZero"],["Ideal","instHasQuotient"],["Ring","toSemiring"],["Nat"],["AddMonoidWithOne","toNatCast"],["Iff"],["Ideal","Quotient","eq"],["id"],["NegZeroClass","toZero"],["Eq","mpr"],["AddZero","toZero"],["Nat","instDvd"],["AddMonoid","toAddZeroClass"],["Nat","cast"],["HasQuotient","Quotient"],["RingHom","instFunLike"],["SubNegZeroMonoid","toSubNegMonoid"],["SubtractionCommMonoid","toSubtractionMonoid"],["CommRing","toNonUnitalCommRing"],["DFunLike","coe"],["Ideal","Quotient","ring"],["SubNegZeroMonoid","toNegZeroClass"],["Iff","intro"],["congrArg"],["Ideal"],["Ideal","Quotient","mk"],["Ideal","zero_mem"],["Zero","toOfNat0"],["Ideal","Quotient","instRingQuotient"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["propext"],["Iff","trans"],["CommRing","toCommSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["map_natCast"],["OfNat","ofNat"],["Ring","toAddCommGroup"],["CommRing","toRing"],["AddGroupWithOne","toAddGroup"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["Submodule","setLike"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHSub"]]},{"isProp":true,"kind":"theorem","name":["Ideal","natCast_mem_of_charP_quotient"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3 : CommRing.{u_1} R] (p : Nat) (I : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))) [inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.11 : CharP.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3) I) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3) I) (Ring.toAddGroupWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3) I) (Ideal.Quotient.instRingQuotient.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3 I))) p], Membership.mem.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))) (SetLike.instMembership.{u_1, u_1} (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))) R (Submodule.setLike.{u_1, u_1} R R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))))) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3))))) I (Nat.cast.{u_1} R (AddMonoidWithOne.toNatCast.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3132357513._hygCtx._hyg.3)))) p)","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (p : ℕ) (I : Ideal R) [CharP (R ⧸ I) p], ↑p ∈ I","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (p : ℕ) (I : Ideal R) [CharP (R ⧸ I) p], ↑p ∈ I","typeReferences":[["CommRing","toCommSemiring"],["Nat","cast"],["SetLike","instMembership"],["HasQuotient","Quotient"],["CharP"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["CommSemiring","toSemiring"],["Membership","mem"],["AddGroupWithOne","toAddMonoidWithOne"],["CommRing"],["Ideal","instHasQuotient_1"],["CommRing","toRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["Semiring","toNonAssocSemiring"],["Ideal"],["Ring","toAddGroupWithOne"],["Submodule","setLike"],["Ideal","Quotient","instRingQuotient"],["Semiring","toModule"]],"valueReferences":[["RingHom"],["Nat","cast"],["Eq","trans"],["HasQuotient","Quotient"],["Membership","mem"],["Iff","mp"],["Submodule","Quotient","instZeroQuotient"],["RingHom","instFunLike"],["AddGroupWithOne","toAddMonoidWithOne"],["Ideal","Quotient","eq_zero_iff_mem"],["RingHom","instRingHomClass"],["AddMonoidWithOne","toAddMonoid"],["Ideal","Quotient","ring"],["DFunLike","coe"],["congrArg"],["Semiring","toNonAssocSemiring"],["Ideal"],["Ring","toAddGroupWithOne"],["Ideal","instIsTwoSided_1"],["Ideal","Quotient","mk"],["congrFun'"],["Ideal","Quotient","instRingQuotient"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Ideal","Quotient","semiring"],["CommRing","toCommSemiring"],["SetLike","instMembership"],["True"],["CommSemiring","toSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["map_natCast"],["AddZeroClass","toAddZero"],["Ideal","instHasQuotient"],["OfNat","ofNat"],["Ring","toSemiring"],["Ring","toAddCommGroup"],["eq_self"],["CommRing","toRing"],["AddMonoidWithOne","toNatCast"],["of_eq_true"],["Submodule","setLike"],["CharP","cast_eq_zero"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Ideal","Quotient","index_eq_zero"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3 : CommRing.{u_1} R] (I : Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))), Eq.{succ u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) I) (Nat.cast.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) I) (AddMonoidWithOne.toNatCast.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) I) (AddGroupWithOne.toAddMonoidWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) I) (Ring.toAddGroupWithOne.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) I) (Ideal.Quotient.instRingQuotient.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3 I)))) (AddSubgroup.index.{u_1} R (AddCommGroup.toAddGroup.{u_1} R (Ring.toAddCommGroup.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Submodule.toAddSubgroup.{u_1, u_1} R R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) (Ring.toAddCommGroup.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) I))) (OfNat.ofNat.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) I) 0 (Zero.toOfNat0.{u_1} (HasQuotient.Quotient.{u_1, u_1} R (Ideal.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) (Ideal.instHasQuotient_1.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) I) (Submodule.Quotient.instZeroQuotient.{u_1, u_1} R R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3) (Ring.toAddCommGroup.{u_1} R (CommRing.toRing.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3)) (Semiring.toModule.{u_1} R (CommSemiring.toSemiring.{u_1} R (CommRing.toCommSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.2656154848._hygCtx._hyg.3))) I)))","typeFull":"∀ {R : Type u_1} [inst : CommRing R] (I : Ideal R), ↑(Submodule.toAddSubgroup I).index = 0","typeReadable":"∀ {R : Type u_1} [inst : CommRing R] (I : Ideal R), ↑(Submodule.toAddSubgroup I).index = 0","typeReferences":[["CommRing","toCommSemiring"],["Nat","cast"],["HasQuotient","Quotient"],["AddCommGroup","toAddGroup"],["CommSemiring","toSemiring"],["Submodule","Quotient","instZeroQuotient"],["AddGroupWithOne","toAddMonoidWithOne"],["Submodule","toAddSubgroup"],["CommRing"],["OfNat","ofNat"],["Ideal","instHasQuotient_1"],["Ring","toAddCommGroup"],["CommRing","toRing"],["AddMonoidWithOne","toNatCast"],["Ideal"],["Ring","toAddGroupWithOne"],["Zero","toOfNat0"],["Ideal","Quotient","instRingQuotient"],["AddSubgroup","index"],["Semiring","toModule"],["Eq"]],"valueReferences":[["Nat","cast"],["Eq","trans"],["HasQuotient","Quotient"],["AddCommGroup","toAddGroup"],["Classical","propDecidable"],["Submodule","Quotient","instZeroQuotient"],["AddGroupWithOne","toAddMonoidWithOne"],["AddMonoidWithOne","toAddMonoid"],["AddSubgroup","index","eq_1"],["Ideal","instHasQuotient_1"],["congrArg"],["Fintype","ofFinite"],["Fintype","card"],["dif_pos"],["Nat","cast_card_eq_zero"],["Ring","toAddGroupWithOne"],["Ideal"],["instOfNatNat"],["congrFun'"],["Ideal","Quotient","instRingQuotient"],["Zero","toOfNat0"],["dif_neg"],["Finite"],["Eq"],["Semiring","toModule"],["CommRing","toCommSemiring"],["True"],["CommSemiring","toSemiring"],["Submodule","toAddSubgroup"],["AddZeroClass","toAddZero"],["Nat","card"],["OfNat","ofNat"],["Ring","toAddCommGroup"],["eq_self"],["CommRing","toRing"],["AddSubgroup"],["Nat"],["AddMonoidWithOne","toNatCast"],["of_eq_true"],["QuotientAddGroup","instHasQuotientAddSubgroup"],["Nat","cast_zero"],["id"],["AddSubgroup","index"],["Eq","mpr"],["dite"],["AddZero","toZero"],["Nat","card_eq"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["CharP","ker_intAlgebraMap_eq_span"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3 : Ring.{u_1} R] (p : Nat) [inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.9 : CharP.{u_1} R (AddGroupWithOne.toAddMonoidWithOne.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3)) p], Eq.{1} (Ideal.{0} Int Int.instSemiring) (RingHom.ker.{0, u_1, u_1} Int R (RingHom.{0, u_1} Int R (Semiring.toNonAssocSemiring.{0} Int (CommSemiring.toSemiring.{0} Int Int.instCommSemiring)) (Semiring.toNonAssocSemiring.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3))) Int.instSemiring (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3) (RingHom.instFunLike.{0, u_1} Int R (Semiring.toNonAssocSemiring.{0} Int (CommSemiring.toSemiring.{0} Int Int.instCommSemiring)) (Semiring.toNonAssocSemiring.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3))) (RingHom.instRingHomClass.{0, u_1} Int R (Semiring.toNonAssocSemiring.{0} Int (CommSemiring.toSemiring.{0} Int Int.instCommSemiring)) (Semiring.toNonAssocSemiring.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3))) (algebraMap.{0, u_1} Int R Int.instCommSemiring (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3) (Ring.toIntAlgebra.{u_1} R inst._@.Mathlib.Algebra.CharP.Quotient.3961732859._hygCtx._hyg.3))) (Ideal.span.{0} Int Int.instSemiring (Singleton.singleton.{0, 0} Int (Set.{0} Int) (Set.instSingletonSet.{0} Int) (Nat.cast.{0} Int instNatCastInt p)))","typeFull":"∀ {R : Type u_1} [inst : Ring R] (p : ℕ) [CharP R p], RingHom.ker (algebraMap ℤ R) = Ideal.span {↑p}","typeReadable":"∀ {R : Type u_1} [inst : Ring R] (p : ℕ) [CharP R p], RingHom.ker (algebraMap ℤ R) = Ideal.span {↑p}","typeReferences":[["RingHom"],["Ideal","span"],["Nat","cast"],["Singleton","singleton"],["AddGroupWithOne","toAddMonoidWithOne"],["RingHom","instFunLike"],["RingHom","instRingHomClass"],["Ring","toIntAlgebra"],["RingHom","ker"],["Int","instCommSemiring"],["Semiring","toNonAssocSemiring"],["Ideal"],["Ring","toAddGroupWithOne"],["Eq"],["instNatCastInt"],["Set"],["CharP"],["CommSemiring","toSemiring"],["Set","instSingletonSet"],["Int"],["Ring","toSemiring"],["Nat"],["Int","instSemiring"],["Ring"],["algebraMap"]],"valueReferences":[["SubtractionMonoid","toSubNegZeroMonoid"],["RingHom"],["semigroupDvd"],["Ring","toNonAssocRing"],["Eq","trans"],["Singleton","singleton"],["Membership","mem"],["NonUnitalSemiring","toSemigroupWithZero"],["RingHom","instRingHomClass"],["Ring","toIntAlgebra"],["AddGroup","toSubtractionMonoid"],["Int","instCommSemiring"],["Semiring","toNonAssocSemiring"],["iff_self"],["Ring","toAddGroupWithOne"],["Semiring","toModule"],["CharP","intCast_eq_zero_iff"],["SetLike","instMembership"],["Dvd","dvd"],["NonUnitalCommSemiring","toNonUnitalSemiring"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["eq_intCast"],["Ring","toSemiring"],["RingHom","mem_ker","_simp_1"],["Int","instDvd"],["Iff"],["NegZeroClass","toZero"],["algebraMap"],["Ideal","span"],["Nat","cast"],["Int","castRingHom"],["RingHom","instFunLike"],["SubNegZeroMonoid","toNegZeroClass"],["DFunLike","coe"],["Int","cast"],["RingHom","ker"],["congrArg"],["Ideal"],["CommSemiring","toNonUnitalCommSemiring"],["NonAssocRing","toAddCommGroupWithOne"],["Ideal","ext"],["congr"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["propext"],["instNatCastInt"],["True"],["Set"],["CommSemiring","toSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Set","instSingletonSet"],["OfNat","ofNat"],["Int"],["AddCommGroupWithOne","toAddGroupWithOne"],["AddGroupWithOne","toAddGroup"],["of_eq_true"],["Submodule","setLike"],["Int","instSemiring"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddGroupWithOne","toIntCast"],["SemigroupWithZero","toSemigroup"],["_private","Mathlib","Algebra","CharP","Quotient",0,"CharP","ker_intAlgebraMap_eq_span","_simp_1_1"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["CharZero","of_addMonoidHom"],"typeFallback":"forall {M : Type.{u_1}} {N : Type.{u_2}} [inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4 : AddCommMonoidWithOne.{u_1} M] [inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7 : AddCommMonoidWithOne.{u_2} N] [inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.10 : CharZero.{u_1} M (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} M inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4)] (e : AddMonoidHom.{u_1, u_2} M N (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddMonoidWithOne.toAddMonoid.{u_1} M (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} M inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4)))) (AddZeroClass.toAddZero.{u_2} N (AddMonoid.toAddZeroClass.{u_2} N (AddMonoidWithOne.toAddMonoid.{u_2} N (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} N inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7))))), (Eq.{succ u_2} N (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (AddMonoidHom.{u_1, u_2} M N (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddMonoidWithOne.toAddMonoid.{u_1} M (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} M inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4)))) (AddZeroClass.toAddZero.{u_2} N (AddMonoid.toAddZeroClass.{u_2} N (AddMonoidWithOne.toAddMonoid.{u_2} N (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} N inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7))))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (AddMonoidHom.instFunLike.{u_1, u_2} M N (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddMonoidWithOne.toAddMonoid.{u_1} M (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} M inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4)))) (AddZeroClass.toAddZero.{u_2} N (AddMonoid.toAddZeroClass.{u_2} N (AddMonoidWithOne.toAddMonoid.{u_2} N (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} N inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7))))) e (OfNat.ofNat.{u_1} M 1 (One.toOfNat1.{u_1} M (AddMonoidWithOne.toOne.{u_1} M (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} M inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4))))) (OfNat.ofNat.{u_2} N 1 (One.toOfNat1.{u_2} N (AddMonoidWithOne.toOne.{u_2} N (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} N inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7))))) -> (Function.Injective.{succ u_1, succ u_2} M N (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (AddMonoidHom.{u_1, u_2} M N (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddMonoidWithOne.toAddMonoid.{u_1} M (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} M inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4)))) (AddZeroClass.toAddZero.{u_2} N (AddMonoid.toAddZeroClass.{u_2} N (AddMonoidWithOne.toAddMonoid.{u_2} N (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} N inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7))))) M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (AddMonoidHom.instFunLike.{u_1, u_2} M N (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddMonoidWithOne.toAddMonoid.{u_1} M (AddCommMonoidWithOne.toAddMonoidWithOne.{u_1} M inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.4)))) (AddZeroClass.toAddZero.{u_2} N (AddMonoid.toAddZeroClass.{u_2} N (AddMonoidWithOne.toAddMonoid.{u_2} N (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} N inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7))))) e)) -> (CharZero.{u_2} N (AddCommMonoidWithOne.toAddMonoidWithOne.{u_2} N inst._@.Mathlib.Algebra.CharZero.AddMonoidHom.1588163259._hygCtx._hyg.7))","typeFull":"∀ {M : Type u_1} {N : Type u_2} [inst : AddCommMonoidWithOne M] [inst_1 : AddCommMonoidWithOne N] [CharZero M]\n  (e : M →+ N), e 1 = 1 → Function.Injective ⇑e → CharZero N","typeReadable":"∀ {M : Type u_1} {N : Type u_2} [inst : AddCommMonoidWithOne M] [inst_1 : AddCommMonoidWithOne N] [CharZero M]\n  (e : M →+ N), e 1 = 1 → Function.Injective ⇑e → CharZero N","typeReferences":[["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["DFunLike","coe"],["OfNat","ofNat"],["One","toOfNat1"],["AddMonoidHom"],["AddMonoidWithOne","toOne"],["AddMonoidHom","instFunLike"],["AddCommMonoidWithOne"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["CharZero"],["Eq"],["Function","Injective"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Nat","cast_inj"],["Nat","cast"],["Eq","mp"],["AddMonoidHom","instAddMonoidHomClass"],["CharZero","mk"],["AddMonoidWithOne","toAddMonoid"],["AddZeroClass","toAddZero"],["DFunLike","coe"],["congrArg"],["Function","Injective","eq_iff"],["Nat"],["AddMonoidWithOne","toNatCast"],["AddMonoidHom"],["Eq","symm"],["AddMonoidHom","instFunLike"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["propext"],["map_natCast'"],["AddMonoid","toAddZeroClass"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["mulRightEmbedding"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3], G -> (Function.Embedding.{succ u_1, succ u_1} G G)","typeFull":"{G : Type u_1} → [inst : Mul G] → [IsRightCancelMul G] → G → G ↪ G","typeReadable":"{G : Type u_1} → [inst : Mul G] → [IsRightCancelMul G] → G → G ↪ G","typeReferences":[["IsRightCancelMul"],["Function","Embedding"],["Mul"]],"valueReferences":[["Function","Embedding","mk"],["instHMul"],["HMul","hMul"],["mul_left_injective"]]},{"isProp":true,"kind":"theorem","name":["mulRightEmbedding_apply"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3] (g : G) (h : G), Eq.{succ u_1} G (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) G (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : G) => G) (Function.instFunLikeEmbedding.{succ u_1, succ u_1} G G) (mulRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g) h) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3) h g)","typeFull":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsRightCancelMul G] (g h : G), (mulRightEmbedding g) h = h * g","typeReadable":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsRightCancelMul G] (g h : G), (mulRightEmbedding g) h = h * g","typeReferences":[["IsRightCancelMul"],["mulRightEmbedding"],["Function","Embedding"],["Mul"],["instHMul"],["HMul","hMul"],["Function","instFunLikeEmbedding"],["Eq"],["DFunLike","coe"]],"valueReferences":[["mulRightEmbedding"],["Eq","refl"],["Function","Embedding"],["Function","instFunLikeEmbedding"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["addRightEmbedding","congr_simp"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3] (g : G) (g_1 : G), (Eq.{succ u_1} G g g_1) -> (Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (addRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g) (addRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g_1))","typeFull":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsRightCancelAdd G] (g g_1 : G),\n  g = g_1 → addRightEmbedding g = addRightEmbedding g_1","typeReadable":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsRightCancelAdd G] (g g_1 : G),\n  g = g_1 → addRightEmbedding g = addRightEmbedding g_1","typeReferences":[["Add"],["Function","Embedding"],["IsRightCancelAdd"],["addRightEmbedding"],["Eq"]],"valueReferences":[["Eq","refl"],["Function","Embedding"],["addRightEmbedding"],["Eq"],["Eq","rec"]]},{"isProp":true,"kind":"theorem","name":["mulLeftEmbedding_eq_mulRightEmbedding"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3 : CommMagma.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.6 : IsCancelMul.{u_1} G (CommMagma.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3)] (g : G), Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (mulLeftEmbedding.{u_1} G (CommMagma.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) (IsCancelMul.toIsLeftCancelMul.{u_1} G (CommMagma.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.6) g) (mulRightEmbedding.{u_1} G (CommMagma.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) (IsCancelMul.toIsRightCancelMul.{u_1} G (CommMagma.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.6) g)","typeFull":"∀ {G : Type u_1} [inst : CommMagma G] [inst_1 : IsCancelMul G] (g : G), mulLeftEmbedding g = mulRightEmbedding g","typeReadable":"∀ {G : Type u_1} [inst : CommMagma G] [inst_1 : IsCancelMul G] (g : G), mulLeftEmbedding g = mulRightEmbedding g","typeReferences":[["mulRightEmbedding"],["CommMagma","toMul"],["Function","Embedding"],["IsCancelMul"],["IsCancelMul","toIsLeftCancelMul"],["IsCancelMul","toIsRightCancelMul"],["Eq"],["mulLeftEmbedding"],["CommMagma"]],"valueReferences":[["Function","Embedding","ext"],["mulRightEmbedding"],["CommMagma","toMul"],["mul_comm"],["IsCancelMul","toIsLeftCancelMul"],["IsCancelMul","toIsRightCancelMul"],["mulLeftEmbedding"]]},{"isProp":false,"kind":"definition","name":["mulLeftEmbedding"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3], G -> (Function.Embedding.{succ u_1, succ u_1} G G)","typeFull":"{G : Type u_1} → [inst : Mul G] → [IsLeftCancelMul G] → G → G ↪ G","typeReadable":"{G : Type u_1} → [inst : Mul G] → [IsLeftCancelMul G] → G → G ↪ G","typeReferences":[["Function","Embedding"],["Mul"],["IsLeftCancelMul"]],"valueReferences":[["Function","Embedding","mk"],["mul_right_injective"],["instHMul"],["HMul","hMul"]]},{"isProp":true,"kind":"theorem","name":["mulLeftEmbedding","eq_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3] (g : G), Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (mulLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g) (Function.Embedding.mk.{succ u_1, succ u_1} G G (fun (h : G) => HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3) g h) (mul_right_injective.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g))","typeFull":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsLeftCancelMul G] (g : G),\n  mulLeftEmbedding g = { toFun := fun h => g * h, inj' := ⋯ }","typeReadable":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsLeftCancelMul G] (g : G),\n  mulLeftEmbedding g = { toFun := fun h => g * h, inj' := ⋯ }","typeReferences":[["Function","Embedding","mk"],["mul_right_injective"],["Function","Embedding"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"],["mulLeftEmbedding"],["IsLeftCancelMul"]],"valueReferences":[["Eq","refl"],["Function","Embedding"],["mulLeftEmbedding"]]},{"isProp":true,"kind":"theorem","name":["mulRightEmbedding","eq_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3] (g : G), Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (mulRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g) (Function.Embedding.mk.{succ u_1, succ u_1} G G (fun (h : G) => HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3) h g) (mul_left_injective.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g))","typeFull":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsRightCancelMul G] (g : G),\n  mulRightEmbedding g = { toFun := fun h => h * g, inj' := ⋯ }","typeReadable":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsRightCancelMul G] (g : G),\n  mulRightEmbedding g = { toFun := fun h => h * g, inj' := ⋯ }","typeReferences":[["IsRightCancelMul"],["Function","Embedding","mk"],["mulRightEmbedding"],["Function","Embedding"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"],["mul_left_injective"]],"valueReferences":[["mulRightEmbedding"],["Eq","refl"],["Function","Embedding"]]},{"isProp":true,"kind":"theorem","name":["addRightEmbedding","eq_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3] (g : G), Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (addRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g) (Function.Embedding.mk.{succ u_1, succ u_1} G G (fun (h : G) => HAdd.hAdd.{u_1, u_1, u_1} G G G (instHAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3) h g) (add_left_injective.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g))","typeFull":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsRightCancelAdd G] (g : G),\n  addRightEmbedding g = { toFun := fun h => h + g, inj' := ⋯ }","typeReadable":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsRightCancelAdd G] (g : G),\n  addRightEmbedding g = { toFun := fun h => h + g, inj' := ⋯ }","typeReferences":[["HAdd","hAdd"],["Function","Embedding","mk"],["instHAdd"],["Add"],["Function","Embedding"],["IsRightCancelAdd"],["addRightEmbedding"],["Eq"],["add_left_injective"]],"valueReferences":[["Eq","refl"],["Function","Embedding"],["addRightEmbedding"]]},{"isProp":true,"kind":"theorem","name":["mulLeftEmbedding_apply"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3] (g : G) (h : G), Eq.{succ u_1} G (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) G (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : G) => G) (Function.instFunLikeEmbedding.{succ u_1, succ u_1} G G) (mulLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g) h) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3) g h)","typeFull":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsLeftCancelMul G] (g h : G), (mulLeftEmbedding g) h = g * h","typeReadable":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsLeftCancelMul G] (g h : G), (mulLeftEmbedding g) h = g * h","typeReferences":[["Function","Embedding"],["Mul"],["instHMul"],["HMul","hMul"],["Function","instFunLikeEmbedding"],["Eq"],["mulLeftEmbedding"],["DFunLike","coe"],["IsLeftCancelMul"]],"valueReferences":[["Eq","refl"],["Function","Embedding"],["Function","instFunLikeEmbedding"],["mulLeftEmbedding"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["addRightEmbedding_apply"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3] (g : G) (h : G), Eq.{succ u_1} G (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) G (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : G) => G) (Function.instFunLikeEmbedding.{succ u_1, succ u_1} G G) (addRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g) h) (HAdd.hAdd.{u_1, u_1, u_1} G G G (instHAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3) h g)","typeFull":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsRightCancelAdd G] (g h : G), (addRightEmbedding g) h = h + g","typeReadable":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsRightCancelAdd G] (g h : G), (addRightEmbedding g) h = h + g","typeReferences":[["HAdd","hAdd"],["instHAdd"],["Add"],["Function","Embedding"],["IsRightCancelAdd"],["addRightEmbedding"],["Function","instFunLikeEmbedding"],["Eq"],["DFunLike","coe"]],"valueReferences":[["Eq","refl"],["Function","Embedding"],["addRightEmbedding"],["Function","instFunLikeEmbedding"],["DFunLike","coe"]]},{"isProp":false,"kind":"definition","name":["addRightEmbedding"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3], G -> (Function.Embedding.{succ u_1, succ u_1} G G)","typeFull":"{G : Type u_1} → [inst : Add G] → [IsRightCancelAdd G] → G → G ↪ G","typeReadable":"{G : Type u_1} → [inst : Add G] → [IsRightCancelAdd G] → G → G ↪ G","typeReferences":[["Add"],["Function","Embedding"],["IsRightCancelAdd"]],"valueReferences":[["HAdd","hAdd"],["Function","Embedding","mk"],["instHAdd"],["add_left_injective"]]},{"isProp":true,"kind":"theorem","name":["addLeftEmbedding_apply"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3] (g : G) (h : G), Eq.{succ u_1} G (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) G (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : G) => G) (Function.instFunLikeEmbedding.{succ u_1, succ u_1} G G) (addLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g) h) (HAdd.hAdd.{u_1, u_1, u_1} G G G (instHAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3) g h)","typeFull":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsLeftCancelAdd G] (g h : G), (addLeftEmbedding g) h = g + h","typeReadable":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsLeftCancelAdd G] (g h : G), (addLeftEmbedding g) h = g + h","typeReferences":[["HAdd","hAdd"],["addLeftEmbedding"],["IsLeftCancelAdd"],["instHAdd"],["Add"],["Function","Embedding"],["Function","instFunLikeEmbedding"],["Eq"],["DFunLike","coe"]],"valueReferences":[["addLeftEmbedding"],["Eq","refl"],["Function","Embedding"],["Function","instFunLikeEmbedding"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["mulRightEmbedding","congr_simp"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 : IsRightCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3] (g : G) (g_1 : G), (Eq.{succ u_1} G g g_1) -> (Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (mulRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g) (mulRightEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.2159921533._hygCtx._hyg.6 g_1))","typeFull":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsRightCancelMul G] (g g_1 : G),\n  g = g_1 → mulRightEmbedding g = mulRightEmbedding g_1","typeReadable":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsRightCancelMul G] (g g_1 : G),\n  g = g_1 → mulRightEmbedding g = mulRightEmbedding g_1","typeReferences":[["IsRightCancelMul"],["mulRightEmbedding"],["Function","Embedding"],["Mul"],["Eq"]],"valueReferences":[["mulRightEmbedding"],["Eq","refl"],["Function","Embedding"],["Eq"],["Eq","rec"]]},{"isProp":true,"kind":"theorem","name":["addLeftEmbedding","eq_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3] (g : G), Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (addLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g) (Function.Embedding.mk.{succ u_1, succ u_1} G G (fun (h : G) => HAdd.hAdd.{u_1, u_1, u_1} G G G (instHAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3) g h) (add_right_injective.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g))","typeFull":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsLeftCancelAdd G] (g : G),\n  addLeftEmbedding g = { toFun := fun h => g + h, inj' := ⋯ }","typeReadable":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsLeftCancelAdd G] (g : G),\n  addLeftEmbedding g = { toFun := fun h => g + h, inj' := ⋯ }","typeReferences":[["HAdd","hAdd"],["Function","Embedding","mk"],["addLeftEmbedding"],["IsLeftCancelAdd"],["instHAdd"],["Add"],["add_right_injective"],["Function","Embedding"],["Eq"]],"valueReferences":[["addLeftEmbedding"],["Eq","refl"],["Function","Embedding"]]},{"isProp":true,"kind":"theorem","name":["addLeftEmbedding_eq_addRightEmbedding"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3 : AddCommMagma.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.6 : IsCancelAdd.{u_1} G (AddCommMagma.toAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3)] (g : G), Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (addLeftEmbedding.{u_1} G (AddCommMagma.toAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) (IsCancelAdd.toIsLeftCancelAdd.{u_1} G (AddCommMagma.toAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.6) g) (addRightEmbedding.{u_1} G (AddCommMagma.toAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) (IsCancelAdd.toIsRightCancelAdd.{u_1} G (AddCommMagma.toAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Group.Embedding.362261539._hygCtx._hyg.6) g)","typeFull":"∀ {G : Type u_1} [inst : AddCommMagma G] [inst_1 : IsCancelAdd G] (g : G), addLeftEmbedding g = addRightEmbedding g","typeReadable":"∀ {G : Type u_1} [inst : AddCommMagma G] [inst_1 : IsCancelAdd G] (g : G), addLeftEmbedding g = addRightEmbedding g","typeReferences":[["AddCommMagma"],["IsCancelAdd","toIsRightCancelAdd"],["addLeftEmbedding"],["IsCancelAdd"],["Function","Embedding"],["addRightEmbedding"],["AddCommMagma","toAdd"],["IsCancelAdd","toIsLeftCancelAdd"],["Eq"]],"valueReferences":[["IsCancelAdd","toIsRightCancelAdd"],["addLeftEmbedding"],["Function","Embedding","ext"],["addRightEmbedding"],["AddCommMagma","toAdd"],["IsCancelAdd","toIsLeftCancelAdd"],["add_comm"]]},{"isProp":false,"kind":"definition","name":["addLeftEmbedding"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3], G -> (Function.Embedding.{succ u_1, succ u_1} G G)","typeFull":"{G : Type u_1} → [inst : Add G] → [IsLeftCancelAdd G] → G → G ↪ G","typeReadable":"{G : Type u_1} → [inst : Add G] → [IsLeftCancelAdd G] → G → G ↪ G","typeReferences":[["IsLeftCancelAdd"],["Add"],["Function","Embedding"]],"valueReferences":[["HAdd","hAdd"],["Function","Embedding","mk"],["instHAdd"],["add_right_injective"]]},{"isProp":true,"kind":"theorem","name":["mulLeftEmbedding","congr_simp"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Mul.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelMul.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3] (g : G) (g_1 : G), (Eq.{succ u_1} G g g_1) -> (Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (mulLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g) (mulLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g_1))","typeFull":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsLeftCancelMul G] (g g_1 : G),\n  g = g_1 → mulLeftEmbedding g = mulLeftEmbedding g_1","typeReadable":"∀ {G : Type u_1} [inst : Mul G] [inst_1 : IsLeftCancelMul G] (g g_1 : G),\n  g = g_1 → mulLeftEmbedding g = mulLeftEmbedding g_1","typeReferences":[["Function","Embedding"],["Mul"],["Eq"],["mulLeftEmbedding"],["IsLeftCancelMul"]],"valueReferences":[["Eq","refl"],["Function","Embedding"],["Eq"],["Eq","rec"],["mulLeftEmbedding"]]},{"isProp":true,"kind":"theorem","name":["addLeftEmbedding","congr_simp"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 : Add.{u_1} G] [inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 : IsLeftCancelAdd.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3] (g : G) (g_1 : G), (Eq.{succ u_1} G g g_1) -> (Eq.{succ u_1} (Function.Embedding.{succ u_1, succ u_1} G G) (addLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g) (addLeftEmbedding.{u_1} G inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Group.Embedding.1360568216._hygCtx._hyg.6 g_1))","typeFull":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsLeftCancelAdd G] (g g_1 : G),\n  g = g_1 → addLeftEmbedding g = addLeftEmbedding g_1","typeReadable":"∀ {G : Type u_1} [inst : Add G] [inst_1 : IsLeftCancelAdd G] (g g_1 : G),\n  g = g_1 → addLeftEmbedding g = addLeftEmbedding g_1","typeReferences":[["addLeftEmbedding"],["IsLeftCancelAdd"],["Add"],["Function","Embedding"],["Eq"]],"valueReferences":[["addLeftEmbedding"],["Eq","refl"],["Function","Embedding"],["Eq"],["Eq","rec"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Equiv.Finite.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Fintype","decidableEqMulEquivFintype","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 : Mul.{u_1} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 : Mul.{u_2} β] (a : MulEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) (b : MulEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16), Iff (Eq.{max (succ u_1) (succ u_2)} (α -> β) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (MulEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (MulEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (MulEquiv.instEquivLike.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) a) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (MulEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (MulEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (MulEquiv.instEquivLike.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) b)) (Eq.{max (succ u_1) (succ u_2)} (MulEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) a b)","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Mul α] [inst_1 : Mul β] (a b : α ≃* β), ⇑a = ⇑b ↔ a = b","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Mul α] [inst_1 : Mul β] (a b : α ≃* β), ⇑a = ⇑b ↔ a = b","typeReferences":[["MulEquiv"],["Iff"],["EquivLike","toFunLike"],["Mul"],["MulEquiv","instEquivLike"],["Eq"],["DFunLike","coe"]],"valueReferences":[["Function","Injective","eq_iff"],["MulEquiv"],["EquivLike","toFunLike"],["MulEquiv","instEquivLike"],["DFunLike","coe"],["DFunLike","coe_injective"]]},{"isProp":false,"kind":"definition","name":["Fintype","decidableEqMulEquivFintype"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7 : DecidableEq.{succ u_5} β] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 : Fintype.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 : Mul.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 : Mul.{u_5} β], DecidableEq.{max (succ u_5) (succ u_4)} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)","typeFull":"{α : Type u_4} →\n  {β : Type u_5} → [DecidableEq β] → [Fintype α] → [inst : Mul α] → [inst_1 : Mul β] → DecidableEq (α ≃* β)","typeReadable":"{α : Type u_4} →\n  {β : Type u_5} → [DecidableEq β] → [Fintype α] → [inst : Mul α] → [inst_1 : Mul β] → DecidableEq (α ≃* β)","typeReferences":[["MulEquiv"],["DecidableEq"],["Mul"],["Fintype"]],"valueReferences":[["decidable_of_iff"],["MulEquiv"],["EquivLike","toFunLike"],["Fintype","decidableEqMulEquivFintype","_proof_1"],["Fintype","decidablePiFintype"],["MulEquiv","instEquivLike"],["Eq"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["Fintype","decidableEqAddEquivFintype","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} {β : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 : Add.{u_1} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 : Add.{u_2} β] (a : AddEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) (b : AddEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16), Iff (Eq.{max (succ u_1) (succ u_2)} (α -> β) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (AddEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (AddEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (AddEquiv.instEquivLike.