Admit nat cases for now

theories/Core/Soundness/NatCases.v

@@ -3,7 +3,7 @@ From Coq Require Import Morphisms Morphisms_Prop Morphisms_Relations Relation_De
 From Mcltt Require Import Base LibTactics.
 From Mcltt.Core.Completeness Require Import FundamentalTheorem.
 From Mcltt.Core.Semantic Require Import Realizability.
-From Mcltt.Core.Soundness Require Import LogicalRelation Realizability UniverseCases.
+From Mcltt.Core.Soundness Require Import ContextCases LogicalRelation Realizability SubstitutionCases TermStructureCases UniverseCases.
 From Mcltt.Core.Syntactic Require Import Corollaries.
 Import Domain_Notations.
 
@@ -67,7 +67,7 @@ Proof.
   destruct_conjs.
   eapply glu_rel_exp_of_nat; mauto.
   intros.
-  destruct_glu_rel_exp_sub.
+  destruct_glu_rel_exp_with_sub.
   simplify_evals.
   match_by_head1 glu_univ_elem invert_glu_univ_elem.
   apply_predicate_equivalence.
@@ -80,13 +80,109 @@ Qed.
 #[export]
 Hint Resolve glu_rel_exp_succ : mcltt.
 
-Lemma glu_rel_exp_natrec : forall {Γ A i MZ MS M},
-    {{ Γ , ℕ ⊩ A : Type@i }} ->
+Lemma glu_rel_exp_natrec : forall {Γ i A MZ MS M},
+    {{ Γ, ℕ ⊩ A : Type@i }} ->
     {{ Γ ⊩ MZ : A[Id,,zero] }} ->
-    {{ Γ , ℕ , A ⊩ MS : A[Wk∘Wk,,succ(#1)] }} ->
+    {{ Γ, ℕ, A ⊩ MS : A[Wk∘Wk,,succ #1] }} ->
     {{ Γ ⊩ M : ℕ }} ->
     {{ Γ ⊩ rec M return A | zero -> MZ | succ -> MS end : A[Id,,M] }}.
+Proof.
+  intros * HA HMZ HMS HM.
+  assert {{ ⊩ Γ }} as [SbΓ] by mauto 2.
+  assert {{ Γ ⊩ ℕ : Type@i }} by mauto 3.
+  invert_glu_rel_exp HM.
+  (* assert {{ Γ, ℕ ⊢ A : Type@i }} by mauto 3. *)
+  (* assert {{ Γ ⊢ MZ : A[Id,,zero] }} by mauto 3. *)
+  (* assert {{ Γ, ℕ, A ⊢ MS : A[Wk∘Wk,,succ #1] }} by mauto 3. *)
+  (* assert {{ Γ ⊢ M : ℕ }} by mauto 3. *)
+  (* pose proof HMZ as [] *)
+  (* assert {{ ⊩ Γ }} by (econstructor; mauto 3). *)
+  (* assert {{ Γ ⊩ zero : ℕ }} by mauto. *)
+  (* assert {{ Γ ⊩ A[Id,,zero] : Type@i }}  *)
+  (* invert_glu_rel_exp HMZ. *)
+  (* rename x into *)
+  (* assert {{ ⊩ Γ }} by (econstructor; mauto 2). *)
+  (* assert {{ Γ ⊩ ℕ : Type@jMZ }} as Hℕ by mauto 3. *)
+  (* assert {{ ⊩ Γ, ℕ }} by mauto 3. *)
+  (* pose (SbΓℕ := cons_glu_sub_pred jMZ Γ {{{ ℕ }}} SbΓ). *)
+  (* assert {{ EG Γ, ℕ ∈ glu_ctx_env ↘ SbΓℕ }} by (invert_glu_rel_exp Hℕ; econstructor; mauto 3; try reflexivity). *)
+  (* clear Hℕ. *)
+  (* invert_glu_rel_exp HA. *)
+  (* rename x into k. *)
+  (* pose (SbΓℕA := cons_glu_sub_pred i {{{ Γ, ℕ }}} A SbΓℕ). *)
+  (* assert {{ EG Γ, ℕ, A ∈ glu_ctx_env ↘ SbΓℕA }} by (econstructor; mauto 3; try reflexivity). *)
+  (* invert_glu_rel_exp HMS. *)
+  (* rename x into jMS. *)
+  (* invert_glu_rel_exp HM. *)
+  (* rename x into jM. *)
+  (* assert {{ Γ ⊨ rec M return A | zero -> MZ | succ -> MS end : A[Id,,M] }} as [env_relΓ] by mauto using completeness_fundamental_exp. *)
+  (* destruct_conjs. *)
+  (* eexists; split; mauto. *)
+  (* eexists. *)
+  (* intros. *)
+  (* assert {{ Dom ρ ≈ ρ ∈ env_relΓ }} by (eapply glu_ctx_env_per_env; revgoals; eassumption). *)
+  (* destruct_glu_rel_exp_sub. *)
+  (* (on_all_hyp: destruct_rel_by_assumption env_relΓ). *)
+  (* destruct_by_head rel_typ. *)
+  (* destruct_by_head rel_exp. *)
+  (* simplify_evals. *)
+  (* match_by_head glu_univ_elem ltac:(fun H => directed invert_glu_univ_elem H). *)
+  (* apply_predicate_equivalence. *)
+  (* dir_inversion_clear_by_head nat_glu_exp_pred. *)
+  (* rename p'0 into p. *)
+  (* rename m0 into mz. *)
+  (* rename a1 into am. *)
+  (* rename a0 into az. *)
+  (* rename m1 into r. *)
+  (* eapply mk_glu_rel_exp_sub''; mauto 3. *)
+  (* intros. *)
+  (* This part requires a separate lemma for a clean induction *)
 Admitted.
 
