replace UJudg with Arrange

[coq-hetmet.git] / src / HaskStrongToProof.v

diff --git a/src/HaskStrongToProof.v b/src/HaskStrongToProof.v
index 79e16cd..b0c5b11 100644 (file)
--- a/src/HaskStrongToProof.v
+++ b/src/HaskStrongToProof.v
@@ -16,51 +16,24 @@ Require Import HaskProof.
 
 Section HaskStrongToProof.
 
 
 Section HaskStrongToProof.
 

-(* Whereas RLeft/RRight perform left and right context expansion on a single uniform rule, these functions perform
- * expansion on an entire uniform proof *)
-Definition ext_left  {Γ}{Δ}(ctx:Tree ??(LeveledHaskType Γ ★))
-  := @nd_map' _ _ _ _ (ext_tree_left ctx)  (fun h c r => nd_rule (@RLeft Γ Δ h c ctx r)).
-Definition ext_right {Γ}{Δ}(ctx:Tree ??(LeveledHaskType Γ ★))
-  := @nd_map' _ _ _ _ (ext_tree_right ctx) (fun h c r => nd_rule (@RRight Γ Δ h c ctx r)).
-
-Definition pivotContext {Γ}{Δ} a b c τ :
-  @ND _ (@URule Γ Δ)
-    [ Γ >> Δ > (a,,b),,c |- τ]
-    [ Γ >> Δ > (a,,c),,b |- τ].
-  set (ext_left a _ _ (nd_rule (@RExch Γ Δ τ c b))) as q.
-  simpl in *.
-  eapply nd_comp ; [ eapply nd_rule; apply RCossa | idtac ]. 
-  eapply nd_comp ; [ idtac | eapply nd_rule; apply RAssoc ].
-  apply q.
+Definition pivotContext {T} a b c : @Arrange T ((a,,b),,c) ((a,,c),,b) :=
+  RComp (RComp (RCossa _ _ _) (RLeft a (RExch c b))) (RAssoc _ _ _).
+
+Definition copyAndPivotContext {T} a b c : @Arrange T ((a,,b),,(c,,b)) ((a,,c),,b).
+  eapply RComp; [ idtac | apply (RLeft (a,,c) (RCont b)) ].
+  eapply RComp; [ idtac | apply RCossa ]. 
+  eapply RComp; [ idtac | apply (RRight b (pivotContext a b c)) ].
+  apply RAssoc.

   Defined.
   Defined.

-
-Definition copyAndPivotContext {Γ}{Δ} a b c τ :
-  @ND _ (@URule Γ Δ)
-    [ Γ >> Δ > (a,,b),,(c,,b) |- τ]
-    [ Γ >> Δ > (a,,c),,b |- τ].
-    set (ext_left (a,,c) _ _ (nd_rule (@RCont Γ Δ τ b))) as q.
-    simpl in *.
-    eapply nd_comp; [ idtac | apply q ].
-    clear q.
-    eapply nd_comp ; [ idtac | eapply nd_rule; apply RCossa ].
-    set (ext_right b _ _ (@pivotContext _ Δ a b c τ)) as q.
-    simpl in *.
-    eapply nd_comp ; [ idtac | apply q ]. 
-    clear q.
-    apply nd_rule.
-    apply RAssoc.
-    Defined.
-
-

   
 Context {VV:Type}{eqd_vv:EqDecidable VV}.
 
   
 Context {VV:Type}{eqd_vv:EqDecidable VV}.
 

-  (* maintenance of Xi *)
-  Fixpoint dropVar (lv:list VV)(v:VV) : ??VV :=
-    match lv with
-      | nil     => Some v
-      | v'::lv' => if eqd_dec v v' then None else dropVar lv' v
-    end.
+(* maintenance of Xi *)
+Fixpoint dropVar (lv:list VV)(v:VV) : ??VV :=
+  match lv with
+    | nil     => Some v
+    | v'::lv' => if eqd_dec v v' then None else dropVar lv' v
+  end.

 
 Fixpoint mapOptionTree' {a b:Type}(f:a->??b)(t:@Tree ??a) : @Tree ??b :=
   match t with 
 
 Fixpoint mapOptionTree' {a b:Type}(f:a->??b)(t:@Tree ??a) : @Tree ??b :=
   match t with 

@@ -422,23 +395,22 @@ Lemma stripping_nothing_is_inert
     reflexivity.
     Qed.
 
     reflexivity.
     Qed.
 