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) a) (DFunLike.coe.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (AddEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_1) (succ u_2), succ u_1, succ u_2} (AddEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (AddEquiv.instEquivLike.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) b)) (Eq.{max (succ u_1) (succ u_2)} (AddEquiv.{u_1, u_2} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) a b)","typeFull":"∀ {α : Type u_1} {β : Type u_2} [inst : Add α] [inst_1 : Add β] (a b : α ≃+ β), ⇑a = ⇑b ↔ a = b","typeReadable":"∀ {α : Type u_1} {β : Type u_2} [inst : Add α] [inst_1 : Add β] (a b : α ≃+ β), ⇑a = ⇑b ↔ a = b","typeReferences":[["Add"],["Iff"],["EquivLike","toFunLike"],["Eq"],["DFunLike","coe"],["AddEquiv"],["AddEquiv","instEquivLike"]],"valueReferences":[["Function","Injective","eq_iff"],["EquivLike","toFunLike"],["DFunLike","coe"],["AddEquiv"],["AddEquiv","instEquivLike"],["DFunLike","coe_injective"]]},{"isProp":true,"kind":"theorem","name":["Fintype","decidableEqMulEquivFintype","eq_1"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7 : DecidableEq.{succ u_5} β] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 : Fintype.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 : Mul.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 : Mul.{u_5} β] (a : MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) (b : MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16), Eq.{1} (Decidable (Eq.{max (succ u_5) (succ u_4)} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) a b)) (Fintype.decidableEqMulEquivFintype.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 a b) (decidable_of_iff (Eq.{max (succ u_5) (succ u_4)} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) a b) (Eq.{max (succ u_4) (succ u_5)} (α -> β) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (MulEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) a) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (MulEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) b)) (Fintype.decidableEqMulEquivFintype._proof_1.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 a b) (Fintype.decidablePiFintype.{u_5, u_4} α (fun (a : α) => β) (fun (a : α) => inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (MulEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) a) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (MulEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (MulEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) b)))","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : DecidableEq β] [inst_1 : Fintype α] [inst_2 : Mul α] [inst_3 : Mul β]\n  (a b : α ≃* β), Fintype.decidableEqMulEquivFintype a b = decidable_of_iff (⇑a = ⇑b) ⋯","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : DecidableEq β] [inst_1 : Fintype α] [inst_2 : Mul α] [inst_3 : Mul β]\n  (a b : α ≃* β), Fintype.decidableEqMulEquivFintype a b = decidable_of_iff (⇑a = ⇑b) ⋯","typeReferences":[["decidable_of_iff"],["MulEquiv"],["Decidable"],["DecidableEq"],["Mul"],["Fintype","decidableEqMulEquivFintype"],["Fintype","decidablePiFintype"],["MulEquiv","instEquivLike"],["Fintype"],["DFunLike","coe"],["EquivLike","toFunLike"],["Fintype","decidableEqMulEquivFintype","_proof_1"],["Eq"]],"valueReferences":[["MulEquiv"],["Decidable"],["Eq","refl"],["Fintype","decidableEqMulEquivFintype"],["Eq"]]},{"isProp":false,"kind":"definition","name":["Fintype","decidableEqAddEquivFintype"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7 : DecidableEq.{succ u_5} β] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 : Fintype.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 : Add.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 : Add.{u_5} β], DecidableEq.{max (succ u_5) (succ u_4)} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)","typeFull":"{α : Type u_4} →\n  {β : Type u_5} → [DecidableEq β] → [Fintype α] → [inst : Add α] → [inst_1 : Add β] → DecidableEq (α ≃+ β)","typeReadable":"{α : Type u_4} →\n  {β : Type u_5} → [DecidableEq β] → [Fintype α] → [inst : Add α] → [inst_1 : Add β] → DecidableEq (α ≃+ β)","typeReferences":[["Add"],["DecidableEq"],["Fintype"],["AddEquiv"]],"valueReferences":[["decidable_of_iff"],["EquivLike","toFunLike"],["Fintype","decidablePiFintype"],["Eq"],["Fintype","decidableEqAddEquivFintype","_proof_1"],["DFunLike","coe"],["AddEquiv"],["AddEquiv","instEquivLike"]]},{"isProp":true,"kind":"theorem","name":["Fintype","decidableEqAddEquivFintype","eq_1"],"typeFallback":"forall {α : Type.{u_4}} {β : Type.{u_5}} [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7 : DecidableEq.{succ u_5} β] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 : Fintype.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 : Add.{u_4} α] [inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 : Add.{u_5} β] (a : AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) (b : AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16), Eq.{1} (Decidable (Eq.{max (succ u_5) (succ u_4)} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) a b)) (Fintype.decidableEqAddEquivFintype.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 a b) (decidable_of_iff (Eq.{max (succ u_5) (succ u_4)} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) a b) (Eq.{max (succ u_4) (succ u_5)} (α -> β) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (AddEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) a) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (AddEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) b)) (Fintype.decidableEqAddEquivFintype._proof_1.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16 a b) (Fintype.decidablePiFintype.{u_5, u_4} α (fun (a : α) => β) (fun (a : α) => inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.7) inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.10 (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (AddEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) a) (DFunLike.coe.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : α) => β) (EquivLike.toFunLike.{max (succ u_4) (succ u_5), succ u_4, succ u_5} (AddEquiv.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16) α β (AddEquiv.instEquivLike.{u_4, u_5} α β inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.13 inst._@.Mathlib.Algebra.Group.Equiv.Finite.3072951324._hygCtx._hyg.16)) b)))","typeFull":"∀ {α : Type u_4} {β : Type u_5} [inst : DecidableEq β] [inst_1 : Fintype α] [inst_2 : Add α] [inst_3 : Add β]\n  (a b : α ≃+ β), Fintype.decidableEqAddEquivFintype a b = decidable_of_iff (⇑a = ⇑b) ⋯","typeReadable":"∀ {α : Type u_4} {β : Type u_5} [inst : DecidableEq β] [inst_1 : Fintype α] [inst_2 : Add α] [inst_3 : Add β]\n  (a b : α ≃+ β), Fintype.decidableEqAddEquivFintype a b = decidable_of_iff (⇑a = ⇑b) ⋯","typeReferences":[["decidable_of_iff"],["Decidable"],["Fintype","decidableEqAddEquivFintype"],["Add"],["DecidableEq"],["Fintype","decidablePiFintype"],["Fintype"],["AddEquiv"],["DFunLike","coe"],["AddEquiv","instEquivLike"],["EquivLike","toFunLike"],["Eq"],["Fintype","decidableEqAddEquivFintype","_proof_1"]],"valueReferences":[["Fintype","decidableEqAddEquivFintype"],["Decidable"],["Eq","refl"],["Eq"],["AddEquiv"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","mem_closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3 : Add.{u_1} M] {s : Set.{u_1} M} {x : M}, Iff (Membership.mem.{u_1, u_1} M (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3))) (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3 s) x) (forall (p : AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3), (HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s (SetLike.coe.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3)) p)) -> (Membership.mem.{u_1, u_1} M (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3))) p x))","typeFull":"∀ {M : Type u_1} [inst : Add M] {s : Set M} {x : M},\n  x ∈ AddSemigroupIdeal.closure s ↔ ∀ (p : AddSemigroupIdeal M), s ⊆ ↑p → x ∈ p","typeReadable":"∀ {M : Type u_1} [inst : Add M] {s : Set M} {x : M},\n  x ∈ AddSemigroupIdeal.closure s ↔ ∀ (p : AddSemigroupIdeal M), s ⊆ ↑p → x ∈ p","typeReferences":[["Set","instHasSubset"],["AddSemigroupIdeal","closure"],["SetLike","coe"],["SetLike","instMembership"],["HasSubset","Subset"],["Set"],["Add"],["Iff"],["Membership","mem"],["Add","toVAdd"],["AddSemigroupIdeal"],["SubAddAction","instSetLike"]],"valueReferences":[["SubAddAction","mem_closure"],["Add","toVAdd"]]},{"isProp":false,"kind":"definition","name":["SemigroupIdeal"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3 : Mul.{u_1} M], Type.{u_1}","typeFull":"(M : Type u_1) → [Mul M] → Type u_1","typeReadable":"(M : Type u_1) → [Mul M] → Type u_1","typeReferences":[["Mul"]],"valueReferences":[["instSMulOfMul"],["SubMulAction"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","eq_1"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3 : Mul.{u_1} M], Eq.{succ (succ u_1)} Type.{u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3) (SubMulAction.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3))","typeFull":"∀ (M : Type u_1) [inst : Mul M], SemigroupIdeal M = SubMulAction M M","typeReadable":"∀ (M : Type u_1) [inst : Mul M], SemigroupIdeal M = SubMulAction M M","typeReferences":[["SemigroupIdeal"],["Mul"],["instSMulOfMul"],["SubMulAction"],["Eq"]],"valueReferences":[["SemigroupIdeal"],["Eq","refl"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","closure_le"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3 : Mul.{u_1} M] {s : Set.{u_1} M} {I : SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3}, Iff (LE.le.{u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) (Preorder.toLE.{u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) (SubMulAction.instPartialOrder.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3)))) (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3 s) I) (HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s (SetLike.coe.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3)) I))","typeFull":"∀ {M : Type u_1} [inst : Mul M] {s : Set M} {I : SemigroupIdeal M}, SemigroupIdeal.closure s ≤ I ↔ s ⊆ ↑I","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {s : Set M} {I : SemigroupIdeal M}, SemigroupIdeal.closure s ≤ I ↔ s ⊆ ↑I","typeReferences":[["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["PartialOrder","toPreorder"],["Set"],["Mul"],["instSMulOfMul"],["SubMulAction","instPartialOrder"],["Set","instHasSubset"],["SemigroupIdeal"],["SetLike","coe"],["HasSubset","Subset"],["Iff"],["LE","le"],["Preorder","toLE"]],"valueReferences":[["SubMulAction","closure_le"],["instSMulOfMul"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","coe_closure'"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.2261310793._hygCtx._hyg.3 : AddMonoid.{u_1} M] {s : Set.{u_1} M}, Eq.{succ u_1} (Set.{u_1} M) (SetLike.coe.{u_1, u_1} (AddSemigroupIdeal.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2261310793._hygCtx._hyg.3)))) M (SubAddAction.instSetLike.{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 inst._@.Mathlib.Algebra.Group.Ideal.2261310793._hygCtx._hyg.3))))) (AddSemigroupIdeal.closure.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2261310793._hygCtx._hyg.3))) s)) (HAdd.hAdd.{u_1, u_1, u_1} (Set.{u_1} M) (Set.{u_1} M) (Set.{u_1} M) (instHAdd.{u_1} (Set.{u_1} M) (Set.add.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2261310793._hygCtx._hyg.3))))) (Set.univ.{u_1} M) s)","typeFull":"∀ {M : Type u_1} [inst : AddMonoid M] {s : Set M}, ↑(AddSemigroupIdeal.closure s) = Set.univ + s","typeReadable":"∀ {M : Type u_1} [inst : AddMonoid M] {s : Set M}, ↑(AddSemigroupIdeal.closure s) = Set.univ + s","typeReferences":[["AddSemigroupIdeal","closure"],["Set"],["instHAdd"],["AddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Set","univ"],["HAdd","hAdd"],["SetLike","coe"],["Set","add"],["AddSemigroupIdeal"],["Add","toVAdd"],["SubAddAction","instSetLike"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Eq","trans"],["Membership","mem"],["Exists","intro"],["Union","union"],["Set","mem_univ"],["congrArg"],["And","intro"],["Set","add"],["congrFun'"],["Zero","toOfNat0"],["Add","toVAdd"],["Eq"],["Set","instUnion"],["propext"],["AddSemigroup","toAdd"],["AddSemigroupIdeal","closure"],["Exists"],["True"],["instHAdd"],["Set"],["And"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddSemigroupIdeal","coe_closure"],["OfNat","ofNat"],["Set","instMembership"],["HAdd","hAdd"],["Set","univ"],["Set","instHasSubset"],["eq_self"],["zero_add"],["SetLike","coe"],["of_eq_true"],["HasSubset","Subset"],["AddMonoid","toAddSemigroup"],["Set","union_eq_right"],["id"],["AddSemigroupIdeal"],["Eq","mpr"],["SubAddAction","instSetLike"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","mem_closure''"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3 : AddMonoid.{u_1} M] {s : Set.{u_1} M} {x : M}, Iff (Membership.mem.{u_1, u_1} M (AddSemigroupIdeal.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))) (SetLike.instMembership.{u_1, u_1} (AddSemigroupIdeal.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))) M (SubAddAction.instSetLike.{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 inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))))) (AddSemigroupIdeal.closure.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{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 (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1472811682._hygCtx._hyg.3)))) y z) x))))","typeFull":"∀ {M : Type u_1} [inst : AddMonoid M] {s : Set M} {x : M}, x ∈ AddSemigroupIdeal.closure s ↔ ∃ y, ∃ z ∈ s, y + z = x","typeReadable":"∀ {M : Type u_1} [inst : AddMonoid M] {s : Set M} {x : M}, x ∈ AddSemigroupIdeal.closure s ↔ ∃ y, ∃ z ∈ s, y + z = x","typeReferences":[["AddSemigroupIdeal","closure"],["SetLike","instMembership"],["Exists"],["instHAdd"],["Set"],["Membership","mem"],["And"],["AddMonoid"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["Set","instMembership"],["HAdd","hAdd"],["Iff"],["AddSemigroupIdeal"],["Add","toVAdd"],["SubAddAction","instSetLike"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Eq","trans"],["Membership","mem"],["eq_true"],["Set","mem_univ"],["congrArg"],["SetLike","mem_coe"],["iff_self"],["Set","add"],["funext"],["Eq","symm"],["congrFun'"],["Add","toVAdd"],["Eq"],["propext"],["AddSemigroupIdeal","closure"],["Exists"],["SetLike","instMembership"],["True"],["Set"],["instHAdd"],["And"],["true_and"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["Set","instMembership"],["AddSemigroupIdeal","coe_closure'"],["Set","univ"],["HAdd","hAdd"],["Set","mem_add"],["SetLike","coe"],["of_eq_true"],["Iff"],["id"],["AddSemigroupIdeal"],["Eq","mpr"],["SubAddAction","instSetLike"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","closure","eq_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3 : Mul.{u_1} M] (s : Set.{u_1} M), Eq.{succ u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3) (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3 s) (SubMulAction.closure.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3) s)","typeFull":"∀ {M : Type u_1} [inst : Mul M] (s : Set M), SemigroupIdeal.closure s = SubMulAction.closure M s","typeReadable":"∀ {M : Type u_1} [inst : Mul M] (s : Set M), SemigroupIdeal.closure s = SubMulAction.closure M s","typeReferences":[["SemigroupIdeal","closure"],["SemigroupIdeal"],["Set"],["Mul"],["instSMulOfMul"],["SubMulAction","closure"],["Eq"]],"valueReferences":[["SemigroupIdeal","closure"],["SemigroupIdeal"],["Eq","refl"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","closure_le"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3 : Add.{u_1} M] {s : Set.{u_1} M} {I : AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3}, Iff (LE.le.{u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) (Preorder.toLE.{u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) (SubAddAction.instPartialOrder.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3)))) (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3 s) I) (HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s (SetLike.coe.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.648911380._hygCtx._hyg.3)) I))","typeFull":"∀ {M : Type u_1} [inst : Add M] {s : Set M} {I : AddSemigroupIdeal M}, AddSemigroupIdeal.closure s ≤ I ↔ s ⊆ ↑I","typeReadable":"∀ {M : Type u_1} [inst : Add M] {s : Set M} {I : AddSemigroupIdeal M}, AddSemigroupIdeal.closure s ≤ I ↔ s ⊆ ↑I","typeReferences":[["AddSemigroupIdeal","closure"],["PartialOrder","toPreorder"],["Add"],["Set"],["Set","instHasSubset"],["SetLike","coe"],["HasSubset","Subset"],["Iff"],["LE","le"],["SubAddAction","instPartialOrder"],["AddSemigroupIdeal"],["Add","toVAdd"],["SubAddAction","instSetLike"],["Preorder","toLE"]],"valueReferences":[["SubAddAction","closure_le"],["Add","toVAdd"]]},{"isProp":false,"kind":"definition","name":["AddSemigroupIdeal"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3 : Add.{u_1} M], Type.{u_1}","typeFull":"(M : Type u_1) → [Add M] → Type u_1","typeReadable":"(M : Type u_1) → [Add M] → Type u_1","typeReferences":[["Add"]],"valueReferences":[["SubAddAction"],["Add","toVAdd"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","FG","eq_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 : Add.{u_1} M] (I : AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3), Eq.{1} Prop (AddSemigroupIdeal.FG.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 I) (Exists.{succ u_1} (Set.{u_1} M) (fun (s : Set.{u_1} M) => And (Set.Finite.{u_1} M s) (Eq.{succ u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3) I (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 s))))","typeFull":"∀ {M : Type u_1} [inst : Add M] (I : AddSemigroupIdeal M), I.FG = ∃ s, s.Finite ∧ I = AddSemigroupIdeal.closure s","typeReadable":"∀ {M : Type u_1} [inst : Add M] (I : AddSemigroupIdeal M), I.FG = ∃ s, s.Finite ∧ I = AddSemigroupIdeal.closure s","typeReferences":[["AddSemigroupIdeal","closure"],["Exists"],["Set"],["Add"],["And"],["AddSemigroupIdeal"],["Set","Finite"],["AddSemigroupIdeal","FG"],["Eq"]],"valueReferences":[["Eq","refl"],["AddSemigroupIdeal","FG"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Ideal",0,"SemigroupIdeal","mem_closure''","_simp_1_1"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Pointwise.Set.Basic.3586886705._hygCtx._hyg.10 : Mul.{u_2} α] {s : Set.{u_2} α} {t : Set.{u_2} α} {a : α}, Eq.{1} Prop (Membership.mem.{u_2, u_2} α (Set.{u_2} α) (Set.instMembership.{u_2} α) (HMul.hMul.{u_2, u_2, u_2} (Set.{u_2} α) (Set.{u_2} α) (Set.{u_2} α) (instHMul.{u_2} (Set.{u_2} α) (Set.mul.{u_2} α inst._@.Mathlib.Algebra.Group.Pointwise.Set.Basic.3586886705._hygCtx._hyg.10)) s t) a) (Exists.{succ u_2} α (fun (x : α) => And (Membership.mem.{u_2, u_2} α (Set.{u_2} α) (Set.instMembership.{u_2} α) s x) (Exists.{succ u_2} α (fun (y : α) => And (Membership.mem.{u_2, u_2} α (Set.{u_2} α) (Set.instMembership.{u_2} α) t y) (Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Group.Pointwise.Set.Basic.3586886705._hygCtx._hyg.10) x y) a)))))","typeFull":"∀ {α : Type u_2} [inst : Mul α] {s t : Set α} {a : α}, (a ∈ s * t) = ∃ x ∈ s, ∃ y ∈ t, x * y = a","typeReadable":"∀ {α : Type u_2} [inst : Mul α] {s t : Set α} {a : α}, (a ∈ s * t) = ∃ x ∈ s, ∃ y ∈ t, x * y = a","typeReferences":[["Set","mul"],["Exists"],["Set"],["Membership","mem"],["Mul"],["And"],["instHMul"],["HMul","hMul"],["Eq"],["Set","instMembership"]],"valueReferences":[["Set","mul"],["Exists"],["Set"],["Membership","mem"],["Set","mem_mul"],["And"],["instHMul"],["HMul","hMul"],["Eq"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Ideal",0,"SemigroupIdeal","mem_closure'","_simp_1_1"],"typeFallback":"forall {α : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Pointwise.Set.Basic.3586886705._hygCtx._hyg.10 : Mul.{u_2} α] {s : Set.{u_2} α} {t : Set.{u_2} α} {a : α}, Eq.{1} Prop (Membership.mem.{u_2, u_2} α (Set.{u_2} α) (Set.instMembership.{u_2} α) (HMul.hMul.{u_2, u_2, u_2} (Set.{u_2} α) (Set.{u_2} α) (Set.{u_2} α) (instHMul.{u_2} (Set.{u_2} α) (Set.mul.{u_2} α inst._@.Mathlib.Algebra.Group.Pointwise.Set.Basic.3586886705._hygCtx._hyg.10)) s t) a) (Exists.{succ u_2} α (fun (x : α) => And (Membership.mem.{u_2, u_2} α (Set.{u_2} α) (Set.instMembership.{u_2} α) s x) (Exists.{succ u_2} α (fun (y : α) => And (Membership.mem.{u_2, u_2} α (Set.{u_2} α) (Set.instMembership.{u_2} α) t y) (Eq.{succ u_2} α (HMul.hMul.{u_2, u_2, u_2} α α α (instHMul.{u_2} α inst._@.Mathlib.Algebra.Group.Pointwise.Set.Basic.3586886705._hygCtx._hyg.10) x y) a)))))","typeFull":"∀ {α : Type u_2} [inst : Mul α] {s t : Set α} {a : α}, (a ∈ s * t) = ∃ x ∈ s, ∃ y ∈ t, x * y = a","typeReadable":"∀ {α : Type u_2} [inst : Mul α] {s t : Set α} {a : α}, (a ∈ s * t) = ∃ x ∈ s, ∃ y ∈ t, x * y = a","typeReferences":[["Set","mul"],["Exists"],["Set"],["Membership","mem"],["Mul"],["And"],["instHMul"],["HMul","hMul"],["Eq"],["Set","instMembership"]],"valueReferences":[["Set","mul"],["Exists"],["Set"],["Membership","mem"],["Set","mem_mul"],["And"],["instHMul"],["HMul","hMul"],["Eq"],["propext"],["Set","instMembership"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","mem_closure_of_mem"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3 : Mul.{u_1} M] {s : Set.{u_1} M} {x : M}, (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s x) -> (Membership.mem.{u_1, u_1} M (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3))) (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3 s) x)","typeFull":"∀ {M : Type u_1} [inst : Mul M] {s : Set M} {x : M}, x ∈ s → x ∈ SemigroupIdeal.closure s","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {s : Set M} {x : M}, x ∈ s → x ∈ SemigroupIdeal.closure s","typeReferences":[["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["SemigroupIdeal"],["SetLike","instMembership"],["Set"],["Membership","mem"],["Mul"],["instSMulOfMul"],["Set","instMembership"]],"valueReferences":[["SubMulAction","mem_closure_of_mem"],["instSMulOfMul"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","eq_1"],"typeFallback":"forall (M : Type.{u_1}) [inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3 : Add.{u_1} M], Eq.{succ (succ u_1)} Type.{u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3) (SubAddAction.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.261416278._hygCtx._hyg.3))","typeFull":"∀ (M : Type u_1) [inst : Add M], AddSemigroupIdeal M = SubAddAction M M","typeReadable":"∀ (M : Type u_1) [inst : Add M], AddSemigroupIdeal M = SubAddAction M M","typeReferences":[["SubAddAction"],["Add"],["Add","toVAdd"],["AddSemigroupIdeal"],["Eq"]],"valueReferences":[["Eq","refl"],["AddSemigroupIdeal"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","mul_mem"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3 : Mul.{u_1} M] (I : SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) (x : M) {y : M}, (Membership.mem.{u_1, u_1} M (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3))) I y) -> (Membership.mem.{u_1, u_1} M (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3))) I (HMul.hMul.{u_1, u_1, u_1} M M M (instHMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) x y))","typeFull":"∀ {M : Type u_1} [inst : Mul M] (I : SemigroupIdeal M) (x : M) {y : M}, y ∈ I → x * y ∈ I","typeReadable":"∀ {M : Type u_1} [inst : Mul M] (I : SemigroupIdeal M) (x : M) {y : M}, y ∈ I → x * y ∈ I","typeReferences":[["SubMulAction","instSetLike"],["SemigroupIdeal"],["SetLike","instMembership"],["Membership","mem"],["Mul"],["instHMul"],["HMul","hMul"],["instSMulOfMul"]],"valueReferences":[["instSMulOfMul"],["SubMulAction","smul_mem"]]},{"isProp":false,"kind":"definition","name":["AddSemigroupIdeal","FG"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 : Add.{u_1} M], (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3) -> Prop","typeFull":"{M : Type u_1} → [inst : Add M] → AddSemigroupIdeal M → Prop","typeReadable":"{M : Type u_1} → [inst : Add M] → AddSemigroupIdeal M → Prop","typeReferences":[["Add"],["AddSemigroupIdeal"]],"valueReferences":[["AddSemigroupIdeal","closure"],["Exists"],["Set"],["And"],["AddSemigroupIdeal"],["Eq"],["Set","Finite"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","coe_closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3 : Semigroup.{u_1} M] {s : Set.{u_1} M}, Eq.{succ u_1} (Set.{u_1} M) (SetLike.coe.{u_1, u_1} (SemigroupIdeal.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3)) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3))) (SemigroupIdeal.closure.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3) s)) (Union.union.{u_1} (Set.{u_1} M) (Set.instUnion.{u_1} M) s (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} M) (Set.{u_1} M) (Set.{u_1} M) (instHMul.{u_1} (Set.{u_1} M) (Set.mul.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3))) (Set.univ.{u_1} M) s))","typeFull":"∀ {M : Type u_1} [inst : Semigroup M] {s : Set M}, ↑(SemigroupIdeal.closure s) = s ∪ Set.univ * s","typeReadable":"∀ {M : Type u_1} [inst : Semigroup M] {s : Set M}, ↑(SemigroupIdeal.closure s) = s ∪ Set.univ * s","typeReferences":[["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["Set"],["instSMulOfMul"],["HMul","hMul"],["Semigroup"],["Union","union"],["Semigroup","toMul"],["Set","univ"],["Set","mul"],["SemigroupIdeal"],["SetLike","coe"],["instHMul"],["Eq"],["Set","instUnion"]],"valueReferences":[["PartialOrder","toPreorder"],["Eq","trans"],["Membership","mem"],["HMul","hMul"],["Union","union"],["SubMulAction"],["Set","mem_univ"],["Set","mul"],["SemigroupIdeal","closure_le"],["Or"],["LE","le","antisymm"],["Eq","ndrec"],["Set","instUnion"],["_private","Mathlib","Algebra","Group","Ideal",0,"SemigroupIdeal","coe_closure","_simp_1_1"],["Exists"],["SetLike","instMembership"],["And"],["Set","instMembership"],["Exists","casesOn"],["SetLike","coe"],["SemigroupIdeal","mul_mem"],["HasSubset","Subset"],["Eq","refl"],["Iff","mpr"],["HSMul","hSMul"],["Set","mul_mem_mul"],["SubMulAction","mk"],["id"],["instHMul"],["Eq","mpr"],["SemigroupIdeal","closure"],["Or","elim"],["Or","inr"],["SemigroupIdeal","mem_closure_of_mem"],["Semigroup","toMul"],["congrArg"],["Or","inl"],["instHSMul"],["Eq"],["Preorder","toLE"],["SubMulAction","instSetLike"],["Set"],["instSMulOfMul"],["Set","mem_union","_simp_1"],["SubMulAction","instPartialOrder"],["Set","univ"],["Or","casesOn"],["Set","instHasSubset"],["SemigroupIdeal"],["LE","le"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Ideal",0,"SemigroupIdeal","coe_closure","_simp_1_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3 : Semigroup.{u_1} G] (a : G) (b : G) (c : G), Eq.{succ u_1} G (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) b c)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) (HMul.hMul.{u_1, u_1, u_1} G G G (instHMul.{u_1} G (Semigroup.toMul.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.2875213658._hygCtx._hyg.3)) a b) c)","typeFull":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReadable":"∀ {G : Type u_1} [inst : Semigroup G] (a b c : G), a * (b * c) = a * b * c","typeReferences":[["instHMul"],["HMul","hMul"],["Semigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["Eq","symm"],["instHMul"],["HMul","hMul"],["mul_assoc"],["Semigroup","toMul"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","mem_closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3 : Mul.{u_1} M] {s : Set.{u_1} M} {x : M}, Iff (Membership.mem.{u_1, u_1} M (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3))) (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3 s) x) (forall (p : SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3), (HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s (SetLike.coe.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3)) p)) -> (Membership.mem.{u_1, u_1} M (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1334727363._hygCtx._hyg.3))) p x))","typeFull":"∀ {M : Type u_1} [inst : Mul M] {s : Set M} {x : M},\n  x ∈ SemigroupIdeal.closure s ↔ ∀ (p : SemigroupIdeal M), s ⊆ ↑p → x ∈ p","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {s : Set M} {x : M},\n  x ∈ SemigroupIdeal.closure s ↔ ∀ (p : SemigroupIdeal M), s ⊆ ↑p → x ∈ p","typeReferences":[["Set","instHasSubset"],["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["SemigroupIdeal"],["SetLike","coe"],["SetLike","instMembership"],["HasSubset","Subset"],["Set"],["Iff"],["Membership","mem"],["Mul"],["instSMulOfMul"]],"valueReferences":[["instSMulOfMul"],["SubMulAction","mem_closure"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","mem_closure'"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3 : AddSemigroup.{u_1} M] {s : Set.{u_1} M} {x : M}, Iff (Membership.mem.{u_1, u_1} M (AddSemigroupIdeal.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)) (SetLike.instMembership.{u_1, u_1} (AddSemigroupIdeal.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)))) (AddSemigroupIdeal.closure.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3) s) x) (Or (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) 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 (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)) y z) x)))))","typeFull":"∀ {M : Type u_1} [inst : AddSemigroup M] {s : Set M} {x : M},\n  x ∈ AddSemigroupIdeal.closure s ↔ x ∈ s ∨ ∃ y, ∃ z ∈ s, y + z = x","typeReadable":"∀ {M : Type u_1} [inst : AddSemigroup M] {s : Set M} {x : M},\n  x ∈ AddSemigroupIdeal.closure s ↔ x ∈ s ∨ ∃ y, ∃ z ∈ s, y + z = x","typeReferences":[["AddSemigroupIdeal","closure"],["SetLike","instMembership"],["Exists"],["instHAdd"],["Set"],["Membership","mem"],["And"],["Set","instMembership"],["HAdd","hAdd"],["AddSemigroup"],["Or"],["Iff"],["AddSemigroupIdeal"],["Add","toVAdd"],["SubAddAction","instSetLike"],["Eq"],["AddSemigroup","toAdd"]],"valueReferences":[["Eq","trans"],["Membership","mem"],["eq_true"],["Union","union"],["Set","mem_univ"],["congrArg"],["SetLike","mem_coe"],["iff_self"],["Or"],["Set","add"],["funext"],["Eq","symm"],["congrFun'"],["Add","toVAdd"],["Eq"],["Set","instUnion"],["propext"],["AddSemigroup","toAdd"],["AddSemigroupIdeal","closure"],["Exists"],["SetLike","instMembership"],["True"],["Set"],["instHAdd"],["And"],["true_and"],["AddSemigroupIdeal","coe_closure"],["Set","instMembership"],["Set","univ"],["HAdd","hAdd"],["Set","mem_add"],["SetLike","coe"],["of_eq_true"],["Iff"],["Set","mem_union"],["id"],["AddSemigroupIdeal"],["Eq","mpr"],["SubAddAction","instSetLike"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","mem_closure'"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3 : Semigroup.{u_1} M] {s : Set.{u_1} M} {x : M}, Iff (Membership.mem.{u_1, u_1} M (SemigroupIdeal.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)) (SetLike.instMembership.{u_1, u_1} (SemigroupIdeal.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)))) (SemigroupIdeal.closure.{u_1} M (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3) s) x) (Or (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) 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 (Semigroup.toMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3146180754._hygCtx._hyg.3)) y z) x)))))","typeFull":"∀ {M : Type u_1} [inst : Semigroup M] {s : Set M} {x : M},\n  x ∈ SemigroupIdeal.closure s ↔ x ∈ s ∨ ∃ y, ∃ z ∈ s, y * z = x","typeReadable":"∀ {M : Type u_1} [inst : Semigroup M] {s : Set M} {x : M},\n  x ∈ SemigroupIdeal.closure s ↔ x ∈ s ∨ ∃ y, ∃ z ∈ s, y * z = x","typeReferences":[["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["SetLike","instMembership"],["Exists"],["Set"],["Membership","mem"],["And"],["HMul","hMul"],["instSMulOfMul"],["Semigroup"],["Semigroup","toMul"],["Set","instMembership"],["SemigroupIdeal"],["Or"],["Iff"],["instHMul"],["Eq"]],"valueReferences":[["SemigroupIdeal","closure"],["Eq","trans"],["Membership","mem"],["HMul","hMul"],["Union","union"],["_private","Mathlib","Algebra","Group","Ideal",0,"SemigroupIdeal","mem_closure'","_simp_1_1"],["Semigroup","toMul"],["congrArg"],["Set","mul"],["SetLike","mem_coe"],["iff_self"],["Or"],["funext"],["Eq","symm"],["congrFun'"],["Eq"],["Set","instUnion"],["propext"],["SubMulAction","instSetLike"],["Exists"],["SetLike","instMembership"],["True"],["Set"],["And"],["instSMulOfMul"],["Set","mem_univ","_simp_1"],["true_and"],["Set","mem_union","_simp_1"],["Set","instMembership"],["Set","univ"],["SemigroupIdeal"],["SetLike","coe"],["of_eq_true"],["Iff"],["id"],["instHMul"],["Eq","mpr"],["SemigroupIdeal","coe_closure"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","closure_mono"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3 : Add.{u_1} M] {s : Set.{u_1} M} {t : Set.{u_1} M}, (HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s t) -> (LE.le.{u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3) (Preorder.toLE.{u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3) (SubAddAction.instPartialOrder.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3)))) (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3 s) (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3 t))","typeFull":"∀ {M : Type u_1} [inst : Add M] {s t : Set M}, s ⊆ t → AddSemigroupIdeal.closure s ≤ AddSemigroupIdeal.closure t","typeReadable":"∀ {M : Type u_1} [inst : Add M] {s t : Set M}, s ⊆ t → AddSemigroupIdeal.closure s ≤ AddSemigroupIdeal.closure t","typeReferences":[["Set","instHasSubset"],["AddSemigroupIdeal","closure"],["PartialOrder","toPreorder"],["HasSubset","Subset"],["Set"],["Add"],["LE","le"],["SubAddAction","instPartialOrder"],["Add","toVAdd"],["AddSemigroupIdeal"],["Preorder","toLE"]],"valueReferences":[["SubAddAction","closure_mono"],["Add","toVAdd"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","closure_mono"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3 : Mul.{u_1} M] {s : Set.{u_1} M} {t : Set.{u_1} M}, (HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s t) -> (LE.le.{u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3) (Preorder.toLE.{u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3) (PartialOrder.toPreorder.{u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3) (SubMulAction.instPartialOrder.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3)))) (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3 s) (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1205987714._hygCtx._hyg.3 t))","typeFull":"∀ {M : Type u_1} [inst : Mul M] {s t : Set M}, s ⊆ t → SemigroupIdeal.closure s ��� SemigroupIdeal.closure t","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {s t : Set M}, s ⊆ t → SemigroupIdeal.closure s ≤ SemigroupIdeal.closure t","typeReferences":[["SemigroupIdeal","closure"],["Set","instHasSubset"],["SemigroupIdeal"],["PartialOrder","toPreorder"],["HasSubset","Subset"],["Set"],["LE","le"],["Mul"],["instSMulOfMul"],["Preorder","toLE"],["SubMulAction","instPartialOrder"]],"valueReferences":[["SubMulAction","closure_mono"],["instSMulOfMul"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","closure","eq_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3 : Add.{u_1} M] (s : Set.{u_1} M), Eq.{succ u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3) (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3 s) (SubAddAction.closure.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3) s)","typeFull":"∀ {M : Type u_1} [inst : Add M] (s : Set M), AddSemigroupIdeal.closure s = SubAddAction.closure M s","typeReadable":"∀ {M : Type u_1} [inst : Add M] (s : Set M), AddSemigroupIdeal.closure s = SubAddAction.closure M s","typeReferences":[["SubAddAction","closure"],["AddSemigroupIdeal","closure"],["Set"],["Add"],["Add","toVAdd"],["AddSemigroupIdeal"],["Eq"]],"valueReferences":[["AddSemigroupIdeal","closure"],["Eq","refl"],["AddSemigroupIdeal"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","subset_closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3 : Mul.{u_1} M] {s : Set.{u_1} M}, HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s (SetLike.coe.{u_1, u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3) M (SubMulAction.instSetLike.{u_1, u_1} M M (instSMulOfMul.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3)) (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3 s))","typeFull":"∀ {M : Type u_1} [inst : Mul M] {s : Set M}, s ⊆ ↑(SemigroupIdeal.closure s)","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {s : Set M}, s ⊆ ↑(SemigroupIdeal.closure s)","typeReferences":[["SemigroupIdeal","closure"],["Set","instHasSubset"],["SubMulAction","instSetLike"],["SemigroupIdeal"],["SetLike","coe"],["HasSubset","Subset"],["Set"],["Mul"],["instSMulOfMul"]],"valueReferences":[["SubMulAction","subset_closure"],["instSMulOfMul"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","coe_closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3 : AddSemigroup.