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

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic.sym.json

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

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[inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.18 : AddCommGroup.{u_1} M] [inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.21 : Module.{u, u_1} R M (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.4)) (AddCommGroup.toAddCommMonoid.{u_1} M inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.18)] [inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.25 : LieRingModule.{v, u_1} L M inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.7 inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.18] [inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.29 : LieModule.{u, v, u_1} R L M inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4104274711._hygCtx._hyg.7 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R L M → Prop","typeReadable":"{R : Type u} →\n  {L : Type v} →\n    [inst : CommRing R] →\n      [inst_1 : LieRing L] →\n        [inst_2 : LieAlgebra R L] →\n          {M : Type u_1} →\n            [inst_3 : AddCommGroup M] →\n              [inst_4 : Module R M] → [inst_5 : LieRingModule L M] → [LieModule R L M] → LieSubmodule R L M → Prop","typeReferences":[["LieRingModule"],["CommRing","toCommSemiring"],["Module"],["CommSemiring","toSemiring"],["AddCommGroup","toAddCommMonoid"],["AddCommGroup"],["LieAlgebra"],["CommRing"],["LieModule"],["LieSubmodule"],["LieRing"]],"valueReferences":[["Nat"],["Exists"],["LE","le"],["LieSubmodule","ucs"],["Eq"],["Bot","bot"],["instLENat"],["LieSubmodule","instBot"],["LieSubmodule"]]},{"isProp":true,"kind":"constructor","name":["LieSubalgebra","IsCartanSubalgebra","mk"],"typeFallback":"forall {R : Type.{u}} {L : Type.{v}} [inst._@.Mathlib.Algebra.Lie.CartanSubalgebra.4074352571._hygCtx._hyg.19 : CommRing.{u} R] 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.Submodule.IterateMapComap.sym.json

@@ -0,0 +1 @@
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⇑i → f.ker ≤ 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{R : Type.{u_1}} {N : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8 : AddCommMonoid.{u_2} N] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.11 : Module.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.15 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.18 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.15] (f : LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5)) N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.18) (i : LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5)) N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.18) (K : Submodule.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.11), (LE.le.{u_3} (Submodule.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.18) (Preorder.toLE.{u_3} (Submodule.{u_1, u_3} R M 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inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.18 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5) f K) (Submodule.map.{u_1, u_1, u_2, u_3} R R N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.18 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5)) (RingHomSurjective.ids.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5) i K)) -> (forall (n : Nat), LE.le.{u_2} (Submodule.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.11) (Preorder.toLE.{u_2} (Submodule.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.1937883463._hygCtx._hyg.5 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M] [inst_4 : Module R M] (f i : N →ₗ[R] M) (K : Submodule R N),\n  Submodule.map f K ≤ Submodule.map i K → ∀ (n : ℕ), f.iterateMapComap i n K ≤ f.iterateMapComap i (n + 1) K","typeReadable":"∀ {R : Type u_1} {N : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid N] [inst_2 : Module R N]\n  [inst_3 : AddCommMonoid M] [inst_4 : Module R M] (f i : N →ₗ[R] M) (K : Submodule R N),\n  Submodule.map f K ≤ Submodule.map i K → ∀ (n : ℕ), f.iterateMapComap i n K ≤ f.iterateMapComap i (n + 1) 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{R : Type.{u_1}} {N : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.8 : AddCommMonoid.