-Definition arrangeContext 
+Definition arrangeContext

      (Γ:TypeEnv)(Δ:CoercionEnv Γ)
      v      (* variable to be pivoted, if found *)
      ctx    (* initial context *)
      (Γ:TypeEnv)(Δ:CoercionEnv Γ)
      v      (* variable to be pivoted, if found *)
      ctx    (* initial context *)

-     τ      (* type of succedent *)

      (ξ:VV -> LeveledHaskType Γ ★)
      :
  
     (* a proof concluding in a context where that variable does not appear *)
      (ξ:VV -> LeveledHaskType Γ ★)
      :
  
     (* a proof concluding in a context where that variable does not appear *)

-    sum (ND (@URule Γ Δ)
-          [Γ >> Δ > mapOptionTree ξ                        ctx                        |- τ]
-          [Γ >> Δ > mapOptionTree ξ (stripOutVars (v::nil) ctx),,[]                   |- τ])
+     sum (Arrange
+          (mapOptionTree ξ                        ctx                        )
+          (mapOptionTree ξ (stripOutVars (v::nil) ctx),,[]                   ))

    
     (* or a proof concluding in a context where that variable appears exactly once in the left branch *)
    
     (* or a proof concluding in a context where that variable appears exactly once in the left branch *)

-        (ND (@URule Γ Δ)
-          [Γ >> Δ > mapOptionTree ξ                         ctx                       |- τ]
-          [Γ >> Δ > mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v])                |- τ]).
+        (Arrange
+          (mapOptionTree ξ                         ctx                       )
+          (mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v])                )).

 
   induction ctx.
   
 
   induction ctx.
   

@@ -455,18 +427,15 @@ Definition arrangeContext
             (* where the leaf is v *)
             apply inr.
               subst.
             (* where the leaf is v *)
             apply inr.
               subst.

-              apply nd_rule.

               apply RuCanL.
 
             (* where the leaf is NOT v *)
             apply inl.
               apply RuCanL.
 
             (* where the leaf is NOT v *)
             apply inl.

-              apply nd_rule.

               apply RuCanR.
   
         (* empty leaf *)
         destruct case_None.
           apply inl; simpl in *.
               apply RuCanR.
   
         (* empty leaf *)
         destruct case_None.
           apply inl; simpl in *.

-          apply nd_rule.

           apply RuCanR.
   
       (* branch *)
           apply RuCanR.
   
       (* branch *)

@@ -486,87 +455,77 @@ Definition arrangeContext
 
     destruct case_Neither.
       apply inl.
 
     destruct case_Neither.
       apply inl.

-      eapply nd_comp; [idtac | eapply nd_rule; apply RuCanR ].
-        exact (nd_comp
+      eapply RComp; [idtac | apply RuCanR ].
+        exact (RComp

           (* order will not matter because these are central as morphisms *)
           (* order will not matter because these are central as morphisms *)

-          (ext_right _ _ _ (nd_comp lpf (nd_rule (@RCanR _ _ _ _))))
-          (ext_left  _ _ _ (nd_comp rpf (nd_rule (@RCanR _ _ _ _))))).
+          (RRight _ (RComp lpf (RCanR _)))
+          (RLeft  _ (RComp rpf (RCanR _)))).

 
 
     destruct case_Right.
       apply inr.
       fold (stripOutVars (v::nil)).
 
 
     destruct case_Right.
       apply inr.
       fold (stripOutVars (v::nil)).

-      set (ext_right (mapOptionTree ξ ctx2) _ _ (nd_comp lpf (nd_rule (@RCanR _ _ _ _)))) as q.
+      set (RRight (mapOptionTree ξ ctx2)  (RComp lpf ((RCanR _)))) as q.

       simpl in *.
       simpl in *.

-      eapply nd_comp.
+      eapply RComp.

       apply q.
       clear q.
       clear lpf.
       unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
       apply q.
       clear q.
       clear lpf.
       unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.