+(* Goal forall {i Γ SbΓ A MZ MS}, *)
+(*     {{ EG Γ ∈ glu_ctx_env ↘ SbΓ }} -> *)
+(*     {{ Γ, ℕ ⊩ A : Type@i }} -> *)
+(*     {{ Γ ⊩ MZ : A[Id,,zero] }} -> *)
+(*     {{ Γ, ℕ, A ⊩ MS : A[Wk∘Wk,,succ #1] }} -> *)
+(*     forall {Δ M m}, *)
+(*       glu_nat Δ M m -> *)
+(*       forall {σ p am P El r}, *)
+(*         {{ Δ ⊢s σ ® p ∈ SbΓ }} -> *)
+(*         {{ ⟦ A ⟧ p ↦ m ↘ am }} -> *)
+(*         {{ DG am ∈ glu_univ_elem i ↘ P ↘ El }} -> *)
+(*         {{ rec m ⟦return A | zero -> MZ | succ -> MS end⟧ p ↘ r }} -> *)
+(*         {{ Δ ⊢ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[σ,,M] ® r ∈ El }}. *)
+(* Proof. *)
+  (* intros * ? HA HMZ HMS. *)
+  (* (* common things *) *)
+  (* assert {{ Γ, ℕ ⊢ A : Type@i }} by mauto 3. *)
+  (* assert {{ Γ ⊢ MZ : A[Id,,zero] }} by mauto 3. *)
+  (* assert {{ Γ, ℕ, A ⊢ MS : A[Wk∘Wk,,succ #1] }} by mauto 3. *)
+  (* invert_glu_rel_exp HMZ. *)
+  (* assert {{ Γ ⊩ ℕ : Type@jMZ }} as Hℕ by mauto 3. *)
+  (* assert {{ ⊩ Γ, ℕ }} by mauto 3. *)
+
+  (* induction 1; intros; rename m into M; rename Γ0 into Δ. *)
+  (* - (* glu_nat_zero *) *)
+  (*   match_by_head eval_natrec ltac:(fun H => directed inversion H; subst; clear H). *)
+  (*   destruct_by_head glu_rel_exp_sub. *)
+  (*   simplify_evals. *)
+  (*   handle_functional_glu_univ_elem. *)
+  (*   assert {{ Δ ⊢ A[σ,,M] ≈ A[Id,,zero][σ] : Type@i }} as -> by admit. *)
+  (*   assert *)
+  (*     {{ Δ ⊢ rec M return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end ≈ MZ[σ] : A[Id,,zero][σ] }} as -> by admit. *)
+  (*   eassumption. *)
+  (* - (* glu_nat_succ *) *)
+  (*   rename m' into M'. *)
+  (*   rename a into m'. *)
+  (*   match_by_head eval_natrec ltac:(fun H => directed inversion H; subst; clear H). *)
+  (*   destruct_by_head glu_rel_exp_sub. *)
+  (*   simplify_evals. *)
+  (*   handle_functional_glu_univ_elem. *)
+  (*   (* assert {{ Δ ⊢ rec M' return A[q σ] | zero -> MZ[σ] | succ -> MS[q (q σ)] end : A[σ,, M'] ® r0 ∈ ? }} by admit. *) *)
+  (*   admit. *)
+  (* - (* glu_nat_neut *) *)
+  (*   admit. *)
+
 #[export]
 Hint Resolve glu_rel_exp_natrec : mcltt.