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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Algebra.StrictPositivity.sym.json

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+[{"isProp":true,"kind":"theorem","name":["isStrictlyPositive_algebraMap"],"typeFallback":"forall {A : Type.{u_1}} {𝕜 : Type.{u_2}} [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.4 : Ring.{u_1} A] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.7 : PartialOrder.{u_1} A] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.10 : ZeroLEOneClass.{u_1} A (MulZeroClass.toZero.{u_1} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u_1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_1} A (NonAssocRing.toNonUnitalNonAssocRing.{u_1} A (Ring.toNonAssocRing.{u_1} A inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.4))))) (AddMonoidWithOne.toOne.{u_1} A (AddGroupWithOne.toAddMonoidWithOne.{u_1} A (Ring.toAddGroupWithOne.{u_1} A inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.4))) (Preorder.toLE.{u_1} A (PartialOrder.toPreorder.{u_1} A inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.7))] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.13 : Semifield.{u_2} 𝕜] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.16 : PartialOrder.{u_2} 𝕜] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.19 : Algebra.{u_2, u_1} 𝕜 A (Semifield.toCommSemiring.{u_2} 𝕜 inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.13) (Ring.toSemiring.{u_1} A inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.4)] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.23 : PosSMulMono.{u_2, u_1} 𝕜 A (Algebra.toSMul.{u_2, u_1} 𝕜 A (Semifield.toCommSemiring.{u_2} 𝕜 inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.13) (Ring.toSemiring.{u_1} A inst._@.Mathlib.Algebra.Algebra.StrictPositivity.1589461516._hygCtx._hyg.4) 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inst._@.Mathlib.Algebra.Algebra.StrictPositivity.2880894046._hygCtx._hyg.4 inst._@.Mathlib.Algebra.Algebra.StrictPositivity.2880894046._hygCtx._hyg.16 a) x) -> (LT.lt.{u_2} 𝕜 (Preorder.toLT.{u_2} 𝕜 (PartialOrder.toPreorder.{u_2} 𝕜 inst._@.Mathlib.Algebra.Algebra.StrictPositivity.2880894046._hygCtx._hyg.13)) (OfNat.ofNat.{u_2} 𝕜 0 (Zero.toOfNat0.{u_2} 𝕜 (MulZeroClass.toZero.{u_2} 𝕜 (NonUnitalNonAssocSemiring.toMulZeroClass.{u_2} 𝕜 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_2} 𝕜 (Semiring.toNonAssocSemiring.{u_2} 𝕜 (CommSemiring.toSemiring.{u_2} 𝕜 inst._@.Mathlib.Algebra.Algebra.StrictPositivity.2880894046._hygCtx._hyg.10))))))) x))","typeFull":"∀ {A : Type u_1} {𝕜 : Type u_2} [inst : Ring A] [inst_1 : PartialOrder A] [inst_2 : CommSemiring 𝕜]\n  [inst_3 : PartialOrder 𝕜] [inst_4 : Algebra 𝕜 A] [NonnegSpectrumClass 𝕜 A] {a : A},\n  IsStrictlyPositive a → ∀ {x : 𝕜}, x ∈ spectrum 𝕜 a → 0 < x","typeReadable":"∀ {A : Type u_1} {𝕜 : Type u_2} [inst : Ring A] [inst_1 : PartialOrder A] [inst_2 : CommSemiring 𝕜]\n  [inst_3 : PartialOrder 𝕜] [inst_4 : Algebra 𝕜 A] [NonnegSpectrumClass 𝕜 A] {a : A},\n  IsStrictlyPositive a → ∀ {x : 𝕜}, x ∈ spectrum 𝕜 a → 0 < 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{A : Type.{u_1}} [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.3 : Semiring.{u_1} A] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.6 : PartialOrder.{u_1} A] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.9 : StarRing.{u_1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_1} A (Semiring.toNonAssocSemiring.{u_1} A inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.3))] [inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.12 : StarOrderedRing.{u_1} A (Semiring.toNonUnitalSemiring.{u_1} A inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.3) inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.6 inst._@.Mathlib.Algebra.Algebra.StrictPositivity.3669994791._hygCtx._hyg.9] {a : A}, (IsStrictlyPositive.{u_1} A (Preorder.toLE.{u_1} A (PartialOrder.toPreorder.{u_1} A 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.BigOperators.Group.Finset.Pi.sym.json