-      eapply nd_comp; [ idtac | eapply nd_rule; apply RAssoc ].
-      set (ext_left (mapOptionTree ξ (stripOutVars (v :: nil) ctx1)) [Γ >> Δ>mapOptionTree ξ ctx2 |- τ]
-                                                            [Γ >> Δ> (mapOptionTree ξ (stripOutVars (v :: nil) ctx2),,[ξ v])  |- τ]) as qq.
-      apply qq.
-      clear qq.
+      eapply RComp; [ idtac | apply RAssoc ].
+      apply RLeft.

       apply rpf.
 
     destruct case_Left.
       apply inr.
       unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
       fold (stripOutVars (v::nil)).
       apply rpf.
 
     destruct case_Left.
       apply inr.
       unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
       fold (stripOutVars (v::nil)).

-      eapply nd_comp; [ idtac | eapply pivotContext ].
-      set (nd_comp rpf (nd_rule (@RCanR _ _ _ _ ) ) ) as rpf'.
-      set (ext_left ((mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v])) _ _ rpf') as qq.
+      eapply RComp; [ idtac | eapply pivotContext ].
+      set (RComp rpf (RCanR _ )) as rpf'.
+      set (RLeft ((mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v])) rpf') as qq.

       simpl in *.
       simpl in *.

-      eapply nd_comp; [ idtac | apply qq ].
+      eapply RComp; [ idtac | apply qq ].

       clear qq rpf' rpf.
       clear qq rpf' rpf.

-      set (ext_right (mapOptionTree ξ ctx2) [Γ >>Δ> mapOptionTree ξ ctx1 |- τ] [Γ >>Δ> (mapOptionTree ξ (stripOutVars (v :: nil) ctx1),, [ξ v]) |- τ]) as q.
-      apply q.
-      clear q.
+      apply (RRight (mapOptionTree ξ ctx2)).

       apply lpf.
 
     destruct case_Both.
       apply inr.
       unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
       fold (stripOutVars (v::nil)).
       apply lpf.
 
     destruct case_Both.
       apply inr.
       unfold mapOptionTree in *; simpl; fold (mapOptionTree ξ) in *.
       fold (stripOutVars (v::nil)).

-      eapply nd_comp; [ idtac | eapply copyAndPivotContext ].
-        exact (nd_comp
-          (* order will not matter because these are central as morphisms *)
-          (ext_right _ _ _ lpf)
-          (ext_left  _ _ _ rpf)).
+      eapply RComp; [ idtac | eapply copyAndPivotContext ].
+        (* order will not matter because these are central as morphisms *)
+        exact (RComp (RRight _ lpf) (RLeft _ rpf)).

 
     Defined.
 
 (* same as before, but use RWeak if necessary *)
 
     Defined.
 
 (* same as before, but use RWeak if necessary *)

-Definition arrangeContextAndWeaken v ctx Γ Δ τ ξ : 
-       ND (@URule Γ Δ)
-          [Γ >> Δ>mapOptionTree ξ                        ctx                |- τ]
-          [Γ >> Δ>mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v])        |- τ].
-  set (arrangeContext Γ Δ v ctx τ ξ) as q.
+Definition arrangeContextAndWeaken  
+     (Γ:TypeEnv)(Δ:CoercionEnv Γ)
+     v      (* variable to be pivoted, if found *)
+     ctx    (* initial context *)
+     (ξ:VV -> LeveledHaskType Γ ★) :
+       Arrange
+          (mapOptionTree ξ                        ctx                )
+          (mapOptionTree ξ ((stripOutVars (v::nil) ctx),,[v])        ).
+  set (arrangeContext Γ Δ v ctx ξ) as q.

   destruct q; auto.
   destruct q; auto.

-  eapply nd_comp; [ apply n | idtac ].
-  clear n.
-  refine (ext_left _ _ _ (nd_rule (RWeak _ _))).
+  eapply RComp; [ apply a | idtac ].
+  refine (RLeft _ (RWeak _)).

   Defined.
 
   Defined.
 

-Lemma updating_stripped_tree_is_inert {Γ} (ξ:VV -> LeveledHaskType Γ ★) v tree t lev :
-      mapOptionTree (update_ξ ξ lev ((v,t)::nil)) (stripOutVars (v :: nil) tree)
-    = mapOptionTree ξ (stripOutVars (v :: nil) tree).
-  set (@updating_stripped_tree_is_inert' Γ lev ξ ((v,t)::nil)) as p.
-  rewrite p.
-  simpl.
-  reflexivity.
-  Qed.
-

 Lemma cheat : forall {T}(a b:list T), distinct (app a b) -> distinct (app b a).
   admit.
   Qed.
 