{u_1} M] {s : Set.{u_1} M}, Eq.{succ u_1} (Set.{u_1} M) (SetLike.coe.{u_1, u_1} (AddSemigroupIdeal.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3)) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3))) (AddSemigroupIdeal.closure.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3) s)) (Union.union.{u_1} (Set.{u_1} M) (Set.instUnion.{u_1} M) s (HAdd.hAdd.{u_1, u_1, u_1} (Set.{u_1} M) (Set.{u_1} M) (Set.{u_1} M) (instHAdd.{u_1} (Set.{u_1} M) (Set.add.{u_1} M (AddSemigroup.toAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3499520070._hygCtx._hyg.3))) (Set.univ.{u_1} M) s))","typeFull":"∀ {M : Type u_1} [inst : AddSemigroup M] {s : Set M}, ↑(AddSemigroupIdeal.closure s) = s ∪ (Set.univ + s)","typeReadable":"∀ {M : Type u_1} [inst : AddSemigroup M] {s : Set M}, ↑(AddSemigroupIdeal.closure s) = s ∪ (Set.univ + s)","typeReferences":[["AddSemigroupIdeal","closure"],["Set"],["instHAdd"],["Union","union"],["Set","univ"],["HAdd","hAdd"],["AddSemigroup"],["SetLike","coe"],["Set","add"],["AddSemigroupIdeal"],["Add","toVAdd"],["SubAddAction","instSetLike"],["Eq"],["Set","instUnion"],["AddSemigroup","toAdd"]],"valueReferences":[["SubAddAction"],["PartialOrder","toPreorder"],["AddSemigroupIdeal","closure_le"],["Eq","trans"],["Membership","mem"],["Union","union"],["Set","mem_univ"],["Or"],["Eq","symm"],["LE","le","antisymm"],["Eq","ndrec"],["Set","instUnion"],["AddSemigroup","toAdd"],["AddSemigroupIdeal","closure"],["SetLike","instMembership"],["Exists"],["instHVAdd"],["And"],["Set","instMembership"],["Exists","casesOn"],["SetLike","coe"],["HasSubset","Subset"],["Eq","refl"],["Set","add_mem_add"],["Iff","mpr"],["Set","mem_union"],["id"],["Eq","mpr"],["SubAddAction","instSetLike"],["AddSemigroupIdeal","add_mem"],["Or","elim"],["Or","inr"],["congrArg"],["SubAddAction","mk"],["Or","inl"],["Set","add"],["SubAddAction","instPartialOrder"],["Add","toVAdd"],["Eq"],["Preorder","toLE"],["propext"],["AddSemigroupIdeal","mem_closure_of_mem"],["Set"],["instHAdd"],["HVAdd","hVAdd"],["HAdd","hAdd"],["Set","univ"],["Or","casesOn"],["Set","instHasSubset"],["add_assoc"],["LE","le"],["AddSemigroupIdeal"],["And","casesOn"]]},{"isProp":false,"kind":"definition","name":["SemigroupIdeal","FG"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 : Mul.{u_1} M], (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3) -> Prop","typeFull":"{M : Type u_1} → [inst : Mul M] → SemigroupIdeal M → Prop","typeReadable":"{M : Type u_1} → [inst : Mul M] → SemigroupIdeal M → Prop","typeReferences":[["SemigroupIdeal"],["Mul"]],"valueReferences":[["SemigroupIdeal","closure"],["SemigroupIdeal"],["Exists"],["Set"],["And"],["Eq"],["Set","Finite"]]},{"isProp":false,"kind":"definition","name":["AddSemigroupIdeal","closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3 : Add.{u_1} M], (Set.{u_1} M) -> (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3)","typeFull":"{M : Type u_1} → [inst : Add M] → Set M → AddSemigroupIdeal M","typeReadable":"{M : Type u_1} → [inst : Add M] → Set M → AddSemigroupIdeal M","typeReferences":[["Set"],["Add"],["AddSemigroupIdeal"]],"valueReferences":[["SubAddAction","closure"],["Add","toVAdd"]]},{"isProp":false,"kind":"definition","name":["SemigroupIdeal","closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3 : Mul.{u_1} M], (Set.{u_1} M) -> (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2641984536._hygCtx._hyg.3)","typeFull":"{M : Type u_1} → [inst : Mul M] → Set M → SemigroupIdeal M","typeReadable":"{M : Type u_1} → [inst : Mul M] → Set M → SemigroupIdeal M","typeReferences":[["SemigroupIdeal"],["Set"],["Mul"]],"valueReferences":[["instSMulOfMul"],["SubMulAction","closure"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","mem_closure''"],"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}, Iff (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"],["Iff"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["Eq"]],"valueReferences":[["SemigroupIdeal","closure"],["MulOneClass","toMulOne"],["Eq","trans"],["Membership","mem"],["HMul","hMul"],["congrArg"],["Set","mul"],["MulOne","toMul"],["SetLike","mem_coe"],["iff_self"],["funext"],["Monoid","toMulOneClass"],["Eq","symm"],["_private","Mathlib","Algebra","Group","Ideal",0,"SemigroupIdeal","mem_closure''","_simp_1_1"],["congrFun'"],["Eq"],["propext"],["SubMulAction","instSetLike"],["Exists"],["SetLike","instMembership"],["True"],["Set"],["And"],["instSMulOfMul"],["Set","mem_univ","_simp_1"],["true_and"],["SemigroupIdeal","coe_closure'"],["Set","instMembership"],["Set","univ"],["SemigroupIdeal"],["SetLike","coe"],["of_eq_true"],["Iff"],["id"],["instHMul"],["Eq","mpr"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","fg_iff"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3 : Add.{u_1} M] {I : AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3}, Iff (AddSemigroupIdeal.FG.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3 I) (Exists.{succ u_1} (Finset.{u_1} M) (fun (s : Finset.{u_1} M) => Eq.{succ u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3) I (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Finset.{u_1} M) M (Finset.instSetLike.{u_1} M) s))))","typeFull":"∀ {M : Type u_1} [inst : Add M] {I : AddSemigroupIdeal M}, I.FG ↔ ∃ s, I = AddSemigroupIdeal.closure ↑s","typeReadable":"∀ {M : Type u_1} [inst : Add M] {I : AddSemigroupIdeal M}, I.FG ↔ ∃ s, I = AddSemigroupIdeal.closure ↑s","typeReferences":[["Finset","instSetLike"],["AddSemigroupIdeal","closure"],["SetLike","coe"],["Finset"],["Exists"],["Add"],["Iff"],["AddSemigroupIdeal"],["Eq"],["AddSemigroupIdeal","FG"]],"valueReferences":[["Add","toVAdd"],["SubAddAction","fg_iff"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","FG","eq_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 : Mul.{u_1} M] (I : SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3), Eq.{1} Prop (SemigroupIdeal.FG.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 I) (Exists.{succ u_1} (Set.{u_1} M) (fun (s : Set.{u_1} M) => And (Set.Finite.{u_1} M s) (Eq.{succ u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3) I (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.882695844._hygCtx._hyg.3 s))))","typeFull":"∀ {M : Type u_1} [inst : Mul M] (I : SemigroupIdeal M), I.FG = ∃ s, s.Finite ∧ I = SemigroupIdeal.closure s","typeReadable":"∀ {M : Type u_1} [inst : Mul M] (I : SemigroupIdeal M), I.FG = ∃ s, s.Finite ∧ I = SemigroupIdeal.closure s","typeReferences":[["SemigroupIdeal","closure"],["SemigroupIdeal"],["Exists"],["SemigroupIdeal","FG"],["Set"],["Mul"],["And"],["Set","Finite"],["Eq"]],"valueReferences":[["SemigroupIdeal","FG"],["Eq","refl"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","fg_iff"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3 : Mul.{u_1} M] {I : SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3}, Iff (SemigroupIdeal.FG.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3 I) (Exists.{succ u_1} (Finset.{u_1} M) (fun (s : Finset.{u_1} M) => Eq.{succ u_1} (SemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3) I (SemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.3946582115._hygCtx._hyg.3 (SetLike.coe.{u_1, u_1} (Finset.{u_1} M) M (Finset.instSetLike.{u_1} M) s))))","typeFull":"∀ {M : Type u_1} [inst : Mul M] {I : SemigroupIdeal M}, I.FG ↔ ∃ s, I = SemigroupIdeal.closure ↑s","typeReadable":"∀ {M : Type u_1} [inst : Mul M] {I : SemigroupIdeal M}, I.FG ↔ ∃ s, I = SemigroupIdeal.closure ↑s","typeReferences":[["SemigroupIdeal","closure"],["Finset","instSetLike"],["SemigroupIdeal"],["SetLike","coe"],["Finset"],["Exists"],["SemigroupIdeal","FG"],["Iff"],["Mul"],["Eq"]],"valueReferences":[["SubMulAction","fg_iff"],["instSMulOfMul"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","subset_closure"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3 : Add.{u_1} M] {s : Set.{u_1} M}, HasSubset.Subset.{u_1} (Set.{u_1} M) (Set.instHasSubset.{u_1} M) s (SetLike.coe.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3)) (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.4067780068._hygCtx._hyg.3 s))","typeFull":"∀ {M : Type u_1} [inst : Add M] {s : Set M}, s ⊆ ↑(AddSemigroupIdeal.closure s)","typeReadable":"∀ {M : Type u_1} [inst : Add M] {s : Set M}, s ⊆ ↑(AddSemigroupIdeal.closure s)","typeReferences":[["Set","instHasSubset"],["AddSemigroupIdeal","closure"],["SetLike","coe"],["HasSubset","Subset"],["Set"],["Add"],["Add","toVAdd"],["AddSemigroupIdeal"],["SubAddAction","instSetLike"]],"valueReferences":[["Add","toVAdd"],["SubAddAction","subset_closure"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","mem_closure_of_mem"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3 : Add.{u_1} M] {s : Set.{u_1} M} {x : M}, (Membership.mem.{u_1, u_1} M (Set.{u_1} M) (Set.instMembership.{u_1} M) s x) -> (Membership.mem.{u_1, u_1} M (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3))) (AddSemigroupIdeal.closure.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1375402882._hygCtx._hyg.3 s) x)","typeFull":"∀ {M : Type u_1} [inst : Add M] {s : Set M} {x : M}, x ∈ s → x ∈ AddSemigroupIdeal.closure s","typeReadable":"∀ {M : Type u_1} [inst : Add M] {s : Set M} {x : M}, x ∈ s → x ∈ AddSemigroupIdeal.closure s","typeReferences":[["AddSemigroupIdeal","closure"],["SetLike","instMembership"],["Set"],["Add"],["Membership","mem"],["Add","toVAdd"],["AddSemigroupIdeal"],["SubAddAction","instSetLike"],["Set","instMembership"]],"valueReferences":[["Add","toVAdd"],["SubAddAction","mem_closure_of_mem"]]},{"isProp":true,"kind":"theorem","name":["SemigroupIdeal","coe_closure'"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.2261310793._hygCtx._hyg.3 : Monoid.{u_1} M] {s : Set.{u_1} M}, Eq.{succ u_1} (Set.{u_1} M) (SetLike.coe.{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.2261310793._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.2261310793._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.2261310793._hygCtx._hyg.3))) s)) (HMul.hMul.{u_1, u_1, u_1} (Set.{u_1} M) (Set.{u_1} M) (Set.{u_1} M) (instHMul.{u_1} (Set.{u_1} M) (Set.mul.{u_1} M (MulOne.toMul.{u_1} M (MulOneClass.toMulOne.{u_1} M (Monoid.toMulOneClass.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.2261310793._hygCtx._hyg.3))))) (Set.univ.{u_1} M) s)","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {s : Set M}, ↑(SemigroupIdeal.closure s) = Set.univ * s","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {s : Set M}, ↑(SemigroupIdeal.closure s) = Set.univ * s","typeReferences":[["MulOneClass","toMulOne"],["SemigroupIdeal","closure"],["SubMulAction","instSetLike"],["Set"],["instSMulOfMul"],["HMul","hMul"],["Set","univ"],["Set","mul"],["SemigroupIdeal"],["MulOne","toMul"],["SetLike","coe"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["Eq"]],"valueReferences":[["SemigroupIdeal","closure"],["MulOneClass","toMulOne"],["Eq","trans"],["Membership","mem"],["Exists","intro"],["HMul","hMul"],["Union","union"],["Set","mem_univ"],["Semigroup","toMul"],["congrArg"],["Set","mul"],["And","intro"],["MulOne","toMul"],["Monoid","toMulOneClass"],["congrFun'"],["Monoid","toSemigroup"],["Eq"],["Set","instUnion"],["propext"],["SubMulAction","instSetLike"],["Exists"],["True"],["MulOne","toOne"],["Set"],["And"],["instSMulOfMul"],["OfNat","ofNat"],["Set","instMembership"],["Set","univ"],["Set","instHasSubset"],["eq_self"],["SemigroupIdeal"],["SetLike","coe"],["of_eq_true"],["One","toOfNat1"],["HasSubset","Subset"],["Set","union_eq_right"],["id"],["instHMul"],["Eq","mpr"],["SemigroupIdeal","coe_closure"],["one_mul"]]},{"isProp":true,"kind":"theorem","name":["AddSemigroupIdeal","add_mem"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3 : Add.{u_1} M] (I : AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) (x : M) {y : M}, (Membership.mem.{u_1, u_1} M (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3))) I y) -> (Membership.mem.{u_1, u_1} M (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) (SetLike.instMembership.{u_1, u_1} (AddSemigroupIdeal.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) M (SubAddAction.instSetLike.{u_1, u_1} M M (Add.toVAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3))) I (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M inst._@.Mathlib.Algebra.Group.Ideal.1358877491._hygCtx._hyg.3) x y))","typeFull":"∀ {M : Type u_1} [inst : Add M] (I : AddSemigroupIdeal M) (x : M) {y : M}, y ∈ I → x + y ∈ I","typeReadable":"∀ {M : Type u_1} [inst : Add M] (I : AddSemigroupIdeal M) (x : M) {y : M}, y ∈ I → x + y ∈ I","typeReferences":[["HAdd","hAdd"],["SetLike","instMembership"],["instHAdd"],["Add"],["Membership","mem"],["Add","toVAdd"],["AddSemigroupIdeal"],["SubAddAction","instSetLike"]],"valueReferences":[["SubAddAction","vadd_mem"],["Add","toVAdd"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Irreducible.Lemmas.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Even","not_addIrreducible"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1006633933._hygCtx._hyg.5 : AddMonoid.{u_2} M] {x : M}, (Even.{u_2} M (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1006633933._hygCtx._hyg.5))) x) -> (Not (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1006633933._hygCtx._hyg.5 x))","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {x : M}, Even x → ¬AddIrreducible x","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {x : M}, Even x → ¬AddIrreducible x","typeReferences":[["Not"],["AddIrreducible"],["Even"],["AddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddIrreducible","not_even"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_units","_simp_1_2"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Units.Defs.3091346513._hygCtx._hyg.5 : Monoid.{u_1} M] (a : M) (u : Units.{u_1} M inst._@.Mathlib.Algebra.Group.Units.Defs.3091346513._hygCtx._hyg.5), Eq.{1} Prop (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Units.Defs.3091346513._hygCtx._hyg.5 (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.Units.Defs.3091346513._hygCtx._hyg.5)))) a (Units.val.{u_1} M inst._@.Mathlib.Algebra.Group.Units.Defs.3091346513._hygCtx._hyg.5 u))) (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Units.Defs.3091346513._hygCtx._hyg.5 a)","typeFull":"∀ {M : Type u_1} [inst : Monoid M] (a : M) (u : Mˣ), IsUnit (a * ↑u) = IsUnit a","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] (a : M) (u : Mˣ), IsUnit (a * ↑u) = IsUnit a","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","val"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"],["Eq"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","isUnit_mul_units"],["Units","val"],["Monoid","toMulOneClass"],["instHMul"],["HMul","hMul"],["IsUnit"],["propext"]]},{"isProp":true,"kind":"theorem","name":["AddEquiv","addIrreducible_iff"],"typeFallback":"forall {F : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 : AddMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 : AddMonoid.{u_3} N] {x : M} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14 : EquivLike.{succ u_1, succ u_2, succ u_3} F M N] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.19 : AddEquivClass.{u_1, u_2, u_3} F M N (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5))) (AddZero.toAdd.{u_3} N (AddZeroClass.toAddZero.{u_3} N (AddMonoid.toAddZeroClass.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8))) inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14] (f : F), Iff (AddIrreducible.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 (DFunLike.coe.{succ u_1, succ u_2, succ u_3} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (EquivLike.toFunLike.{succ u_1, succ u_2, succ u_3} F M N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14) f x)) (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 x)","typeFull":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : AddMonoid M] [inst_1 : AddMonoid N] {x : M}\n  [inst_2 : EquivLike F M N] [AddEquivClass F M N] (f : F), AddIrreducible (f x) ↔ AddIrreducible x","typeReadable":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : AddMonoid M] [inst_1 : AddMonoid N] {x : M}\n  [inst_2 : EquivLike F M N] [AddEquivClass F M N] (f : F), AddIrreducible (f x) ↔ AddIrreducible x","typeReferences":[["AddIrreducible"],["Iff"],["EquivLike","toFunLike"],["AddEquivClass"],["AddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["DFunLike","coe"],["EquivLike"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["implies_congr"],["Eq","trans"],["AddIrreducible"],["AddEquivClass","instAddMonoidHomClass"],["DFunLike","coe"],["congrArg"],["iff_self"],["Or"],["congr"],["EquivLike","toFunLike"],["forall_congr"],["Eq","symm"],["AddEquiv","isAddUnit_map"],["congrFun'"],["Eq"],["AddMonoidHomClass","toAddHomClass"],["propext"],["Not"],["True"],["instHAdd"],["And"],["IsAddUnit"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["EquivLike","surjective"],["map_add"],["HAdd","hAdd"],["EmbeddingLike","apply_eq_iff_eq"],["Function","Surjective","forall"],["of_eq_true"],["Eq","refl"],["Iff"],["EquivLike","toEmbeddingLike"],["addIrreducible_iff"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["irreducible_mul_units"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 : Monoid.{u_2} M] {y : M} (u : Units.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5), Iff (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} M M M (instHMul.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5)))) y (Units.val.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 u))) (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 y)","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {y : M} (u : Mˣ), Irreducible (y * ↑u) ↔ Irreducible y","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {y : M} (u : Mˣ), Irreducible (y * ↑u) ↔ Irreducible y","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Irreducible"],["Units","val"],["Iff"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["HMul","hMul"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["Eq","trans"],["Units","instInv"],["HMul","hMul"],["mul_assoc"],["Semigroup","toMul"],["congrArg"],["Iff","intro"],["MulOne","toMul"],["Irreducible"],["Units","mul_inv_cancel_right"],["Or"],["congr"],["Monoid","toMulOneClass"],["Eq","symm"],["congrFun'"],["Monoid","toSemigroup"],["Eq"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_units","_simp_1_1"],["Units"],["propext"],["Not"],["Inv","inv"],["And"],["IsUnit"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_units","_simp_1_3"],["Units","isUnit_mul_units"],["Units","val"],["Eq","refl"],["Iff"],["id"],["instHMul"],["Eq","mpr"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_units","_simp_1_2"]]},{"isProp":true,"kind":"theorem","name":["addIrreducible_isAddUnit_add"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 : AddMonoid.{u_2} M] {x : M} {y : M}, (IsAddUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x) -> (Iff (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} M M M (instHAdd.{u_2} M (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5)))) x y)) (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 y))","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {x y : M}, IsAddUnit x → (AddIrreducible (x + y) ↔ AddIrreducible y)","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {x y : M}, IsAddUnit x → (AddIrreducible (x + y) ↔ AddIrreducible y)","typeReferences":[["HAdd","hAdd"],["AddIrreducible"],["instHAdd"],["Iff"],["AddMonoid"],["IsAddUnit"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["HAdd","hAdd"],["addIrreducible_addUnits_add"],["AddIrreducible"],["instHAdd"],["Iff"],["AddUnits","val"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"addIrreducible_isAddUnit_add","match_1_1"],["Eq","rec"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["AddIrreducible","not_even"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.562513752._hygCtx._hyg.5 : AddMonoid.{u_2} M] {x : M}, (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.562513752._hygCtx._hyg.5 x) -> (Not (Even.{u_2} M (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.562513752._hygCtx._hyg.5))) x))","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {x : M}, AddIrreducible x → ¬Even x","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {x : M}, AddIrreducible x → ¬Even x","typeReferences":[["Not"],["AddIrreducible"],["Even"],["AddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["even_iff_exists_two_nsmul"],["Bool"],["AddIrreducible"],["Decidable","decide"],["congrArg"],["instOfNatNat"],["Even"],["Eq","symm"],["instHSMul"],["Eq"],["Eq","ndrec"],["Bool","true"],["propext"],["of_decide_eq_true"],["Not"],["Exists"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["OfNat","ofNat"],["Exists","casesOn"],["Nat"],["AddMonoid","toNSMul"],["Eq","refl"],["HSMul","hSMul"],["id"],["instDecidableEqNat"],["False"],["Eq","mpr"],["not_addIrreducible_nsmul"],["Ne"],["instDecidableNot"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Irreducible","of_map"],"typeFallback":"forall {F : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.5 : Monoid.{u_2} M] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.8 : Monoid.{u_3} N] {f : F} {x : M} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.14 : FunLike.{succ u_1, succ u_2, succ u_3} F M N] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.19 : MonoidHomClass.{u_1, u_2, u_3} F M N (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.5)) (MulOneClass.toMulOne.{u_3} N (Monoid.toMulOneClass.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.8)) inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.14] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.24 : IsLocalHom.{u_2, u_3, u_1} M N F inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.14 f], (Irreducible.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.8 (DFunLike.coe.{succ u_1, succ u_2, succ u_3} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.14 f x)) -> (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1196463279._hygCtx._hyg.5 x)","typeFull":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Monoid M] [inst_1 : Monoid N] {f : F} {x : M}\n  [inst_2 : FunLike F M N] [MonoidHomClass F M N] [IsLocalHom f], Irreducible (f x) → Irreducible x","typeReadable":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Monoid M] [inst_1 : Monoid N] {f : F} {x : M}\n  [inst_2 : FunLike F M N] [MonoidHomClass F M N] [IsLocalHom f], Irreducible (f x) → Irreducible x","typeReferences":[["MulOneClass","toMulOne"],["FunLike"],["Irreducible"],["MonoidHomClass"],["IsLocalHom"],["Monoid","toMulOneClass"],["Monoid"],["DFunLike","coe"]],"valueReferences":[["MulOneClass","toMulOne"],["MonoidHomClass","toMulHomClass"],["Irreducible","not_isUnit"],["HMul","hMul"],["IsUnit"],["DFunLike","coe"],["Irreducible","mk"],["Irreducible","isUnit_or_isUnit"],["IsUnit","map"],["MulOne","toMul"],["Irreducible"],["Or"],["IsUnit","of_map"],["Or","imp"],["Monoid","toMulOneClass"],["Eq","symm"],["map_mul"],["instHMul"],["Eq","ndrec"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_addIrreducible_nsmul","match_1_1"],"typeFallback":"forall (motive : forall (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 : Nat), (Ne.{1} Nat x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) -> Prop) (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 : Nat) (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.37.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.59 : Ne.{1} Nat x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))), (forall (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.66 : Ne.{1} Nat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))), motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.66) -> (forall (n : Nat) (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.79 : Ne.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))), motive (Nat.succ (Nat.succ n)) x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.79) -> (motive x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.37.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.59)","typeFull":"∀ (motive : (x : ℕ) → x ≠ 1 → Prop) (x : ℕ) (x_1 : x ≠ 1),\n  (∀ (x : 0 ≠ 1), motive 0 x) → (∀ (n : ℕ) (x : n + 2 ≠ 1), motive n.succ.succ x) → motive x x_1","typeReadable":"∀ (motive : (x : ℕ) → x ≠ 1 → Prop) (x : ℕ) (x_1 : x ≠ 1),\n  (∀ (x : 0 ≠ 1), motive 0 x) → (∀ (n : ℕ) (x : n + 2 ≠ 1), motive n.succ.succ x) → motive x x_1","typeReferences":[["instAddNat"],["HAdd","hAdd"],["Nat"],["Nat","succ"],["instHAdd"],["instOfNatNat"],["Ne"],["OfNat","ofNat"]],"valueReferences":[["absurd"],["Nat"],["False","elim"],["Nat","casesOn"],["Nat","succ"],["instOfNatNat"],["Eq","refl"],["False"],["Ne"],["Eq"],["Nat","zero"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["irreducible_units_mul"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 : Monoid.{u_2} M] {y : M} (u : Units.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5), Iff (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} M M M (instHMul.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5)))) (Units.val.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 u) y)) (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 y)","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {y : M} (u : Mˣ), Irreducible (↑u * y) ↔ Irreducible y","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {y : M} (u : Mˣ), Irreducible (↑u * y) ↔ Irreducible y","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Irreducible"],["Units","val"],["Iff"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["HMul","hMul"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_units_mul","_simp_1_2"],["Eq","trans"],["Units","instInv"],["HMul","hMul"],["mul_assoc"],["Semigroup","toMul"],["congrArg"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_units_mul","_simp_1_1"],["Iff","intro"],["MulOne","toMul"],["Irreducible"],["Or"],["Units","inv_mul_cancel_left"],["congr"],["Monoid","toMulOneClass"],["Eq","symm"],["congrFun'"],["Monoid","toSemigroup"],["Eq"],["Units"],["propext"],["Not"],["Inv","inv"],["And"],["IsUnit"],["Units","isUnit_units_mul"],["Units","val"],["Eq","refl"],["Iff"],["id"],["instHMul"],["Eq","mpr"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_units_mul","_simp_1_3"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"addIrreducible_add_isAddUnit","match_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 : AddMonoid.{u_1} M] {x : M} (motive : (IsAddUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x) -> Prop) (h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.34 : IsAddUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x), (forall (x_1 : AddUnits.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5) (hx : Eq.{succ u_1} M (AddUnits.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x_1) x), motive (Exists.intro.{succ u_1} (AddUnits.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5) (fun (u : AddUnits.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5) => Eq.{succ u_1} M (AddUnits.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 u) x) x_1 hx)) -> (motive h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.34)","typeFull":"∀ {M : Type u_1} [inst : AddMonoid M] {x : M} (motive : IsAddUnit x → Prop) (h : IsAddUnit x),\n  (∀ (x_1 : AddUnits M) (hx : ↑x_1 = x), motive ⋯) → motive h","typeReadable":"∀ {M : Type u_1} [inst : AddMonoid M] {x : M} (motive : IsAddUnit x → Prop) (h : IsAddUnit x),\n  (∀ (x_1 : AddUnits M) (hx : ↑x_1 = x), motive ⋯) → motive h","typeReferences":[["AddUnits"],["AddUnits","val"],["Exists","intro"],["AddMonoid"],["IsAddUnit"],["Eq"]],"valueReferences":[["Exists","casesOn"],["AddUnits"],["AddUnits","val"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"MulEquiv","irreducible_iff","_simp_1_2"],"typeFallback":"forall {M : Type.{u_4}} {N : Type.{u_5}} {F : Type.{u_9}} [inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.11 : Mul.{u_4} M] [inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.14 : Mul.{u_5} N] [inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.17 : FunLike.{succ u_9, succ u_4, succ u_5} F M N] [inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.22 : MulHomClass.{u_9, u_4, u_5} F M N inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.17] (f : F) (x : M) (y : M), Eq.{succ u_5} N (HMul.hMul.{u_5, u_5, u_5} N N N (instHMul.{u_5} N inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.14) (DFunLike.coe.{succ u_9, succ u_4, succ u_5} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.17 f x) (DFunLike.coe.{succ u_9, succ u_4, succ u_5} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.17 f y)) (DFunLike.coe.{succ u_9, succ u_4, succ u_5} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.17 f (HMul.hMul.{u_4, u_4, u_4} M M M (instHMul.{u_4} M inst._@.Mathlib.Algebra.Group.Hom.Defs.2143170598._hygCtx._hyg.11) x y))","typeFull":"∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]\n  [MulHomClass F M N] (f : F) (x y : M), f x * f y = f (x * y)","typeReadable":"∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : Mul M] [inst_1 : Mul N] [inst_2 : FunLike F M N]\n  [MulHomClass F M N] (f : F) (x y : M), f x * f y = f (x * y)","typeReferences":[["FunLike"],["Mul"],["instHMul"],["HMul","hMul"],["Eq"],["DFunLike","coe"],["MulHomClass"]],"valueReferences":[["map_mul"],["Eq","symm"],["instHMul"],["HMul","hMul"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["IsSquare","not_irreducible"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1006633933._hygCtx._hyg.5 : Monoid.{u_2} M] {x : M}, (IsSquare.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1006633933._hygCtx._hyg.5))) x) -> (Not (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1006633933._hygCtx._hyg.5 x))","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {x : M}, IsSquare x → ¬Irreducible x","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {x : M}, IsSquare x → ¬Irreducible x","typeReferences":[["Not"],["MulOneClass","toMulOne"],["IsSquare"],["Irreducible"],["MulOne","toMul"],["Monoid","toMulOneClass"],["Monoid"]],"valueReferences":[["Irreducible","not_isSquare"]]},{"isProp":true,"kind":"theorem","name":["AddIrreducible","map"],"typeFallback":"forall {F : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 : AddMonoid.{u_2} M] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 : AddMonoid.{u_3} N] {x : M} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14 : EquivLike.{succ u_1, succ u_2, succ u_3} F M N] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.19 : AddEquivClass.{u_1, u_2, u_3} F M N (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5))) (AddZero.toAdd.{u_3} N (AddZeroClass.toAddZero.{u_3} N (AddMonoid.toAddZeroClass.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8))) inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14] (f : F), (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 x) -> (AddIrreducible.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 (DFunLike.coe.{succ u_1, succ u_2, succ u_3} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (EquivLike.toFunLike.{succ u_1, succ u_2, succ u_3} F M N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14) f x))","typeFull":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : AddMonoid M] [inst_1 : AddMonoid N] {x : M}\n  [inst_2 : EquivLike F M N] [AddEquivClass F M N] (f : F), AddIrreducible x → AddIrreducible (f x)","typeReadable":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : AddMonoid M] [inst_1 : AddMonoid N] {x : M}\n  [inst_2 : EquivLike F M N] [AddEquivClass F M N] (f : F), AddIrreducible x → AddIrreducible (f x)","typeReferences":[["AddIrreducible"],["EquivLike","toFunLike"],["AddEquivClass"],["AddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["DFunLike","coe"],["EquivLike"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddEquiv","addIrreducible_iff"],["AddIrreducible"],["Iff","mpr"],["EquivLike","toFunLike"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_units","_simp_1_3"],"typeFallback":"forall {a : Prop} {b : Prop} {c : Prop}, Eq.{1} Prop (Iff (And a b) (And a c)) (a -> (Iff b c))","typeFull":"∀ {a b c : Prop}, (a ∧ b ↔ a ∧ c) = (a → (b ↔ c))","typeReadable":"∀ {a b c : Prop}, (a ∧ b ↔ a ∧ c) = (a → (b ↔ c))","typeReferences":[["Iff"],["And"],["Eq"]],"valueReferences":[["Iff"],["and_congr_right_iff"],["And"],["propext"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_irreducible_pow","match_1_1"],"typeFallback":"forall (motive : forall (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 : Nat), (Ne.{1} Nat x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) -> Prop) (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 : Nat) (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.37.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.59 : Ne.{1} Nat x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))), (forall (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.66 : Ne.{1} Nat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))), motive (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.66) -> (forall (n : Nat) (x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.79 : Ne.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))), motive (Nat.succ (Nat.succ n)) x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.79) -> (motive x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.36.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.56 x._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.37.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.59)","typeFull":"∀ (motive : (x : ℕ) → x ≠ 1 → Prop) (x : ℕ) (x_1 : x ≠ 1),\n  (∀ (x : 0 ≠ 1), motive 0 x) → (∀ (n : ℕ) (x : n + 2 ≠ 1), motive n.succ.succ x) → motive x x_1","typeReadable":"∀ (motive : (x : ℕ) → x ≠ 1 → Prop) (x : ℕ) (x_1 : x ≠ 1),\n  (∀ (x : 0 ≠ 1), motive 0 x) → (∀ (n : ℕ) (x : n + 2 ≠ 1), motive n.succ.succ x) → motive x x_1","typeReferences":[["instAddNat"],["HAdd","hAdd"],["Nat"],["Nat","succ"],["instHAdd"],["instOfNatNat"],["Ne"],["OfNat","ofNat"]],"valueReferences":[["absurd"],["Nat"],["False","elim"],["Nat","casesOn"],["Nat","succ"],["instOfNatNat"],["Eq","refl"],["False"],["Ne"],["Eq"],["Nat","zero"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["Irreducible","map"],"typeFallback":"forall {F : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 : Monoid.{u_2} M] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 : Monoid.{u_3} N] {x : M} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14 : EquivLike.{succ u_1, succ u_2, succ u_3} F M N] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.19 : MulEquivClass.{u_1, u_2, u_3} F M N (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5))) (MulOne.toMul.{u_3} N (MulOneClass.toMulOne.{u_3} N (Monoid.toMulOneClass.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8))) inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14] (f : F), (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 x) -> (Irreducible.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 (DFunLike.coe.{succ u_1, succ u_2, succ u_3} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (EquivLike.toFunLike.{succ u_1, succ u_2, succ u_3} F M N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14) f x))","typeFull":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Monoid M] [inst_1 : Monoid N] {x : M} [inst_2 : EquivLike F M N]\n  [MulEquivClass F M N] (f : F), Irreducible x → Irreducible (f x)","typeReadable":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Monoid M] [inst_1 : Monoid N] {x : M} [inst_2 : EquivLike F M N]\n  [MulEquivClass F M N] (f : F), Irreducible x → Irreducible (f x)","typeReferences":[["MulOneClass","toMulOne"],["Irreducible"],["MulOne","toMul"],["MulEquivClass"],["EquivLike","toFunLike"],["Monoid","toMulOneClass"],["Monoid"],["DFunLike","coe"],["EquivLike"]],"valueReferences":[["Irreducible"],["MulEquiv","irreducible_iff"],["Iff","mpr"],["EquivLike","toFunLike"],["DFunLike","coe"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_units_mul","_simp_1_2"],"typeFallback":"forall {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Units.Defs.709793645._hygCtx._hyg.6 : Monoid.{u_3} M] (u : Units.{u_3} M inst._@.Mathlib.Algebra.Group.Units.Defs.709793645._hygCtx._hyg.6) (a : M), Eq.{1} Prop (IsUnit.{u_3} M inst._@.Mathlib.Algebra.Group.Units.Defs.709793645._hygCtx._hyg.6 (HMul.hMul.{u_3, u_3, u_3} M M M (instHMul.{u_3} M (MulOne.toMul.{u_3} M (MulOneClass.toMulOne.{u_3} M (Monoid.toMulOneClass.{u_3} M inst._@.Mathlib.Algebra.Group.Units.Defs.709793645._hygCtx._hyg.6)))) (Units.val.