X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FHaskProofToStrong.v;h=f907514838bd9106a2e0af3b59ac8f4c8ea6ff4a;hp=89a0919c5d53cf3066c47c0b368935ba13aa9bc8;hb=b3214686b18b2d6f6905394494da8d1c17587bdb;hpb=8c26722a1ee110077968a8a166eb7130266b2035

diff --git a/src/HaskProofToStrong.v b/src/HaskProofToStrong.v
index 89a0919..f907514 100644
--- a/src/HaskProofToStrong.v
+++ b/src/HaskProofToStrong.v
@@ -16,144 +16,861 @@ Require Import HaskProof.
 
 Section HaskProofToStrong.
 
-  Context
-    {VV:Type}
-    {eqdec_vv:EqDecidable VV}
-    {fresh: forall (l:list VV), { lf:VV & distinct (lf::l) }}.
+  Context {VV:Type} {eqdec_vv:EqDecidable VV} {freshM:FreshMonad VV}.
 
-  Definition Exprs (Γ:TypeEnv)(Δ:CoercionEnv Γ)(ξ:VV -> LeveledHaskType Γ ★)(τ:Tree ??(LeveledHaskType Γ ★)) :=
-    ITree _ (fun τ => Expr Γ Δ ξ τ) τ.
+  Definition fresh := FMT_fresh freshM.
+  Definition FreshM := FMT freshM.
+  Definition FreshMon := FMT_Monad freshM.
+  Existing Instance FreshMon.
+
+  Definition ExprVarResolver Γ := VV -> LeveledHaskType Γ ★.
+
+  Definition ujudg2exprType {Γ}{Δ}(ξ:ExprVarResolver Γ)(j:UJudg Γ Δ) : Type :=
+    match j with
+      mkUJudg Σ τ => forall vars, Σ = mapOptionTree ξ vars ->
+        FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ)
+      end.
 
   Definition judg2exprType (j:Judg) : Type :=
     match j with
-      (Γ > Δ > Σ |- τ) => { ξ:VV -> LeveledHaskType Γ ★ & Exprs Γ Δ ξ τ }
+      (Γ > Δ > Σ |- τ) => forall (ξ:ExprVarResolver Γ) vars, Σ = mapOptionTree ξ vars ->
+        FreshM (ITree _ (fun t => Expr Γ Δ ξ t) τ)
       end.
 
-  (* reminder: need to pass around uniq-supplies *)
-  Definition rule2expr
-    : forall h j
-      (r:Rule h [j]),
-      ITree _ judg2exprType h ->
-      judg2exprType j.
-  
-      intros.
-      destruct j.
-      refine (match r as R in Rule H C return C=[Γ > Δ > t |- t0] -> _ with
-      | RURule a b c d e    => let case_RURule := tt in _
-      | RNote   x n z        => let case_RNote := tt in _
-      | RLit    Γ Δ l     _    => let case_RLit := tt in _
-      | RVar    Γ Δ σ         p => let case_RVar := tt in _
-      | RLam    Γ Δ Σ tx te     x => let case_RLam := tt in _
-      | RCast   Γ Δ Σ σ τ γ     x => let case_RCast := tt in _
-      | RAbsT   Γ Δ Σ κ σ a    => let case_RAbsT := tt in _
-      | RAppT   Γ Δ Σ κ σ τ     y => let case_RAppT := tt in _
-      | RAppCo  Γ Δ Σ κ σ₁ σ₂ γ σ l => let case_RAppCo := tt in _
-      | RAbsCo  Γ Δ Σ κ σ  σ₁ σ₂  y => let case_RAbsCo := tt in _
-      | RApp    Γ Δ Σ₁ Σ₂ tx te p => let case_RApp := tt in _
-      | RLet    Γ Δ Σ₁ Σ₂ σ₁ σ₂ p => let case_RLet := tt in _
-      | RLetRec Γ p lri x y => let case_RLetRec := tt in _
-      | RBindingGroup Γ p lri m x q => let case_RBindingGroup := tt in _
-      | REmptyGroup _ _ => let case_REmptyGroup := tt in _
-      | RCase   Σ Γ T κlen κ θ ldcd τ  => let case_RCase := tt in _
-      | RBrak   Σ a b c n m => let case_RBrak := tt in _
-      | REsc    Σ a b c n m => let case_REsc := tt in _
-              end (refl_equal _)  ); intros.
-
-    destruct case_RURule.
-      destruct d; [ destruct o | idtac ]; inversion H; subst.
-      clear H.
-      destruct u.
-      refine (match e as R in URule H C return C=[a >> b > t1 |- t2] -> _ with
-      | RLeft   h c ctx r   => let case_RLeft := tt in _
-      | RRight  h c ctx r   => let case_RRight := tt in _
-      | RCanL   t a       => let case_RCanL := tt in _
-      | RCanR   t a       => let case_RCanR := tt in _
-      | RuCanL  t a       => let case_RuCanL := tt in _
-      | RuCanR  t a       => let case_RuCanR := tt in _
-      | RAssoc  t a b c   => let case_RAssoc := tt in _
-      | RCossa  t a b c   => let case_RCossa := tt in _
-      | RExch   t a b     => let case_RExch := tt in _
-      | RWeak   t a       => let case_RWeak := tt in _
-      | RCont   t a       => let case_RCont := tt in _
-              end (refl_equal _)  ); intros.
-
-      destruct case_RCanL. admit.
-      destruct case_RCanR. admit.
-      destruct case_RuCanL. admit.
-      destruct case_RuCanR. admit.
-      destruct case_RAssoc. admit.
-      destruct case_RCossa. admit.
-      destruct case_RLeft. admit.
-      destruct case_RRight. admit.
-      destruct case_RExch. admit.
-      destruct case_RWeak. admit.
-      destruct case_RCont. admit.
-      destruct case_RBrak. admit.
-      destruct case_REsc. admit.
-      destruct case_RNote. admit.
-      destruct case_RLit. admit.
-      destruct case_RVar. admit.
-      destruct case_RLam. admit.
-      destruct case_RCast. admit.
-      destruct case_RBindingGroup. admit.
-      destruct case_RApp. admit.
-      destruct case_RLet. admit.
-      destruct case_REmptyGroup. admit.
-      destruct case_RAppT. admit.
-      destruct case_RAbsT. admit.
-      destruct case_RAppCo. admit.
-      destruct case_RAbsCo. admit.
-      destruct case_RLetRec. admit.
-      destruct case_RCase. admit.
-      Defined.
+  Definition justOne Γ Δ ξ τ : ITree _ (fun t => Expr Γ Δ ξ t) [τ] -> Expr Γ Δ ξ τ.
+    intros.
+    inversion X; auto.
+    Defined.
 