{u_2} N] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.11 : Module.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.15 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.18 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.15], (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5)) N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.18) -> (LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5)) N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.18) -> Nat -> (Submodule.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.11) -> (Submodule.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.2783040150._hygCtx._hyg.11)","typeFull":"{R : Type u_1} →\n  {N : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid N] →\n          [inst_2 : Module R N] →\n            [inst_3 : AddCommMonoid M] →\n              [inst_4 : Module R M] → (N →ₗ[R] M) → (N →ₗ[R] M) → ℕ → Submodule R N → Submodule R N","typeReadable":"{R : Type u_1} →\n  {N : Type u_2} →\n    {M : Type u_3} →\n      [inst : Semiring R] →\n        [inst_1 : AddCommMonoid N] →\n          [inst_2 : Module R N] →\n            [inst_3 : AddCommMonoid M] →\n              [inst_4 : Module R M] → (N →ₗ[R] M) → (N →ₗ[R] M) → ℕ → Submodule R N → Submodule R N","typeReferences":[["Nat"],["AddCommMonoid"],["Semiring","toNonAssocSemiring"],["RingHom","id"],["Module"],["LinearMap"],["Submodule"],["Semiring"]],"valueReferences":[["Semiring","toNonAssocSemiring"],["Nat","iterate"],["Submodule","comap"],["RingHom","id"],["Submodule","map"],["RingHomSurjective","ids"],["Submodule"]]},{"isProp":true,"kind":"theorem","name":["LinearMap","iterateMapComap_eq_succ"],"typeFallback":"forall {R : Type.{u_1}} {N : Type.{u_2}} {M : Type.{u_3}} [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 : Semiring.{u_1} R] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8 : AddCommMonoid.{u_2} N] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.11 : Module.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.15 : AddCommMonoid.{u_3} M] [inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.18 : Module.{u_1, u_3} R M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.15] (f : LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5)) N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.18) (i : LinearMap.{u_1, u_1, u_2, u_3} R R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 (RingHom.id.{u_1} R (Semiring.toNonAssocSemiring.{u_1} R inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5)) N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.18) (K : Submodule.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.11) (m : Nat), (Eq.{succ u_2} (Submodule.{u_1, u_2} R N inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.11) (LinearMap.iterateMapComap.{u_1, u_2, u_3} R N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.18 f i m K) (LinearMap.iterateMapComap.{u_1, u_2, u_3} R N M inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.5 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.8 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.11 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.15 inst._@.Mathlib.Algebra.Module.Submodule.IterateMapComap.3805489164._hygCtx._hyg.18 f i (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 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f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K →\n    Function.Surjective ⇑f → Function.Injective ⇑i → ∀ (n : ℕ), f.iterateMapComap i n K = f.iterateMapComap i (n + 1) K","typeReadable":"∀ {R : Type u_1} {N : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommMonoid N] [inst_2 : Module R N]\n  [inst_3 : AddCommMonoid M] [inst_4 : Module R M] (f i : N →ₗ[R] M) (K : Submodule R N) (m : ℕ),\n  f.iterateMapComap i m K = f.iterateMapComap i (m + 1) K →\n    Function.Surjective ⇑f → Function.Injective ⇑i → ∀ (n : ℕ), f.iterateMapComap i n K = f.iterateMapComap i (n + 1) 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[inst_2 : Module R N]\n  [inst_3 : AddCommMonoid M] [inst_4 : Module R M] (f i : N →ₗ[R] M) (n : ℕ),\n  f.iterateMapComap i n = (fun K => Submodule.comap f (Submodule.map i K))^[n]","typeReferences":[["Module"],["Submodule","map"],["LinearMap","iterateMapComap"],["LinearMap"],["RingHomSurjective","ids"],["Submodule"],["AddCommMonoid"],["Nat"],["Semiring","toNonAssocSemiring"],["Nat","iterate"],["Submodule","comap"],["RingHom","id"],["Eq"],["Semiring"]],"valueReferences":[["Eq","refl"],["LinearMap","iterateMapComap"],["Submodule"]]}]

