Proposition per_subobject_forall_elim : per_subobject_mor_law (id_partial_setoid_morphism A) (per_subobject_subst φ per_subobject_forall) ψ. Proof. do 2 use forall_intro. use impl_intro. use weaken_right. use impl_intro. cbn. simplify_form. rewrite partial_setoid_subst. simplify. use hyp_sym. use (exists_elim (weaken_left (hyperdoctrine_hyp _) _)). rewrite !conj_subst. use hyp_ltrans. use weaken_right. rewrite !partial_setoid_subst. simplify. pose (a₁ := π₂ (π₁ (π₁ (tm_var (((𝟙 ×h A) ×h A) ×h B))))). pose (a₂ := π₂ (π₁ (tm_var (((𝟙 ×h A) ×h A) ×h B)))). pose (b := π₂ (tm_var (((𝟙 ×h A) ×h A) ×h B))). fold a₁ a₂ b. use per_subobject_forall_form_all. - exact b. - do 2 use weaken_right. apply hyperdoctrine_hyp. - use partial_setoid_mor_eq_defined. + exact a₁. + exact b. + use weaken_left. apply hyperdoctrine_hyp. + use weaken_right. use weaken_left. use (partial_setoid_mor_cod_defined φ a₁ b). apply hyperdoctrine_hyp. + 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.Adjunctions.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.CategoryTheory.DisplayedCats.Fiber. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DependentProducts. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DependentSums. Require Import UniMath.CategoryTheory.DisplayedCats.Fiberwise.DualBeckChevalley. Require Import UniMath.CategoryTheory.Limits.Pullbacks. 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.Existential.