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

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["List","prod_eq_zero_iff"],"typeFallback":"forall {M₀ : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx._hyg.7 : MonoidWithZero.{u_4} M₀] [inst._@.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx._hyg.12 : Nontrivial.{u_4} M₀] [inst._@.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx._hyg.15 : NoZeroDivisors.{u_4} M₀ (MulZeroClass.toMul.{u_4} M₀ (MulZeroOneClass.toMulZeroClass.{u_4} M₀ (MonoidWithZero.toMulZeroOneClass.{u_4} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx._hyg.7))) (MulZeroClass.toZero.{u_4} M₀ (MulZeroOneClass.toMulZeroClass.{u_4} M₀ (MonoidWithZero.toMulZeroOneClass.{u_4} M₀ inst._@.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx._hyg.7)))] {l : List.{u_4} M₀}, Iff (Eq.{succ u_4} M₀ (List.prod.{u_4} M₀ (MulZeroClass.toMul.{u_4} M₀ (MulZeroOneClass.toMulZeroClass.{u_4} M₀ (MonoidWithZero.toMulZeroOneClass.{u_4} M₀ 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(l.prod = 0) = (0 ∈ l)","typeReferences":[["MulOneClass","toMulOne"],["NoZeroDivisors"],["MulOne","toOne"],["Membership","mem"],["MulZeroClass","toMul"],["List","instMembership"],["MonoidWithZero","toMulZeroOneClass"],["List"],["MulZeroOneClass","toMulOneClass"],["List","prod"],["OfNat","ofNat"],["MonoidWithZero"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Nontrivial"],["Zero","toOfNat0"],["Eq"]],"valueReferences":[["MulOneClass","toMulOne"],["MulOne","toOne"],["MulZeroClass","toMul"],["Membership","mem"],["List","instMembership"],["List"],["MonoidWithZero","toMulZeroOneClass"],["MulZeroOneClass","toMulOneClass"],["List","prod"],["OfNat","ofNat"],["MulZeroOneClass","toMulZeroClass"],["MulZeroClass","toZero"],["Zero","toOfNat0"],["Eq"],["List","prod_eq_zero_iff"],["propext"]]},{"isProp":true,"kind":"theorem","name":["List","prod_ne_zero"],"typeFallback":"forall {M₀ : Type.{u_4}} [inst._@.Mathlib.Algebra.BigOperators.Ring.List.3370944760._hygCtx._hyg.7 : 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{M₀ : Type.{u_1}} (motive : (List.{u_1} M₀) -> Prop) (x._@.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx.38.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx._hyg.50 : List.{u_1} M₀), (Unit -> (motive (List.nil.{u_1} M₀))) -> (forall (a : M₀) (l : List.{u_1} M₀), motive (List.cons.{u_1} M₀ a l)) -> (motive x._@.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx.38.Mathlib.Algebra.BigOperators.Ring.List.1456635985._hygCtx._hyg.50)","typeFull":"∀ {M₀ : Type u_1} (motive : List M₀ → Prop) (x : List M₀),\n  (∀ (a : Unit), motive []) → (∀ (a : M₀) (l : List M₀), motive (a :: l)) → motive x","typeReadable":"∀ {M₀ : Type u_1} (motive : List M₀ → Prop) (x : List M₀),\n  (∀ (a : Unit), motive []) → (∀ (a : M₀) (l : List M₀), motive (a :: l)) → motive x","typeReferences":[["List","nil"],["List"],["List","cons"],["Unit"]],"valueReferences":[["List","casesOn"],["Unit","unit"]]},{"isProp":true,"kind":"theorem","name":["Commute","list_sum_right"],"typeFallback":"forall {R 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.sym.json

