Proposition per_subobject_impl_elim : per_subobject_mor_law (id_partial_setoid_morphism Γ) (per_subobject_conj H ψ₁ per_subobject_impl) ψ₂. Proof. do 2 use forall_intro. use impl_intro. use weaken_right. use impl_intro. cbn ; unfold ζ. simplify_form. rewrite !partial_setoid_subst. simplify. pose (γ₁ := π₂ (π₁ (tm_var ((𝟙 ×h Γ) ×h Γ)))). pose (γ₂ := π₂ (tm_var ((𝟙 ×h Γ) ×h Γ))). fold γ₁ γ₂. use per_subobject_eq. - exact γ₁. - use weaken_left. apply hyperdoctrine_hyp. - use impl_elim. + exact (ψ₁ [ γ₁ ]). + use weaken_right. use weaken_left. apply hyperdoctrine_hyp. + do 2 use weaken_right. use weaken_left. apply hyperdoctrine_hyp. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.FiberwiseCartesianClosed. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Hyperdoctrines.Hyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.FirstOrderHyperdoctrine. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERs. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERMorphisms. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.PERCategory. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.Logic.SubobjectDispCat. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.Logic.Conjunction.