Add semantic rules for function beta and eta

theories/structural.v

@@ -24,6 +24,71 @@ Proof.
   by asimpl.
 Qed.
 
+Lemma Su_Wt n Γ a i :
+  Γ ⊢ a ∈ @PUniv n i ->
+  Γ ⊢ a ≲ a.
+Proof. hauto lq:on ctrs:LEq, Eq. Qed.
+
+Lemma Wt_Univ n Γ a A i
+  (h : Γ ⊢ a ∈ A) :
+  Γ ⊢ @PUniv n i ∈ PUniv (S i).
+Proof.
+  hauto lq:on ctrs:Wt use:wff_mutual.
+Qed.
+
+
+Lemma Pi_Inv n Γ (A : PTm n) B U :
+  Γ ⊢ PBind PPi A B ∈ U ->
+  exists i, Γ ⊢ A ∈ PUniv i /\
+  funcomp (ren_PTm shift) (scons A Γ) ⊢ B ∈ PUniv i /\
+  Γ ⊢ PUniv i ≲ U.
+Proof.
+  move E :(PBind PPi A B) => T h.
+  move : A B E.
+  elim : n Γ T U / h => //=.
+  - hauto lq:on ctrs:Wt,LEq,Eq use:Wt_Univ.
+  - hauto lq:on rew:off ctrs:LEq.
+Qed.
+
+(* Lemma regularity : *)
+(*   (forall n (Γ : fin n -> PTm n), ⊢ Γ -> forall i, exists j, Γ ⊢ Γ i ∈ PUniv j) /\ *)
+(*   (forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> exists i, Γ ⊢ A ∈ PUniv i)  /\ *)
+(*   (forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> Γ ⊢ a ∈ A /\ Γ ⊢ b ∈ A /\ exists i, Γ ⊢ A ∈ PUniv i) /\ *)
+(*   (forall n Γ (A B : PTm n), Γ ⊢ A ≲ B -> exists i, Γ ⊢ A ∈ PUniv i /\ Γ ⊢ A ∈ PUniv i). *)
+(* Proof. *)
+(*   apply wt_mutual => //=. *)
+(*   - admit. *)
+(*   - admit. *)
+
+Lemma T_App' n Γ (b a : PTm n) A B U :
+  U = subst_PTm (scons a VarPTm) B ->
+  Γ ⊢ b ∈ PBind PPi A B ->
+  Γ ⊢ a ∈ A ->
+  Γ ⊢ PApp b a ∈ U.
+Proof. move => ->. apply T_App. Qed.
+
+Lemma T_Pair' n Γ (a b : PTm n) A B i U :
+  U = subst_PTm (scons a VarPTm) B ->
+  Γ ⊢ a ∈ A ->
+  Γ ⊢ b ∈ U ->
+  Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
+  Γ ⊢ PPair a b ∈ PBind PSig A B.
+Proof.
+  move => ->. eauto using T_Pair.
+Qed.
+
+Lemma T_Proj2' n Γ (a : PTm n) A B U :
+  U = subst_PTm (scons (PProj PL a) VarPTm) B ->
+  Γ ⊢ a ∈ PBind PSig A B ->
+  Γ ⊢ PProj PR a ∈ U.
+Proof. move => ->. apply T_Proj2. Qed.
+
+Lemma E_Bind' n Γ i p (A0 A1 : PTm n) B0 B1 :
+  Γ ⊢ A0 ≡ A1 ∈ PUniv i ->
+  funcomp (ren_PTm shift) (scons A0 Γ) ⊢ B0 ≡ B1 ∈ PUniv i ->
+  Γ ⊢ PBind p A0 B0 ≡ PBind p A1 B1 ∈ PUniv i.
+Proof. hauto lq:on use:E_Bind, wff_mutual. Qed.
+
 Lemma renaming :
   (forall n (Γ : fin n -> PTm n), ⊢ Γ -> True) /\
   (forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> forall m Δ (ξ : fin n -> fin m), ⊢ Δ -> renaming_ok Δ Γ ξ ->
@@ -40,6 +105,14 @@ Proof.
   - hauto lq:on rew:off ctrs:Wt, Wff  use:renaming_up.
   - move => n Γ a A B i hP ihP ha iha m Δ ξ hΔ hξ.
     apply : T_Abs; eauto.
-    apply iha; last by apply renaming_up.
-    econstructor; eauto.
-    by apply renaming_up.
+    move : ihP(hΔ) (hξ); repeat move/[apply]. move/Pi_Inv.
+    hauto lq:on rew:off ctrs:Wff,Wt use:renaming_up.
+  - move => *. apply : T_App'; eauto. by asimpl.
+  - move => n Γ a A b B i hA ihA hB ihB hS ihS m Δ ξ hξ hΔ.
+    eapply T_Pair' with (U := ren_PTm ξ (subst_PTm (scons a VarPTm) B));eauto. by asimpl.
+  - move => n Γ a A B ha iha m Δ ξ hΔ hξ. apply : T_Proj2'; eauto. by asimpl.
+  - hauto lq:on ctrs:Wt,LEq.
+  - hauto lq:on ctrs:Eq.
+  - move => n Γ i p A0 A1 B0 B1 hΓ _ hA ihA hB ihB m Δ ξ hΔ hξ.
+    apply E_Bind'; eauto.
+    apply ihB; last by hauto l:on use:renaming_up.