proposition: Theorem EquivalenceOfTwoDefinitions (D:ExactCategoryData) : ExactCategoryProperties D ⇔ ExactCategoryProperties_Quillen D.
proof: Proof. split. { intros prop. set (M := (D,,prop) : ExactCategory). split. { exact (@EC_IsomorphicToExact M). } split. { exact (@DirectSumToExact M). } split. { exact (@EC_ExactToKernelCokernel M). } split. { split. { exact (@EC_ComposeMono M). } { exact (@EC_ComposeEpi M). } } split. { split. { exact (@EC_PullbackEpi M). } { exact (@EC_PushoutMono M). } } { split. { exact (@AdmMonoFromComposite M). } { exact (@AdmEpiFromComposite M). } } } { intros [P1 [P2 [P3 [[P4 P4'] [[P5 P5'] [P6 P7]]]]]]. split. { split. { exact P1. } { intros P Q i. exact (P1 P Q (InverseMorphismPairIsomorphism i)). } } { split. { split. { intros A. apply (squash_to_hProp (to_hasZero D)); intros Z. apply hinhpr. exists Z. set (Q := TrivialDirectSum Z A). exists (to_Pr2 Q). exact (P2 A Z Q). } { intros A. apply (squash_to_hProp (to_hasZero D)); intros Z. apply hinhpr. exists Z. set (Q := TrivialDirectSum' Z A). exists (to_In1 Q). exact (P2 Z A Q). } } split. { exact P3. } split. { split. { exact P4. } { exact P4'. } } split. { exact P5. } { exact P5'. } } } Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.Notations UniMath.MoreFoundations.PartA UniMath.Algebra.BinaryOperations UniMath.Algebra.Monoids UniMath.Algebra.Groups UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Limits.Zero UniMath.CategoryTheory.Limits.Kernels UniMath.CategoryTheory.Limits.Cokernels UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.Pushouts UniMath.CategoryTheory.Limits.BinDirectSums UniMath.CategoryTheory.Limits.Opp UniMath.CategoryTheory.CategoriesWithBinOps UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.PrecategoriesWithAbgrops UniMath.CategoryTheory.PreAdditive UniMath.CategoryTheory.Morphisms UniMath.CategoryTheory.Additive UniMath.CategoryTheory.Subcategory.Full UniMath.MoreFoundations.Propositions