Proposition per_subobject_truth_laws (Γ : partial_setoid H) : per_subobject_laws (tm_var Γ ~ tm_var Γ). Proof. split. - use forall_intro. use impl_intro. use weaken_right. rewrite partial_setoid_subst. simplify. apply hyperdoctrine_hyp. - do 2 use forall_intro. use impl_intro. use weaken_right. use impl_intro. 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 partial_setoid_refl_r. { exact γ₁. } 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.FiberwiseTerminal. 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.