 Lemma cheat : forall {T}(a b:list T), distinct (app a b) -> distinct (app b a).
   admit.
   Qed.
 

-Definition arrangeContextAndWeaken'' Γ Δ ξ v : forall ctx z, 
+Definition arrangeContextAndWeaken''
+     (Γ:TypeEnv)(Δ:CoercionEnv Γ)
+     v      (* variable to be pivoted, if found *)
+     (ξ:VV -> LeveledHaskType Γ ★) : forall ctx,

   distinct (leaves v) ->
   distinct (leaves v) ->

-  ND (@URule Γ Δ)
-    [Γ >> Δ>(mapOptionTree ξ ctx)                                       |-  z]
-    [Γ >> Δ>(mapOptionTree ξ (stripOutVars (leaves v) ctx)),,(mapOptionTree ξ v) |-  z].
+  Arrange
+    ((mapOptionTree ξ ctx)                                       )
+    ((mapOptionTree ξ (stripOutVars (leaves v) ctx)),,(mapOptionTree ξ v)).

 
   induction v; intros.
     destruct a.
 
   induction v; intros.
     destruct a.

@@ -575,11 +534,11 @@ Definition arrangeContextAndWeaken'' Γ Δ ξ v : forall ctx z,
     fold (mapOptionTree ξ) in *.
     intros.
     apply arrangeContextAndWeaken.
     fold (mapOptionTree ξ) in *.
     intros.
     apply arrangeContextAndWeaken.

+    apply Δ.

 
   unfold mapOptionTree; simpl in *.
     intros.
     rewrite (@stripping_nothing_is_inert Γ); auto.
 
   unfold mapOptionTree; simpl in *.
     intros.
     rewrite (@stripping_nothing_is_inert Γ); auto.

-    apply nd_rule.

     apply RuCanR.
     intros.
     unfold mapOptionTree in *.
     apply RuCanR.
     intros.
     unfold mapOptionTree in *.

@@ -587,7 +546,7 @@ Definition arrangeContextAndWeaken'' Γ Δ ξ v : forall ctx z,
     fold (mapOptionTree ξ) in *.
     set (mapOptionTree ξ) as X in *.
 
     fold (mapOptionTree ξ) in *.
     set (mapOptionTree ξ) as X in *.
 

-    set (IHv2 ((stripOutVars (leaves v1) ctx),, v1) z) as IHv2'.
+    set (IHv2 ((stripOutVars (leaves v1) ctx),, v1)) as IHv2'.

     unfold stripOutVars in IHv2'.
     simpl in IHv2'.
     fold (stripOutVars (leaves v2)) in IHv2'.
     unfold stripOutVars in IHv2'.
     simpl in IHv2'.
     fold (stripOutVars (leaves v2)) in IHv2'.

@@ -599,19 +558,27 @@ Definition arrangeContextAndWeaken'' Γ Δ ξ v : forall ctx z,
     fold X in IHv2'.
     set (distinct_app _ _ _ H) as H'.
     destruct H' as [H1 H2].
     fold X in IHv2'.
     set (distinct_app _ _ _ H) as H'.
     destruct H' as [H1 H2].

-    set (nd_comp (IHv1 _ _ H1) (IHv2' H2)) as qq.
-    eapply nd_comp.
+    set (RComp (IHv1 _ H1) (IHv2' H2)) as qq.
+    eapply RComp.

       apply qq.
       clear qq IHv2' IHv2 IHv1.
       rewrite strip_twice_lemma.
 
       rewrite (strip_distinct' v1 (leaves v2)).
       apply qq.
       clear qq IHv2' IHv2 IHv1.
       rewrite strip_twice_lemma.
 
       rewrite (strip_distinct' v1 (leaves v2)).

-        apply nd_rule.

         apply RCossa.
         apply cheat.
         auto.
     Defined.
 
         apply RCossa.
         apply cheat.
         auto.
     Defined.
 