{u_3} M inst._@.Mathlib.Algebra.Group.Units.Defs.709793645._hygCtx._hyg.6 u) a)) (IsUnit.{u_3} M inst._@.Mathlib.Algebra.Group.Units.Defs.709793645._hygCtx._hyg.6 a)","typeFull":"∀ {M : Type u_3} [inst : Monoid M] (u : Mˣ) (a : M), IsUnit (↑u * a) = IsUnit a","typeReadable":"∀ {M : Type u_3} [inst : Monoid M] (u : Mˣ) (a : M), IsUnit (↑u * a) = IsUnit a","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","val"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"],["Eq"],["Units"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Units","val"],["Monoid","toMulOneClass"],["instHMul"],["HMul","hMul"],["IsUnit"],["Units","isUnit_units_mul"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"MulEquiv","irreducible_iff","_simp_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 : Monoid.{u_1} M] {p : M}, Eq.{1} Prop (Irreducible.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 p) (And (Not (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 p)) (forall {{a : M}} {{b : M}}, (Eq.{succ u_1} M p (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.Irreducible.Defs.2447818553._hygCtx._hyg.3)))) a b)) -> (Or (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 a) (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 b))))","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p = (¬IsUnit p ∧ ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b)","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p = (¬IsUnit p ∧ ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b)","typeReferences":[["MulOneClass","toMulOne"],["Not"],["MulOne","toMul"],["Irreducible"],["Or"],["Monoid","toMulOneClass"],["And"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"],["Eq"]],"valueReferences":[["Not"],["MulOneClass","toMulOne"],["And"],["HMul","hMul"],["IsUnit"],["Irreducible"],["MulOne","toMul"],["Or"],["irreducible_iff"],["Monoid","toMulOneClass"],["instHMul"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_units","_simp_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 : Monoid.{u_1} M] {p : M}, Eq.{1} Prop (Irreducible.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 p) (And (Not (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 p)) (forall {{a : M}} {{b : M}}, (Eq.{succ u_1} M p (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.Irreducible.Defs.2447818553._hygCtx._hyg.3)))) a b)) -> (Or (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 a) (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 b))))","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p = (¬IsUnit p ∧ ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b)","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p = (¬IsUnit p ∧ ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b)","typeReferences":[["MulOneClass","toMulOne"],["Not"],["MulOne","toMul"],["Irreducible"],["Or"],["Monoid","toMulOneClass"],["And"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"],["Eq"]],"valueReferences":[["Not"],["MulOneClass","toMulOne"],["And"],["HMul","hMul"],["IsUnit"],["Irreducible"],["MulOne","toMul"],["Or"],["irreducible_iff"],["Monoid","toMulOneClass"],["instHMul"],["Eq"],["propext"]]},{"isProp":true,"kind":"theorem","name":["irreducible_mul_iff"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 : Monoid.{u_2} M] {x : M} {y : M}, Iff (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} M M M (instHMul.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5)))) x y)) (Or (And (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 x) (IsUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 y)) (And (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 y) (IsUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 x)))","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {x y : M}, Irreducible (x * y) ↔ Irreducible x ∧ IsUnit y ∨ Irreducible y ∧ IsUnit x","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {x y : M}, Irreducible (x * y) ↔ Irreducible x ∧ IsUnit y ∨ Irreducible y ∧ IsUnit x","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Irreducible"],["Or"],["Iff"],["Monoid","toMulOneClass"],["And"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"]],"valueReferences":[["MulOneClass","toMulOne"],["irreducible_mul_isUnit"],["Or","symm"],["Eq","mp"],["HMul","hMul"],["Iff","intro"],["congrArg"],["Irreducible","isUnit_or_isUnit"],["And","intro"],["MulOne","toMul"],["Irreducible"],["irreducible_isUnit_mul"],["Or"],["Or","imp"],["Monoid","toMulOneClass"],["Eq"],["propext"],["rfl"],["And"],["IsUnit"],["Or","casesOn"],["id"],["instHMul"],["Eq","mpr"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["not_irreducible_pow"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 : Monoid.{u_2} M] {x : M} {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) -> (Not (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 (HPow.hPow.{u_2, 0, u_2} M Nat M (instHPow.{u_2, 0} M Nat (Monoid.toPow.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) x n)))","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {x : M} {n : ℕ}, n ≠ 1 → ¬Irreducible (x ^ n)","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {x : M} {n : ℕ}, n ≠ 1 → ¬Irreducible (x ^ n)","typeReferences":[["instHPow"],["Not"],["Irreducible"],["Nat"],["Monoid","toPow"],["instOfNatNat"],["Monoid"],["Ne"],["HPow","hPow"],["OfNat","ofNat"]],"valueReferences":[["MulOneClass","toMulOne"],["instAddNat"],["Eq","trans"],["Eq","mp"],["congrArg"],["Irreducible"],["not_false_eq_true"],["isUnit_pow_iff"],["Or"],["pow_succ"],["Monoid","toPow"],["instOfNatNat"],["Nat","succ_ne_zero"],["Monoid","toMulOneClass"],["not_irreducible_one","_simp_2"],["propext"],["instHPow"],["Not"],["True"],["MulOne","toOne"],["instHAdd"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_irreducible_pow","match_1_3"],["IsUnit"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_irreducible_pow","match_1_1"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["IsUnit","pow"],["or_self"],["Nat"],["of_eq_true"],["One","toOfNat1"],["Nat","succ"],["False"],["pow_zero"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_irreducible_pow","match_1_3"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 : Monoid.{u_1} M] {x : M} (n : Nat) (motive : (Irreducible.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._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.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) x (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))))) -> Prop) (h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.95.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.108 : Irreducible.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._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.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) x (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))))), (forall (h₁ : Not (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._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.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) x (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))))) (h₂ : forall {{a : M}} {{b : M}}, (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.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) x (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) (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.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)))) a b)) -> (Or (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 a) (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 b))), motive (Irreducible.mk.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._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.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) x (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)))) h₁ h₂)) -> (motive h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.95.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.108)","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {x : M} (n : ℕ) (motive : Irreducible (x ^ (n + 2)) → Prop)\n  (h : Irreducible (x ^ (n + 2))),\n  (∀ (h₁ : ¬IsUnit (x ^ (n + 2))) (h₂ : ∀ ⦃a b : M⦄, x ^ (n + 2) = a * b → IsUnit a ∨ IsUnit b), motive ⋯) → motive h","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {x : M} (n : ℕ) (motive : Irreducible (x ^ (n + 2)) → Prop)\n  (h : Irreducible (x ^ (n + 2))),\n  (∀ (h₁ : ¬IsUnit (x ^ (n + 2))) (h₂ : ∀ ⦃a b : M⦄, x ^ (n + 2) = a * b → IsUnit a ∨ IsUnit b), motive ⋯) → motive h","typeReferences":[["MulOneClass","toMulOne"],["instHPow"],["instAddNat"],["Not"],["instHAdd"],["HMul","hMul"],["IsUnit"],["HPow","hPow"],["OfNat","ofNat"],["Irreducible","mk"],["HAdd","hAdd"],["MulOne","toMul"],["Nat"],["Irreducible"],["Or"],["Monoid","toPow"],["instOfNatNat"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["Eq"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["instHPow"],["Nat"],["Monoid","toPow"],["instOfNatNat"],["instHAdd"],["HPow","hPow"],["OfNat","ofNat"],["Irreducible","casesOn"]]},{"isProp":true,"kind":"theorem","name":["MulEquiv","irreducible_iff"],"typeFallback":"forall {F : Type.{u_1}} {M : Type.{u_2}} {N : Type.{u_3}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 : Monoid.{u_2} M] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 : Monoid.{u_3} N] {x : M} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14 : EquivLike.{succ u_1, succ u_2, succ u_3} F M N] [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.19 : MulEquivClass.{u_1, u_2, u_3} F M N (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5))) (MulOne.toMul.{u_3} N (MulOneClass.toMulOne.{u_3} N (Monoid.toMulOneClass.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8))) inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14] (f : F), Iff (Irreducible.{u_3} N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.8 (DFunLike.coe.{succ u_1, succ u_2, succ u_3} F M (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : M) => N) (EquivLike.toFunLike.{succ u_1, succ u_2, succ u_3} F M N inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.14) f x)) (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2447818553._hygCtx._hyg.5 x)","typeFull":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Monoid M] [inst_1 : Monoid N] {x : M} [inst_2 : EquivLike F M N]\n  [MulEquivClass F M N] (f : F), Irreducible (f x) ↔ Irreducible x","typeReadable":"∀ {F : Type u_1} {M : Type u_2} {N : Type u_3} [inst : Monoid M] [inst_1 : Monoid N] {x : M} [inst_2 : EquivLike F M N]\n  [MulEquivClass F M N] (f : F), Irreducible (f x) ↔ Irreducible x","typeReferences":[["MulOneClass","toMulOne"],["Irreducible"],["MulOne","toMul"],["MulEquivClass"],["Iff"],["EquivLike","toFunLike"],["Monoid","toMulOneClass"],["Monoid"],["DFunLike","coe"],["EquivLike"]],"valueReferences":[["implies_congr"],["MulOneClass","toMulOne"],["Eq","trans"],["MulEquivClass","instMonoidHomClass"],["HMul","hMul"],["DFunLike","coe"],["congrArg"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"MulEquiv","irreducible_iff","_simp_1_2"],["MulOne","toMul"],["Irreducible"],["iff_self"],["Or"],["congr"],["EquivLike","toFunLike"],["Monoid","toMulOneClass"],["forall_congr"],["congrFun'"],["Eq"],["MulEquiv","isUnit_map","_simp_2"],["propext"],["Not"],["MonoidHomClass","toMulHomClass"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"MulEquiv","irreducible_iff","_simp_1_1"],["True"],["And"],["IsUnit"],["EquivLike","surjective"],["EmbeddingLike","apply_eq_iff_eq","_simp_1"],["Function","Surjective","forall"],["of_eq_true"],["Eq","refl"],["Iff"],["instHMul"],["EquivLike","toEmbeddingLike"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_units_mul","_simp_1_3"],"typeFallback":"forall {a : Prop} {b : Prop} {c : Prop}, Eq.{1} Prop (Iff (And a b) (And a c)) (a -> (Iff b c))","typeFull":"∀ {a b c : Prop}, (a ∧ b ↔ a ∧ c) = (a → (b ↔ c))","typeReadable":"∀ {a b c : Prop}, (a ∧ b ↔ a ∧ c) = (a → (b ↔ c))","typeReferences":[["Iff"],["And"],["Eq"]],"valueReferences":[["Iff"],["and_congr_right_iff"],["And"],["propext"]]},{"isProp":true,"kind":"theorem","name":["addIrreducible_addUnits_add"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 : AddMonoid.{u_2} M] {y : M} (u : AddUnits.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5), Iff (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} M M M (instHAdd.{u_2} M (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5)))) (AddUnits.val.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 u) y)) (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4208400645._hygCtx._hyg.5 y)","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {y : M} (u : AddUnits M), AddIrreducible (↑u + y) ↔ AddIrreducible y","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {y : M} (u : AddUnits M), AddIrreducible (↑u + y) ↔ AddIrreducible y","typeReferences":[["HAdd","hAdd"],["AddUnits"],["AddIrreducible"],["instHAdd"],["Iff"],["AddUnits","val"],["AddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddIrreducible"],["Eq","trans"],["AddUnits","neg_add_cancel_left"],["and_congr_right_iff"],["AddUnits","isAddUnit_addUnits_add"],["congrArg"],["Iff","intro"],["Or"],["congr"],["AddUnits","val"],["Eq","symm"],["congrFun'"],["Eq"],["propext"],["AddSemigroup","toAdd"],["Not"],["AddUnits"],["Neg","neg"],["instHAdd"],["AddUnits","instNeg"],["And"],["IsAddUnit"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["HAdd","hAdd"],["Eq","refl"],["add_assoc"],["AddMonoid","toAddSemigroup"],["Iff"],["id"],["Eq","mpr"],["addIrreducible_iff"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"addIrreducible_isAddUnit_add","match_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 : AddMonoid.{u_1} M] {x : M} (motive : (IsAddUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x) -> Prop) (h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.34 : IsAddUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x), (forall (x_1 : AddUnits.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5) (ha : Eq.{succ u_1} M (AddUnits.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x_1) x), motive (Exists.intro.{succ u_1} (AddUnits.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5) (fun (u : AddUnits.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5) => Eq.{succ u_1} M (AddUnits.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 u) x) x_1 ha)) -> (motive h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.34)","typeFull":"∀ {M : Type u_1} [inst : AddMonoid M] {x : M} (motive : IsAddUnit x → Prop) (h : IsAddUnit x),\n  (∀ (x_1 : AddUnits M) (ha : ↑x_1 = x), motive ⋯) → motive h","typeReadable":"∀ {M : Type u_1} [inst : AddMonoid M] {x : M} (motive : IsAddUnit x → Prop) (h : IsAddUnit x),\n  (∀ (x_1 : AddUnits M) (ha : ↑x_1 = x), motive ⋯) → motive h","typeReferences":[["AddUnits"],["AddUnits","val"],["Exists","intro"],["AddMonoid"],["IsAddUnit"],["Eq"]],"valueReferences":[["Exists","casesOn"],["AddUnits"],["AddUnits","val"],["Eq"]]},{"isProp":true,"kind":"theorem","name":["irreducible_mul_isUnit"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 : Monoid.{u_2} M] {x : M} {y : M}, (IsUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x) -> (Iff (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} M M M (instHMul.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5)))) y x)) (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 y))","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {x y : M}, IsUnit x → (Irreducible (y * x) ↔ Irreducible y)","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {x y : M}, IsUnit x → (Irreducible (y * x) ↔ Irreducible y)","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Irreducible"],["Iff"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"]],"valueReferences":[["MulOneClass","toMulOne"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_isUnit","match_1_1"],["MulOne","toMul"],["Irreducible"],["Units","val"],["Iff"],["Monoid","toMulOneClass"],["instHMul"],["HMul","hMul"],["irreducible_mul_units"],["Eq","rec"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_isUnit_mul","match_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 : Monoid.{u_1} M] {x : M} (motive : (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x) -> Prop) (h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.34 : IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x), (forall (x_1 : Units.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5) (ha : Eq.{succ u_1} M (Units.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x_1) x), motive (Exists.intro.{succ u_1} (Units.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5) (fun (u : Units.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5) => Eq.{succ u_1} M (Units.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 u) x) x_1 ha)) -> (motive h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.34)","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {x : M} (motive : IsUnit x → Prop) (h : IsUnit x),\n  (∀ (x_1 : Mˣ) (ha : ↑x_1 = x), motive ⋯) → motive h","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {x : M} (motive : IsUnit x → Prop) (h : IsUnit x),\n  (∀ (x_1 : Mˣ) (ha : ↑x_1 = x), motive ⋯) → motive h","typeReferences":[["Units","val"],["Monoid"],["Exists","intro"],["IsUnit"],["Eq"],["Units"]],"valueReferences":[["Exists","casesOn"],["Units","val"],["Eq"],["Units"]]},{"isProp":true,"kind":"theorem","name":["addIrreducible_add_isAddUnit"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 : AddMonoid.{u_2} M] {x : M} {y : M}, (IsAddUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x) -> (Iff (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} M M M (instHAdd.{u_2} M (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5)))) y x)) (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 y))","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {x y : M}, IsAddUnit x → (AddIrreducible (y + x) ↔ AddIrreducible y)","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {x y : M}, IsAddUnit x → (AddIrreducible (y + x) ↔ AddIrreducible y)","typeReferences":[["HAdd","hAdd"],["AddIrreducible"],["instHAdd"],["Iff"],["AddMonoid"],["IsAddUnit"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["HAdd","hAdd"],["AddIrreducible"],["instHAdd"],["Iff"],["AddUnits","val"],["addIrreducible_add_addUnits"],["Eq","rec"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"addIrreducible_add_isAddUnit","match_1_1"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_mul_isUnit","match_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 : Monoid.{u_1} M] {x : M} (motive : (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x) -> Prop) (h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.34 : IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x), (forall (x_1 : Units.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5) (hx : Eq.{succ u_1} M (Units.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 x_1) x), motive (Exists.intro.{succ u_1} (Units.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5) (fun (u : Units.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5) => Eq.{succ u_1} M (Units.val.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.5 u) x) x_1 hx)) -> (motive h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.1528427244._hygCtx._hyg.34)","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {x : M} (motive : IsUnit x → Prop) (h : IsUnit x),\n  (∀ (x_1 : Mˣ) (hx : ↑x_1 = x), motive ⋯) → motive h","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {x : M} (motive : IsUnit x → Prop) (h : IsUnit x),\n  (∀ (x_1 : Mˣ) (hx : ↑x_1 = x), motive ⋯) → motive h","typeReferences":[["Units","val"],["Monoid"],["Exists","intro"],["IsUnit"],["Eq"],["Units"]],"valueReferences":[["Exists","casesOn"],["Units","val"],["Eq"],["Units"]]},{"isProp":true,"kind":"theorem","name":["Irreducible","not_isSquare"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.562513752._hygCtx._hyg.5 : Monoid.{u_2} M] {x : M}, (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.562513752._hygCtx._hyg.5 x) -> (Not (IsSquare.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.562513752._hygCtx._hyg.5))) x))","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {x : M}, Irreducible x → ¬IsSquare x","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {x : M}, Irreducible x → ¬IsSquare x","typeReferences":[["MulOneClass","toMulOne"],["Not"],["IsSquare"],["MulOne","toMul"],["Irreducible"],["Monoid","toMulOneClass"],["Monoid"]],"valueReferences":[["MulOneClass","toMulOne"],["Bool"],["Decidable","decide"],["congrArg"],["IsSquare"],["MulOne","toMul"],["Irreducible"],["Monoid","toPow"],["instOfNatNat"],["Monoid","toMulOneClass"],["Eq","symm"],["Eq"],["Eq","ndrec"],["Bool","true"],["propext"],["of_decide_eq_true"],["instHPow"],["Not"],["not_irreducible_pow"],["Exists"],["isSquare_iff_exists_sq"],["HPow","hPow"],["OfNat","ofNat"],["Exists","casesOn"],["Nat"],["Eq","refl"],["id"],["instDecidableEqNat"],["False"],["Eq","mpr"],["Ne"],["instDecidableNot"]]},{"isProp":true,"kind":"theorem","name":["addIrreducible_add_addUnits"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 : AddMonoid.{u_2} M] {y : M} (u : AddUnits.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5), Iff (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} M M M (instHAdd.{u_2} M (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5)))) y (AddUnits.val.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 u))) (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.102323637._hygCtx._hyg.5 y)","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {y : M} (u : AddUnits M), AddIrreducible (y + ↑u) ↔ AddIrreducible y","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {y : M} (u : AddUnits M), AddIrreducible (y + ↑u) ↔ AddIrreducible y","typeReferences":[["HAdd","hAdd"],["AddUnits"],["AddIrreducible"],["instHAdd"],["Iff"],["AddUnits","val"],["AddMonoid"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddUnits","isAddUnit_add_addUnits"],["AddIrreducible"],["Eq","trans"],["and_congr_right_iff"],["congrArg"],["Iff","intro"],["Or"],["congr"],["AddUnits","val"],["Eq","symm"],["congrFun'"],["Eq"],["propext"],["AddSemigroup","toAdd"],["AddUnits","add_neg_cancel_right"],["Not"],["AddUnits"],["Neg","neg"],["instHAdd"],["AddUnits","instNeg"],["And"],["IsAddUnit"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["HAdd","hAdd"],["Eq","refl"],["add_assoc"],["AddMonoid","toAddSemigroup"],["Iff"],["id"],["Eq","mpr"],["addIrreducible_iff"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["irreducible_isUnit_mul"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 : Monoid.{u_2} M] {x : M} {y : M}, (IsUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 x) -> (Iff (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 (HMul.hMul.{u_2, u_2, u_2} M M M (instHMul.{u_2} M (MulOne.toMul.{u_2} M (MulOneClass.toMulOne.{u_2} M (Monoid.toMulOneClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5)))) x y)) (Irreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.81635122._hygCtx._hyg.5 y))","typeFull":"∀ {M : Type u_2} [inst : Monoid M] {x y : M}, IsUnit x → (Irreducible (x * y) ↔ Irreducible y)","typeReadable":"∀ {M : Type u_2} [inst : Monoid M] {x y : M}, IsUnit x → (Irreducible (x * y) ↔ Irreducible y)","typeReferences":[["MulOneClass","toMulOne"],["MulOne","toMul"],["Irreducible"],["Iff"],["Monoid","toMulOneClass"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"]],"valueReferences":[["MulOneClass","toMulOne"],["irreducible_units_mul"],["MulOne","toMul"],["Irreducible"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_isUnit_mul","match_1_1"],["Units","val"],["Iff"],["Monoid","toMulOneClass"],["instHMul"],["HMul","hMul"],["Eq","rec"]]},{"isProp":true,"kind":"definition","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_addIrreducible_nsmul","match_1_3"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 : AddMonoid.{u_1} M] {x : M} (n : Nat) (motive : (AddIrreducible.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 (HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) x)) -> Prop) (h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.95.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.108 : AddIrreducible.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 (HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) x)), (forall (h₁ : Not (IsAddUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 (HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) x))) (h₂ : forall {{a : M}} {{b : M}}, (Eq.{succ u_1} M (HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) x) (HAdd.hAdd.{u_1, u_1, u_1} M M M (instHAdd.{u_1} M (AddZero.toAdd.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)))) a b)) -> (Or (IsAddUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 a) (IsAddUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 b))), motive (AddIrreducible.mk.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 (HSMul.hSMul.{0, u_1, u_1} Nat M M (instHSMul.{0, u_1} Nat M (AddMonoid.toNSMul.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2))) x) h₁ h₂)) -> (motive h._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx.95.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.108)","typeFull":"∀ {M : Type u_1} [inst : AddMonoid M] {x : M} (n : ℕ) (motive : AddIrreducible ((n + 2) • x) → Prop)\n  (h : AddIrreducible ((n + 2) • x)),\n  (∀ (h₁ : ¬IsAddUnit ((n + 2) • x)) (h₂ : ∀ ⦃a b : M⦄, (n + 2) • x = a + b → IsAddUnit a ∨ IsAddUnit b), motive ⋯) →\n    motive h","typeReadable":"∀ {M : Type u_1} [inst : AddMonoid M] {x : M} (n : ℕ) (motive : AddIrreducible ((n + 2) • x) → Prop)\n  (h : AddIrreducible ((n + 2) • x)),\n  (∀ (h₁ : ¬IsAddUnit ((n + 2) • x)) (h₂ : ∀ ⦃a b : M⦄, (n + 2) • x = a + b → IsAddUnit a ∨ IsAddUnit b), motive ⋯) →\n    motive h","typeReferences":[["AddIrreducible","mk"],["instAddNat"],["Not"],["AddIrreducible"],["instHAdd"],["AddMonoid"],["IsAddUnit"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["OfNat","ofNat"],["HAdd","hAdd"],["Nat"],["Or"],["AddMonoid","toNSMul"],["instOfNatNat"],["HSMul","hSMul"],["instHSMul"],["Eq"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["Nat"],["AddMonoid","toNSMul"],["instOfNatNat"],["instHAdd"],["HSMul","hSMul"],["AddIrreducible","casesOn"],["instHSMul"],["OfNat","ofNat"]]},{"isProp":true,"kind":"theorem","name":["addIrreducible_add_iff"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 : AddMonoid.{u_2} M] {x : M} {y : M}, Iff (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 (HAdd.hAdd.{u_2, u_2, u_2} M M M (instHAdd.{u_2} M (AddZero.toAdd.{u_2} M (AddZeroClass.toAddZero.{u_2} M (AddMonoid.toAddZeroClass.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5)))) x y)) (Or (And (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 x) (IsAddUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 y)) (And (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 y) (IsAddUnit.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.2262315617._hygCtx._hyg.5 x)))","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {x y : M},\n  AddIrreducible (x + y) ↔ AddIrreducible x ∧ IsAddUnit y ∨ AddIrreducible y ∧ IsAddUnit x","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {x y : M},\n  AddIrreducible (x + y) ↔ AddIrreducible x ∧ IsAddUnit y ∨ AddIrreducible y ∧ IsAddUnit x","typeReferences":[["HAdd","hAdd"],["Or"],["AddIrreducible"],["instHAdd"],["Iff"],["And"],["AddMonoid"],["IsAddUnit"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["AddIrreducible"],["Or","symm"],["Eq","mp"],["Iff","intro"],["congrArg"],["And","intro"],["addIrreducible_isAddUnit_add"],["Or"],["Or","imp"],["Eq"],["propext"],["rfl"],["instHAdd"],["AddIrreducible","isAddUnit_or_isAddUnit"],["And"],["addIrreducible_add_isAddUnit"],["IsAddUnit"],["AddZeroClass","toAddZero"],["AddZero","toAdd"],["HAdd","hAdd"],["Or","casesOn"],["id"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["And","casesOn"]]},{"isProp":true,"kind":"theorem","name":["not_addIrreducible_nsmul"],"typeFallback":"forall {M : Type.{u_2}} [inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 : AddMonoid.{u_2} M] {x : M} {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) -> (Not (AddIrreducible.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5 (HSMul.hSMul.{0, u_2, u_2} Nat M M (instHSMul.{0, u_2} Nat M (AddMonoid.toNSMul.{u_2} M inst._@.Mathlib.Algebra.Group.Irreducible.Lemmas.4286791179._hygCtx._hyg.5)) n x)))","typeFull":"∀ {M : Type u_2} [inst : AddMonoid M] {x : M} {n : ℕ}, n ≠ 1 → ¬AddIrreducible (n • x)","typeReadable":"∀ {M : Type u_2} [inst : AddMonoid M] {x : M} {n : ℕ}, n ≠ 1 → ¬AddIrreducible (n • x)","typeReferences":[["Not"],["Nat"],["AddMonoid","toNSMul"],["AddIrreducible"],["instOfNatNat"],["HSMul","hSMul"],["instHSMul"],["Ne"],["AddMonoid"],["OfNat","ofNat"]],"valueReferences":[["instAddNat"],["Eq","trans"],["AddIrreducible"],["zero_nsmul"],["Eq","mp"],["congrArg"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_addIrreducible_nsmul","match_1_1"],["not_addIrreducible_zero"],["not_false_eq_true"],["Or"],["instOfNatNat"],["Nat","succ_ne_zero"],["instHSMul"],["Zero","toOfNat0"],["propext"],["IsAddUnit","nsmul"],["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"not_addIrreducible_nsmul","match_1_3"],["Not"],["True"],["instHAdd"],["IsAddUnit"],["succ_nsmul"],["AddZeroClass","toAddZero"],["OfNat","ofNat"],["HAdd","hAdd"],["or_self"],["Nat"],["of_eq_true"],["AddMonoid","toNSMul"],["Nat","succ"],["eq_false"],["HSMul","hSMul"],["False"],["AddZero","toZero"],["isAddUnit_nsmul_iff"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Group","Irreducible","Lemmas",0,"irreducible_units_mul","_simp_1_1"],"typeFallback":"forall {M : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 : Monoid.{u_1} M] {p : M}, Eq.{1} Prop (Irreducible.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 p) (And (Not (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 p)) (forall {{a : M}} {{b : M}}, (Eq.{succ u_1} M p (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.Irreducible.Defs.2447818553._hygCtx._hyg.3)))) a b)) -> (Or (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 a) (IsUnit.{u_1} M inst._@.Mathlib.Algebra.Group.Irreducible.Defs.2447818553._hygCtx._hyg.3 b))))","typeFull":"∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p = (¬IsUnit p ∧ ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b)","typeReadable":"∀ {M : Type u_1} [inst : Monoid M] {p : M}, Irreducible p = (¬IsUnit p ∧ ∀ ⦃a b : M⦄, p = a * b → IsUnit a ∨ IsUnit b)","typeReferences":[["MulOneClass","toMulOne"],["Not"],["MulOne","toMul"],["Irreducible"],["Or"],["Monoid","toMulOneClass"],["And"],["Monoid"],["instHMul"],["HMul","hMul"],["IsUnit"],["Eq"]],"valueReferences":[["Not"],["MulOneClass","toMulOne"],["And"],["HMul","hMul"],["IsUnit"],["Irreducible"],["MulOne","toMul"],["Or"],["irreducible_iff"],["Monoid","toMulOneClass"],["instHMul"],["Eq"],["propext"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Lie.CartanMatrix.sym.json

@@ -0,0 +1 @@
+[]

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

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Module_LinearMap_Star___unexpand_LinearEquiv_1"],"typeFallback":"Lean.PrettyPrinter.Unexpander","typeFull":"Lean.PrettyPrinter.Unexpander","typeReadable":"Lean.PrettyPrinter.Unexpander","typeReferences":[["Lean","PrettyPrinter","Unexpander"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Bool","false"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","MonadQuotation","toMonadRef"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["List","cons"],["Bool","true"],["Unit","unit"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","PrettyPrinter","instMonadQuotationUnexpandM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["Bool","or"],["cond"],["Unit"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","withRef"],["Bool"],["Applicative","toPure"],["Lean","PrettyPrinter","UnexpandM"],["ReaderT","instMonadExceptOf"],["EStateM"],["Lean","Syntax","isOfKind"],["Lean","Syntax","matchesNull"],["PUnit"],["instOfNatNat"],["Lean","Syntax","node5"],["Lean","MonadQuotation","getContext"],["Eq"],["List","nil"],["Bind","bind"],["EStateM","instMonadExceptOfOfBacktrackable"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","matchesIdent"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Module_LinearMap_Star___macroRules_term_≃ₗ⋆[_]__1"],"typeFallback":"Lean.Macro","typeFull":"Lean.Macro","typeReadable":"Lean.Macro","typeReferences":[["Lean","Macro"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Lean","Macro","State"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","Syntax","ident"],["String"],["Lean","Macro","Context"],["Lean","Macro","Exception","unsupportedSyntax"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["Lean","Syntax","Preresolved"],["List","cons"],["Bool","true"],["Lean","Syntax","Preresolved","decl"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["instMonadExceptOfMonadExceptOf"],["EStateM","instMonad"],["Lean","Macro","instMonadQuotationMacroM"],["Lean","Syntax"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","Name","anonymous"],["Bool"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["EStateM"],["String","toRawSubstring'"],["Lean","Syntax","isOfKind"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Lean","addMacroScope"],["Lean","MacroM"],["Eq"],["Lean","Syntax","node2"],["EStateM","instMonadExceptOfOfBacktrackable"],["Bind","bind"],["List","nil"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","node1"],["Lean","Syntax","Preresolved","namespace"],["Lean","Macro","instMonadRefMacroM"],["Lean","Syntax","node3"],["Lean","Macro","Exception"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Module_LinearMap_Star___macroRules_term_→ₗ⋆[_]__1"],"typeFallback":"Lean.Macro","typeFull":"Lean.Macro","typeReadable":"Lean.Macro","typeReferences":[["Lean","Macro"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Lean","Macro","State"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","Syntax","ident"],["String"],["Lean","Macro","Context"],["Lean","Macro","Exception","unsupportedSyntax"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["Lean","Syntax","Preresolved"],["List","cons"],["Bool","true"],["Lean","Syntax","Preresolved","decl"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["instMonadExceptOfMonadExceptOf"],["EStateM","instMonad"],["Lean","Macro","instMonadQuotationMacroM"],["Lean","Syntax"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","Name","anonymous"],["Bool"],["Applicative","toPure"],["ReaderT","instMonadExceptOf"],["EStateM"],["String","toRawSubstring'"],["Lean","Syntax","isOfKind"],["PUnit"],["instOfNatNat"],["Lean","MonadQuotation","getContext"],["Lean","addMacroScope"],["Lean","MacroM"],["Eq"],["Lean","Syntax","node2"],["EStateM","instMonadExceptOfOfBacktrackable"],["Bind","bind"],["List","nil"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","node1"],["Lean","Syntax","Preresolved","namespace"],["Lean","Macro","instMonadRefMacroM"],["Lean","Syntax","node3"],["Lean","Macro","Exception"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["term_→ₗ⋆[_]_"],"typeFallback":"Lean.TrailingParserDescr","typeFull":"Lean.TrailingParserDescr","typeReadable":"Lean.TrailingParserDescr","typeReferences":[["Lean","TrailingParserDescr"]],"valueReferences":[["Nat"],["Lean","ParserDescr","cat"],["Lean","Name","mkStr1"],["instOfNatNat"],["Lean","ParserDescr","trailingNode"],["Lean","ParserDescr","symbol"],["OfNat","ofNat"],["Lean","ParserDescr","binary"]]},{"isProp":false,"kind":"definition","name":["_aux_Mathlib_Algebra_Module_LinearMap_Star___unexpand_LinearMap_1"],"typeFallback":"Lean.PrettyPrinter.Unexpander","typeFull":"Lean.PrettyPrinter.Unexpander","typeReadable":"Lean.PrettyPrinter.Unexpander","typeReferences":[["Lean","PrettyPrinter","Unexpander"]],"valueReferences":[["Lean","Name"],["Lean","TSyntax","raw"],["Bool","false"],["Lean","MonadQuotation","getCurrMacroScope"],["Lean","MacroScope"],["Lean","SourceInfo"],["Lean","MonadQuotation","toMonadRef"],["Lean","MonadRef","mkInfoFromRefPos"],["ReaderT","instApplicativeOfMonad"],["List","cons"],["Bool","true"],["Unit","unit"],["MonadExcept","throw"],["Lean","Name","mkStr4"],["Lean","SyntaxNodeKind"],["Lean","TSyntax","mk"],["EStateM","instMonad"],["Lean","PrettyPrinter","instMonadQuotationUnexpandM"],["instMonadExceptOfMonadExceptOf"],["Lean","Syntax"],["Bool","or"],["cond"],["Unit"],["instDecidableEqBool"],["Nat"],["Monad","toBind"],["Lean","Syntax","atom"],["Lean","withRef"],["Bool"],["Applicative","toPure"],["Lean","PrettyPrinter","UnexpandM"],["ReaderT","instMonadExceptOf"],["EStateM"],["Lean","Syntax","isOfKind"],["Lean","Syntax","matchesNull"],["PUnit"],["instOfNatNat"],["Lean","Syntax","node5"],["Lean","MonadQuotation","getContext"],["Eq"],["List","nil"],["Bind","bind"],["EStateM","instMonadExceptOfOfBacktrackable"],["Lean","Name","mkStr1"],["ite"],["Lean","Syntax","matchesIdent"],["ReaderT","instMonad"],["OfNat","ofNat"],["EStateM","nonBacktrackable"],["Lean","Syntax","getArg"],["Pure","pure"]]},{"isProp":false,"kind":"definition","name":["term_≃ₗ⋆[_]_"],"typeFallback":"Lean.TrailingParserDescr","typeFull":"Lean.TrailingParserDescr","typeReadable":"Lean.TrailingParserDescr","typeReferences":[["Lean","TrailingParserDescr"]],"valueReferences":[["Nat"],["Lean","ParserDescr","cat"],["Lean","Name","mkStr1"],["instOfNatNat"],["Lean","ParserDescr","trailingNode"],["Lean","ParserDescr","symbol"],["OfNat","ofNat"],["Lean","ParserDescr","binary"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["one_le_ite"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 : LE.