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

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Star.Pi.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.Center.Preadditive.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.CategoryTheory.ConcreteCategory.ReflectsIso.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Combinatorics.Additive.CovBySMul.sym.json

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[inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.5 : Monoid.{u_1} M] [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.11 : MulAction.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.5], Real -> (Set.{u_3} X) -> (Set.{u_3} X) -> Prop","typeFull":"(M : Type u_1) → {X : Type u_3} → [inst : Monoid M] → [MulAction M X] → ℝ → Set X → Set X → Prop","typeReadable":"(M : Type u_1) → {X : Type u_3} → [inst : Monoid M] → [MulAction M X] → ℝ → Set X → Set X → 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{M : Type.{u_1}} {X : Type.{u_3}} [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.5 : Monoid.{u_1} M] [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.11 : MulAction.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.5] {K : Real} {A₁ : Set.{u_3} X} {A₂ : Set.{u_3} X} {B : Set.{u_3} X}, (HasSubset.Subset.{u_3} (Set.{u_3} X) (Set.instHasSubset.{u_3} X) A₁ A₂) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.11 K A₂ B) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.11 K A₁ B)","typeFull":"∀ {M : Type u_1} {X : Type u_3} [inst : Monoid M] [inst_1 : MulAction M X] {K : ℝ} {A₁ A₂ B : Set X},\n  A₁ ⊆ A₂ → CovBySMul M K A₂ B → CovBySMul M K A₁ 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[inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1178164624._hygCtx._hyg.5 : Monoid.{u_1} M] [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1178164624._hygCtx._hyg.11 : MulAction.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1178164624._hygCtx._hyg.5] {K : Real} {L : Real} {A : Set.{u_3} X} {B : Set.{u_3} X}, (LE.le.{0} Real Real.instLE K L) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1178164624._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1178164624._hygCtx._hyg.11 K A B) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1178164624._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1178164624._hygCtx._hyg.11 L A B)","typeFull":"∀ {M : Type u_1} {X : Type u_3} [inst : Monoid M] [inst_1 : MulAction M X] {K L : ℝ} {A B : Set X},\n  K ≤ L → CovBySMul M K A B → CovBySMul M L A B","typeReadable":"∀ {M : Type u_1} {X : Type u_3} [inst : Monoid M] [inst_1 : MulAction M X] {K L : 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inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.5] {K : Real} {A₁ : Set.{u_3} X} {A₂ : Set.{u_3} X} {B : Set.{u_3} X}, (HasSubset.Subset.{u_3} (Set.{u_3} X) (Set.instHasSubset.{u_3} X) A₁ A₂) -> (CovByVAdd.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.11 K A₂ B) -> (CovByVAdd.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1105760300._hygCtx._hyg.11 K A₁ B)","typeFull":"∀ {M : Type u_1} {X : Type u_3} [inst : AddMonoid M] [inst_1 : AddAction M X] {K : ℝ} {A₁ A₂ B : Set X},\n  A₁ ⊆ A₂ → CovByVAdd M K A₂ B → CovByVAdd M K A₁ B","typeReadable":"∀ {M : Type u_1} {X : Type u_3} [inst : AddMonoid M] [inst_1 : AddAction M X] {K : ℝ} {A₁ A₂ B : Set X},\n  A₁ ⊆ A₂ → CovByVAdd M K A₂ B → CovByVAdd M K A₁ 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inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1536434454._hygCtx._hyg.5] {K : Real} {A : Set.{u_3} X} {B₁ : Set.{u_3} X} {B₂ : Set.{u_3} X}, (HasSubset.Subset.{u_3} (Set.{u_3} X) (Set.instHasSubset.{u_3} X) B₁ B₂) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1536434454._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1536434454._hygCtx._hyg.11 K A B₁) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1536434454._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.1536434454._hygCtx._hyg.11 K A B₂)","typeFull":"∀ {M : Type u_1} {X : Type u_3} [inst : Monoid M] [inst_1 : MulAction M X] {K : ℝ} {A B₁ B₂ : Set X},\n  B₁ ⊆ B₂ → CovBySMul M K A B₁ → CovBySMul M K A B₂","typeReadable":"∀ {M : Type u_1} {X : Type u_3} [inst : Monoid M] [inst_1 : MulAction M X] {K : ℝ} {A B₁ B₂ : Set X},\n  B₁ ⊆ B₂ → CovBySMul M K A B₁ → CovBySMul M K A 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inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3015053212._hygCtx._hyg.5] {K : Real} {A₁ : Set.