Proposition per_subobject_conj_subst {Γ₁ Γ₂ : partial_setoid H} (s : partial_setoid_morphism Γ₁ Γ₂) (ψ₁ ψ₂ : per_subobject Γ₂) : per_subobject_mor_law (id_partial_setoid_morphism Γ₁) (per_subobject_conj (per_subobject_subst s ψ₁) (per_subobject_subst s ψ₂)) (per_subobject_subst s (per_subobject_conj ψ₁ ψ₂)). Proof. do 2 use forall_intro. use impl_intro. use weaken_right. use impl_intro. cbn. rewrite partial_setoid_subst. rewrite conj_subst. rewrite !exists_subst. use hyp_sym. use hyp_ltrans. use (exists_elim (weaken_left (hyperdoctrine_hyp _) _)). rewrite !conj_subst. use hyp_ltrans. use weaken_right. use hyp_ltrans. rewrite exists_subst. use (exists_elim (weaken_left (hyperdoctrine_hyp _) _)). rewrite !conj_subst. use hyp_ltrans. use weaken_right. simplify_form. rewrite !partial_setoid_subst. simplify. pose (γ₁ := π₂ (π₁ (π₁ (π₁ (tm_var ((((𝟙 ×h Γ₁) ×h Γ₁) ×h Γ₂) ×h Γ₂)))))). pose (γ₁' := π₂ (π₁ (π₁ (tm_var ((((𝟙 ×h Γ₁) ×h Γ₁) ×h Γ₂) ×h Γ₂))))). pose (γ₂ := π₂ (π₁ (tm_var ((((𝟙 ×h Γ₁) ×h Γ₁) ×h Γ₂) ×h Γ₂)))). pose (γ₂' := π₂ (tm_var ((((𝟙 ×h Γ₁) ×h Γ₁) ×h Γ₂) ×h Γ₂))). use exists_intro. { exact γ₂. } simplify. fold γ₁ γ₁' γ₂ γ₂'. assert ((γ₁ ~ γ₁' ∧ s [⟨ γ₁, γ₂ ⟩] ∧ ψ₁ [γ₂]) ∧ s [⟨ γ₁, γ₂' ⟩] ∧ ψ₂ [γ₂'] ⊢ γ₂ ~ γ₂') as r. { use (partial_setoid_mor_unique_im s). - exact γ₁. - use weaken_left. use weaken_right. use weaken_left. apply hyperdoctrine_hyp. - use weaken_right. use weaken_left. apply hyperdoctrine_hyp. } repeat (use conj_intro). - use weaken_left. use (partial_setoid_mor_eq_defined s). + exact γ₁. + exact γ₂. + use weaken_left. apply hyperdoctrine_hyp. + use weaken_right. use weaken_left. exact (partial_setoid_mor_cod_defined s _ _ (hyperdoctrine_hyp _)). + use weaken_right. use weaken_left. apply hyperdoctrine_hyp. - use weaken_left. do 2 use weaken_right. apply hyperdoctrine_hyp. - use per_subobject_eq. + exact γ₂'. + use partial_setoid_sym. exact r. + do 2 use weaken_right. 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.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.