{u_1} α], (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3)) a) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3)) b) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3)) (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  1 ≤ a → 1 ≤ b → 1 ≤ if p then a else b","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  1 ≤ a → 1 ≤ b → 1 ≤ if p then a else b","typeReferences":[["One","toOfNat1"],["Decidable"],["ite"],["LE","le"],["LE"],["One"],["OfNat","ofNat"]],"valueReferences":[["if_neg"],["One","toOfNat1"],["ite"],["LE","le"],["id"],["Eq","mpr"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["ite_lt_one"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 : LT.{u_1} α], (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 a (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3))) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 b (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3))) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  a < 1 → b < 1 → (if p then a else b) < 1","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  a < 1 → b < 1 → (if p then a else b) < 1","typeReferences":[["LT","lt"],["One","toOfNat1"],["Decidable"],["ite"],["One"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["if_neg"],["One","toOfNat1"],["ite"],["id"],["congrFun'"],["Eq","mpr"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["ite_neg"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 : LT.{u_1} α], (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3))) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 b (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3))) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.12 (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2132176656._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  a < 0 → b < 0 → (if p then a else b) < 0","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  a < 0 → b < 0 → (if p then a else b) < 0","typeReferences":[["LT","lt"],["Decidable"],["ite"],["Zero","toOfNat0"],["Zero"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["if_neg"],["ite"],["id"],["congrFun'"],["Eq","mpr"],["Zero","toOfNat0"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_nonneg"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 : LE.{u_1} α], (forall (h : p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3)) (a h)) -> (forall (h : Not p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3)) (b h)) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3)) (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), 0 ≤ a h) → (∀ (h : ¬p), 0 ≤ b h) → 0 ≤ dite p a b","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), 0 ≤ a h) → (∀ (h : ¬p), 0 ≤ b h) → 0 ≤ dite p a b","typeReferences":[["Not"],["Decidable"],["LE","le"],["LE"],["Zero","toOfNat0"],["Zero"],["dite"],["OfNat","ofNat"]],"valueReferences":[["dif_pos"],["LE","le"],["id"],["Eq","mpr"],["Zero","toOfNat0"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["ite_nonpos"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 : LE.{u_1} α], (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 a (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3))) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 b (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3))) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  a ≤ 0 → b ≤ 0 → (if p then a else b) ≤ 0","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  a ≤ 0 → b ≤ 0 → (if p then a else b) ≤ 0","typeReferences":[["Decidable"],["ite"],["LE","le"],["LE"],["Zero","toOfNat0"],["Zero"],["OfNat","ofNat"]],"valueReferences":[["if_neg"],["ite"],["LE","le"],["id"],["congrFun'"],["Eq","mpr"],["Zero","toOfNat0"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["one_lt_ite"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 : LT.{u_1} α], (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3)) a) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3)) b) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3)) (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  1 < a → 1 < b → 1 < if p then a else b","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  1 < a → 1 < b → 1 < if p then a else b","typeReferences":[["LT","lt"],["One","toOfNat1"],["Decidable"],["ite"],["One"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["if_neg"],["One","toOfNat1"],["ite"],["id"],["Eq","mpr"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["ite_le_one"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 : LE.{u_1} α], (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 a (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3))) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 b (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3))) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.12 (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.43517200._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  a ≤ 1 → b ≤ 1 → (if p then a else b) ≤ 1","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  a ≤ 1 → b ≤ 1 → (if p then a else b) ≤ 1","typeReferences":[["One","toOfNat1"],["Decidable"],["ite"],["LE","le"],["LE"],["One"],["OfNat","ofNat"]],"valueReferences":[["if_neg"],["One","toOfNat1"],["ite"],["LE","le"],["id"],["congrFun'"],["Eq","mpr"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_pos"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 : LT.{u_1} α], (forall (h : p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3)) (a h)) -> (forall (h : Not p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3)) (b h)) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3)) (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), 0 < a h) → (∀ (h : ¬p), 0 < b h) → 0 < dite p a b","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), 0 < a h) → (∀ (h : ¬p), 0 < b h) → 0 < dite p a b","typeReferences":[["LT","lt"],["Not"],["Decidable"],["Zero","toOfNat0"],["Zero"],["dite"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["dif_pos"],["id"],["Eq","mpr"],["Zero","toOfNat0"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["one_lt_dite"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 : LT.{u_1} α], (forall (h : p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3)) (a h)) -> (forall (h : Not p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3)) (b h)) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.3)) (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.418862788._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), 1 < a h) → (∀ (h : ¬p), 1 < b h) → 1 < dite p a b","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), 1 < a h) → (∀ (h : ¬p), 1 < b h) → 1 < dite p a b","typeReferences":[["LT","lt"],["Not"],["One","toOfNat1"],["Decidable"],["One"],["dite"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["dif_pos"],["One","toOfNat1"],["id"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["ite_pos"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 : LT.{u_1} α], (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3)) a) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3)) b) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.3)) (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.185473315._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  0 < a → 0 < b → 0 < if p then a else b","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LT α],\n  0 < a → 0 < b → 0 < if p then a else b","typeReferences":[["LT","lt"],["Decidable"],["ite"],["Zero","toOfNat0"],["Zero"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["if_neg"],["ite"],["id"],["Eq","mpr"],["Zero","toOfNat0"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["ite_nonneg"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.7 : Decidable p] {a : α} {b : α} [inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 : LE.{u_1} α], (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3)) a) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3)) b) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.12 (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.3)) (ite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.3562914014._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  0 ≤ a → 0 ≤ b → 0 ≤ if p then a else b","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a b : α} [inst_2 : LE α],\n  0 ≤ a → 0 ≤ b → 0 ≤ if p then a else b","typeReferences":[["Decidable"],["ite"],["LE","le"],["LE"],["Zero","toOfNat0"],["Zero"],["OfNat","ofNat"]],"valueReferences":[["if_neg"],["ite"],["LE","le"],["id"],["Eq","mpr"],["Zero","toOfNat0"],["Decidable","casesOn"],["Eq"],["OfNat","ofNat"],["if_pos"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_le_one"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 : LE.{u_1} α], (forall (h : p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 (a h) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3))) -> (forall (h : Not p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 (b h) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3))) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), a h ≤ 1) → (∀ (h : ¬p), b h ≤ 1) → dite p a b ≤ 1","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), a h ≤ 1) → (∀ (h : ¬p), b h ≤ 1) → dite p a b ≤ 1","typeReferences":[["Not"],["One","toOfNat1"],["Decidable"],["LE","le"],["LE"],["One"],["dite"],["OfNat","ofNat"]],"valueReferences":[["dif_pos"],["One","toOfNat1"],["LE","le"],["id"],["congrFun'"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_nonpos"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 : LE.{u_1} α], (forall (h : p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 (a h) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3))) -> (forall (h : Not p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 (b h) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3))) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.19 (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.781777767._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), a h ≤ 0) → (∀ (h : ¬p), b h ≤ 0) → dite p a b ≤ 0","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), a h ≤ 0) → (∀ (h : ¬p), b h ≤ 0) → dite p a b ≤ 0","typeReferences":[["Not"],["Decidable"],["LE","le"],["LE"],["Zero","toOfNat0"],["Zero"],["dite"],["OfNat","ofNat"]],"valueReferences":[["dif_pos"],["LE","le"],["id"],["congrFun'"],["Eq","mpr"],["Zero","toOfNat0"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_lt_one"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 : LT.{u_1} α], (forall (h : p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 (a h) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3))) -> (forall (h : Not p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 (b h) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3))) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), a h < 1) → (∀ (h : ¬p), b h < 1) → dite p a b < 1","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), a h < 1) → (∀ (h : ¬p), b h < 1) → dite p a b < 1","typeReferences":[["LT","lt"],["Not"],["One","toOfNat1"],["Decidable"],["One"],["dite"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["dif_pos"],["One","toOfNat1"],["id"],["congrFun'"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["dite_neg"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3 : Zero.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 : LT.{u_1} α], (forall (h : p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 (a h) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3))) -> (forall (h : Not p), LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 (b h) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3))) -> (LT.lt.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.19 (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.7 a b) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2509838743._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), a h < 0) → (∀ (h : ¬p), b h < 0) → dite p a b < 0","typeReadable":"∀ {α : Type u_1} [inst : Zero α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LT α],\n  (∀ (h : p), a h < 0) → (∀ (h : ¬p), b h < 0) → dite p a b < 0","typeReferences":[["LT","lt"],["Not"],["Decidable"],["Zero","toOfNat0"],["Zero"],["dite"],["LT"],["OfNat","ofNat"]],"valueReferences":[["LT","lt"],["dif_pos"],["id"],["congrFun'"],["Eq","mpr"],["Zero","toOfNat0"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]},{"isProp":true,"kind":"theorem","name":["one_le_dite"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3 : One.{u_1} α] {p : Prop} [inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.7 : Decidable p] {a : p -> α} {b : (Not p) -> α} [inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 : LE.{u_1} α], (forall (h : p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3)) (a h)) -> (forall (h : Not p), LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3)) (b h)) -> (LE.le.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.19 (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.3)) (dite.{succ u_1} α p inst._@.Mathlib.Algebra.Notation.Lemmas.2934996218._hygCtx._hyg.7 a b))","typeFull":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), 1 ≤ a h) → (∀ (h : ¬p), 1 ≤ b h) → 1 ≤ dite p a b","typeReadable":"∀ {α : Type u_1} [inst : One α] {p : Prop} [inst_1 : Decidable p] {a : p → α} {b : ¬p → α} [inst_2 : LE α],\n  (∀ (h : p), 1 ≤ a h) → (∀ (h : ¬p), 1 ≤ b h) → 1 ≤ dite p a b","typeReferences":[["Not"],["One","toOfNat1"],["Decidable"],["LE","le"],["LE"],["One"],["dite"],["OfNat","ofNat"]],"valueReferences":[["dif_pos"],["One","toOfNat1"],["LE","le"],["id"],["Eq","mpr"],["dif_neg"],["Decidable","casesOn"],["Eq"],["dite"],["OfNat","ofNat"],["congrArg"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":false,"kind":"definition","name":["Flag","instSMulOrderIso"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Action.Flag.2611996235._hygCtx._hyg.3 : Preorder.{u_1} α], SMul.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.2611996235._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.2611996235._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.2611996235._hygCtx._hyg.3))","typeFull":"{α : Type u_1} → [inst : Preorder α] → SMul (α ≃o α) (Flag α)","typeReadable":"{α : Type u_1} → [inst : Preorder α] → SMul (α ≃o α) (Flag α)","typeReferences":[["Flag"],["SMul"],["Preorder"],["OrderIso"],["Preorder","toLE"]],"valueReferences":[["SMul","mk"],["Flag"],["Equiv","instEquivLike"],["EquivLike","toFunLike"],["OrderIso"],["Preorder","toLE"],["DFunLike","coe"],["Equiv"],["Flag","map"]]},{"isProp":true,"kind":"theorem","name":["Flag","instMulActionOrderIso","_proof_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3 : Preorder.{u_1} α], Function.Injective.{succ u_1, succ u_1} (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3)) (Set.{u_1} α) (SetLike.coe.{u_1, u_1} (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3)) α (Flag.instSetLike.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3)))","typeFull":"∀ {α : Type u_1} [inst : Preorder α], Function.Injective SetLike.coe","typeReadable":"∀ {α : Type u_1} [inst : Preorder α], Function.Injective SetLike.coe","typeReferences":[["Flag"],["Preorder"],["SetLike","coe"],["Flag","instSetLike"],["Set"],["Preorder","toLE"],["Function","Injective"]],"valueReferences":[["Flag"],["SetLike","coe_injective"],["Flag","instSetLike"],["Preorder","toLE"]]},{"isProp":true,"kind":"theorem","name":["Flag","coe_smul"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3 : Preorder.{u_1} α] (e : OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (s : Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)), Eq.{succ u_1} (Set.{u_1} α) (SetLike.coe.{u_1, u_1} (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (Flag.instSetLike.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (HSMul.hSMul.{u_1, u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (instHSMul.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.instSMulOrderIso.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) e s)) (HSMul.hSMul.{u_1, u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Set.{u_1} α) (Set.{u_1} α) (instHSMul.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Set.{u_1} α) (Set.smulSet.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (SemigroupAction.toSMul.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (Monoid.toSemigroup.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (DivInvMonoid.toMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Group.toDivInvMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (RelIso.instGroup.{u_1} α (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21))))) (MulAction.toSemigroupAction.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (DivInvMonoid.toMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Group.toDivInvMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (RelIso.instGroup.{u_1} α (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21)))) (RelIso.applyMulAction.{u_1} α (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21)))))) e (SetLike.coe.{u_1, u_1} (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (Flag.instSetLike.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) s))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] (e : α ≃o α) (s : Flag α), ↑(e • s) = e • ↑s","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] (e : α ≃o α) (s : Flag α), ↑(e • s) = e • ↑s","typeReferences":[["Flag","instSMulOrderIso"],["Flag","instSetLike"],["Set"],["OrderIso"],["SemigroupAction","toSMul"],["RelIso","instGroup"],["Preorder"],["Flag"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["RelIso","applyMulAction"],["LE","le"],["HSMul","hSMul"],["Set","smulSet"],["instHSMul"],["Monoid","toSemigroup"],["Preorder","toLE"],["Eq"],["MulAction","toSemigroupAction"],["Group","toDivInvMonoid"]],"valueReferences":[["rfl"],["Flag"],["Flag","instSMulOrderIso"],["SetLike","coe"],["Flag","instSetLike"],["Set"],["OrderIso"],["HSMul","hSMul"],["instHSMul"],["Preorder","toLE"]]},{"isProp":false,"kind":"definition","name":["Flag","instMulActionOrderIso"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3 : Preorder.{u_1} α], MulAction.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3)) (DivInvMonoid.toMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3)) (Group.toDivInvMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3)) (RelIso.instGroup.{u_1} α (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.3955836709._hygCtx._hyg.3) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21))))","typeFull":"{α : Type u_1} → [inst : Preorder α] → MulAction (α ≃o α) (Flag α)","typeReadable":"{α : Type u_1} → [inst : Preorder α] → MulAction (α ≃o α) (Flag α)","typeReferences":[["RelIso","instGroup"],["Flag"],["Preorder"],["DivInvMonoid","toMonoid"],["MulAction"],["LE","le"],["OrderIso"],["Preorder","toLE"],["Group","toDivInvMonoid"]],"valueReferences":[["Function","Injective","mulAction"],["Flag","instSMulOrderIso"],["Flag","instSetLike"],["Set"],["OrderIso"],["Flag","coe_smul"],["RelIso","instGroup"],["Flag"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["Flag","instMulActionOrderIso","_proof_1"],["LE","le"],["Set","mulActionSet"],["RelIso","applyMulAction"],["Preorder","toLE"],["Group","toDivInvMonoid"]]},{"isProp":true,"kind":"theorem","name":["Flag","coe_smul","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3 : Preorder.{u_1} α] (e : OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (s : Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)), Eq.{succ u_1} (Set.{u_1} α) (HSMul.hSMul.{u_1, u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Set.{u_1} α) (Set.{u_1} α) (instHSMul.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Set.{u_1} α) (Set.smulSet.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (SemigroupAction.toSMul.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (Monoid.toSemigroup.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (DivInvMonoid.toMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Group.toDivInvMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (RelIso.instGroup.{u_1} α (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21))))) (MulAction.toSemigroupAction.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (DivInvMonoid.toMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Group.toDivInvMonoid.{u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (RelIso.instGroup.{u_1} α (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21)))) (RelIso.applyMulAction.{u_1} α (fun (x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) (x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 : α) => LE.le.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) x1._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21 x2._@.Mathlib.Order.Hom.Basic.294108515._hygCtx._hyg.21)))))) e (SetLike.coe.{u_1, u_1} (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (Flag.instSetLike.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) s)) (SetLike.coe.{u_1, u_1} (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) α (Flag.instSetLike.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (HSMul.hSMul.{u_1, u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (instHSMul.{u_1, u_1} (OrderIso.{u_1, u_1} α α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3) (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.{u_1} α (Preorder.toLE.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) (Flag.instSMulOrderIso.{u_1} α inst._@.Mathlib.Algebra.Order.Group.Action.Flag.1393146857._hygCtx._hyg.3)) e s))","typeFull":"∀ {α : Type u_1} [inst : Preorder α] (e : α ≃o α) (s : Flag α), e • ↑s = ↑(e • s)","typeReadable":"∀ {α : Type u_1} [inst : Preorder α] (e : α ≃o α) (s : Flag α), e • ↑s = ↑(e • s)","typeReferences":[["Flag","instSMulOrderIso"],["Flag","instSetLike"],["Set"],["OrderIso"],["SemigroupAction","toSMul"],["RelIso","instGroup"],["Preorder"],["Flag"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["RelIso","applyMulAction"],["LE","le"],["HSMul","hSMul"],["Set","smulSet"],["instHSMul"],["Monoid","toSemigroup"],["Preorder","toLE"],["Eq"],["MulAction","toSemigroupAction"],["Group","toDivInvMonoid"]],"valueReferences":[["Flag","instSMulOrderIso"],["Flag","instSetLike"],["Set"],["OrderIso"],["SemigroupAction","toSMul"],["Flag","coe_smul"],["RelIso","instGroup"],["Flag"],["SetLike","coe"],["DivInvMonoid","toMonoid"],["RelIso","applyMulAction"],["LE","le"],["HSMul","hSMul"],["Eq","symm"],["Set","smulSet"],["instHSMul"],["Monoid","toSemigroup"],["Preorder","toLE"],["MulAction","toSemigroupAction"],["Group","toDivInvMonoid"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.GroupWithZero.Submonoid.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Submonoid","pos","_proof_1"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3 : MulZeroOneClass.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.9 : PosMulStrictMono.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6)] {a._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.31 : α} {b._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.32 : α}, (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))) a._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.31) -> (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))) b._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.32) -> (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))) (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3))) a._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.31 b._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.32))","typeFull":"∀ (α : Type u_1) [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [PosMulStrictMono α] {a b : α},\n  0 < a → 0 < b → 0 < a * b","typeReadable":"∀ (α : Type u_1) [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [PosMulStrictMono α] {a b : α},\n  0 < a → 0 < b → 0 < a * b","typeReferences":[["PartialOrder","toPreorder"],["MulZeroClass","toMul"],["Preorder","toLT"],["MulZeroOneClass"],["HMul","hMul"],["OfNat","ofNat"],["LT","lt"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["PartialOrder"],["instHMul"],["Zero","toOfNat0"],["PosMulStrictMono"]],"valueReferences":[["PartialOrder","toPreorder"],["MulZeroOneClass","toMulZeroClass"],["mul_pos"]]},{"isProp":true,"kind":"theorem","name":["Submonoid","pos","_proof_2"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3 : MulZeroOneClass.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.12 : ZeroLEOneClass.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3))) (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.15 : NeZero.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))))], LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))))","typeFull":"∀ (α : Type u_1) [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [ZeroLEOneClass α] [NeZero 1], 0 < 1","typeReadable":"∀ (α : Type u_1) [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [ZeroLEOneClass α] [NeZero 1], 0 < 1","typeReferences":[["MulOneClass","toMulOne"],["PartialOrder","toPreorder"],["MulOne","toOne"],["Preorder","toLT"],["MulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["OfNat","ofNat"],["ZeroLEOneClass"],["LT","lt"],["NeZero"],["MulZeroOneClass","toMulZeroClass"],["One","toOfNat1"],["MulZeroClass","toZero"],["PartialOrder"],["Zero","toOfNat0"],["Preorder","toLE"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toOne"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["MulZeroOneClass","toMulOneClass"],["zero_lt_one"]]},{"isProp":true,"kind":"theorem","name":["Submonoid","mem_pos"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3 : MulZeroOneClass.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.9 : PosMulStrictMono.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.12 : ZeroLEOneClass.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3))) (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.15 : NeZero.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)))))] {a : α}, Iff (Membership.mem.{u_1, u_1} α (Submonoid.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) α (Submonoid.instSetLike.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3))) (Submonoid.pos.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.15) a) (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)))) a)","typeFull":"∀ {α : Type u_1} [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [inst_2 : PosMulStrictMono α]\n  [inst_3 : ZeroLEOneClass α] [inst_4 : NeZero 1] {a : α}, a ∈ Submonoid.pos α ↔ 0 < a","typeReadable":"∀ {α : Type u_1} [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [inst_2 : PosMulStrictMono α]\n  [inst_3 : ZeroLEOneClass α] [inst_4 : NeZero 1] {a : α}, a ∈ Submonoid.pos α ↔ 0 < a","typeReferences":[["MulOneClass","toMulOne"],["SetLike","instMembership"],["PartialOrder","toPreorder"],["MulOne","toOne"],["Submonoid","instSetLike"],["Membership","mem"],["MulZeroClass","toMul"],["Preorder","toLT"],["MulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["OfNat","ofNat"],["ZeroLEOneClass"],["LT","lt"],["Submonoid"],["NeZero"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Iff"],["PartialOrder"],["Zero","toOfNat0"],["PosMulStrictMono"],["Preorder","toLE"],["Submonoid","pos"]],"valueReferences":[["Submonoid"],["SetLike","instMembership"],["Submonoid","instSetLike"],["Membership","mem"],["MulZeroOneClass","toMulOneClass"],["Iff","rfl"],["Submonoid","pos"]]},{"isProp":true,"kind":"theorem","name":["Submonoid","mem_pos","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3 : MulZeroOneClass.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.9 : PosMulStrictMono.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.12 : ZeroLEOneClass.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3))) (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.15 : NeZero.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)))))] {a : α}, Eq.{1} Prop (Membership.mem.{u_1, u_1} α (Submonoid.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) (SetLike.instMembership.{u_1, u_1} (Submonoid.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)) α (Submonoid.instSetLike.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3))) (Submonoid.pos.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.15) a) (LT.lt.{u_1} α (Preorder.toLT.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.6)) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.2015895844._hygCtx._hyg.3)))) a)","typeFull":"∀ {α : Type u_1} [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [inst_2 : PosMulStrictMono α]\n  [inst_3 : ZeroLEOneClass α] [inst_4 : NeZero 1] {a : α}, (a ∈ Submonoid.pos α) = (0 < a)","typeReadable":"∀ {α : Type u_1} [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [inst_2 : PosMulStrictMono α]\n  [inst_3 : ZeroLEOneClass α] [inst_4 : NeZero 1] {a : α}, (a ∈ Submonoid.pos α) = (0 < a)","typeReferences":[["MulOneClass","toMulOne"],["SetLike","instMembership"],["PartialOrder","toPreorder"],["MulOne","toOne"],["Submonoid","instSetLike"],["Membership","mem"],["MulZeroClass","toMul"],["Preorder","toLT"],["MulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["OfNat","ofNat"],["ZeroLEOneClass"],["LT","lt"],["Submonoid"],["NeZero"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["PartialOrder"],["Zero","toOfNat0"],["Eq"],["PosMulStrictMono"],["Preorder","toLE"],["Submonoid","pos"]],"valueReferences":[["SetLike","instMembership"],["PartialOrder","toPreorder"],["Submonoid","instSetLike"],["Membership","mem"],["Preorder","toLT"],["MulZeroOneClass","toMulOneClass"],["OfNat","ofNat"],["Submonoid"],["LT","lt"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Zero","toOfNat0"],["Submonoid","mem_pos"],["propext"],["Submonoid","pos"]]},{"isProp":false,"kind":"definition","name":["Submonoid","pos"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3 : MulZeroOneClass.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.9 : PosMulStrictMono.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.12 : ZeroLEOneClass.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3))) (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.15 : NeZero.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))))], Submonoid.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)","typeFull":"(α : Type u_1) →\n  [inst : MulZeroOneClass α] →\n    [inst_1 : PartialOrder α] → [PosMulStrictMono α] → [ZeroLEOneClass α] → [NeZero 1] → Submonoid α","typeReadable":"(α : Type u_1) →\n  [inst : MulZeroOneClass α] →\n    [inst_1 : PartialOrder α] → [PosMulStrictMono α] → [ZeroLEOneClass α] → [NeZero 1] → Submonoid α","typeReferences":[["MulOneClass","toMulOne"],["PartialOrder","toPreorder"],["MulOne","toOne"],["MulZeroClass","toMul"],["MulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["OfNat","ofNat"],["ZeroLEOneClass"],["Submonoid"],["NeZero"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["PartialOrder"],["PosMulStrictMono"],["Preorder","toLE"]],"valueReferences":[["MulOneClass","toMulOne"],["PartialOrder","toPreorder"],["Submonoid","pos","_proof_1"],["MulZeroOneClass","toMulOneClass"],["Subsemigroup","mk"],["OfNat","ofNat"],["Submonoid","mk"],["MulOne","toMul"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Set","Ioi"],["Zero","toOfNat0"],["Submonoid","pos","_proof_2"]]},{"isProp":true,"kind":"theorem","name":["Submonoid","coe_pos"],"typeFallback":"forall (α : Type.{u_1}) [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3 : MulZeroOneClass.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6 : PartialOrder.{u_1} α] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.9 : PosMulStrictMono.{u_1} α (MulZeroClass.toMul.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6)] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.12 : ZeroLEOneClass.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3))) (Preorder.toLE.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6))] [inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.15 : NeZero.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (OfNat.ofNat.{u_1} α 1 (One.toOfNat1.{u_1} α (MulOne.toOne.{u_1} α (MulOneClass.toMulOne.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))))], Eq.{succ u_1} (Set.{u_1} α) (SetLike.coe.{u_1, u_1} (Submonoid.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) α (Submonoid.instSetLike.{u_1} α (MulZeroOneClass.toMulOneClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)) (Submonoid.pos.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.9 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.12 inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.15)) (Set.Ioi.{u_1} α (PartialOrder.toPreorder.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.6) (OfNat.ofNat.{u_1} α 0 (Zero.toOfNat0.{u_1} α (MulZeroClass.toZero.{u_1} α (MulZeroOneClass.toMulZeroClass.{u_1} α inst._@.Mathlib.Algebra.Order.GroupWithZero.Submonoid.3837012676._hygCtx._hyg.3)))))","typeFull":"∀ (α : Type u_1) [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [inst_2 : PosMulStrictMono α]\n  [inst_3 : ZeroLEOneClass α] [inst_4 : NeZero 1], ↑(Submonoid.pos α) = Set.Ioi 0","typeReadable":"∀ (α : Type u_1) [inst : MulZeroOneClass α] [inst_1 : PartialOrder α] [inst_2 : PosMulStrictMono α]\n  [inst_3 : ZeroLEOneClass α] [inst_4 : NeZero 1], ↑(Submonoid.pos α) = Set.Ioi 0","typeReferences":[["MulOneClass","toMulOne"],["PartialOrder","toPreorder"],["MulOne","toOne"],["Submonoid","instSetLike"],["Set"],["MulZeroClass","toMul"],["MulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["OfNat","ofNat"],["ZeroLEOneClass"],["Submonoid"],["NeZero"],["SetLike","coe"],["One","toOfNat1"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["PartialOrder"],["Set","Ioi"],["Zero","toOfNat0"],["Eq"],["PosMulStrictMono"],["Preorder","toLE"],["Submonoid","pos"]],"valueReferences":[["Submonoid"],["SetLike","coe"],["Submonoid","instSetLike"],["Set"],["Eq","refl"],["MulZeroOneClass","toMulOneClass"],["Submonoid","pos"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Order.Sum.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Sum","one_le_elim_iff"],"typeFallback":"forall {α₁ : Type.{u_1}} {α₂ : Type.{u_2}} {β : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5 : LE.{u_3} β] [inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8 : One.{u_3} β] {v₁ : α₁ -> β} {v₂ : α₂ -> β}, Iff (LE.le.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.hasLe.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5)) (OfNat.ofNat.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) 1 (One.toOfNat1.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.instOne.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8)))) (Sum.elim.{u_1, u_2, succ u_3} α₁ α₂ β v₁ v₂)) (And (LE.le.{max u_1 u_3} (α₁ -> β) (Pi.hasLe.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5)) (OfNat.ofNat.{max u_1 u_3} (α₁ -> β) 1 (One.toOfNat1.{max u_1 u_3} (α₁ -> β) (Pi.instOne.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8)))) v₁) (LE.le.{max u_2 u_3} (α₂ -> β) (Pi.hasLe.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5)) (OfNat.ofNat.{max u_2 u_3} (α₂ -> β) 1 (One.toOfNat1.{max u_2 u_3} (α₂ -> β) (Pi.instOne.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8)))) v₂))","typeFull":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : One β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  1 ≤ Sum.elim v₁ v₂ ↔ 1 ≤ v₁ ∧ 1 ≤ v₂","typeReadable":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : One β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  1 ≤ Sum.elim v₁ v₂ ↔ 1 ≤ v₁ ∧ 1 ≤ v₂","typeReferences":[["Pi","instOne"],["One","toOfNat1"],["Sum","elim"],["Iff"],["Pi","hasLe"],["LE","le"],["And"],["One"],["LE"],["OfNat","ofNat"],["Sum"]],"valueReferences":[["Sum","const_le_elim_iff"]]},{"isProp":true,"kind":"theorem","name":["Sum","elim_le_one_iff"],"typeFallback":"forall {α₁ : Type.