@@ -0,0 +1 @@
+[{"isProp":true,"kind":"theorem","name":["CategoryTheory","projectiveDimension_eq_of_semiLinearEquiv"],"typeFallback":"forall {R : Type.{u}} [inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.1745785768._hygCtx._hyg.3 : CommRing.{u} R] [inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.1745785768._hygCtx._hyg.6 : Small.{v, u} R] {R' : Type.{u'}} [inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.1745785768._hygCtx._hyg.10 : CommRing.{u'} R'] [inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.1745785768._hygCtx._hyg.13 : Small.{v', u'} R'] (e : RingEquiv.{u, u'} R R' (Distrib.toMul.{u} R (NonUnitalNonAssocSemiring.toDistrib.{u} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u} R (NonUnitalNonAssocCommRing.toNonUnitalNonAssocRing.{u} R (NonUnitalCommRing.toNonUnitalNonAssocCommRing.{u} R (CommRing.toNonUnitalCommRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.1745785768._hygCtx._hyg.3)))))) 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[inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3 : CommRing.{u} R] [inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.6 : Small.{v, u} R] [inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.9 : Small.{v', u} R] {M : ModuleCat.{v, u} R (CommRing.toRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3)} {N : ModuleCat.{v', u} R (CommRing.toRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3)}, (LinearEquiv.{u, u, v, v'} R R (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3)) (CommSemiring.toSemiring.{u} R (CommRing.toCommSemiring.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3)) (RingHom.id.{u} R (Semiring.toNonAssocSemiring.{u} R 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inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3) N) (AddCommGroup.toAddCommMonoid.{v} (ModuleCat.carrier.{v, u} R (CommRing.toRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3) M) (ModuleCat.isAddCommGroup.{v, u} R (CommRing.toRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3) M)) (AddCommGroup.toAddCommMonoid.{v'} (ModuleCat.carrier.{v', u} R (CommRing.toRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3) N) (ModuleCat.isAddCommGroup.{v', u} R (CommRing.toRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3) N)) (ModuleCat.isModule.{v, u} R (CommRing.toRing.{u} R inst._@.Mathlib.Algebra.Category.ModuleCat.ProjectiveDimension.205620553._hygCtx._hyg.3) M) (ModuleCat.isModule.{v', u} R (CommRing.toRing.{u} R 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{C : Type.{u}} [inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2549173404._hygCtx._hyg.3 : CategoryTheory.Category.{v, u} C] [inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2549173404._hygCtx._hyg.6 : CategoryTheory.Abelian.{v, u} C inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2549173404._hygCtx._hyg.3] (X : C), Eq.{1} Prop (Eq.{1} (WithBot.{0} ENat) (CategoryTheory.projectiveDimension.{v, u} C inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2549173404._hygCtx._hyg.3 inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2549173404._hygCtx._hyg.6 X) (Bot.bot.{0} (WithBot.{0} ENat) (WithBot.bot.{0} ENat))) (CategoryTheory.Limits.IsZero.{v, u} C inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2549173404._hygCtx._hyg.3 X)","typeFull":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] (X : C),\n  (CategoryTheory.projectiveDimension X = ⊥) = CategoryTheory.Limits.IsZero X","typeReadable":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] (X : C),\n  (CategoryTheory.projectiveDimension X = ⊥) = CategoryTheory.Limits.IsZero X","typeReferences":[["CategoryTheory","Abelian"],["WithBot"],["CategoryTheory","Category"],["ENat"],["CategoryTheory","Limits","IsZero"],["CategoryTheory","projectiveDimension"],["WithBot","bot"],["Eq"],["Bot","bot"]],"valueReferences":[["WithBot"],["CategoryTheory","projectiveDimension_eq_bot_iff"],["ENat"],["CategoryTheory","Limits","IsZero"],["CategoryTheory","projectiveDimension"],["WithBot","bot"],["Eq"],["Bot","bot"],["propext"]]},{"isProp":true,"kind":"theorem","name":["_private","Mathlib","Algebra","Category","ModuleCat","ProjectiveDimension",0,"CategoryTheory","projectiveDimension_eq_of_semiLinearEquiv","_simp_1_3"],"typeFallback":"forall {C : Type.{u}} [inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2049256411._hygCtx._hyg.3 : CategoryTheory.Category.{v, u} C] [inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2049256411._hygCtx._hyg.6 : CategoryTheory.Abelian.{v, u} C inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2049256411._hygCtx._hyg.3] (X : C) (n : Nat), Eq.{1} Prop (LE.le.{0} (WithBot.{0} ENat) (Preorder.toLE.{0} (WithBot.{0} ENat) (WithBot.instPreorder.{0} ENat instPreorderENat)) (CategoryTheory.projectiveDimension.{v, u} C inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2049256411._hygCtx._hyg.3 inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2049256411._hygCtx._hyg.6 X) (Nat.cast.{0} (WithBot.{0} ENat) (WithBot.natCast.{0} ENat ENat.instNatCast) n)) (CategoryTheory.HasProjectiveDimensionLE.{v, u} C inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2049256411._hygCtx._hyg.3 inst._@.Mathlib.CategoryTheory.Abelian.Projective.Dimension.2049256411._hygCtx._hyg.6 X n)","typeFull":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] (X : C) (n : ℕ),\n  (CategoryTheory.projectiveDimension X ≤ ↑n) = CategoryTheory.HasProjectiveDimensionLE X n","typeReadable":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] (X : C) (n : ℕ),\n  (CategoryTheory.projectiveDimension X ≤ ↑n) = CategoryTheory.HasProjectiveDimensionLE X 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Category.Ring.Epi.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Divisibility.Hom.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.EuclideanDomain.Int.sym.json

