More cases for completeness (#88)

* Clean up proofs

* Fix a tactic

* Finish context and subst

* Add some cases for nat

* Improve completeness proofs

* Update proofs

* Update PER tactic

* Defer some nat cases

theories/Core/Completeness/ContextCases.v

@@ -0,0 +1,42 @@
+From Coq Require Import Morphisms_Relations RelationClasses.
+From Mcltt Require Import Base LibTactics.
+From Mcltt.Core Require Import Completeness.LogicalRelation System.
+Import Domain_Notations.
+
+Proposition valid_ctx_empty :
+  {{ ⊨ ⋅ }}.
+Proof.
+  do 2 econstructor.
+  apply Equivalence_Reflexive.
+Qed.
+
+Lemma rel_ctx_extend : forall {Γ Γ' A A' i},
+    {{ ⊨ Γ ≈ Γ' }} ->
+    {{ Γ ⊨ A ≈ A' : Type@i }} ->
+    {{ ⊨ Γ, A ≈ Γ', A' }}.
+Proof with intuition.
+  pose proof (@relation_equivalence_pointwise domain).
+  pose proof (@relation_equivalence_pointwise env).
+  intros * [env_relΓΓ'] [env_relΓ].
+  pose (env_relΓ0 := env_relΓ).
+  destruct_conjs.
+  handle_per_ctx_env_irrel.
+  eexists.
+  per_ctx_env_econstructor; eauto.
+  - instantiate (1 := fun p p' (equiv_p_p' : env_relΓ p p') m m' =>
+                        forall i R,
+                          rel_typ i A p A' p' R ->
+                          R m m').
+    intros.
+    (on_all_hyp: destruct_rel_by_assumption env_relΓ).
+    destruct_by_head rel_typ.
+    invert_rel_typ_body.
+    destruct_by_head rel_exp.
+    destruct_conjs.
+    econstructor; eauto.
+    apply -> per_univ_elem_morphism_iff; eauto.
+    split; intros; destruct_by_head rel_typ; handle_per_univ_elem_irrel...
+    eapply H12.
+    econstructor...
+  - apply Equivalence_Reflexive.
+Qed.