{u_1}} {α₂ : Type.{u_2}} {β : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5 : LE.{u_3} β] [inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8 : One.{u_3} β] {v₁ : α₁ -> β} {v₂ : α₂ -> β}, Iff (LE.le.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.hasLe.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5)) (Sum.elim.{u_1, u_2, succ u_3} α₁ α₂ β v₁ v₂) (OfNat.ofNat.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) 1 (One.toOfNat1.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.instOne.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8))))) (And (LE.le.{max u_1 u_3} (α₁ -> β) (Pi.hasLe.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5)) v₁ (OfNat.ofNat.{max u_1 u_3} (α₁ -> β) 1 (One.toOfNat1.{max u_1 u_3} (α₁ -> β) (Pi.instOne.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8))))) (LE.le.{max u_2 u_3} (α₂ -> β) (Pi.hasLe.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5)) v₂ (OfNat.ofNat.{max u_2 u_3} (α₂ -> β) 1 (One.toOfNat1.{max u_2 u_3} (α₂ -> β) (Pi.instOne.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8))))))","typeFull":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : One β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  Sum.elim v₁ v₂ ≤ 1 ↔ v₁ ≤ 1 ∧ v₂ ≤ 1","typeReadable":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : One β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  Sum.elim v₁ v₂ ≤ 1 ↔ v₁ ≤ 1 ∧ v₂ ≤ 1","typeReferences":[["Pi","instOne"],["One","toOfNat1"],["Sum","elim"],["Iff"],["Pi","hasLe"],["LE","le"],["And"],["One"],["LE"],["OfNat","ofNat"],["Sum"]],"valueReferences":[["Sum","elim_le_const_iff"]]},{"isProp":true,"kind":"theorem","name":["Sum","nonneg_elim_iff"],"typeFallback":"forall {α₁ : Type.{u_1}} {α₂ : Type.{u_2}} {β : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5 : LE.{u_3} β] [inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8 : Zero.{u_3} β] {v₁ : α₁ -> β} {v₂ : α₂ -> β}, Iff (LE.le.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.hasLe.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5)) (OfNat.ofNat.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) 0 (Zero.toOfNat0.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.instZero.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8)))) (Sum.elim.{u_1, u_2, succ u_3} α₁ α₂ β v₁ v₂)) (And (LE.le.{max u_1 u_3} (α₁ -> β) (Pi.hasLe.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5)) (OfNat.ofNat.{max u_1 u_3} (α₁ -> β) 0 (Zero.toOfNat0.{max u_1 u_3} (α₁ -> β) (Pi.instZero.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8)))) v₁) (LE.le.{max u_2 u_3} (α₂ -> β) (Pi.hasLe.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.5)) (OfNat.ofNat.{max u_2 u_3} (α₂ -> β) 0 (Zero.toOfNat0.{max u_2 u_3} (α₂ -> β) (Pi.instZero.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.3050040001._hygCtx._hyg.8)))) v₂))","typeFull":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : Zero β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  0 ≤ Sum.elim v₁ v₂ ↔ 0 ≤ v₁ ∧ 0 ≤ v₂","typeReadable":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : Zero β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  0 ≤ Sum.elim v₁ v₂ ↔ 0 ≤ v₁ ∧ 0 ≤ v₂","typeReferences":[["Sum","elim"],["Iff"],["Pi","hasLe"],["LE","le"],["And"],["LE"],["Zero","toOfNat0"],["Zero"],["OfNat","ofNat"],["Pi","instZero"],["Sum"]],"valueReferences":[["Sum","const_le_elim_iff"],["Zero","zero"]]},{"isProp":true,"kind":"theorem","name":["Sum","elim_nonpos_iff"],"typeFallback":"forall {α₁ : Type.{u_1}} {α₂ : Type.{u_2}} {β : Type.{u_3}} [inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5 : LE.{u_3} β] [inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8 : Zero.{u_3} β] {v₁ : α₁ -> β} {v₂ : α₂ -> β}, Iff (LE.le.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.hasLe.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5)) (Sum.elim.{u_1, u_2, succ u_3} α₁ α₂ β v₁ v₂) (OfNat.ofNat.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) 0 (Zero.toOfNat0.{max (max u_1 u_2) u_3} ((Sum.{u_1, u_2} α₁ α₂) -> β) (Pi.instZero.{max u_1 u_2, u_3} (Sum.{u_1, u_2} α₁ α₂) (fun (a._@._internal._hyg.0 : Sum.{u_1, u_2} α₁ α₂) => β) (fun (i : Sum.{u_1, u_2} α₁ α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8))))) (And (LE.le.{max u_1 u_3} (α₁ -> β) (Pi.hasLe.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5)) v₁ (OfNat.ofNat.{max u_1 u_3} (α₁ -> β) 0 (Zero.toOfNat0.{max u_1 u_3} (α₁ -> β) (Pi.instZero.{u_1, u_3} α₁ (fun (a._@._internal._hyg.0 : α₁) => β) (fun (i : α₁) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8))))) (LE.le.{max u_2 u_3} (α₂ -> β) (Pi.hasLe.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.5)) v₂ (OfNat.ofNat.{max u_2 u_3} (α₂ -> β) 0 (Zero.toOfNat0.{max u_2 u_3} (α₂ -> β) (Pi.instZero.{u_2, u_3} α₂ (fun (a._@._internal._hyg.0 : α₂) => β) (fun (i : α₂) => inst._@.Mathlib.Algebra.Order.Sum.2892999._hygCtx._hyg.8))))))","typeFull":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : Zero β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  Sum.elim v₁ v₂ ≤ 0 ↔ v₁ ≤ 0 ∧ v₂ ≤ 0","typeReadable":"∀ {α₁ : Type u_1} {α₂ : Type u_2} {β : Type u_3} [inst : LE β] [inst_1 : Zero β] {v₁ : α₁ → β} {v₂ : α₂ → β},\n  Sum.elim v₁ v₂ ≤ 0 ↔ v₁ ≤ 0 ∧ v₂ ≤ 0","typeReferences":[["Sum","elim"],["Iff"],["Pi","hasLe"],["LE","le"],["And"],["LE"],["Zero","toOfNat0"],["Zero"],["OfNat","ofNat"],["Pi","instZero"],["Sum"]],"valueReferences":[["Sum","elim_le_const_iff"],["Zero","zero"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Polynomial.Cardinal.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Polynomial","cardinalMk_le_max"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Polynomial.Cardinal.2753875788._hygCtx._hyg.3 : Semiring.{u} R], LE.le.{succ u} Cardinal.{u} Cardinal.instLE.{u} (Cardinal.mk.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Cardinal.2753875788._hygCtx._hyg.3)) (Max.max.{succ u} Cardinal.{u} (SemilatticeSup.toMax.{succ u} Cardinal.{u} (Lattice.toSemilatticeSup.{succ u} Cardinal.{u} (ConditionallyCompleteLattice.toLattice.{succ u} Cardinal.{u} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u} Cardinal.{u} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u} Cardinal.{u} Cardinal.instConditionallyCompleteLinearOrderBot.{u}))))) (Cardinal.mk.{u} R) Cardinal.aleph0.{u})","typeFull":"∀ {R : Type u} [inst : Semiring R], Cardinal.mk (Polynomial R) ≤ max (Cardinal.mk R) Cardinal.aleph0","typeReadable":"∀ {R : Type u} [inst : Semiring R], Cardinal.mk (Polynomial R) ≤ max (Cardinal.mk R) Cardinal.aleph0","typeReferences":[["Lattice","toSemilatticeSup"],["Cardinal","aleph0"],["Cardinal"],["Cardinal","mk"],["Polynomial"],["Max","max"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["LE","le"],["SemilatticeSup","toMax"],["Cardinal","instConditionallyCompleteLinearOrderBot"],["ConditionallyCompleteLattice","toLattice"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["Cardinal","instLE"],["Semiring"]],"valueReferences":[["Cardinal","one_le_aleph0"],["Lattice","toSemilatticeSup"],["PartialOrder","toPreorder"],["Cardinal","aleph0"],["le_max_of_le_right"],["Cardinal","linearOrder"],["Cardinal","partialOrder"],["Cardinal","mk"],["Subsingleton"],["Or"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["Unique","instSubsingleton"],["Eq"],["Cardinal","instLE"],["Polynomial","inhabited"],["Polynomial","cardinalMk_eq_max"],["Cardinal"],["OfNat","ofNat"],["Or","casesOn"],["Polynomial"],["Eq","trans_le"],["Max","max"],["Cardinal","mk_eq_one"],["One","toOfNat1"],["Cardinal","instOne"],["instNonemptyOfInhabited"],["Eq","refl"],["SemilatticeSup","toMax"],["LE","le"],["Nontrivial"],["Cardinal","instConditionallyCompleteLinearOrderBot"],["ConditionallyCompleteLattice","toLattice"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["Eq","le"],["Polynomial","unique"],["subsingleton_or_nontrivial"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","cardinalMk_eq_max"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Polynomial.Cardinal.3607116813._hygCtx._hyg.3 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Polynomial.Cardinal.3607116813._hygCtx._hyg.6 : Nontrivial.{u} R], Eq.{succ (succ u)} Cardinal.{u} (Cardinal.mk.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Cardinal.3607116813._hygCtx._hyg.3)) (Max.max.{succ u} Cardinal.{u} (SemilatticeSup.toMax.{succ u} Cardinal.{u} (Lattice.toSemilatticeSup.{succ u} Cardinal.{u} (ConditionallyCompleteLattice.toLattice.{succ u} Cardinal.{u} (ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice.{succ u} Cardinal.{u} (ConditionallyCompleteLinearOrderBot.toConditionallyCompleteLinearOrder.{succ u} Cardinal.{u} Cardinal.instConditionallyCompleteLinearOrderBot.{u}))))) (Cardinal.mk.{u} R) Cardinal.aleph0.{u})","typeFull":"∀ {R : Type u} [inst : Semiring R] [Nontrivial R], Cardinal.mk (Polynomial R) = max (Cardinal.mk R) Cardinal.aleph0","typeReadable":"∀ {R : Type u} [inst : Semiring R] [Nontrivial R], Cardinal.mk (Polynomial R) = max (Cardinal.mk R) Cardinal.aleph0","typeReferences":[["Lattice","toSemilatticeSup"],["Cardinal","aleph0"],["Cardinal"],["Cardinal","mk"],["Polynomial"],["Max","max"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["Nontrivial"],["SemilatticeSup","toMax"],["Cardinal","instConditionallyCompleteLinearOrderBot"],["ConditionallyCompleteLattice","toLattice"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["Eq"],["Semiring"]],"valueReferences":[["instAddNat"],["Polynomial","instAdd"],["Lattice","toSemilatticeSup"],["Polynomial","toFinsuppIso"],["Cardinal","lift"],["AddMonoidAlgebra","eq_1"],["Eq","trans"],["Cardinal","aleph0"],["instInfiniteNat"],["Cardinal","linearOrder"],["congrArg"],["Cardinal","mk"],["Semiring","toNonAssocSemiring"],["Equiv","cardinal_eq"],["RingEquiv","toEquiv"],["ConditionallyCompleteLinearOrder","toConditionallyCompleteLattice"],["Polynomial","instMul"],["Cardinal","mk_finsupp_lift_of_infinite"],["AddMonoidAlgebra","nonUnitalNonAssocSemiring"],["Eq"],["AddMonoidAlgebra","instMul"],["max_comm"],["Distrib","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["Cardinal","lift_uzero"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Cardinal"],["Polynomial"],["Max","max"],["Nat"],["MulZeroClass","toZero"],["Eq","refl"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["SemilatticeSup","toMax"],["id"],["Cardinal","instConditionallyCompleteLinearOrderBot"],["Eq","mpr"],["ConditionallyCompleteLinearOrderBot","toConditionallyCompleteLinearOrder"],["ConditionallyCompleteLattice","toLattice"],["AddMonoidAlgebra"],["Finsupp"],["LinearOrder","toMax"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Polynomial.Degree.Monomial.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Polynomial","C_mul_X_pow_eq_self"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14 : Semiring.{u} R] {p : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14}, (LE.le.{0} Nat instLENat (Finset.card.{0} Nat (Polynomial.support.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14 p)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) -> (Eq.{succ u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (HMul.hMul.{u, u, u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (instHMul.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.instMul.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14)) (DFunLike.coe.{succ u, succ u, succ u} (RingHom.{u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.semiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (RingHom.instFunLike.{u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.semiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14))) (Polynomial.C.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.leadingCoeff.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14 p)) (HPow.hPow.{u, 0, u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) Nat (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (instHPow.{u, 0} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) Nat (Monoid.toPow.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (MonoidWithZero.toMonoid.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Semiring.toMonoidWithZero.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.semiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14))))) (Polynomial.X.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14) (Polynomial.natDegree.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3830686741._hygCtx._hyg.14 p))) p)","typeFull":"∀ {R : Type u} [inst : Semiring R] {p : Polynomial R},\n  p.support.card ≤ 1 → Polynomial.C p.leadingCoeff * Polynomial.X ^ p.natDegree = p","typeReadable":"∀ {R : Type u} [inst : Semiring R] {p : Polynomial R},\n  p.support.card ≤ 1 → Polynomial.C p.leadingCoeff * Polynomial.X ^ p.natDegree = p","typeReferences":[["RingHom"],["Polynomial","natDegree"],["HMul","hMul"],["RingHom","instFunLike"],["DFunLike","coe"],["Polynomial","support"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["instOfNatNat"],["MonoidWithZero","toMonoid"],["Polynomial","instMul"],["Eq"],["instHPow"],["Polynomial","X"],["Polynomial","C"],["Finset","card"],["Semiring","toMonoidWithZero"],["Polynomial","semiring"],["HPow","hPow"],["OfNat","ofNat"],["Polynomial"],["Nat"],["LE","le"],["Polynomial","leadingCoeff"],["instHMul"],["instLENat"],["Semiring"]],"valueReferences":[["RingHom"],["LinearMap","instFunLike"],["Polynomial","natDegree"],["RingHom","instFunLike"],["HMul","hMul"],["DFunLike","coe"],["congrArg"],["Polynomial","module"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["RingHom","id"],["MonoidWithZero","toMonoid"],["Polynomial","instMul"],["Semiring","toModule"],["Eq"],["instHPow"],["Polynomial","X"],["Polynomial","C"],["Polynomial","monomial"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["Polynomial","semiring"],["Polynomial","C_mul_X_pow_eq_monomial"],["HPow","hPow"],["LinearMap"],["Polynomial"],["Nat"],["Eq","refl"],["Polynomial","leadingCoeff"],["id"],["instHMul"],["Eq","mpr"],["Polynomial","monomial_natDegree_leadingCoeff_eq_self"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","monomial_natDegree_leadingCoeff_eq_self"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 : Semiring.{u} R] {p : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14}, (LE.le.{0} Nat instLENat (Finset.card.{0} Nat (Polynomial.support.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 p)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) -> (Eq.{succ u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (DFunLike.coe.{succ u, succ u, succ u} (LinearMap.{u, u, u, u} R R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14)) R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (Polynomial.semiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14)))) (Semiring.toModule.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (Polynomial.module.{u, u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 (Semiring.toModule.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14))) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (LinearMap.instFunLike.{u, u, u, u} R R R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (Polynomial.semiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14)))) (Semiring.toModule.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14) (Polynomial.module.{u, u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 (Semiring.toModule.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14))) (Polynomial.monomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 (Polynomial.natDegree.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 p)) (Polynomial.leadingCoeff.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3045275217._hygCtx._hyg.14 p)) p)","typeFull":"∀ {R : Type u} [inst : Semiring R] {p : Polynomial R},\n  p.support.card ≤ 1 → (Polynomial.monomial p.natDegree) p.leadingCoeff = p","typeReadable":"∀ {R : Type u} [inst : Semiring R] {p : Polynomial R},\n  p.support.card ≤ 1 → (Polynomial.monomial p.natDegree) p.leadingCoeff = p","typeReferences":[["LinearMap","instFunLike"],["Polynomial","monomial"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Finset","card"],["Polynomial","natDegree"],["Polynomial","semiring"],["LinearMap"],["DFunLike","coe"],["OfNat","ofNat"],["Polynomial","module"],["Polynomial"],["Nat"],["Polynomial","support"],["Semiring","toNonAssocSemiring"],["instOfNatNat"],["RingHom","id"],["Polynomial","leadingCoeff"],["LE","le"],["Semiring","toModule"],["Eq"],["instLENat"],["Semiring"]],"valueReferences":[["Polynomial","monomial_zero_right"],["Eq","trans"],["LinearMap","instFunLike"],["Classical","propDecidable"],["Polynomial","natDegree"],["Iff","mp"],["Polynomial","leadingCoeff_monomial"],["DFunLike","coe"],["congrArg"],["Polynomial","natDegree_monomial"],["Polynomial","module"],["Polynomial","support"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["instOfNatNat"],["congr"],["Eq","symm"],["congrFun'"],["Zero","toOfNat0"],["Eq","ndrec"],["Semiring","toModule"],["Eq"],["Exists"],["True"],["ite"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Polynomial","monomial"],["Finset","card"],["Polynomial","instZero"],["Polynomial","semiring"],["LinearMap"],["ite_cond_eq_false"],["OfNat","ofNat"],["Polynomial","card_support_le_one_iff_monomial"],["Exists","casesOn"],["eq_self"],["Polynomial"],["Nat"],["of_eq_true"],["MulZeroClass","toZero"],["eq_false"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Polynomial","leadingCoeff"],["LE","le"],["dite"],["instLENat"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","natDegree_le_pred"],"typeFallback":"forall {R : Type.{u}} {n : Nat} [inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3240046117._hygCtx._hyg.14 : Semiring.{u} R] {p : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3240046117._hygCtx._hyg.14}, (LE.le.{0} Nat instLENat (Polynomial.natDegree.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3240046117._hygCtx._hyg.14 p) n) -> (Eq.{succ u} R (Polynomial.coeff.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3240046117._hygCtx._hyg.14 p n) (OfNat.ofNat.{u} R 0 (Zero.toOfNat0.{u} R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3240046117._hygCtx._hyg.14))))))) -> (LE.le.{0} Nat instLENat (Polynomial.natDegree.{u} R inst._@.Mathlib.Algebra.Polynomial.Degree.Monomial.3240046117._hygCtx._hyg.14 p) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))","typeFull":"∀ {R : Type u} {n : ℕ} [inst : Semiring R] {p : Polynomial R}, p.natDegree ≤ n → p.coeff n = 0 → p.natDegree ≤ n - 1","typeReadable":"∀ {R : Type u} {n : ℕ} [inst : Semiring R] {p : Polynomial R}, p.natDegree ≤ n → p.coeff n = 0 → p.natDegree ≤ n - 1","typeReferences":[["Polynomial","coeff"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Polynomial","natDegree"],["OfNat","ofNat"],["Polynomial"],["Nat"],["Semiring","toNonAssocSemiring"],["instSubNat"],["instOfNatNat"],["MulZeroClass","toZero"],["LE","le"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSub","hSub"],["Zero","toOfNat0"],["instHSub"],["Eq"],["instLENat"],["Semiring"]],"valueReferences":[["instAddNat"],["Nat","le_succ_iff_eq_or_le"],["Eq","trans"],["Eq","mp"],["Polynomial","natDegree"],["Iff","mp"],["Nat","add_eq_zero_iff","_simp_1"],["congrArg"],["Nat","instNeZeroSucc"],["Nat","succ_eq_add_one"],["one_ne_zero","_simp_1"],["Semiring","toNonAssocSemiring"],["False","elim"],["Nat","instOne"],["Or"],["instOfNatNat"],["Eq","symm"],["HSub","hSub"],["Polynomial","coeff_natDegree"],["congrFun'"],["Zero","toOfNat0"],["Nat","instMulZeroClass"],["Eq"],["propext"],["Polynomial","leadingCoeff_eq_zero"],["Nat","casesAuxOn"],["Polynomial","coeff"],["instHAdd"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["And"],["Or","resolve_left"],["Polynomial","instZero"],["OfNat","ofNat"],["HAdd","hAdd"],["Polynomial"],["Nat"],["instSubNat"],["MulZeroClass","toZero"],["Nat","succ"],["Polynomial","leadingCoeff"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["LE","le"],["and_false"],["Nat","right_eq_add","_simp_1"],["False"],["instHSub"],["instLENat"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Polynomial","map_smul"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14 : Semiring.{u} R] {p : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29 : Semiring.{v} S] (f : RingHom.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29)) (r : R), Eq.{succ v} (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29) (Polynomial.map.{u, v} R S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29 f (HSMul.hSMul.{u, u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (instHSMul.{u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (SMulZeroClass.toSMul.{u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (Polynomial.instZero.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (Polynomial.smulZeroClass.{u, u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14 R (DistribSMul.toSMulZeroClass.{u, u} R R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14))))) (instDistribSMul.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14))))))) r p)) (HSMul.hSMul.{v, v, v} S (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29) (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29) (instHSMul.{v, v} S (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29) (SMulZeroClass.toSMul.{v, v} S (Polynomial.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29) (Polynomial.instZero.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29) (Polynomial.smulZeroClass.{v, v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29 S (DistribSMul.toSMulZeroClass.{v, v} S S (AddMonoid.toAddZeroClass.{v} S (AddMonoidWithOne.toAddMonoid.{v} S (AddCommMonoidWithOne.toAddMonoidWithOne.{v} S (NonAssocSemiring.toAddCommMonoidWithOne.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29))))) (instDistribSMul.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29))))))) (DFunLike.coe.{max (succ u) (succ v), succ u, succ v} (RingHom.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29)) f r) (Polynomial.map.{u, v} R S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.4062404420._hygCtx._hyg.29 f p))","typeFull":"∀ {R : Type u} {S : Type v} [inst : Semiring R] {p : Polynomial R} [inst_1 : Semiring S] (f : R →+* S) (r : R),\n  Polynomial.map f (r • p) = f r • Polynomial.map f p","typeReadable":"∀ {R : Type u} {S : Type v} [inst : Semiring R] {p : Polynomial R} [inst_1 : Semiring S] (f : R →+* S) (r : R),\n  Polynomial.map f (r • p) = f r • Polynomial.map f p","typeReferences":[["RingHom"],["Polynomial","map"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["RingHom","instFunLike"],["Polynomial","instZero"],["SMulZeroClass","toSMul"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","smulZeroClass"],["DFunLike","coe"],["Polynomial"],["Semiring","toNonAssocSemiring"],["HSMul","hSMul"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Semiring"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["RingHom"],["Polynomial","map"],["HMul","hMul"],["RingHom","instFunLike"],["SMulZeroClass","toSMul"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","smulZeroClass"],["instDistribSMul"],["DFunLike","coe"],["congrArg"],["Polynomial","eval₂_smul"],["Polynomial","eval₂"],["Semiring","toNonAssocSemiring"],["Polynomial","instMul"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["RingHom","comp"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["Polynomial","X"],["Polynomial","map","eq_1"],["NonUnitalNonAssocSemiring","toDistrib"],["Polynomial","C"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Polynomial","instZero"],["Polynomial","semiring"],["Polynomial"],["Eq","refl"],["HSMul","hSMul"],["id"],["instHMul"],["Eq","mpr"],["Polynomial","C_mul'"],["RingHom","comp_apply"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Polynomial","leval"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 : Semiring.{u_1} R], R -> (LinearMap.{u_1, u_1, u_1, u_1} R R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)) (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Semiring.toNonAssocSemiring.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Polynomial.semiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))) (Polynomial.module.{u_1, u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)) (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))","typeFull":"{R : Type u_1} → [inst : Semiring R] → R → Polynomial R →ₗ[R] R","typeReadable":"{R : Type u_1} → [inst : Semiring R] → R → Polynomial R →ₗ[R] R","typeReferences":[["Polynomial","module"],["Polynomial"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Polynomial","semiring"],["Semiring","toModule"],["LinearMap"],["Semiring"]],"valueReferences":[["LinearMap","mk"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["AddHom","mk"],["Polynomial","semiring"],["Polynomial","leval","_proof_1"],["Polynomial","eval"],["Polynomial"],["Polynomial","module"],["Semiring","toNonAssocSemiring"],["AddCommMonoid","toAddCommSemigroup"],["RingHom","id"],["Polynomial","eval_add"],["AddCommMagma","toAdd"],["AddCommSemigroup","toAddCommMagma"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","smul_comp"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.29 : SMulZeroClass.{v, u} S R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14))))] [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.33 : IsScalarTower.{v, u, u} S R R (SMulZeroClass.toSMul.{v, u} S R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14)))) inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.29) (SMulZeroClass.toSMul.{u, u} R R (AddZero.toZero.{u} R (AddZeroClass.toAddZero.{u} R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14))))))) (DistribSMul.toSMulZeroClass.{u, u} R R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14))))) (instDistribSMul.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14))))) (SMulZeroClass.toSMul.{v, u} S R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14)))) inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.29)] (s : S) (p : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (q : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14), Eq.{succ u} (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (Polynomial.comp.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14 (HSMul.hSMul.{v, u, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (instHSMul.{v, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (SMulZeroClass.toSMul.{v, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (Polynomial.instZero.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (Polynomial.smulZeroClass.{u, v} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14 S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.29))) s p) q) (HSMul.hSMul.{v, u, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (instHSMul.{v, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (SMulZeroClass.toSMul.{v, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (Polynomial.instZero.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14) (Polynomial.smulZeroClass.{u, v} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14 S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.29))) s (Polynomial.comp.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.379536305._hygCtx._hyg.14 p q))","typeFull":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : SMulZeroClass S R] [IsScalarTower S R R] (s : S)\n  (p q : Polynomial R), (s • p).comp q = s • p.comp q","typeReadable":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : SMulZeroClass S R] [IsScalarTower S R R] (s : S)\n  (p q : Polynomial R), (s • p).comp q = s • p.comp q","typeReferences":[["IsScalarTower"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Polynomial","instZero"],["SMulZeroClass","toSMul"],["Polynomial","smulZeroClass"],["instDistribSMul"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["Polynomial"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["HSMul","hSMul"],["SMulZeroClass"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHSMul"],["Polynomial","comp"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["RingHom"],["SemigroupAction","toSMul"],["HMul","hMul"],["SMulZeroClass","toSMul"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","smulZeroClass"],["Polynomial","eval₂_smul"],["Polynomial","eval₂"],["Semiring","toNonAssocSemiring"],["Polynomial","comp","eq_1"],["Polynomial","instMul"],["DistribMulAction","toMulAction"],["Eq","symm"],["Monoid","toSemigroup"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Polynomial"],["Eq","refl"],["one_smul"],["HSMul","hSMul"],["id"],["instHMul"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Polynomial","isScalarTower"],["RingHom","instFunLike"],["DFunLike","coe"],["congrArg"],["smul_one_smul"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Polynomial","comp"],["Eq"],["Polynomial","C"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Polynomial","instZero"],["Semiring","toMonoidWithZero"],["Polynomial","semiring"],["OfNat","ofNat"],["Module","toDistribMulAction"],["Polynomial","smul_eq_C_mul"],["One","toOfNat1"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["smul_assoc"],["Polynomial","distribMulAction"],["MulAction","toSemigroupAction"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","leval","_proof_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 : Semiring.{u_1} R] (r : R) (c : R) (f : Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31), Eq.{succ u_1} R (Polynomial.eval.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 r (HSMul.hSMul.{u_1, u_1, u_1} R (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (instHSMul.{u_1, u_1} R (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (SMulZeroClass.toSMul.{u_1, u_1} R (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Polynomial.instZero.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Polynomial.smulZeroClass.{u_1, u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 R (DistribSMul.toSMulZeroClass.{u_1, u_1} R 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.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))))))) c f)) (HSMul.hSMul.{u_1, u_1, u_1} R R R (instHSMul.{u_1, u_1} R R (SMulZeroClass.toSMul.{u_1, u_1} R R (MulZeroClass.toZero.{u_1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)))) (DistribSMul.toSMulZeroClass.{u_1, u_1} R 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.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))))) (instDistribSMul.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)))))) c (Polynomial.eval.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 r f))","typeFull":"∀ {R : Type u_1} [inst : Semiring R] (r c : R) (f : Polynomial R), Polynomial.eval r (c • f) = c • Polynomial.eval r f","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] (r c : R) (f : Polynomial R), Polynomial.eval r (c • f) = c • Polynomial.eval r f","typeReferences":[["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Polynomial","instZero"],["SMulZeroClass","toSMul"],["instDistribSMul"],["Polynomial","smulZeroClass"],["AddMonoidWithOne","toAddMonoid"],["Polynomial"],["Polynomial","eval"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["HSMul","hSMul"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Semiring"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Polynomial","eval_smul"],["IsScalarTower","left"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Semiring","toMonoidWithZero"],["AddCommMonoid","toAddMonoid"],["AddMonoidWithOne","toAddMonoid"],["instDistribSMul"],["Module","toDistribMulAction"],["Semiring","toNonAssocSemiring"],["MonoidWithZero","toMonoid"],["DistribMulAction","toMulAction"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","leval_apply"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 : Semiring.{u_1} R] (r : R) (f : Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31), Eq.{succ u_1} R (DFunLike.coe.{succ u_1, succ u_1, succ u_1} (LinearMap.{u_1, u_1, u_1, u_1} R R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)) (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Semiring.toNonAssocSemiring.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Polynomial.semiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))) (Polynomial.module.{u_1, u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)) (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)) (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) => R) (LinearMap.instFunLike.{u_1, u_1, u_1, u_1} R R (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Semiring.toNonAssocSemiring.{u_1} (Polynomial.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (Polynomial.semiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))) (Polynomial.module.{u_1, u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31)) (Semiring.toModule.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31) (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31))) (Polynomial.leval.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 r) f) (Polynomial.eval.{u_1} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.2869681963._hygCtx._hyg.31 r f)","typeFull":"∀ {R : Type u_1} [inst : Semiring R] (r : R) (f : Polynomial R), (Polynomial.leval r) f = Polynomial.eval r f","typeReadable":"∀ {R : Type u_1} [inst : Semiring R] (r : R) (f : Polynomial R), (Polynomial.leval r) f = Polynomial.eval r f","typeReferences":[["LinearMap","instFunLike"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Polynomial","semiring"],["Polynomial","leval"],["LinearMap"],["DFunLike","coe"],["Polynomial","eval"],["Polynomial"],["Polynomial","module"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq"],["Semiring","toModule"],["Semiring"]],"valueReferences":[["LinearMap","instFunLike"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Polynomial","semiring"],["LinearMap"],["Polynomial","leval"],["DFunLike","coe"],["Polynomial"],["Polynomial","module"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Eq","refl"],["Semiring","toModule"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","eval₂_smul"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.29 : Semiring.{v} S] (g : RingHom.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.29)) (p : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (x : S) {s : R}, Eq.{succ v} S (Polynomial.eval₂.{u, v} R S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.29 g x (HSMul.hSMul.{u, u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (instHSMul.{u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (SMulZeroClass.toSMul.{u, u} R (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (Polynomial.instZero.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (Polynomial.smulZeroClass.{u, u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14 R (DistribSMul.toSMulZeroClass.{u, u} R R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14))))) (instDistribSMul.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14))))))) s p)) (HMul.hMul.{v, v, v} S S S (instHMul.{v} S (Distrib.toMul.{v} S (NonUnitalNonAssocSemiring.toDistrib.{v} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{v} S (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.29))))) (DFunLike.coe.{max (succ u) (succ v), succ u, succ v} (RingHom.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.29)) R (fun (x._@.Mathlib.Data.FunLike.Basic.2582841819._hygCtx._hyg.11 : R) => S) (RingHom.instFunLike.{u, v} R S (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14) (Semiring.toNonAssocSemiring.{v} S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.29)) g s) (Polynomial.eval₂.