Proposition subobject_to_per_axioms : per_axioms χ. Proof. unfold χ. split. - do 2 use forall_intro. use impl_intro. use weaken_right. simplify_form. rewrite !partial_setoid_subst. simplify. pose (x₁ := π₂ (π₁ (tm_var ((𝟙 ×h X) ×h X)))). pose (x₂ := π₂ (tm_var ((𝟙 ×h X) ×h X))). fold x₁ x₂. use conj_intro. + use weaken_left. use partial_setoid_sym. apply hyperdoctrine_hyp. + use per_subobject_eq. * exact x₁. * use weaken_left. apply hyperdoctrine_hyp. * use weaken_right. apply hyperdoctrine_hyp. - do 3 use forall_intro. use impl_intro. use weaken_right. use impl_intro. simplify_form. rewrite !partial_setoid_subst. simplify. pose (x₁ := π₂ (π₁ (π₁ (tm_var (((𝟙 ×h X) ×h X) ×h X))))). pose (x₂ := π₂ (π₁ (tm_var (((𝟙 ×h X) ×h X) ×h X)))). pose (x₃ := π₂ (tm_var (((𝟙 ×h X) ×h X) ×h X))). fold x₁ x₂ x₃. use conj_intro. + use partial_setoid_trans. * exact x₂. * do 2 use weaken_left. apply hyperdoctrine_hyp. * use weaken_right. use weaken_left. apply hyperdoctrine_hyp. + use weaken_left. use weaken_right. apply hyperdoctrine_hyp. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Monics. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Examples.MonoCodomain. Require Import UniMath.CategoryTheory.DisplayedCats.MonoCodomain.FiberMonoCod. 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.PERMonomorphisms. Require Import UniMath.CategoryTheory.Hyperdoctrines.PartialEqRels.Logic.SubobjectDispCat.