{u_3} X} {A₂ : Set.{u_3} X} {B₁ : Set.{u_3} X} {B₂ : Set.{u_3} X}, (HasSubset.Subset.{u_3} (Set.{u_3} X) (Set.instHasSubset.{u_3} X) A₁ A₂) -> (HasSubset.Subset.{u_3} (Set.{u_3} X) (Set.instHasSubset.{u_3} X) B₁ B₂) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3015053212._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3015053212._hygCtx._hyg.11 K A₂ B₁) -> (CovBySMul.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3015053212._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3015053212._hygCtx._hyg.11 K A₁ B₂)","typeFull":"∀ {M : Type u_1} {X : Type u_3} [inst : Monoid M] [inst_1 : MulAction M X] {K : ℝ} {A₁ A₂ B₁ B₂ : Set X},\n  A₁ ⊆ A₂ → B₁ ⊆ B₂ → CovBySMul M K A₂ B₁ → CovBySMul M K A₁ B₂","typeReadable":"∀ {M : Type u_1} {X : Type u_3} [inst : Monoid M] [inst_1 : MulAction M X] {K : ℝ} {A₁ A₂ B₁ B₂ : Set X},\n  A₁ ⊆ A₂ → B₁ ⊆ B₂ → CovBySMul M K A₂ B₁ → CovBySMul M K A₁ B₂","typeReferences":[["Set","instHasSubset"],["HasSubset","Subset"],["Real"],["MulAction"],["Set"],["Monoid"],["CovBySMul"]],"valueReferences":[["CovBySMul","subset_right"],["CovBySMul","subset_left"]]},{"isProp":true,"kind":"theorem","name":["covByVAdd_zero"],"typeFallback":"forall {M : Type.{u_1}} {X : Type.{u_3}} [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3413396302._hygCtx._hyg.5 : AddMonoid.{u_1} M] [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3413396302._hygCtx._hyg.11 : AddAction.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3413396302._hygCtx._hyg.5] {A : Set.{u_3} X} {B : Set.{u_3} X}, Iff (CovByVAdd.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3413396302._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3413396302._hygCtx._hyg.11 (OfNat.ofNat.{0} Real 0 (Zero.toOfNat0.{0} Real Real.instZero)) A B) (Eq.{succ u_3} (Set.{u_3} X) A 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(M : Type.{u_1}) {X : Type.{u_3}} [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.5 : AddMonoid.{u_1} M] [inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.11 : AddAction.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.5] (K : Real) (A : Set.{u_3} X) (B : Set.{u_3} X), Eq.{1} Prop (CovByVAdd.{u_1, u_3} M X inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.5 inst._@.Mathlib.Combinatorics.Additive.CovBySMul.3756826125._hygCtx._hyg.11 K A B) (Exists.{succ u_1} (Finset.{u_1} M) (fun (F : Finset.{u_1} M) => And (LE.le.{0} Real Real.instLE (Nat.cast.{0} Real Real.instNatCast (Finset.card.{u_1} M F)) K) (HasSubset.Subset.{u_3} (Set.{u_3} X) (Set.instHasSubset.{u_3} X) A (HVAdd.hVAdd.{u_1, u_3, u_3} (Set.{u_1} M) (Set.{u_3} X) (Set.{u_3} X) (instHVAdd.{u_1, u_3} (Set.{u_1} M) (Set.{u_3} X) (Set.vadd.{u_1, u_3} M X (AddSemigroupAction.toVAdd.{u_1, u_3} M X 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.List.OfFn.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["List","find?_ofFn_eq_some_of_injective"],"typeFallback":"forall {α : Type.{u}} {n : Nat} {f : (Fin n) -> α} {p : α -> Bool} {i : Fin n}, (Function.Injective.{1, succ u} (Fin n) α f) -> (Iff (Eq.{succ u} (Option.{u} α) (List.find?.{u} α p (List.ofFn.{u} α n f)) (Option.some.{u} α (f i))) (And (Eq.{1} Bool (p (f i)) Bool.true) (forall (j : Fin n), (LT.lt.{0} (Fin n) (instLTFin n) j i) -> (Not (Eq.{1} Bool (p (f j)) Bool.true)))))","typeFull":"∀ {α : Type u} {n : ℕ} {f : Fin n → α} {p : α → Bool} {i : Fin n},\n  Function.Injective f →\n    (List.find? p (List.ofFn f) = some (f i) ↔ p (f i) = true ∧ ∀ (j : Fin n), j < i → ¬p (f j) = true)","typeReadable":"∀ {α : Type u} {n : ℕ} {f : Fin n → α} {p : α → Bool} {i : Fin n},\n  Function.Injective f →\n    (List.find? p (List.ofFn f) = some (f i) ↔ p (f i) = true ∧ ∀ (j : Fin n), j < i → ¬p (f j) = 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Multiset.NatAntidiagonal.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Nat.Basic.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Nat.Cast.WithTop.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Real.CompleteField.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Data.Set.Pointwise.Support.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.GroupTheory.MonoidLocalization.DivPairs.sym.json