+Lemma updating_stripped_tree_is_inert {Γ} (ξ:VV -> LeveledHaskType Γ ★) v tree t lev :
+      mapOptionTree (update_ξ ξ lev ((v,t)::nil)) (stripOutVars (v :: nil) tree)
+    = mapOptionTree ξ (stripOutVars (v :: nil) tree).
+  set (@updating_stripped_tree_is_inert' Γ lev ξ ((v,t)::nil)) as p.
+  rewrite p.
+  simpl.
+  reflexivity.
+  Qed.
+

 (* IDEA: use multi-conclusion proofs instead *)
 Inductive LetRecSubproofs Γ Δ ξ lev : forall tree, ELetRecBindings Γ Δ ξ lev tree -> Type := 
   | lrsp_nil  : LetRecSubproofs Γ Δ ξ lev [] (ELR_nil _ _ _ _)
 (* IDEA: use multi-conclusion proofs instead *)
 Inductive LetRecSubproofs Γ Δ ξ lev : forall tree, ELetRecBindings Γ Δ ξ lev tree -> Type := 
   | lrsp_nil  : LetRecSubproofs Γ Δ ξ lev [] (ELR_nil _ _ _ _)

@@ -665,23 +632,25 @@ Lemma letRecSubproofsToND' Γ Δ ξ lev τ tree  :
   eapply nd_comp; [ idtac | eapply nd_rule; apply z ].
   clear z.
 
   eapply nd_comp; [ idtac | eapply nd_rule; apply z ].
   clear z.
 

-  set (@arrangeContextAndWeaken''  Γ Δ ξ' pctx (expr2antecedent body,,eLetRecContext branches)) as q'.
+  set (@arrangeContextAndWeaken''  Γ Δ  pctx ξ' (expr2antecedent body,,eLetRecContext branches)) as q'.

   unfold passback in *; clear passback.
   unfold pctx in *; clear pctx.
   unfold passback in *; clear passback.
   unfold pctx in *; clear pctx.

-  eapply UND_to_ND in q'.
+  rewrite <- mapleaves in disti.
+  set (q' disti) as q''.

 
   unfold ξ' in *.
   set (@updating_stripped_tree_is_inert' Γ lev ξ (leaves tree)) as zz.
   rewrite <- mapleaves in zz.
 
   unfold ξ' in *.
   set (@updating_stripped_tree_is_inert' Γ lev ξ (leaves tree)) as zz.
   rewrite <- mapleaves in zz.

-  rewrite zz in q'.
+  rewrite zz in q''.

   clear zz.
   clear ξ'.
   Opaque stripOutVars.
   simpl.
   clear zz.
   clear ξ'.
   Opaque stripOutVars.
   simpl.

-  rewrite <- mapOptionTree_compose in q'.
+  rewrite <- mapOptionTree_compose in q''.

   rewrite <- ξlemma.
   rewrite <- ξlemma.

-  eapply nd_comp; [ idtac | apply q' ].
+  eapply nd_comp; [ idtac | eapply nd_rule; apply (RArrange _ _ _ _ _ q'') ].

   clear q'.
   clear q'.

+  clear q''.

   simpl.
 
   set (letRecSubproofsToND _ _ _ _ _ branches lrsp) as q.
   simpl.
 
   set (letRecSubproofsToND _ _ _ _ _ branches lrsp) as q.

@@ -690,10 +659,6 @@ Lemma letRecSubproofsToND' Γ Δ ξ lev τ tree  :
     apply nd_prod; auto.
     rewrite ξlemma.
     apply q.
     apply nd_prod; auto.
     rewrite ξlemma.
     apply q.

-    clear q'.
-
-  rewrite <- mapleaves in disti.
-    apply disti.

     Defined.
 
 Lemma scbwv_coherent {tc}{Γ}{atypes:IList _ (HaskType Γ) _}{sac} :
     Defined.
 
 Lemma scbwv_coherent {tc}{Γ}{atypes:IList _ (HaskType Γ) _}{sac} :

@@ -843,15 +808,14 @@ Definition expr2proof  :
       unfold mapOptionTree; simpl; fold (mapOptionTree ξ).
       eapply nd_comp; [ idtac | eapply nd_rule; apply RLam ].
       set (update_ξ ξ lev ((v,t1)::nil)) as ξ'.
       unfold mapOptionTree; simpl; fold (mapOptionTree ξ).
       eapply nd_comp; [ idtac | eapply nd_rule; apply RLam ].
       set (update_ξ ξ lev ((v,t1)::nil)) as ξ'.