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Group.Pointwise.Finset.Density.sym.json

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

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

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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Homology.GrothendieckAbelian.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Homology.ShortComplex.Retract.sym.json

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(CategoryTheory.ShortComplex.homologyMap.{v, u} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.6 S₁ S₂ φ₁ inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.29) (CategoryTheory.ShortComplex.homologyMap.{v, u} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.6 S₂ S₃ φ₂ inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.29 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.32)) (CategoryTheory.ShortComplex.homologyMap.{v, u} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.6 S₁ S₃ (CategoryTheory.CategoryStruct.comp.{v, max u v} (CategoryTheory.ShortComplex.{v, u} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.6) (CategoryTheory.Category.toCategoryStruct.{v, max u v} (CategoryTheory.ShortComplex.{v, u} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.6) (CategoryTheory.ShortComplex.instCategory.{v, u} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.6)) S₁ S₂ S₃ φ₁ φ₂) inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.26 inst._@.Mathlib.Algebra.Homology.ShortComplex.Homology.189995688._hygCtx._hyg.32)","typeFull":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]\n  {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} [inst_2 : S₁.HasHomology] [inst_3 : S₂.HasHomology]\n  [inst_4 : S₃.HasHomology] (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃),\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ₁)\n      (CategoryTheory.ShortComplex.homologyMap φ₂) =\n    CategoryTheory.ShortComplex.homologyMap (CategoryTheory.CategoryStruct.comp φ₁ φ₂)","typeReadable":"∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]\n  {S₁ S₂ S₃ : CategoryTheory.ShortComplex C} [inst_2 : S₁.HasHomology] [inst_3 : S₂.HasHomology]\n  [inst_4 : S₃.HasHomology] (φ₁ : S₁ ⟶ S₂) (φ₂ : S₂ ⟶ S₃),\n  CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap φ₁)\n      (CategoryTheory.ShortComplex.homologyMap φ₂) =\n    CategoryTheory.ShortComplex.homologyMap (CategoryTheory.CategoryStruct.comp φ₁ φ₂)","typeReferences":[["CategoryTheory","CategoryStruct","toQuiver"],["CategoryTheory","ShortComplex"],["CategoryTheory","CategoryStruct","comp"],["Quiver","Hom"],["CategoryTheory","ShortComplex","instCategory"],["CategoryTheory","Category"],["CategoryTheory","ShortComplex","homologyMap"],["CategoryTheory","ShortComplex","homology"],["CategoryTheory","ShortComplex","HasHomology"],["Eq"],["CategoryTheory","Limits","HasZeroMorphisms"],["CategoryTheory","Category","toCategoryStruct"]],"valueReferences":[["CategoryTheory","CategoryStruct","toQuiver"],["CategoryTheory","ShortComplex"],["CategoryTheory","CategoryStruct","comp"],["Quiver","Hom"],["CategoryTheory","ShortComplex","instCategory"],["CategoryTheory","ShortComplex","homologyMap"],["CategoryTheory","ShortComplex","homology"],["Eq","symm"],["CategoryTheory","Category","toCategoryStruct"],["CategoryTheory","ShortComplex","homologyMap_comp"]]},{"isProp":true,"kind":"theorem","name":["CategoryTheory","ShortComplex","quasiIso_of_retract"],"typeFallback":"forall {C : Type.{u_1}} [inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 : CategoryTheory.Category.{v_1, u_1} C] [inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6 : CategoryTheory.Limits.HasZeroMorphisms.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3] {S₁ : CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6} {T₁ : CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6} {S₂ : CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6} {T₂ : CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6} [inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.17 : CategoryTheory.ShortComplex.HasHomology.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6 S₁] [inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.19 : CategoryTheory.ShortComplex.HasHomology.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6 T₁] [inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.21 : CategoryTheory.ShortComplex.HasHomology.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6 S₂] [inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.23 : CategoryTheory.ShortComplex.HasHomology.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6 T₂] {f₁ : Quiver.Hom.{v_1, max u_1 v_1} (CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) (CategoryTheory.CategoryStruct.toQuiver.{v_1, max u_1 v_1} (CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) (CategoryTheory.Category.toCategoryStruct.{v_1, max u_1 v_1} (CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) (CategoryTheory.ShortComplex.instCategory.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6))) S₁ T₁} {f₂ : Quiver.Hom.{v_1, max u_1 v_1} (CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) (CategoryTheory.CategoryStruct.toQuiver.{v_1, max u_1 v_1} (CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) (CategoryTheory.Category.toCategoryStruct.{v_1, max u_1 v_1} (CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) (CategoryTheory.ShortComplex.instCategory.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6))) S₂ T₂}, (CategoryTheory.RetractArrow.{v_1, max u_1 v_1} (CategoryTheory.ShortComplex.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) (CategoryTheory.ShortComplex.instCategory.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6) S₁ T₁ S₂ T₂ f₁ f₂) -> (forall [hf₂ : CategoryTheory.ShortComplex.QuasiIso.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6 S₂ T₂ inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.21 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.23 f₂], CategoryTheory.ShortComplex.QuasiIso.{v_1, u_1} C inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.3 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.6 S₁ T₁ inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.17 inst._@.Mathlib.Algebra.Homology.ShortComplex.Retract.1483905909._hygCtx._hyg.19 f₁)","typeFull":"∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]\n  {S₁ T₁ S₂ T₂ : CategoryTheory.ShortComplex C} [inst_2 : S₁.HasHomology] [inst_3 : T₁.HasHomology]\n  [inst_4 : S₂.HasHomology] [inst_5 : T₂.HasHomology] {f₁ : S₁ ⟶ T₁} {f₂ : S₂ ⟶ T₂}\n  (h : CategoryTheory.RetractArrow f₁ f₂) [hf₂ : CategoryTheory.ShortComplex.QuasiIso f₂],\n  CategoryTheory.ShortComplex.QuasiIso f₁","typeReadable":"∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]\n  {S₁ T₁ S₂ T₂ : CategoryTheory.ShortComplex C} [inst_2 : S₁.HasHomology] [inst_3 : T₁.HasHomology]\n  [inst_4 : S₂.HasHomology] [inst_5 : T₂.HasHomology] {f₁ : S₁ ⟶ T₁} {f₂ : S₂ ⟶ T₂}\n  (h : CategoryTheory.RetractArrow f₁ f₂) [hf₂ : CategoryTheory.ShortComplex.QuasiIso f₂],\n  CategoryTheory.ShortComplex.QuasiIso 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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.IsPrimePow.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.End.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.LinearMap.Prod.sym.json