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inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8))))) -> (Submonoid.{u_1} (Prod.{u_1, u_1} M M) (Prod.instMulOneClass.{u_1, u_1} M M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)) (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))))","typeFull":"{M : Type u_1} →\n  {G : Type u_2} →\n    [inst : CommMonoid M] → [inst_1 : CommGroup G] → ⊤.LocalizationMap G → Submonoid G → Submonoid (M × M)","typeReadable":"{M : Type u_1} →\n  {G : Type u_2} →\n    [inst : CommMonoid M] → [inst_1 : CommGroup G] → ⊤.LocalizationMap G → Submonoid G → Submonoid (M × M)","typeReferences":[["CommMonoid","toMonoid"],["Submonoid","LocalizationMap"],["Submonoid","instTop"],["CommGroup"],["Submonoid"],["CommGroup","toCommMonoid"],["Prod"],["DivInvMonoid","toMonoid"],["Monoid","toMulOneClass"],["Prod","instMulOneClass"],["Top","top"],["CommMonoid"],["Group","toDivInvMonoid"],["CommGroup","toGroup"]],"valueReferences":[["MulOneClass","toMulOne"],["divMonoidHom"],["CommMonoid","toMonoid"],["Submonoid","LocalizationMap"],["MonoidHom"],["MonoidHom","instFunLike"],["Submonoid","instTop"],["MonoidHom","comp"],["CommGroup","toDivisionCommMonoid"],["CommGroup","toCommMonoid"],["DivisionMonoid","toDivInvMonoid"],["Monoid","toMulOneClass"],["Prod","instMulOneClass"],["DivisionCommMonoid","toDivisionMonoid"],["CommGroup","toGroup"],["Group","toDivInvMonoid"],["Submonoid","LocalizationMap","instFunLike"],["MonoidHom","prodMap"],["Submonoid","divPairs","_proof_2"],["Submonoid","comap"],["Submonoid","divPairs","_proof_1"],["Prod"],["Submonoid"],["DivInvMonoid","toMonoid"],["Top","top"],["MonoidHomClass","toMonoidHom"]]},{"isProp":true,"kind":"theorem","name":["Submonoid","divPairs","eq_1"],"typeFallback":"forall 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(DivInvMonoid.toMonoid.{u_2} G (DivisionMonoid.toDivInvMonoid.{u_2} G (DivisionCommMonoid.toDivisionMonoid.{u_2} G (CommGroup.toDivisionCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8))))) (Monoid.toMulOneClass.{u_2} G (DivInvMonoid.toMonoid.{u_2} G (DivisionMonoid.toDivInvMonoid.{u_2} G (DivisionCommMonoid.toDivisionMonoid.{u_2} G (CommGroup.toDivisionCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8))))) (MonoidHomClass.toMonoidHom.{u_1, u_2, max u_1 u_2} M G (Submonoid.LocalizationMap.{u_1, u_2} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 (Top.top.{u_1} (Submonoid.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))) (Submonoid.instTop.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M 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(Submonoid.instTop.{u_1} M (Monoid.toMulOneClass.{u_1} M (CommMonoid.toMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)))) G (CommGroup.toCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8)) (Submonoid.divPairs._proof_2.{u_2, u_1} M G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8) f))) s)","typeFull":"∀ {M : Type u_1} {G : Type u_2} [inst : CommMonoid M] [inst_1 : CommGroup G] (f : ⊤.LocalizationMap G)\n  (s : Submonoid G), Submonoid.divPairs f s = Submonoid.comap (divMonoidHom.comp ((↑f).prodMap ↑f)) s","typeReadable":"∀ {M : Type u_1} {G : Type u_2} [inst : CommMonoid M] [inst_1 : CommGroup G] (f : ⊤.LocalizationMap G)\n  (s : Submonoid G), Submonoid.divPairs f s = Submonoid.comap (divMonoidHom.comp ((↑f).prodMap ↑f)) 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{M : Type.{u_2}} {G : Type.{u_1}} [inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 : CommMonoid.{u_2} M] [inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8 : CommGroup.{u_1} G], MonoidHomClass.{max u_1 u_2, u_2, u_1} (MonoidHom.{u_2, u_1} (Prod.{u_2, u_2} M M) G (MulOneClass.toMulOne.{u_2} (Prod.{u_2, u_2} M M) (Prod.instMulOneClass.{u_2, u_2} M M (Monoid.toMulOneClass.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)) (Monoid.toMulOneClass.{u_2} M (CommMonoid.toMonoid.{u_2} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)))) (MulOneClass.toMulOne.