-      set (arrangeContextAndWeaken v (expr2antecedent e) Γ Δ [t2 @@ lev] ξ') as pfx.
-        apply UND_to_ND in pfx.
-        unfold mapOptionTree in pfx; simpl in pfx; fold (mapOptionTree ξ) in pfx.
+      set (arrangeContextAndWeaken Γ Δ v (expr2antecedent e) ξ') as pfx.
+        eapply RArrange in pfx.
+        unfold mapOptionTree in pfx; simpl in pfx.

         unfold ξ' in pfx.
         unfold ξ' in pfx.

-        fold (mapOptionTree (update_ξ ξ lev ((v,t1)::nil))) in pfx.

         rewrite updating_stripped_tree_is_inert in pfx.
         unfold update_ξ in pfx.
         destruct (eqd_dec v v).
         rewrite updating_stripped_tree_is_inert in pfx.
         unfold update_ξ in pfx.
         destruct (eqd_dec v v).

-        eapply nd_comp; [ idtac | apply pfx ].
+        eapply nd_comp; [ idtac | apply (nd_rule pfx) ].

         clear pfx.
         apply e'.
         assert False.
         clear pfx.
         apply e'.
         assert False.

@@ -869,7 +833,7 @@ Definition expr2proof  :
       fold (@mapOptionTree VV).
       fold (mapOptionTree ξ).
       set (update_ξ ξ v ((lev,tv)::nil)) as ξ'.
       fold (@mapOptionTree VV).
       fold (mapOptionTree ξ).
       set (update_ξ ξ v ((lev,tv)::nil)) as ξ'.

-      set (arrangeContextAndWeaken lev (expr2antecedent ebody) Γ Δ [t@@v] ξ') as n.
+      set (arrangeContextAndWeaken Γ Δ lev (expr2antecedent ebody) ξ') as n.

       unfold mapOptionTree in n; simpl in n; fold (mapOptionTree ξ') in n.
       unfold ξ' in n.
       rewrite updating_stripped_tree_is_inert in n.
       unfold mapOptionTree in n; simpl in n; fold (mapOptionTree ξ') in n.
       unfold ξ' in n.
       rewrite updating_stripped_tree_is_inert in n.

@@ -877,8 +841,8 @@ Definition expr2proof  :
       destruct (eqd_dec lev lev).
       unfold ξ'.
       unfold update_ξ.
       destruct (eqd_dec lev lev).
       unfold ξ'.
       unfold update_ξ.

-      apply UND_to_ND in n.
-      apply n.
+      eapply RArrange in n.
+      apply (nd_rule n).

       assert False. apply n0; auto. inversion H.
 
     destruct case_EEsc.
       assert False. apply n0; auto. inversion H.
 
     destruct case_EEsc.

@@ -925,11 +889,9 @@ Definition expr2proof  :
       clear o x alts alts' e.
       eapply nd_comp; [ apply e' | idtac ].
       clear e'.
       clear o x alts alts' e.
       eapply nd_comp; [ apply e' | idtac ].
       clear e'.

-      set (fun q r x1 x2 y1 y2 => @UND_to_ND q r [q >> r > x1 |- y1] [q >> r > x2 |- y2]).
-      simpl in n.
-      apply n.
-      clear n.
-
+      apply nd_rule.
+      apply RArrange.
+      simpl.

       rewrite mapleaves'.
       simpl.
       rewrite <- mapOptionTree_compose.
       rewrite mapleaves'.
       simpl.
       rewrite <- mapOptionTree_compose.

@@ -951,6 +913,7 @@ Definition expr2proof  :
       replace (stripOutVars (vec2list (scbwv_exprvars scbx))) with
         (stripOutVars (leaves (unleaves (vec2list (scbwv_exprvars scbx))))).
       apply q.
       replace (stripOutVars (vec2list (scbwv_exprvars scbx))) with
         (stripOutVars (leaves (unleaves (vec2list (scbwv_exprvars scbx))))).
       apply q.

+      apply (sac_Δ sac Γ atypes (weakCK'' Δ)).

       rewrite leaves_unleaves.
       apply (scbwv_exprvars_distinct scbx).
       rewrite leaves_unleaves.
       rewrite leaves_unleaves.
       apply (scbwv_exprvars_distinct scbx).
       rewrite leaves_unleaves.