@@ -0,0 +1 @@
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data_5e932f97dd25535344f80f9dd8da3aab83df0fe6/Mathlib.Algebra.Module.MinimalAxioms.sym.json

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inst._@.Mathlib.Algebra.Module.MinimalAxioms.2724524948._hygCtx._hyg.10) r x._@.Mathlib.Algebra.Module.MinimalAxioms.2724524948._hygCtx._hyg.113) (smul_add r)) (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.MinimalAxioms.2724524948._hygCtx._hyg.7))))))))) (OfNat.ofNat.{u_1} M 0 (Zero.toOfNat0.{u_1} M (AddZero.toZero.{u_1} M (AddZeroClass.toAddZero.{u_1} M (AddMonoid.toAddZeroClass.{u_1} M (SubNegMonoid.toAddMonoid.{u_1} M (AddGroup.toSubNegMonoid.{u_1} M (AddCommGroup.toAddGroup.{u_1} M inst._@.Mathlib.Algebra.Module.MinimalAxioms.2724524948._hygCtx._hyg.7))))))))","typeFull":"∀ {R : Type u_2} {M : Type u_1} [inst : AddCommGroup M] [inst_1 : SMul R M]\n  (smul_add : ∀ (r : R) (x y : M), r • (x + y) = r • x + r • y) (r : R), (AddMonoidHom.mk' (fun x => r • x) ⋯) 0 = 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