{u, v} R S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.14 inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.1168276109._hygCtx._hyg.29 g x p))","typeFull":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (g : R →+* S) (p : Polynomial R) (x : S) {s : R},\n  Polynomial.eval₂ g x (s • p) = g s * Polynomial.eval₂ g x p","typeReadable":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : Semiring S] (g : R →+* S) (p : Polynomial R) (x : S) {s : R},\n  Polynomial.eval₂ g x (s • p) = g s * Polynomial.eval₂ g x p","typeReferences":[["RingHom"],["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["RingHom","instFunLike"],["HMul","hMul"],["Polynomial","instZero"],["SMulZeroClass","toSMul"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","smulZeroClass"],["DFunLike","coe"],["Polynomial","eval₂"],["Polynomial"],["Semiring","toNonAssocSemiring"],["HSMul","hSMul"],["instHMul"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["Semiring"],["DistribSMul","toSMulZeroClass"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instAddNat"],["RingHom"],["Finset"],["Eq","trans"],["NonUnitalSemiring","toSemigroupWithZero"],["HMul","hMul"],["SMulZeroClass","toSMul"],["Polynomial","eval₂_eq_sum"],["NonUnitalRingHomClass","toMulHomClass"],["RingHom","instRingHomClass"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","smulZeroClass"],["Polynomial","eval₂"],["Semiring","toNonAssocSemiring"],["Monoid","toPow"],["forall_congr"],["Finset","sum"],["Finset","mul_sum"],["Semiring","toNonUnitalSemiring"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["Polynomial","coeff_smul"],["DistribSMul","toSMulZeroClass"],["instHPow"],["Polynomial","coeff"],["NonUnitalNonAssocSemiring","toDistrib"],["Polynomial","sum"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["MulZeroClass","zero_mul"],["AddZeroClass","toAddZero"],["implies_true"],["MonoidWithZeroHomClass","toZeroHomClass"],["Polynomial"],["Nat"],["Eq","refl"],["HSMul","hSMul"],["map_mul"],["id"],["instHMul"],["Eq","mpr"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["RingHomClass","toNonUnitalRingHomClass"],["Polynomial","natDegree"],["RingHom","instFunLike"],["mul_assoc"],["AddCommMonoid","toAddMonoid"],["DFunLike","coe"],["Semigroup","toMul"],["Nat","instPreorder"],["congrArg"],["Finset","sum_congr"],["LE","le","trans_lt"],["instOfNatNat"],["congr"],["RingHomClass","toMonoidWithZeroHomClass"],["MonoidWithZero","toMonoid"],["Nat","lt_succ_self"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["map_zero"],["Eq"],["Finset","range"],["True"],["instHAdd"],["Distrib","toMul"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Polynomial","instZero"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["HAdd","hAdd"],["eq_self"],["Polynomial","sum_over_range'"],["of_eq_true"],["Polynomial","natDegree_smul_le"],["Nat","succ"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["SemigroupWithZero","toSemigroup"]]},{"isProp":true,"kind":"theorem","name":["Polynomial","eval_smul"],"typeFallback":"forall {R : Type.{u}} {S : Type.{v}} [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14 : Semiring.{u} R] [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.30 : SMulZeroClass.{v, u} S R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14))))] [inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.34 : IsScalarTower.{v, u, u} S R R (SMulZeroClass.toSMul.{v, u} S R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14)))) inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.30) (SMulZeroClass.toSMul.{u, u} R R (AddZero.toZero.{u} R (AddZeroClass.toAddZero.{u} R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14))))))) (DistribSMul.toSMulZeroClass.{u, u} R R (AddMonoid.toAddZeroClass.{u} R (AddMonoidWithOne.toAddMonoid.{u} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u} R (NonAssocSemiring.toAddCommMonoidWithOne.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14))))) (instDistribSMul.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14))))) (SMulZeroClass.toSMul.{v, u} S R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14)))) inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.30)] (s : S) (p : Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14) (x : R), Eq.{succ u} R (Polynomial.eval.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14 x (HSMul.hSMul.{v, u, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14) (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14) (instHSMul.{v, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14) (SMulZeroClass.toSMul.{v, u} S (Polynomial.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14) (Polynomial.instZero.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14) (Polynomial.smulZeroClass.{u, v} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14 S inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.30))) s p)) (HSMul.hSMul.{v, u, u} S R R (instHSMul.{v, u} S R (SMulZeroClass.toSMul.{v, u} S R (MulZeroClass.toZero.{u} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u} R (Semiring.toNonAssocSemiring.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14)))) inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.30)) s (Polynomial.eval.{u} R inst._@.Mathlib.Algebra.Polynomial.Eval.SMul.3097354570._hygCtx._hyg.14 x p))","typeFull":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : SMulZeroClass S R] [IsScalarTower S R R] (s : S)\n  (p : Polynomial R) (x : R), Polynomial.eval x (s • p) = s • Polynomial.eval x p","typeReadable":"∀ {R : Type u} {S : Type v} [inst : Semiring R] [inst_1 : SMulZeroClass S R] [IsScalarTower S R R] (s : S)\n  (p : Polynomial R) (x : R), Polynomial.eval x (s • p) = s • Polynomial.eval x p","typeReferences":[["IsScalarTower"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["Polynomial","instZero"],["SMulZeroClass","toSMul"],["Polynomial","smulZeroClass"],["instDistribSMul"],["AddZeroClass","toAddZero"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","eval"],["Polynomial"],["Semiring","toNonAssocSemiring"],["MulZeroClass","toZero"],["HSMul","hSMul"],["SMulZeroClass"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Eq"],["NonAssocSemiring","toAddCommMonoidWithOne"],["AddZero","toZero"],["DistribSMul","toSMulZeroClass"],["Semiring"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["RingHom"],["RingHom","id_apply"],["SemigroupAction","toSMul"],["HMul","hMul"],["SMulZeroClass","toSMul"],["instDistribSMul"],["AddMonoidWithOne","toAddMonoid"],["Polynomial","smulZeroClass"],["smul_one_mul"],["Polynomial","eval₂_smul"],["Polynomial","eval₂"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["DistribMulAction","toMulAction"],["Eq","symm"],["Monoid","toSemigroup"],["Semiring","toModule"],["NonAssocSemiring","toAddCommMonoidWithOne"],["NonAssocSemiring","toMulZeroOneClass"],["DistribSMul","toSMulZeroClass"],["MulOne","toOne"],["NonUnitalNonAssocSemiring","toDistrib"],["NonAssocSemiring","toNonUnitalNonAssocSemiring"],["DistribMulAction","toDistribSMul"],["Polynomial","eval"],["Polynomial"],["Eq","refl"],["HSMul","hSMul"],["id"],["instHMul"],["Eq","mpr"],["AddMonoid","toAddZeroClass"],["MulOneClass","toMulOne"],["Polynomial","isScalarTower"],["RingHom","instFunLike"],["MulZeroOneClass","toMulOneClass"],["DFunLike","coe"],["congrArg"],["MulOne","toMul"],["smul_one_smul"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["instHSMul"],["AddCommMonoidWithOne","toAddMonoidWithOne"],["Polynomial","eval","eq_1"],["Eq"],["NonUnitalNonAssocSemiring","toAddCommMonoid"],["Distrib","toMul"],["Semiring","toMonoidWithZero"],["Polynomial","instZero"],["Polynomial","semiring"],["OfNat","ofNat"],["Module","toDistribMulAction"],["One","toOfNat1"],["MulZeroClass","toZero"],["Polynomial","eval₂_id"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Polynomial","distribMulAction"],["MulAction","toSemigroupAction"]]}]

data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Ring.Int.Defs.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_4"],"typeFallback":"Eq.{1} Int (Nat.cast.{0} Int instNatCastInt (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (Nat.cast.{0} Int instNatCastInt (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))","typeFull":"↑0 = ↑0","typeReadable":"↑0 = ↑0","typeReferences":[["Nat"],["Nat","cast"],["instOfNatNat"],["Eq"],["OfNat","ofNat"],["instNatCastInt"],["Int"]],"valueReferences":[["rfl"],["Nat"],["Nat","cast"],["instOfNatNat"],["OfNat","ofNat"],["instNatCastInt"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_12"],"typeFallback":"forall (x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.74 : Nat), Eq.{1} Int (Int.negSucc x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.74) (Int.negSucc x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.74)","typeFull":"∀ (x : ℕ), Int.negSucc x = Int.negSucc x","typeReadable":"∀ (x : ℕ), Int.negSucc x = Int.negSucc x","typeReferences":[["Nat"],["Int","negSucc"],["Eq"],["Int"]],"valueReferences":[["rfl"],["Int","negSucc"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_11"],"typeFallback":"forall (x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.69 : Nat), Eq.{1} Int (Nat.cast.{0} Int instNatCastInt x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.69) (Nat.cast.{0} Int instNatCastInt x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.69)","typeFull":"∀ (x : ℕ), ↑x = ↑x","typeReadable":"∀ (x : ℕ), ↑x = ↑x","typeReferences":[["Nat"],["Nat","cast"],["Eq"],["instNatCastInt"],["Int"]],"valueReferences":[["rfl"],["Nat","cast"],["instNatCastInt"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_9"],"typeFallback":"forall (n : Nat) (a : Int), Eq.{1} Int (SubNegMonoid.zsmul.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)) (Int.negSucc n) a) (Neg.neg.{0} Int (SubNegMonoid.toNeg.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup))) (SubNegMonoid.zsmul.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a))","typeFull":"∀ (n : ℕ) (a : ℤ), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a","typeReadable":"∀ (n : ℕ) (a : ℤ), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a","typeReferences":[["Nat","cast"],["SubNegMonoid","zsmul"],["Neg","neg"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["Int","negSucc"],["Int"],["Nat"],["Int","instAddCommGroup"],["Nat","succ"],["AddGroup","toSubNegMonoid"],["Eq"],["instNatCastInt"]],"valueReferences":[["Int","instAddCommGroup"],["SubNegMonoid","zsmul_neg'"],["AddCommGroup","toAddGroup"],["AddGroup","toSubNegMonoid"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","cast_mul","_simp_1"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3 : NonAssocRing.{u_1} α] (m : Int) (n : Int), Eq.{succ u_1} α (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (Distrib.toMul.{u_1} α (NonUnitalNonAssocSemiring.toDistrib.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))))) (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α (AddCommGroupWithOne.toAddGroupWithOne.{u_1} α (NonAssocRing.toAddCommGroupWithOne.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))) m) (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α (AddCommGroupWithOne.toAddGroupWithOne.{u_1} α (NonAssocRing.toAddCommGroupWithOne.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))) n)) (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α (AddCommGroupWithOne.toAddGroupWithOne.{u_1} α (NonAssocRing.toAddCommGroupWithOne.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))) (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.instMul) m n))","typeFull":"∀ {α : Type u_1} [inst : NonAssocRing α] (m n : ℤ), ↑m * ↑n = ↑(m * n)","typeReadable":"∀ {α : Type u_1} [inst : NonAssocRing α] (m n : ℤ), ↑m * ↑n = ↑(m * n)","typeReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonAssocRing"],["HMul","hMul"],["Int","cast"],["Int","instMul"],["Int"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddCommGroupWithOne","toAddGroupWithOne"],["NonAssocRing","toAddCommGroupWithOne"],["AddGroupWithOne","toIntCast"],["instHMul"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"]],"valueReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["HMul","hMul"],["Int","cast"],["Int"],["Int","instMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddCommGroupWithOne","toAddGroupWithOne"],["Int","cast_mul"],["NonAssocRing","toAddCommGroupWithOne"],["Eq","symm"],["AddGroupWithOne","toIntCast"],["instHMul"],["NonAssocRing","toNonUnitalNonAssocRing"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_10"],"typeFallback":"forall (a : Int), Eq.{1} Int (HAdd.hAdd.{0, 0, 0} Int Int Int (instHAdd.{0} Int (AddSemigroup.toAdd.{0} Int (AddMonoid.toAddSemigroup.{0} Int (SubNegMonoid.toAddMonoid.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)))))) (Neg.neg.{0} Int (SubNegMonoid.toNeg.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup))) a) a) (OfNat.ofNat.{0} Int 0 (Zero.toOfNat0.{0} Int (AddMonoid.toZero.{0} Int (SubNegMonoid.toAddMonoid.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup))))))","typeFull":"∀ (a : ℤ), -a + a = 0","typeReadable":"∀ (a : ℤ), -a + a = 0","typeReferences":[["AddMonoid","toZero"],["instHAdd"],["Neg","neg"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["Int","instAddCommGroup"],["AddMonoid","toAddSemigroup"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["AddSemigroup","toAdd"]],"valueReferences":[["Int","instAddCommGroup"],["AddCommGroup","toAddGroup"],["Int"],["AddGroup","neg_add_cancel"]]},{"isProp":false,"kind":"definition","name":["Int","instCommRing"],"typeFallback":"CommRing.{0} Int","typeFull":"CommRing ℤ","typeReadable":"CommRing ℤ","typeReferences":[["CommRing"],["Int"]],"valueReferences":[["Int","instMonoid"],["Int","mul_add"],["NonUnitalNonAssocSemiring","mk"],["AddCommGroup","toAddGroup"],["Int","instCommRing","_proof_3"],["Int","instCommRing","_proof_10"],["Int","one_mul"],["Int","instCommRing","_proof_12"],["Monoid","toPow"],["SubNegMonoid","toSub"],["AddGroup","toSubNegMonoid"],["instHPow"],["MulOne","toOne"],["CommRing","mk"],["SubNegMonoid","zsmul"],["AddCommMonoid","mk"],["Int","instCommRing","_proof_4"],["Int","mul_zero"],["NatCast","mk"],["Nat"],["Int","instCommRing","_proof_8"],["MulOneClass","toMulOne"],["Int","mul_one"],["Nat","cast"],["Int","instCommSemigroup"],["Semigroup","toMul"],["Int","zero_mul"],["Int","instAddCommGroup"],["Monoid","toMulOneClass"],["Int","instCommRing","_proof_5"],["Int","instCommRing","_proof_6"],["Ring","mk"],["instNatCastInt"],["NonUnitalSemiring","mk"],["AddCommGroup","add_comm"],["SubNegMonoid","toNeg"],["Int","instCommRing","_proof_7"],["Int","instCommRing","_proof_9"],["Semiring","mk"],["Int","instCommRing","_proof_1"],["Int","instCommRing","_proof_2"],["HPow","hPow"],["Int"],["Int","add_mul"],["Int","instCommRing","_proof_11"],["CommSemigroup","mul_comm"],["SubNegMonoid","toAddMonoid"],["CommSemigroup","toSemigroup"],["IntCast","mk"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_5"],"typeFallback":"forall (x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.36 : Nat), Eq.{1} Int (Nat.cast.{0} Int instNatCastInt (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.36 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Nat.cast.{0} Int instNatCastInt (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.36 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))","typeFull":"∀ (x : ℕ), ↑(x + 1) = ↑(x + 1)","typeReadable":"∀ (x : ℕ), ↑(x + 1) = ↑(x + 1)","typeReferences":[["instAddNat"],["HAdd","hAdd"],["Nat"],["Nat","cast"],["instOfNatNat"],["instHAdd"],["Eq"],["OfNat","ofNat"],["instNatCastInt"],["Int"]],"valueReferences":[["instAddNat"],["HAdd","hAdd"],["rfl"],["Nat"],["Nat","cast"],["instOfNatNat"],["instHAdd"],["OfNat","ofNat"],["instNatCastInt"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_7"],"typeFallback":"forall (a : Int), Eq.{1} Int (SubNegMonoid.zsmul.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)) (OfNat.ofNat.{0} Int 0 (instOfNat 0)) a) (OfNat.ofNat.{0} Int 0 (Zero.toOfNat0.{0} Int (AddMonoid.toZero.{0} Int (SubNegMonoid.toAddMonoid.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup))))))","typeFull":"∀ (a : ℤ), SubNegMonoid.zsmul 0 a = 0","typeReadable":"∀ (a : ℤ), SubNegMonoid.zsmul 0 a = 0","typeReferences":[["AddMonoid","toZero"],["SubNegMonoid","toAddMonoid"],["instOfNat"],["Int","instAddCommGroup"],["SubNegMonoid","zsmul"],["AddCommGroup","toAddGroup"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["OfNat","ofNat"],["Int"]],"valueReferences":[["Int","instAddCommGroup"],["AddCommGroup","toAddGroup"],["AddGroup","toSubNegMonoid"],["SubNegMonoid","zsmul_zero'"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_6"],"typeFallback":"forall (a : Int) (b : Int), Eq.{1} Int (HSub.hSub.{0, 0, 0} Int Int Int (instHSub.{0} Int (SubNegMonoid.toSub.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)))) a b) (HAdd.hAdd.{0, 0, 0} Int Int Int (instHAdd.{0} Int (AddSemigroup.toAdd.{0} Int (AddMonoid.toAddSemigroup.{0} Int (SubNegMonoid.toAddMonoid.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)))))) a (Neg.neg.{0} Int (SubNegMonoid.toNeg.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup))) b))","typeFull":"∀ (a b : ℤ), a - b = a + -b","typeReadable":"∀ (a b : ℤ), a - b = a + -b","typeReferences":[["Neg","neg"],["instHAdd"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["Int"],["HAdd","hAdd"],["Int","instAddCommGroup"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid","toSub"],["AddMonoid","toAddSemigroup"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["AddSemigroup","toAdd"]],"valueReferences":[["Int","instAddCommGroup"],["AddCommGroup","toAddGroup"],["AddGroup","toSubNegMonoid"],["SubNegMonoid","sub_eq_add_neg"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","cast_pow"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3 : Ring.{u_1} R] (n : Int) (m : Nat), Eq.{succ u_1} R (Int.cast.{u_1} R (AddGroupWithOne.toIntCast.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3)) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) n m)) (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3))))) (Int.cast.{u_1} R (AddGroupWithOne.toIntCast.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3)) n) m)","typeFull":"∀ {R : Type u_1} [inst : Ring R] (n : ℤ) (m : ℕ), ↑(n ^ m) = ↑n ^ m","typeReadable":"∀ {R : Type u_1} [inst : Ring R] (n : ℤ) (m : ℕ), ↑(n ^ m) = ↑n ^ m","typeReferences":[["instHPow"],["Int","instMonoid"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["Int","cast"],["Ring","toSemiring"],["Int"],["Nat"],["Ring","toAddGroupWithOne"],["Monoid","toPow"],["MonoidWithZero","toMonoid"],["AddGroupWithOne","toIntCast"],["Eq"],["Ring"]],"valueReferences":[["instAddNat"],["MulOneClass","toMulOne"],["Ring","toNonAssocRing"],["Int","instMonoid"],["Eq","trans"],["HMul","hMul"],["AddGroupWithOne","toAddMonoidWithOne"],["Int","cast"],["congrArg"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["MulOne","toMul"],["pow_succ"],["Monoid","toPow"],["Ring","toAddGroupWithOne"],["instOfNatNat"],["NonAssocRing","toAddCommGroupWithOne"],["Int","cast_one"],["congr"],["MonoidWithZero","toMonoid"],["Monoid","toMulOneClass"],["congrFun'"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"],["instHPow"],["NonUnitalNonAssocSemiring","toDistrib"],["MulOne","toOne"],["True"],["instHAdd"],["Distrib","toMul"],["Nat","recAux"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["Ring","toSemiring"],["HAdd","hAdd"],["eq_self"],["AddCommGroupWithOne","toAddGroupWithOne"],["Nat"],["of_eq_true"],["One","toOfNat1"],["Int","cast_mul"],["AddMonoidWithOne","toOne"],["AddGroupWithOne","toIntCast"],["instHMul"],["pow_zero"]]},{"isProp":true,"kind":"theorem","name":["Int","instIsDomain"],"typeFallback":"IsDomain.{0} Int (CommSemiring.toSemiring.{0} Int (CommRing.toCommSemiring.{0} Int Int.instCommRing))","typeFull":"IsDomain ℤ","typeReadable":"IsDomain ℤ","typeReferences":[["Int","instCommRing"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["IsDomain"],["Int"]],"valueReferences":[["Int","instCommRing"],["CommRing","toCommSemiring"],["IsDomain","mk"],["Int","instNontrivial"],["CommSemiring","toSemiring"],["Int","instIsCancelMulZero"],["Int"]]},{"isProp":false,"kind":"definition","name":["Int","instSemiring"],"typeFallback":"Semiring.{0} Int","typeFull":"Semiring ℤ","typeReadable":"Semiring ℤ","typeReferences":[["Int"],["Semiring"]],"valueReferences":[["Int","instCommSemiring"],["CommSemiring","toSemiring"],["inferInstance"],["Int"],["Semiring"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Ring","Int","Defs",0,"Int","cast_mul_eq_zsmul_cast","_simp_1_1"],"typeFallback":"forall {G : Type.{u_1}} [inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3 : SubNegMonoid.{u_1} G] (a : G) (b : G), Eq.{succ u_1} G (HAdd.hAdd.{u_1, u_1, u_1} G G G (instHAdd.{u_1} G (AddZero.toAdd.{u_1} G (AddZeroClass.toAddZero.{u_1} G (AddMonoid.toAddZeroClass.{u_1} G (SubNegMonoid.toAddMonoid.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3))))) a (Neg.neg.{u_1} G (SubNegMonoid.toNeg.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3) b)) (HSub.hSub.{u_1, u_1, u_1} G G G (instHSub.{u_1} G (SubNegMonoid.toSub.{u_1} G inst._@.Mathlib.Algebra.Group.Defs.1234330678._hygCtx._hyg.3)) a b)","typeFull":"∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a + -b = a - b","typeReadable":"∀ {G : Type u_1} [inst : SubNegMonoid G] (a b : G), a + -b = a - b","typeReferences":[["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid"],["SubNegMonoid","toSub"],["HSub","hSub"],["Eq"],["instHSub"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["instHAdd"],["Neg","neg"],["SubNegMonoid","toNeg"],["AddZero","toAdd"],["AddZeroClass","toAddZero"],["HAdd","hAdd"],["sub_eq_add_neg"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid","toSub"],["Eq","symm"],["HSub","hSub"],["instHSub"],["AddMonoid","toAddZeroClass"]]},{"isProp":false,"kind":"definition","name":["Int","instRing"],"typeFallback":"Ring.{0} Int","typeFull":"Ring ℤ","typeReadable":"Ring ℤ","typeReferences":[["Int"],["Ring"]],"valueReferences":[["Int","instCommRing"],["CommRing","toRing"],["inferInstance"],["Int"],["Ring"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_1"],"typeFallback":"forall (x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.50 : Int), Eq.{1} Int (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.50 (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (OfNat.ofNat.{0} Int 1 (One.toOfNat1.{0} Int (MulOne.toOne.{0} Int (MulOneClass.toMulOne.{0} Int (Monoid.toMulOneClass.{0} Int Int.instMonoid)))))","typeFull":"∀ (x : ℤ), x ^ 0 = 1","typeReadable":"∀ (x : ℤ), x ^ 0 = 1","typeReferences":[["instHPow"],["MulOneClass","toMulOne"],["Int","instMonoid"],["MulOne","toOne"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["Nat"],["Monoid","toPow"],["One","toOfNat1"],["instOfNatNat"],["Monoid","toMulOneClass"],["Eq"]],"valueReferences":[["instHPow"],["MulOneClass","toMulOne"],["Int","instMonoid"],["True"],["Eq","trans"],["MulOne","toOne"],["HPow","hPow"],["OfNat","ofNat"],["congrArg"],["Int"],["eq_self"],["Nat"],["of_eq_true"],["Monoid","toPow"],["One","toOfNat1"],["instOfNatNat"],["Monoid","toMulOneClass"],["congrFun'"],["Eq"],["pow_zero"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_8"],"typeFallback":"forall (n : Nat) (a : Int), Eq.{1} Int (SubNegMonoid.zsmul.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a) (HAdd.hAdd.{0, 0, 0} Int Int Int (instHAdd.{0} Int (AddSemigroup.toAdd.{0} Int (AddMonoid.toAddSemigroup.{0} Int (SubNegMonoid.toAddMonoid.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)))))) (SubNegMonoid.zsmul.{0} Int (AddGroup.toSubNegMonoid.{0} Int (AddCommGroup.toAddGroup.{0} Int Int.instAddCommGroup)) (Nat.cast.{0} Int instNatCastInt n) a) a)","typeFull":"∀ (n : ℕ) (a : ℤ), SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a","typeReadable":"∀ (n : ℕ) (a : ℤ), SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a","typeReferences":[["Nat","cast"],["SubNegMonoid","zsmul"],["instHAdd"],["AddCommGroup","toAddGroup"],["Int"],["HAdd","hAdd"],["Nat"],["SubNegMonoid","toAddMonoid"],["Int","instAddCommGroup"],["Nat","succ"],["AddMonoid","toAddSemigroup"],["AddGroup","toSubNegMonoid"],["Eq"],["instNatCastInt"],["AddSemigroup","toAdd"]],"valueReferences":[["Int","instAddCommGroup"],["AddCommGroup","toAddGroup"],["SubNegMonoid","zsmul_succ'"],["AddGroup","toSubNegMonoid"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","instCharZero"],"typeFallback":"CharZero.{0} Int (AddGroupWithOne.toAddMonoidWithOne.{0} Int (Ring.toAddGroupWithOne.{0} Int (CommRing.toRing.{0} Int Int.instCommRing)))","typeFull":"CharZero ℤ","typeReadable":"CharZero ℤ","typeReferences":[["Int","instCommRing"],["CommRing","toRing"],["Ring","toAddGroupWithOne"],["AddGroupWithOne","toAddMonoidWithOne"],["CharZero"],["Int"]],"valueReferences":[["Int","instCommRing"],["CommRing","toRing"],["Ring","toAddGroupWithOne"],["AddGroupWithOne","toAddMonoidWithOne"],["CharZero","mk"],["Int","ofNat","inj"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","cast_mul"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3 : NonAssocRing.{u_1} α] (m : Int) (n : Int), Eq.{succ u_1} α (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α (AddCommGroupWithOne.toAddGroupWithOne.{u_1} α (NonAssocRing.toAddCommGroupWithOne.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))) (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.instMul) m n)) (HMul.hMul.{u_1, u_1, u_1} α α α (instHMul.{u_1} α (Distrib.toMul.{u_1} α (NonUnitalNonAssocSemiring.toDistrib.{u_1} α (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} α (NonAssocRing.toNonUnitalNonAssocRing.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))))) (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α (AddCommGroupWithOne.toAddGroupWithOne.{u_1} α (NonAssocRing.toAddCommGroupWithOne.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))) m) (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α (AddCommGroupWithOne.toAddGroupWithOne.{u_1} α (NonAssocRing.toAddCommGroupWithOne.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2276920919._hygCtx._hyg.3))) n))","typeFull":"∀ {α : Type u_1} [inst : NonAssocRing α] (m n : ℤ), ↑(m * n) = ↑m * ↑n","typeReadable":"∀ {α : Type u_1} [inst : NonAssocRing α] (m n : ℤ), ↑(m * n) = ↑m * ↑n","typeReferences":[["NonUnitalNonAssocSemiring","toDistrib"],["Distrib","toMul"],["NonAssocRing"],["HMul","hMul"],["Int","cast"],["Int"],["Int","instMul"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["AddCommGroupWithOne","toAddGroupWithOne"],["NonAssocRing","toAddCommGroupWithOne"],["AddGroupWithOne","toIntCast"],["instHMul"],["Eq"],["NonAssocRing","toNonUnitalNonAssocRing"]],"valueReferences":[["instAddNat"],["SubtractionMonoid","toSubNegZeroMonoid"],["Eq","trans"],["AddGroupWithOne","toAddMonoidWithOne"],["AddGroup","toSubtractionMonoid"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Eq","symm"],["Eq","ndrec"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Nat","recAux"],["MulZeroClass","zero_mul"],["implies_true"],["SubtractionMonoid","toSubNegMonoid"],["AddMonoid","toAddSemigroup"],["AddMonoidWithOne","toOne"],["one_mul"],["AddMonoid","toAddZeroClass"],["MulZeroOneClass","toMulOneClass"],["SubtractionCommMonoid","toSubtractionMonoid"],["Int","cast"],["Int","instMul"],["add_mul"],["Int","instAddCommGroup"],["instOfNatNat"],["congr"],["Int","eq_nat_or_neg"],["neg_mul"],["Eq"],["instNatCastInt"],["Int","cast_add"],["Int","cast_zero"],["SubNegMonoid","toNeg"],["AddZero","toAdd"],["OfNat","ofNat"],["Int","cast_neg"],["Int"],["HAdd","hAdd"],["Nat","cast_add"],["CommRing","toRing"],["eq_self"],["AddGroupWithOne","toAddGroup"],["AddCommGroup","toDivisionAddCommMonoid"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddGroupWithOne","toIntCast"],["Nat","cast_one"],["Int","instCommRing"],["HMul","hMul"],["AddMonoidWithOne","toAddMonoid"],["Distrib","rightDistribClass"],["Semiring","toNonAssocSemiring"],["Or"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["forall_congr"],["NonUnitalNonAssocRing","toHasDistribNeg"],["neg_zero"],["InvolutiveNeg","toNeg"],["NonAssocRing","toNonUnitalNonAssocRing"],["NonAssocSemiring","toMulZeroOneClass"],["AddSemigroup","toAdd"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["AddZeroClass","toAddZero"],["Int","instNegInt"],["Exists","casesOn"],["Nat"],["AddMonoidWithOne","toNatCast"],["NonAssocRing","toNonAssocSemiring"],["Nat","cast_zero"],["NegZeroClass","toZero"],["id"],["instHMul"],["Int","cast_natCast"],["AddZero","toZero"],["neg_inj","_simp_1"],["Nat","cast"],["Eq","mp"],["CommRing","toNonUnitalCommRing"],["SubNegZeroMonoid","toNegZeroClass"],["congrArg"],["NonAssocRing","toAddCommGroupWithOne"],["Int","cast_one"],["Zero","toOfNat0"],["congrFun'"],["CommRing","toCommSemiring"],["True"],["instHAdd"],["Distrib","toMul"],["CommSemiring","toSemiring"],["neg_add_rev"],["Or","casesOn"],["AddCommGroupWithOne","toAddGroupWithOne"],["NegZeroClass","toNeg"],["of_eq_true"],["One","toOfNat1"],["HasDistribNeg","toInvolutiveNeg"],["SubNegMonoid","toAddMonoid"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_2"],"typeFallback":"forall (x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.55 : Nat) (x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.57 : Int), Eq.{1} Int (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.57 (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.55 (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int (Semigroup.toMul.{0} Int (CommSemigroup.toSemigroup.{0} Int Int.instCommSemigroup))) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.57 x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.55) x._@.Mathlib.Algebra.Ring.Int.Defs.3906542404._hygCtx._hyg.57)","typeFull":"∀ (x : ℕ) (x_1 : ℤ), x_1 ^ (x + 1) = x_1 ^ x * x_1","typeReadable":"∀ (x : ℕ) (x_1 : ℤ), x_1 ^ (x + 1) = x_1 ^ x * x_1","typeReferences":[["instHPow"],["instAddNat"],["Int","instMonoid"],["Int","instCommSemigroup"],["instHAdd"],["HMul","hMul"],["HPow","hPow"],["OfNat","ofNat"],["Semigroup","toMul"],["Int"],["HAdd","hAdd"],["Nat"],["Monoid","toPow"],["instOfNatNat"],["CommSemigroup","toSemigroup"],["instHMul"],["Eq"]],"valueReferences":[["instAddNat"],["Int","instMonoid"],["Int","instCommSemigroup"],["Eq","trans"],["Int","instNatPow"],["instPowNat"],["HMul","hMul"],["Semigroup","toMul"],["Int","instMul"],["congrArg"],["Int","pow_succ"],["Monoid","toPow"],["instOfNatNat"],["congrFun'"],["Eq"],["instHPow"],["True"],["instHAdd"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["eq_self"],["Nat"],["of_eq_true"],["CommSemigroup","toSemigroup"],["instHMul"]]},{"isProp":false,"kind":"definition","name":["Int","instCommSemiring"],"typeFallback":"CommSemiring.{0} Int","typeFull":"CommSemiring ℤ","typeReadable":"CommSemiring ℤ","typeReferences":[["CommSemiring"],["Int"]],"valueReferences":[["Int","instCommRing"],["CommRing","toCommSemiring"],["CommSemiring"],["inferInstance"],["Int"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Ring","Int","Defs",0,"Int","cast_mul_eq_zsmul_cast","_simp_1_2"],"typeFallback":"forall {b : Prop} (α : Sort.{u_1}) [i : Nonempty.{u_1} α], Eq.{1} Prop (α -> b) b","typeFull":"∀ {b : Prop} (α : Sort u_1) [i : Nonempty α], (∀ (a : α), b) = b","typeReadable":"∀ {b : Prop} (α : Sort u_1) [i : Nonempty α], (∀ (a : α), b) = b","typeReferences":[["Nonempty"],["Eq"]],"valueReferences":[["forall_const"],["propext"]]},{"isProp":true,"kind":"theorem","name":["Int","instIsCancelMulZero"],"typeFallback":"IsCancelMulZero.{0} Int Int.instMul (MulZeroClass.toZero.{0} Int (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Int (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Int (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Int (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Int (CommRing.toNonUnitalCommRing.{0} Int Int.instCommRing))))))","typeFull":"IsCancelMulZero ℤ","typeReadable":"IsCancelMulZero ℤ","typeReferences":[["Int","instCommRing"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["MulZeroClass","toZero"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["CommRing","toNonUnitalCommRing"],["IsCancelMulZero"],["Int","instMul"],["Int"]],"valueReferences":[["Int","instCommRing"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Iff","mp"],["HMul","hMul"],["IsCancelMulZero","mk"],["CommRing","toNonUnitalCommRing"],["Int"],["Int","instMul"],["IsRightCancelMulZero","mk"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Int","mul_eq_mul_left_iff"],["Int","mul_eq_mul_right_iff"],["instHMul"],["IsLeftCancelMulZero","mk"],["Eq"]]},{"isProp":false,"kind":"definition","name":["Int","instDistrib"],"typeFallback":"Distrib.{0} Int","typeFull":"Distrib ℤ","typeReadable":"Distrib ℤ","typeReferences":[["Distrib"],["Int"]],"valueReferences":[["Int","instCommRing"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["NonUnitalNonAssocSemiring","toDistrib"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Distrib"],["inferInstance"],["CommRing","toNonUnitalCommRing"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","cast_pow","_simp_1"],"typeFallback":"forall {R : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3 : Ring.{u_1} R] (n : Int) (m : Nat), Eq.{succ u_1} R (HPow.hPow.{u_1, 0, u_1} R Nat R (instHPow.{u_1, 0} R Nat (Monoid.toPow.{u_1} R (MonoidWithZero.toMonoid.{u_1} R (Semiring.toMonoidWithZero.{u_1} R (Ring.toSemiring.{u_1} R inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3))))) (Int.cast.{u_1} R (AddGroupWithOne.toIntCast.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3)) n) m) (Int.cast.{u_1} R (AddGroupWithOne.toIntCast.{u_1} R (Ring.toAddGroupWithOne.{u_1} R inst._@.Mathlib.Algebra.Ring.Int.Defs.3649256131._hygCtx._hyg.3)) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) n m))","typeFull":"∀ {R : Type u_1} [inst : Ring R] (n : ℤ) (m : ℕ), ↑n ^ m = ↑(n ^ m)","typeReadable":"∀ {R : Type u_1} [inst : Ring R] (n : ℤ) (m : ℕ), ↑n ^ m = ↑(n ^ m)","typeReferences":[["instHPow"],["Int","instMonoid"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["Int","cast"],["Int"],["Ring","toSemiring"],["Nat"],["Monoid","toPow"],["Ring","toAddGroupWithOne"],["MonoidWithZero","toMonoid"],["AddGroupWithOne","toIntCast"],["Eq"],["Ring"]],"valueReferences":[["Int","cast_pow"],["instHPow"],["Int","instMonoid"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["Int","cast"],["Int"],["Ring","toSemiring"],["Nat"],["Ring","toAddGroupWithOne"],["Monoid","toPow"],["MonoidWithZero","toMonoid"],["Eq","symm"],["AddGroupWithOne","toIntCast"]]},{"isProp":true,"kind":"theorem","name":["Int","instMulDivCancelClass"],"typeFallback":"MulDivCancelClass.{0} Int (Semiring.toMonoidWithZero.{0} Int (CommSemiring.toSemiring.{0} Int (CommRing.toCommSemiring.{0} Int Int.instCommRing))) Int.instDiv","typeFull":"MulDivCancelClass ℤ","typeReadable":"MulDivCancelClass ℤ","typeReferences":[["Int","instCommRing"],["MulDivCancelClass"],["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Int","instDiv"],["Int"]],"valueReferences":[["Int","instCommRing"],["CommRing","toCommSemiring"],["MulDivCancelClass","mk"],["Int","mul_ediv_cancel"],["CommSemiring","toSemiring"],["Semiring","toMonoidWithZero"],["Int","instDiv"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","instCommRing","_proof_3"],"typeFallback":"forall (a : Int) (b : Int) (c : Int), Eq.{1} Int (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int (Semigroup.toMul.{0} Int (CommSemigroup.toSemigroup.{0} Int Int.instCommSemigroup))) (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int (Semigroup.toMul.{0} Int (CommSemigroup.toSemigroup.{0} Int Int.instCommSemigroup))) a b) c) (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int (Semigroup.toMul.{0} Int (CommSemigroup.toSemigroup.{0} Int Int.instCommSemigroup))) a (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int (Semigroup.toMul.{0} Int (CommSemigroup.toSemigroup.{0} Int Int.instCommSemigroup))) b c))","typeFull":"∀ (a b c : ℤ), a * b * c = a * (b * c)","typeReadable":"∀ (a b c : ℤ), a * b * c = a * (b * c)","typeReferences":[["Int","instCommSemigroup"],["CommSemigroup","toSemigroup"],["instHMul"],["HMul","hMul"],["Eq"],["Semigroup","toMul"],["Int"]],"valueReferences":[["Int","instCommSemigroup"],["Semigroup","mul_assoc"],["CommSemigroup","toSemigroup"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Int","cast_mul_eq_zsmul_cast"],"typeFallback":"forall {α : Type.{u_1}} [inst._@.Mathlib.Algebra.Ring.Int.Defs.2962557747._hygCtx._hyg.3 : AddGroupWithOne.{u_1} α] (m : Int) (n : Int), Eq.{succ u_1} α (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2962557747._hygCtx._hyg.3) (HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.instMul) m n)) (HSMul.hSMul.{0, u_1, u_1} Int α α (instHSMul.{0, u_1} Int α (SubNegMonoid.toZSMul.{u_1} α (AddGroup.toSubNegMonoid.{u_1} α (AddGroupWithOne.toAddGroup.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2962557747._hygCtx._hyg.3)))) m (Int.cast.{u_1} α (AddGroupWithOne.toIntCast.{u_1} α inst._@.Mathlib.Algebra.Ring.Int.Defs.2962557747._hygCtx._hyg.3) n))","typeFull":"∀ {α : Type u_1} [inst : AddGroupWithOne α] (m n : ℤ), ↑(m * n) = m • ↑n","typeReadable":"∀ {α : Type u_1} [inst : AddGroupWithOne α] (m n : ℤ), ↑(m * n) = m • ↑n","typeReferences":[["HMul","hMul"],["AddGroupWithOne"],["Int","cast"],["Int"],["Int","instMul"],["AddGroupWithOne","toAddGroup"],["HSMul","hSMul"],["AddGroupWithOne","toIntCast"],["instHMul"],["instHSMul"],["AddGroup","toSubNegMonoid"],["Eq"],["SubNegMonoid","toZSMul"]],"valueReferences":[["NonUnitalNonAssocRing","toAddCommGroup"],["Int","instSub"],["SubtractionMonoid","toSubNegZeroMonoid"],["Int","instCommRing"],["Int","induction_on"],["Eq","trans"],["AddCommGroup","toAddGroup"],["Int","cast_sub"],["AddGroupWithOne","toAddMonoidWithOne"],["HMul","hMul"],["AddMonoidWithOne","toAddMonoid"],["AddGroup","toSubtractionMonoid"],["Distrib","rightDistribClass"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Semiring","toNonAssocSemiring"],["SubNegMonoid","toSub"],["forall_congr"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["NonAssocSemiring","toMulZeroOneClass"],["AddSemigroup","toAdd"],["sub_mul"],["NonUnitalNonAssocSemiring","toDistrib"],["Neg","neg"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["_private","Mathlib","Algebra","Ring","Int","Defs",0,"Int","cast_mul_eq_zsmul_cast","_simp_1_2"],["MulZeroClass","zero_mul"],["zero_zsmul"],["_private","Mathlib","Algebra","Ring","Int","Defs",0,"Int","cast_mul_eq_zsmul_cast","_simp_1_1"],["AddZeroClass","toAddZero"],["Int","instNegInt"],["add_zsmul"],["implies_true"],["Nat"],["AddMonoid","toNSMul"],["instOfNat"],["instNonemptyOfInhabited"],["AddMonoid","toAddSemigroup"],["HSMul","hSMul"],["NegZeroClass","toZero"],["instHMul"],["SubNegMonoid","toZSMul"],["one_mul"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Nat","cast"],["MulZeroOneClass","toMulOneClass"],["CommRing","toNonUnitalCommRing"],["SubNegZeroMonoid","toNegZeroClass"],["Int","cast"],["add_mul"],["Int","instMul"],["congrArg"],["congr"],["Int","instAdd"],["instHSMul"],["congrFun'"],["Zero","toOfNat0"],["one_zsmul"],["Eq"],["instNatCastInt"],["Int","cast_add"],["natCast_zsmul"],["CommRing","toCommSemiring"],["True"],["Int","cast_zero"],["instHAdd"],["Distrib","toMul"],["CommSemiring","toSemiring"],["AddZero","toAdd"],["OfNat","ofNat"],["Int","instInhabited"],["Int"],["HAdd","hAdd"],["eq_self"],["NegZeroClass","toNeg"],["AddGroupWithOne","toAddGroup"],["of_eq_true"],["SubNegMonoid","toAddMonoid"],["MulZeroClass","toZero"],["sub_zsmul"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["AddGroupWithOne","toIntCast"],["instHSub"]]}]