{u_1} G (Monoid.toMulOneClass.{u_1} G (DivInvMonoid.toMonoid.{u_1} G (DivisionMonoid.toDivInvMonoid.{u_1} G (DivisionCommMonoid.toDivisionMonoid.{u_1} G (CommGroup.toDivisionCommMonoid.{u_1} G 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inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 (Top.top.{u_1} (AddSubmonoid.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))) (AddSubmonoid.instTop.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)))) G (AddCommGroup.toAddCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8)) (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))) (AddZeroClass.toAddZero.{u_2} G (AddMonoid.toAddZeroClass.{u_2} G (SubNegMonoid.toAddMonoid.{u_2} G (SubtractionMonoid.toSubNegMonoid.{u_2} G (SubtractionCommMonoid.toSubtractionMonoid.{u_2} G (AddCommGroup.toDivisionAddCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8)))))) (AddSubmonoid.LocalizationMap.instFunLike.{u_1, u_2} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 (Top.top.{u_1} (AddSubmonoid.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))) (AddSubmonoid.instTop.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)))) G (AddCommGroup.toAddCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8)) (AddSubmonoid.subPairs._proof_2.{u_2, u_1} M G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8) f) (AddMonoidHomClass.toAddMonoidHom.{u_1, u_2, max u_1 u_2} M G (AddSubmonoid.LocalizationMap.{u_1, u_2} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 (Top.top.{u_1} (AddSubmonoid.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))) (AddSubmonoid.instTop.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)))) G (AddCommGroup.toAddCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8)) (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))) (AddZeroClass.toAddZero.{u_2} G (AddMonoid.toAddZeroClass.{u_2} G (SubNegMonoid.toAddMonoid.{u_2} G (SubtractionMonoid.toSubNegMonoid.{u_2} G (SubtractionCommMonoid.toSubtractionMonoid.{u_2} G (AddCommGroup.toDivisionAddCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8)))))) (AddSubmonoid.LocalizationMap.instFunLike.{u_1, u_2} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 (Top.top.{u_1} (AddSubmonoid.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5))) (AddSubmonoid.instTop.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (AddCommMonoid.toAddMonoid.{u_1} M inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5)))) G (AddCommGroup.toAddCommMonoid.{u_2} G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8)) (AddSubmonoid.subPairs._proof_2.{u_2, u_1} M G inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.5 inst._@.Mathlib.GroupTheory.MonoidLocalization.DivPairs.1970198191._hygCtx._hyg.8) f))) s)","typeFull":"∀ {M : Type u_1} {G : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommGroup G] (f : ⊤.LocalizationMap G)\n  (s : AddSubmonoid G), AddSubmonoid.subPairs f s = AddSubmonoid.comap (subAddMonoidHom.comp ((↑f).prodMap ↑f)) s","typeReadable":"∀ {M : Type u_1} {G : Type u_2} [inst : AddCommMonoid M] [inst_1 : AddCommGroup G] (f : ⊤.LocalizationMap G)\n  (s : AddSubmonoid G), AddSubmonoid.subPairs f s = AddSubmonoid.comap (subAddMonoidHom.comp ((↑f).prodMap ↑f)) s","typeReferences":[["AddMonoidHomClass","toAddMonoidHom"],["AddCommGroup","toAddGroup"],["AddSubmonoid","LocalizationMap"],["AddSubmonoid","subPairs","_proof_1"],["SubtractionCommMonoid","toSubtractionMonoid"],["AddCommMonoid","toAddMonoid"],["AddSubmonoid","LocalizationMap","instFunLike"],["AddCommMonoid"],["AddSubmonoid","subPairs","_proof_2"],["Prod","instAddZeroClass"],["AddMonoidHom","comp"],["AddMonoidHom"],["AddSubmonoid","subPairs"],["AddMonoidHom","instFunLike"],["AddGroup","toSubNegMonoid"],["Eq"],["AddMonoidHom","prodMap"],["AddSubmonoid","instTop"],["AddCommGroup"],["AddZeroClass","toAddZero"],["AddSubmonoid","comap"],["Prod"],["SubNegMonoid","toAddMonoid"],["AddCommGroup","toDivisionAddCommMonoid"],["AddSubmonoid"],["SubtractionMonoid","toSubNegMonoid"],["AddCommGroup","toAddCommMonoid"],["Top","top"],["subAddMonoidHom"],["AddMonoid","toAddZeroClass"]],"valueReferences":[["Prod"],["Prod","instAddZeroClass"],["AddSubmonoid"],["Eq","refl"],["AddSubmonoid","subPairs"],["AddCommMonoid","toAddMonoid"],["AddMonoid","toAddZeroClass"]]}]