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["Rat","den_mul_eq_num"],"typeFallback":"forall (q : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat Rat.instMul) (Nat.cast.{0} Rat Rat.instNatCast (Rat.den q)) q) (Int.cast.{0} Rat Rat.instIntCast (Rat.num q))","typeFull":"∀ (q : ℚ), ↑q.den * q = ↑q.num","typeReadable":"∀ (q : ℚ), ↑q.den * q = ↑q.num","typeReferences":[["Rat","instIntCast"],["Rat","instNatCast"],["Nat","cast"],["Rat"],["instHMul"],["HMul","hMul"],["Rat","instMul"],["Eq"],["Int","cast"],["Rat","num"],["Rat","den"]],"valueReferences":[["Nat","cast"],["CommMagma","toMul"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Rat","mul_den_eq_num"],["HMul","hMul"],["Rat","instMul"],["CommRing","toNonUnitalCommRing"],["Int","cast"],["Rat","num"],["congrArg"],["Rat","instIntCast"],["Rat","instNatCast"],["Eq","refl"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocCommSemiring"],["mul_comm"],["id"],["Rat"],["instHMul"],["Eq","mpr"],["Eq"],["Rat","commRing"],["Rat","den"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_11"],"typeFallback":"forall (a : Rat), Eq.{1} Rat (SubNegMonoid.zsmul.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)) (OfNat.ofNat.{0} Int 0 (instOfNat 0)) a) (OfNat.ofNat.{0} Rat 0 (Zero.toOfNat0.{0} Rat (AddMonoid.toZero.{0} Rat (SubNegMonoid.toAddMonoid.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup))))))","typeFull":"∀ (a : ℚ), SubNegMonoid.zsmul 0 a = 0","typeReadable":"∀ (a : ℚ), SubNegMonoid.zsmul 0 a = 0","typeReferences":[["AddMonoid","toZero"],["SubNegMonoid","toAddMonoid"],["instOfNat"],["SubNegMonoid","zsmul"],["AddCommGroup","toAddGroup"],["Rat"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Rat","addCommGroup"],["Eq"],["OfNat","ofNat"],["Int"]],"valueReferences":[["AddCommGroup","toAddGroup"],["Rat"],["AddGroup","toSubNegMonoid"],["Rat","addCommGroup"],["SubNegMonoid","zsmul_zero'"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Ring","Rat",0,"Rat","commRing","_simp_1"],"typeFallback":"Eq.{1} Rat (OfNat.ofNat.{0} Rat 1 (Rat.instOfNat 1)) (Rat.divInt (OfNat.ofNat.{0} Int 1 (instOfNat 1)) (OfNat.ofNat.{0} Int 1 (instOfNat 1)))","typeFull":"1 = Rat.divInt 1 1","typeReadable":"1 = Rat.divInt 1 1","typeReferences":[["instOfNat"],["Rat","divInt"],["Rat"],["Eq"],["Rat","instOfNat"],["OfNat","ofNat"],["Int"]],"valueReferences":[["instOfNat"],["Rat","divInt"],["Rat"],["Eq","symm"],["Rat","divInt_one_one"],["Rat","instOfNat"],["OfNat","ofNat"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Rat","divInt_div_divInt_cancel_left"],"typeFallback":"forall {x : Int}, (Ne.{1} Int x (OfNat.ofNat.{0} Int 0 (instOfNat 0))) -> (forall (n : Int) (d : Int), Eq.{1} Rat (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) (Rat.divInt n x) (Rat.divInt d x)) (Rat.divInt n d))","typeFull":"∀ {x : ℤ}, x ≠ 0 → ∀ (n d : ℤ), Rat.divInt n x / Rat.divInt d x = Rat.divInt n d","typeReadable":"∀ {x : ℤ}, x ≠ 0 → ∀ (n d : ℤ), Rat.divInt n x / Rat.divInt d x = Rat.divInt n d","typeReferences":[["HDiv","hDiv"],["instOfNat"],["Rat","divInt"],["Rat"],["Ne"],["Eq"],["instHDiv"],["OfNat","ofNat"],["Rat","instDiv"],["Int"]],"valueReferences":[["MulOneClass","toMulOne"],["DivInvMonoid","toInv"],["div_eq_mul_inv"],["Rat","divInt"],["GroupWithZero","toDivInvMonoid"],["HMul","hMul"],["Rat","instMul"],["Rat","divInt_mul_divInt_cancel"],["instHDiv"],["congrArg"],["HDiv","hDiv"],["MulOne","toMul"],["Rat","commGroupWithZero"],["Monoid","toMulOneClass"],["Rat","instInv"],["Eq"],["Rat","instDiv"],["Rat","inv_divInt"],["Inv","inv"],["DivInvMonoid","toDiv"],["DivInvMonoid","toMonoid"],["Eq","refl"],["Rat"],["id"],["instHMul"],["Eq","mpr"],["CommGroupWithZero","toGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["Rat","num_div_den"],"typeFallback":"forall (r : Rat), Eq.{1} Rat (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) (Int.cast.{0} Rat Rat.instIntCast (Rat.num r)) (Nat.cast.{0} Rat Rat.instNatCast (Rat.den r))) r","typeFull":"∀ (r : ℚ), ↑r.num / ↑r.den = r","typeReadable":"∀ (r : ℚ), ↑r.num / ↑r.den = r","typeReferences":[["Rat","instIntCast"],["HDiv","hDiv"],["Rat","instNatCast"],["Nat","cast"],["Rat"],["Eq"],["Int","cast"],["instHDiv"],["Rat","num"],["Rat","den"],["Rat","instDiv"]],"valueReferences":[["Nat","cast"],["Rat","divInt_eq_div"],["Rat","divInt"],["AddGroupWithOne","toAddMonoidWithOne"],["Int","cast"],["instHDiv"],["Rat","num"],["congrArg"],["Rat","instIntCast"],["HDiv","hDiv"],["Rat","instNatCast"],["Ring","toAddGroupWithOne"],["Rat","num_divInt_den"],["Eq","symm"],["Eq"],["Rat","den"],["Rat","instDiv"],["Rat","commRing"],["instNatCastInt"],["Int"],["CommRing","toRing"],["AddMonoidWithOne","toNatCast"],["Eq","refl"],["AddGroupWithOne","toIntCast"],["Rat"],["id"],["Eq","mpr"],["Int","cast_natCast"]]},{"isProp":true,"kind":"theorem","name":["Rat","commGroupWithZero","_proof_3"],"typeFallback":"forall (n : Nat) (a : Rat), Eq.{1} Rat (HPow.hPow.{0, 0, 0} Rat Int Rat (instHPow.{0, 0} Rat Int Rat.instPowInt) a (Int.negSucc n)) (Inv.inv.{0} Rat Rat.instInv (HPow.hPow.{0, 0, 0} Rat Int Rat (instHPow.{0, 0} Rat Int Rat.instPowInt) a (Nat.cast.{0} Int instNatCastInt (Nat.succ n))))","typeFull":"∀ (n : ℕ) (a : ℚ), a ^ Int.negSucc n = (a ^ ↑n.succ)⁻¹","typeReadable":"∀ (n : ℕ) (a : ℚ), a ^ Int.negSucc n = (a ^ ↑n.succ)⁻¹","typeReferences":[["instHPow"],["Inv","inv"],["Nat","cast"],["Int","negSucc"],["HPow","hPow"],["Int"],["Nat"],["Nat","succ"],["Rat"],["Rat","instPowInt"],["Rat","instInv"],["Eq"],["instNatCastInt"]],"valueReferences":[["instHPow"],["Eq","refl"],["Int","negSucc"],["Rat"],["Rat","instPowInt"],["HPow","hPow"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_7"],"typeFallback":"Eq.{1} Rat (Int.cast.{0} Rat Rat.instIntCast (Nat.cast.{0} Int instNatCastInt (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)))) (Int.cast.{0} Rat Rat.instIntCast (Nat.cast.{0} Int instNatCastInt (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))","typeFull":"↑↑0 = ↑↑0","typeReadable":"↑↑0 = ↑↑0","typeReferences":[["Rat","instIntCast"],["Nat"],["Nat","cast"],["instOfNatNat"],["Rat"],["Eq"],["Int","cast"],["OfNat","ofNat"],["instNatCastInt"],["Int"]],"valueReferences":[["Rat","instIntCast"],["rfl"],["Nat"],["Nat","cast"],["instOfNatNat"],["Rat"],["Int","cast"],["OfNat","ofNat"],["instNatCastInt"],["Int"]]},{"isProp":true,"kind":"theorem","name":["Rat","commGroupWithZero","_proof_6"],"typeFallback":"Nontrivial.{0} Rat","typeFull":"Nontrivial ℚ","typeReadable":"Nontrivial ℚ","typeReferences":[["Nontrivial"],["Rat"]],"valueReferences":[["Exists"],["Rat"],["Exists","intro"],["Ne"],["Nontrivial","mk"],["Rat","instOfNat"],["OfNat","ofNat"],["Rat","zero_ne_one"]]},{"isProp":true,"kind":"theorem","name":["Rat","divInt_div_divInt_cancel_right"],"typeFallback":"forall {x : Int}, (Ne.{1} Int x (OfNat.ofNat.{0} Int 0 (instOfNat 0))) -> (forall (n : Int) (d : Int), Eq.{1} Rat (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) (Rat.divInt x n) (Rat.divInt x d)) (Rat.divInt d n))","typeFull":"∀ {x : ℤ}, x ≠ 0 → ∀ (n d : ℤ), Rat.divInt x n / Rat.divInt x d = Rat.divInt d n","typeReadable":"∀ {x : ℤ}, x ≠ 0 → ∀ (n d : ℤ), Rat.divInt x n / Rat.divInt x d = Rat.divInt d n","typeReferences":[["HDiv","hDiv"],["instOfNat"],["Rat","divInt"],["Rat"],["Ne"],["Eq"],["instHDiv"],["OfNat","ofNat"],["Rat","instDiv"],["Int"]],"valueReferences":[["MulOneClass","toMulOne"],["DivInvMonoid","toInv"],["div_eq_mul_inv"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["Rat","divInt"],["GroupWithZero","toDivInvMonoid"],["HMul","hMul"],["Rat","instMul"],["Rat","divInt_mul_divInt_cancel"],["CommRing","toNonUnitalCommRing"],["instHDiv"],["congrArg"],["HDiv","hDiv"],["MulOne","toMul"],["Rat","commGroupWithZero"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocCommSemiring"],["Monoid","toMulOneClass"],["mul_comm"],["Rat","instInv"],["Eq"],["Rat","commRing"],["Rat","instDiv"],["Rat","inv_divInt"],["Inv","inv"],["CommMagma","toMul"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["DivInvMonoid","toDiv"],["DivInvMonoid","toMonoid"],["Eq","refl"],["Rat"],["id"],["instHMul"],["Eq","mpr"],["CommGroupWithZero","toGroupWithZero"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_14"],"typeFallback":"forall (a : Rat), Eq.{1} Rat (HAdd.hAdd.{0, 0, 0} Rat Rat Rat (instHAdd.{0} Rat (AddSemigroup.toAdd.{0} Rat (AddMonoid.toAddSemigroup.{0} Rat (SubNegMonoid.toAddMonoid.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)))))) (Neg.neg.{0} Rat (SubNegMonoid.toNeg.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup))) a) a) (OfNat.ofNat.{0} Rat 0 (Zero.toOfNat0.{0} Rat (AddMonoid.toZero.{0} Rat (SubNegMonoid.toAddMonoid.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup))))))","typeFull":"∀ (a : ℚ), -a + a = 0","typeReadable":"∀ (a : ℚ), -a + a = 0","typeReferences":[["AddMonoid","toZero"],["instHAdd"],["Neg","neg"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["Rat","addCommGroup"],["OfNat","ofNat"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["AddMonoid","toAddSemigroup"],["Rat"],["Zero","toOfNat0"],["AddGroup","toSubNegMonoid"],["Eq"],["AddSemigroup","toAdd"]],"valueReferences":[["AddCommGroup","toAddGroup"],["Rat"],["Rat","addCommGroup"],["AddGroup","neg_add_cancel"]]},{"isProp":false,"kind":"definition","name":["Rat","commGroupWithZero"],"typeFallback":"CommGroupWithZero.{0} Rat","typeFull":"CommGroupWithZero ℚ","typeReadable":"CommGroupWithZero ℚ","typeReferences":[["Rat"],["CommGroupWithZero"]],"valueReferences":[["CommGroupWithZero","mk"],["Rat","zpow_zero"],["Rat","commGroupWithZero","_proof_5"],["Rat","mul_inv_cancel"],["CommRing","toNonUnitalCommRing"],["Rat","inv_zero"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["Rat","commGroupWithZero","_proof_1"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Rat","instInv"],["Rat","instPowInt"],["Rat","commGroupWithZero","_proof_2"],["Rat","commRing"],["Rat","instDiv"],["instHPow"],["Rat","commGroupWithZero","_proof_4"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Rat","commGroupWithZero","_proof_3"],["CommMonoidWithZero","mk"],["HPow","hPow"],["Int"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Rat","commMonoid"],["Rat"],["Rat","commGroupWithZero","_proof_6"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_4"],"typeFallback":"forall (n : Nat), Eq.{1} Rat (Int.cast.{0} Rat Rat.instIntCast (Int.negSucc n)) (Neg.neg.{0} Rat (SubNegMonoid.toNeg.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup))) (Nat.cast.{0} Rat (NatCast.mk.{0} Rat (fun (n : Nat) => Int.cast.{0} Rat Rat.instIntCast (Nat.cast.{0} Int instNatCastInt n))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))","typeFull":"∀ (n : ℕ), ↑(Int.negSucc n) = -↑(n + 1)","typeReadable":"∀ (n : ℕ), ↑(Int.negSucc n) = -↑(n + 1)","typeReferences":[["instAddNat"],["Nat","cast"],["instHAdd"],["Neg","neg"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["Int","negSucc"],["Rat","addCommGroup"],["Int","cast"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["Rat","instIntCast"],["NatCast","mk"],["Nat"],["instOfNatNat"],["Rat"],["AddGroup","toSubNegMonoid"],["Eq"],["instNatCastInt"]],"valueReferences":[["Rat","instIntCast"],["Eq","refl"],["Int","negSucc"],["Rat"],["Int","cast"]]},{"isProp":true,"kind":"theorem","name":["Rat","commGroupWithZero","_proof_2"],"typeFallback":"forall (x._@.Mathlib.Algebra.Ring.Rat.2336346949._hygCtx._hyg.27 : Nat) (x._@.Mathlib.Algebra.Ring.Rat.2336346949._hygCtx._hyg.29 : Rat), Eq.{1} Rat (HPow.hPow.{0, 0, 0} Rat Int Rat (instHPow.{0, 0} Rat Int Rat.instPowInt) x._@.Mathlib.Algebra.Ring.Rat.2336346949._hygCtx._hyg.29 (Nat.cast.{0} Int instNatCastInt (Nat.succ x._@.Mathlib.Algebra.Ring.Rat.2336346949._hygCtx._hyg.27))) (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) (HPow.hPow.{0, 0, 0} Rat Int Rat (instHPow.{0, 0} Rat Int Rat.instPowInt) x._@.Mathlib.Algebra.Ring.Rat.2336346949._hygCtx._hyg.29 (Nat.cast.{0} Int instNatCastInt x._@.Mathlib.Algebra.Ring.Rat.2336346949._hygCtx._hyg.27)) x._@.Mathlib.Algebra.Ring.Rat.2336346949._hygCtx._hyg.29)","typeFull":"∀ (x : ℕ) (x_1 : ℚ), x_1 ^ ↑x.succ = x_1 ^ ↑x * x_1","typeReadable":"∀ (x : ℕ) (x_1 : ℚ), x_1 ^ ↑x.succ = x_1 ^ ↑x * x_1","typeReferences":[["instHPow"],["Nat","cast"],["CommMonoid","toMonoid"],["HMul","hMul"],["HPow","hPow"],["Int"],["Semigroup","toMul"],["Nat"],["Nat","succ"],["Rat","commMonoid"],["Rat"],["Rat","instPowInt"],["instHMul"],["Monoid","toSemigroup"],["Eq"],["instNatCastInt"]],"valueReferences":[["instAddNat"],["Nat","cast"],["Rat","zpow_natCast"],["CommMonoid","toMonoid"],["HMul","hMul"],["Rat","instMul"],["Rat","instPowNat"],["Semigroup","toMul"],["congrArg"],["instOfNatNat"],["Rat","instPowInt"],["Monoid","toSemigroup"],["Eq"],["instNatCastInt"],["instHPow"],["instHAdd"],["HPow","hPow"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["Nat"],["Nat","succ"],["Eq","refl"],["Rat","commMonoid"],["Rat"],["id"],["instHMul"],["Eq","mpr"],["Rat","pow_succ"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_5"],"typeFallback":"forall (a : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) (OfNat.ofNat.{0} Rat 1 (One.toOfNat1.{0} Rat (Monoid.toOne.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) a) a","typeFull":"∀ (a : ℚ), 1 * a = a","typeReadable":"∀ (a : ℚ), 1 * a = a","typeReferences":[["One","toOfNat1"],["CommMonoid","toMonoid"],["Rat","commMonoid"],["Monoid","toOne"],["Rat"],["instHMul"],["HMul","hMul"],["Monoid","toSemigroup"],["Eq"],["OfNat","ofNat"],["Semigroup","toMul"]],"valueReferences":[["CommMonoid","toMonoid"],["Rat","commMonoid"],["Monoid","one_mul"],["Rat"]]},{"isProp":true,"kind":"theorem","name":["Rat","natCast_eq_divInt"],"typeFallback":"forall (n : Nat), Eq.{1} Rat (Nat.cast.{0} Rat Rat.instNatCast n) (Rat.divInt (Nat.cast.{0} Int instNatCastInt n) (OfNat.ofNat.{0} Int 1 (instOfNat 1)))","typeFull":"∀ (n : ℕ), ↑n = Rat.divInt (↑n) 1","typeReadable":"∀ (n : ℕ), ↑n = Rat.divInt (↑n) 1","typeReferences":[["Nat"],["Rat","instNatCast"],["Nat","cast"],["instOfNat"],["Rat","divInt"],["Rat"],["Eq"],["OfNat","ofNat"],["instNatCastInt"],["Int"]],"valueReferences":[["Nat","cast"],["Rat","divInt"],["Rat","intCast_eq_divInt"],["AddGroupWithOne","toAddMonoidWithOne"],["Int","cast"],["OfNat","ofNat"],["congrArg"],["Int"],["Rat","instIntCast"],["CommRing","toRing"],["AddMonoidWithOne","toNatCast"],["Rat","instNatCast"],["instOfNat"],["Ring","toAddGroupWithOne"],["Eq","refl"],["Eq","symm"],["id"],["Rat"],["AddGroupWithOne","toIntCast"],["Eq","mpr"],["Int","cast_natCast"],["Eq"],["Rat","commRing"],["instNatCastInt"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_2"],"typeFallback":"forall (n : Nat), Eq.{1} Rat (Int.cast.{0} Rat Rat.instIntCast (Nat.cast.{0} Int instNatCastInt (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (HAdd.hAdd.{0, 0, 0} Rat Rat Rat (instHAdd.{0} Rat (AddSemigroup.toAdd.{0} Rat (AddMonoid.toAddSemigroup.{0} Rat (SubNegMonoid.toAddMonoid.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)))))) (Int.cast.{0} Rat Rat.instIntCast (Nat.cast.{0} Int instNatCastInt n)) (OfNat.ofNat.{0} Rat 1 (One.toOfNat1.{0} Rat (Monoid.toOne.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))))","typeFull":"∀ (n : ℕ), ↑↑(n + 1) = ↑↑n + 1","typeReadable":"∀ (n : ℕ), ↑↑(n + 1) = ↑↑n + 1","typeReferences":[["instAddNat"],["Nat","cast"],["CommMonoid","toMonoid"],["instHAdd"],["AddCommGroup","toAddGroup"],["Monoid","toOne"],["Rat","addCommGroup"],["Int","cast"],["OfNat","ofNat"],["Int"],["Rat","instIntCast"],["HAdd","hAdd"],["Nat"],["One","toOfNat1"],["SubNegMonoid","toAddMonoid"],["instOfNatNat"],["AddMonoid","toAddSemigroup"],["Rat","commMonoid"],["Rat"],["AddGroup","toSubNegMonoid"],["Eq"],["AddSemigroup","toAdd"],["instNatCastInt"]],"valueReferences":[["instAddNat"],["Nat","cast"],["CommMonoid","toMonoid"],["Eq","trans"],["Rat","divInt"],["AddCommGroup","toAddGroup"],["Rat","intCast_eq_divInt"],["HMul","hMul"],["MulZeroOneClass","toMulOneClass"],["Int","cast"],["Int","instMul"],["congrArg"],["Rat","instIntCast"],["Semiring","toNonAssocSemiring"],["instOfNatNat"],["congr"],["Int","instAdd"],["AddGroup","toSubNegMonoid"],["Eq"],["NonAssocSemiring","toMulZeroOneClass"],["instNatCastInt"],["AddSemigroup","toAdd"],["True"],["instHAdd"],["Monoid","toOne"],["mul_one"],["Rat","addCommGroup"],["OfNat","ofNat"],["Int"],["HAdd","hAdd"],["eq_self"],["Rat","divInt_add_divInt"],["Nat"],["instOfNat"],["of_eq_true"],["SubNegMonoid","toAddMonoid"],["One","toOfNat1"],["Int","instSemiring"],["AddMonoid","toAddSemigroup"],["Rat","commMonoid"],["Rat"],["instHMul"],["_private","Mathlib","Algebra","Ring","Rat",0,"Rat","commRing","_simp_1"],["Int","one_ne_zero"]]},{"isProp":true,"kind":"theorem","name":["Rat","mkRat_pow"],"typeFallback":"forall (num : Nat) (den : Nat) (n : Nat), Eq.{1} Rat (HPow.hPow.{0, 0, 0} Rat Nat Rat (instHPow.{0, 0} Rat Nat Rat.instPowNat) (mkRat (Nat.cast.{0} Int instNatCastInt num) den) n) (mkRat (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) (Nat.cast.{0} Int instNatCastInt num) n) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.toPow.{0} Nat Nat.instMonoid)) den n))","typeFull":"∀ (num den n : ℕ), mkRat (↑num) den ^ n = mkRat (↑num ^ n) (den ^ n)","typeReadable":"∀ (num den n : ℕ), mkRat (↑num) den ^ n = mkRat (↑num ^ n) (den ^ n)","typeReferences":[["instHPow"],["Nat","cast"],["Int","instMonoid"],["Rat","instPowNat"],["HPow","hPow"],["Int"],["Nat"],["mkRat"],["Nat","instMonoid"],["Monoid","toPow"],["Rat"],["Eq"],["instNatCastInt"]],"valueReferences":[["Rat","mkRat_eq_divInt"],["instHPow"],["Nat","cast"],["Int","instMonoid"],["Int","instNatPow"],["Rat","divInt"],["instPowNat"],["Rat","instPowNat"],["instNatPowNat"],["HPow","hPow"],["congrArg"],["Int"],["Nat"],["mkRat"],["Rat","divInt_pow"],["Nat","instMonoid"],["Monoid","toPow"],["Eq","refl"],["id"],["Rat"],["Eq","mpr"],["Eq"],["Int","natCast_pow"],["instNatCastInt"]]},{"isProp":true,"kind":"theorem","name":["Rat","instCharZero"],"typeFallback":"CharZero.{0} Rat (AddGroupWithOne.toAddMonoidWithOne.{0} Rat (Ring.toAddGroupWithOne.{0} Rat (CommRing.toRing.{0} Rat Rat.commRing)))","typeFull":"CharZero ℚ","typeReadable":"CharZero ℚ","typeReferences":[["CommRing","toRing"],["Ring","toAddGroupWithOne"],["Rat"],["AddGroupWithOne","toAddMonoidWithOne"],["CharZero"],["Rat","commRing"]],"valueReferences":[["Nat","cast_inj","_simp_1"],["Nat","cast"],["Eq","mp"],["congr_arg"],["AddGroupWithOne","toAddMonoidWithOne"],["CharZero","mk"],["Int","instRing"],["Rat","num"],["Int"],["CommRing","toRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["Ring","toAddGroupWithOne"],["Rat"],["Int","instCharZero"],["Eq"],["Rat","commRing"]]},{"isProp":true,"kind":"theorem","name":["Rat","isDomain"],"typeFallback":"IsDomain.{0} Rat (CommSemiring.toSemiring.{0} Rat (CommRing.toCommSemiring.{0} Rat Rat.commRing))","typeFull":"IsDomain ℚ","typeReadable":"IsDomain ℚ","typeReferences":[["CommRing","toCommSemiring"],["CommSemiring","toSemiring"],["Rat"],["IsDomain"],["Rat","commRing"]],"valueReferences":[["CommRing","toRing"],["Rat","nontrivial"],["Rat","commGroupWithZero"],["NoZeroDivisors","to_isDomain"],["Rat"],["CommGroupWithZero","toGroupWithZero"],["GroupWithZero","noZeroDivisors"],["Rat","commRing"]]},{"isProp":true,"kind":"theorem","name":["Rat","commGroupWithZero","_proof_5"],"typeFallback":"forall (a : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (MulZeroClass.toMul.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))) a (OfNat.ofNat.{0} Rat 0 (Zero.toOfNat0.{0} Rat (MulZeroClass.toZero.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))))) (OfNat.ofNat.{0} Rat 0 (Zero.toOfNat0.{0} Rat (MulZeroClass.toZero.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))))","typeFull":"∀ (a : ℚ), a * 0 = 0","typeReadable":"∀ (a : ℚ), a * 0 = 0","typeReferences":[["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["MulZeroClass","toMul"],["HMul","hMul"],["CommRing","toNonUnitalCommRing"],["OfNat","ofNat"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Rat"],["instHMul"],["Zero","toOfNat0"],["Eq"],["Rat","commRing"]],"valueReferences":[["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Rat"],["CommRing","toNonUnitalCommRing"],["MulZeroClass","mul_zero"],["Rat","commRing"]]},{"isProp":true,"kind":"theorem","name":["Rat","commGroupWithZero","_proof_1"],"typeFallback":"forall (a : Rat) (b : Rat), Eq.{1} Rat (HDiv.hDiv.{0, 0, 0} Rat Rat Rat (instHDiv.{0} Rat Rat.instDiv) a b) (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) a (Inv.inv.{0} Rat Rat.instInv b))","typeFull":"∀ (a b : ℚ), a / b = a * b⁻¹","typeReadable":"∀ (a b : ℚ), a / b = a * b⁻¹","typeReferences":[["Inv","inv"],["CommMonoid","toMonoid"],["HMul","hMul"],["instHDiv"],["Semigroup","toMul"],["HDiv","hDiv"],["Rat","commMonoid"],["Rat"],["instHMul"],["Rat","instInv"],["Monoid","toSemigroup"],["Eq"],["Rat","instDiv"]],"valueReferences":[["HDiv","hDiv"],["Eq","refl"],["Rat"],["instHDiv"],["Rat","instDiv"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_13"],"typeFallback":"forall (n : Nat) (a : Rat), Eq.{1} Rat (SubNegMonoid.zsmul.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)) (Int.negSucc n) a) (Neg.neg.{0} Rat (SubNegMonoid.toNeg.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup))) (SubNegMonoid.zsmul.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a))","typeFull":"∀ (n : ℕ) (a : ℚ), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a","typeReadable":"∀ (n : ℕ) (a : ℚ), SubNegMonoid.zsmul (Int.negSucc n) a = -SubNegMonoid.zsmul (↑n.succ) a","typeReferences":[["Nat","cast"],["SubNegMonoid","zsmul"],["Neg","neg"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["Int","negSucc"],["Rat","addCommGroup"],["Int"],["Nat"],["Nat","succ"],["Rat"],["AddGroup","toSubNegMonoid"],["Eq"],["instNatCastInt"]],"valueReferences":[["SubNegMonoid","zsmul_neg'"],["AddCommGroup","toAddGroup"],["Rat"],["AddGroup","toSubNegMonoid"],["Rat","addCommGroup"]]},{"isProp":false,"kind":"definition","name":["Rat","commSemiring"],"typeFallback":"CommSemiring.{0} Rat","typeFull":"CommSemiring ℚ","typeReadable":"CommSemiring ℚ","typeReferences":[["CommSemiring"],["Rat"]],"valueReferences":[["CommRing","toCommSemiring"],["CommSemiring"],["Rat"],["inferInstance"],["Rat","commRing"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_3"],"typeFallback":"forall (n : Nat), Eq.{1} Rat (Int.cast.{0} Rat Rat.instIntCast (Nat.cast.{0} Int instNatCastInt n)) (Nat.cast.{0} Rat (NatCast.mk.{0} Rat (fun (n : Nat) => Int.cast.{0} Rat Rat.instIntCast (Nat.cast.{0} Int instNatCastInt n))) n)","typeFull":"∀ (n : ℕ), ↑↑n = ↑n","typeReadable":"∀ (n : ℕ), ↑↑n = ↑n","typeReferences":[["NatCast","mk"],["Rat","instIntCast"],["Nat"],["Nat","cast"],["Rat"],["Eq"],["Int","cast"],["instNatCastInt"],["Int"]],"valueReferences":[["Rat","instIntCast"],["Nat","cast"],["Eq","refl"],["Rat"],["Int","cast"],["instNatCastInt"],["Int"]]},{"isProp":false,"kind":"definition","name":["Rat","semiring"],"typeFallback":"Semiring.{0} Rat","typeFull":"Semiring ℚ","typeReadable":"Semiring ℚ","typeReferences":[["Rat"],["Semiring"]],"valueReferences":[["CommSemiring","toSemiring"],["Rat"],["inferInstance"],["Rat","commSemiring"],["Semiring"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_12"],"typeFallback":"forall (n : Nat) (a : Rat), Eq.{1} Rat (SubNegMonoid.zsmul.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)) (Nat.cast.{0} Int instNatCastInt (Nat.succ n)) a) (HAdd.hAdd.{0, 0, 0} Rat Rat Rat (instHAdd.{0} Rat (AddSemigroup.toAdd.{0} Rat (AddMonoid.toAddSemigroup.{0} Rat (SubNegMonoid.toAddMonoid.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)))))) (SubNegMonoid.zsmul.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)) (Nat.cast.{0} Int instNatCastInt n) a) a)","typeFull":"∀ (n : ℕ) (a : ℚ), SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a","typeReadable":"∀ (n : ℕ) (a : ℚ), SubNegMonoid.zsmul (↑n.succ) a = SubNegMonoid.zsmul (↑n) a + a","typeReferences":[["Nat","cast"],["SubNegMonoid","zsmul"],["instHAdd"],["AddCommGroup","toAddGroup"],["Rat","addCommGroup"],["Int"],["HAdd","hAdd"],["Nat"],["SubNegMonoid","toAddMonoid"],["Nat","succ"],["AddMonoid","toAddSemigroup"],["Rat"],["AddGroup","toSubNegMonoid"],["Eq"],["AddSemigroup","toAdd"],["instNatCastInt"]],"valueReferences":[["AddCommGroup","toAddGroup"],["SubNegMonoid","zsmul_succ'"],["Rat"],["AddGroup","toSubNegMonoid"],["Rat","addCommGroup"]]},{"isProp":false,"kind":"definition","name":["Rat","commRing"],"typeFallback":"CommRing.{0} Rat","typeFull":"CommRing ℚ","typeReadable":"CommRing ℚ","typeReferences":[["Rat"],["CommRing"]],"valueReferences":[["Rat","commRing","_proof_2"],["Nat","cast"],["Rat","commRing","_proof_1"],["NonUnitalNonAssocSemiring","mk"],["CommMonoid","toMonoid"],["AddCommGroup","toAddGroup"],["Rat","mul_add"],["Rat","commRing","_proof_3"],["Int","cast"],["Semigroup","toMul"],["Rat","commRing","_proof_5"],["Rat","instIntCast"],["Rat","commRing","_proof_13"],["Rat","commRing","_proof_11"],["Rat","zero_mul"],["Rat","commRing","_proof_14"],["SubNegMonoid","toSub"],["AddGroup","toSubNegMonoid"],["Monoid","toSemigroup"],["Ring","mk"],["Monoid","npow"],["instNatCastInt"],["NonUnitalSemiring","mk"],["CommRing","mk"],["SubNegMonoid","zsmul"],["AddCommGroup","add_comm"],["CommMonoid","mul_comm"],["Rat","commRing","_proof_7"],["SubNegMonoid","toNeg"],["Rat","commRing","_proof_9"],["AddCommMonoid","mk"],["Monoid","toOne"],["Semiring","mk"],["Rat","commRing","_proof_8"],["Rat","add_mul"],["Rat","commRing","_proof_12"],["Rat","addCommGroup"],["Int"],["NatCast","mk"],["SubNegMonoid","toAddMonoid"],["Rat","commRing","_proof_10"],["Rat","commRing","_proof_6"],["Rat","commMonoid"],["Rat"],["Rat","mul_zero"],["Rat","commRing","_proof_4"],["IntCast","mk"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_1"],"typeFallback":"forall (a : Rat) (b : Rat) (c : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) a b) c) (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) a (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) b c))","typeFull":"∀ (a b c : ℚ), a * b * c = a * (b * c)","typeReadable":"∀ (a b c : ℚ), a * b * c = a * (b * c)","typeReferences":[["CommMonoid","toMonoid"],["Rat","commMonoid"],["Rat"],["instHMul"],["HMul","hMul"],["Monoid","toSemigroup"],["Eq"],["Semigroup","toMul"]],"valueReferences":[["CommMonoid","toMonoid"],["Rat","commMonoid"],["Semigroup","mul_assoc"],["Rat"],["Monoid","toSemigroup"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_8"],"typeFallback":"forall (x : Rat), Eq.{1} Rat (Monoid.npow.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) x) (OfNat.ofNat.{0} Rat 1 (One.toOfNat1.{0} Rat (Monoid.toOne.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid))))","typeFull":"∀ (x : ℚ), Monoid.npow 0 x = 1","typeReadable":"∀ (x : ℚ), Monoid.npow 0 x = 1","typeReferences":[["Nat"],["One","toOfNat1"],["CommMonoid","toMonoid"],["instOfNatNat"],["Rat","commMonoid"],["Monoid","toOne"],["Rat"],["Eq"],["Monoid","npow"],["OfNat","ofNat"]],"valueReferences":[["Monoid","npow_zero"],["CommMonoid","toMonoid"],["Rat","commMonoid"],["Rat"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_9"],"typeFallback":"forall (n : Nat) (x : Rat), Eq.{1} Rat (Monoid.npow.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) x) (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) (Monoid.npow.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid) n x) x)","typeFull":"∀ (n : ℕ) (x : ℚ), Monoid.npow (n + 1) x = Monoid.npow n x * x","typeReadable":"∀ (n : ℕ) (x : ℚ), Monoid.npow (n + 1) x = Monoid.npow n x * x","typeReferences":[["instAddNat"],["CommMonoid","toMonoid"],["instHAdd"],["HMul","hMul"],["OfNat","ofNat"],["Semigroup","toMul"],["HAdd","hAdd"],["Nat"],["instOfNatNat"],["Rat","commMonoid"],["Rat"],["instHMul"],["Monoid","toSemigroup"],["Eq"],["Monoid","npow"]],"valueReferences":[["CommMonoid","toMonoid"],["Rat","commMonoid"],["Rat"],["Monoid","npow_succ"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_10"],"typeFallback":"forall (a : Rat) (b : Rat), Eq.{1} Rat (HSub.hSub.{0, 0, 0} Rat Rat Rat (instHSub.{0} Rat (SubNegMonoid.toSub.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)))) a b) (HAdd.hAdd.{0, 0, 0} Rat Rat Rat (instHAdd.{0} Rat (AddSemigroup.toAdd.{0} Rat (AddMonoid.toAddSemigroup.{0} Rat (SubNegMonoid.toAddMonoid.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup)))))) a (Neg.neg.{0} Rat (SubNegMonoid.toNeg.{0} Rat (AddGroup.toSubNegMonoid.{0} Rat (AddCommGroup.toAddGroup.{0} Rat Rat.addCommGroup))) b))","typeFull":"∀ (a b : ℚ), a - b = a + -b","typeReadable":"∀ (a b : ℚ), a - b = a + -b","typeReferences":[["Neg","neg"],["instHAdd"],["AddCommGroup","toAddGroup"],["SubNegMonoid","toNeg"],["Rat","addCommGroup"],["HAdd","hAdd"],["SubNegMonoid","toAddMonoid"],["SubNegMonoid","toSub"],["AddMonoid","toAddSemigroup"],["Rat"],["HSub","hSub"],["AddGroup","toSubNegMonoid"],["Eq"],["instHSub"],["AddSemigroup","toAdd"]],"valueReferences":[["AddCommGroup","toAddGroup"],["Rat"],["AddGroup","toSubNegMonoid"],["Rat","addCommGroup"],["SubNegMonoid","sub_eq_add_neg"]]},{"isProp":true,"kind":"theorem","name":["Rat","divInt_pow"],"typeFallback":"forall (num : Nat) (den : Int) (n : Nat), Eq.{1} Rat (HPow.hPow.{0, 0, 0} Rat Nat Rat (instHPow.{0, 0} Rat Nat Rat.instPowNat) (Rat.divInt (Nat.cast.{0} Int instNatCastInt num) den) n) (Rat.divInt (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) (Nat.cast.{0} Int instNatCastInt num) n) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.toPow.{0} Int Int.instMonoid)) den n))","typeFull":"∀ (num : ℕ) (den : ℤ) (n : ℕ), Rat.divInt (↑num) den ^ n = Rat.divInt (↑num ^ n) (den ^ n)","typeReadable":"∀ (num : ℕ) (den : ℤ) (n : ℕ), Rat.divInt (↑num) den ^ n = Rat.divInt (↑num ^ n) (den ^ n)","typeReferences":[["instHPow"],["Nat"],["Int","instMonoid"],["Nat","cast"],["Monoid","toPow"],["Rat","divInt"],["Rat"],["Rat","instPowNat"],["HPow","hPow"],["Eq"],["instNatCastInt"],["Int"]],"valueReferences":[["Int","instMonoid"],["Nat","cast"],["Eq","trans"],["Rat","divInt_eq_div"],["Rat","divInt"],["AddGroupWithOne","toAddMonoidWithOne"],["CommGroupWithZero","toDivisionCommMonoid"],["Rat","instPowNat"],["div_pow"],["Int","cast"],["instHDiv"],["congrArg"],["Rat","instIntCast"],["HDiv","hDiv"],["DivisionMonoid","toDivInvMonoid"],["Monoid","toPow"],["Rat","commGroupWithZero"],["Ring","toAddGroupWithOne"],["congr"],["MonoidWithZero","toMonoid"],["congrFun'"],["Eq"],["DivisionCommMonoid","toDivisionMonoid"],["Rat","instDiv"],["Rat","commRing"],["instNatCastInt"],["Int","cast_pow"],["instHPow"],["True"],["Semiring","toMonoidWithZero"],["HPow","hPow"],["DivInvMonoid","toDiv"],["Ring","toSemiring"],["Int"],["eq_self"],["CommRing","toRing"],["Nat"],["AddMonoidWithOne","toNatCast"],["of_eq_true"],["DivInvMonoid","toMonoid"],["AddGroupWithOne","toIntCast"],["Rat"],["Int","cast_natCast"]]},{"isProp":true,"kind":"theorem","name":["Rat","commGroupWithZero","_proof_4"],"typeFallback":"forall (a : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (MulZeroClass.toMul.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))) (OfNat.ofNat.{0} Rat 0 (Zero.toOfNat0.{0} Rat (MulZeroClass.toZero.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing)))))))) a) (OfNat.ofNat.{0} Rat 0 (Zero.toOfNat0.{0} Rat (MulZeroClass.toZero.{0} Rat (NonUnitalNonAssocSemiring.toMulZeroClass.{0} Rat (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{0} Rat (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{0} Rat (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{0} Rat (CommRing.toNonUnitalCommRing.{0} Rat Rat.commRing))))))))","typeFull":"∀ (a : ℚ), 0 * a = 0","typeReadable":"∀ (a : ℚ), 0 * a = 0","typeReferences":[["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["MulZeroClass","toMul"],["HMul","hMul"],["CommRing","toNonUnitalCommRing"],["OfNat","ofNat"],["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["MulZeroClass","toZero"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["Rat"],["instHMul"],["Zero","toOfNat0"],["Eq"],["Rat","commRing"]],"valueReferences":[["NonUnitalNonAssocRing","toNonUnitalNonAssocSemiring"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocRing"],["NonUnitalNonAssocSemiring","toMulZeroClass"],["Rat"],["CommRing","toNonUnitalCommRing"],["MulZeroClass","zero_mul"],["Rat","commRing"]]},{"isProp":true,"kind":"theorem","name":["Rat","mul_den_eq_num"],"typeFallback":"forall (q : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat Rat.instMul) q (Nat.cast.{0} Rat Rat.instNatCast (Rat.den q))) (Int.cast.{0} Rat Rat.instIntCast (Rat.num q))","typeFull":"∀ (q : ℚ), q * ↑q.den = ↑q.num","typeReadable":"∀ (q : ℚ), q * ↑q.den = ↑q.num","typeReferences":[["Rat","instIntCast"],["Rat","instNatCast"],["Nat","cast"],["Rat"],["instHMul"],["HMul","hMul"],["Rat","instMul"],["Eq"],["Int","cast"],["Rat","num"],["Rat","den"]],"valueReferences":[["Int","instCommRing"],["Eq","trans"],["Rat","divInt"],["AddGroupWithOne","toAddMonoidWithOne"],["HMul","hMul"],["Rat","instMul"],["AddMonoidWithOne","toAddMonoid"],["Rat","instNatCast"],["Ring","toAddGroupWithOne"],["NonUnitalNonAssocCommRing","toNonUnitalNonAssocCommSemiring"],["Eq","symm"],["mul_comm"],["Rat","commRing"],["Rat","mkRat_self"],["CommMagma","toMul"],["NonUnitalCommRing","toNonUnitalNonAssocCommRing"],["Rat","divInt_ofNat"],["AddZeroClass","toAddZero"],["Rat","den_ne_zero"],["Nat"],["AddMonoidWithOne","toNatCast"],["instOfNat"],["Nat","cast_zero"],["Eq","refl"],["Rat"],["id"],["Int","instCharZero"],["instHMul"],["Eq","mpr"],["Int","cast_natCast"],["AddZero","toZero"],["AddMonoid","toAddZeroClass"],["Nat","cast_inj","_simp_1"],["Nat","cast"],["Eq","mp"],["NonUnitalNonAssocCommSemiring","toCommMagma"],["CommRing","toNonUnitalCommRing"],["Int","cast"],["Int","instRing"],["Rat","num"],["congrArg"],["Int","instMul"],["Rat","instIntCast"],["Rat","divInt_mul_divInt"],["mkRat"],["instOfNatNat"],["congr"],["congrFun'"],["Zero","toOfNat0"],["Eq"],["Rat","den"],["instNatCastInt"],["cast"],["Not"],["Rat","divInt_one"],["True"],["Rat","divInt_mul_right"],["OfNat","ofNat"],["Int"],["eq_self"],["CommRing","toRing"],["of_eq_true"],["Ne"]]},{"isProp":true,"kind":"theorem","name":["Rat","commRing","_proof_6"],"typeFallback":"forall (a : Rat), Eq.{1} Rat (HMul.hMul.{0, 0, 0} Rat Rat Rat (instHMul.{0} Rat (Semigroup.toMul.{0} Rat (Monoid.toSemigroup.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid)))) a (OfNat.ofNat.{0} Rat 1 (One.toOfNat1.{0} Rat (Monoid.toOne.{0} Rat (CommMonoid.toMonoid.{0} Rat Rat.commMonoid))))) a","typeFull":"∀ (a : ℚ), a * 1 = a","typeReadable":"∀ (a : ℚ), a * 1 = a","typeReferences":[["One","toOfNat1"],["CommMonoid","toMonoid"],["Rat","commMonoid"],["Monoid","toOne"],["Rat"],["instHMul"],["HMul","hMul"],["Monoid","toSemigroup"],["Eq"],["OfNat","ofNat"],["Semigroup","toMul"]],"valueReferences":[["CommMonoid","toMonoid"],["Rat","commMonoid"],["Rat"],["Monoid","mul_one"]]}]