-  Definition closed2expr : forall c (pn:@ClosedND _ Rule c), ITree _ judg2exprType c.
-    refine ((
-      fix closed2expr' j (pn:@ClosedND _ Rule j) {struct pn} : ITree _ judg2exprType j :=
-      match pn in @ClosedND _ _ J return ITree _ judg2exprType J with
-      | cnd_weak             => let case_nil    := tt in _
-      | cnd_rule h c cnd' r  => let case_rule   := tt in (fun rest => _) (closed2expr' _ cnd')
-      | cnd_branch _ _ c1 c2 => let case_branch := tt in (fun rest1 rest2 => _) (closed2expr' _ c1) (closed2expr' _ c2)
-      end)); clear closed2expr'; intros; subst.
+  Definition ileaf `(it:ITree X F [t]) : F t.
+    inversion it.
+    apply X0.
+    Defined.
 
-      destruct case_nil.
-        apply INone.
+  Lemma update_branches : forall Γ (ξ:VV -> LeveledHaskType Γ ★) lev l1 l2 q,
+    update_ξ ξ lev (app l1 l2) q = update_ξ (update_ξ ξ lev l2) lev l1 q.
+    intros.
+    induction l1.
+      reflexivity.
+      simpl.
+      destruct a; simpl.
+      rewrite IHl1.
+      reflexivity.
+      Qed.
 
-      destruct case_rule.
-        set (@rule2expr h) as q.
-        destruct c.
-        destruct o.
-        apply ILeaf.
-        eapply rule2expr.
-        apply r.
-        apply rest.
+   Lemma quark {T} (l1:list T) l2 vf :
+      (In vf (app l1 l2)) <->
+       (In vf l1) \/ (In vf l2).
+     induction l1.
+     simpl; auto.
+     split; intro.
+     right; auto.
+     inversion H.
+     inversion H0.
+     auto.
+     split.
 
-        apply no_rules_with_empty_conclusion in r.
-        inversion r.
-        auto.
+     destruct IHl1.
+     simpl in *.
+     intro.
+     destruct H1.
+     left; left; auto.
+     set (H H1) as q.
+     destruct q.
+     left; right; auto.
+     right; auto.
+     simpl.
 
-        simpl.
-        apply systemfc_all_rules_one_conclusion in r.
-        inversion r.
+     destruct IHl1.
+     simpl in *.
+     intro.
+     destruct H1.
+     destruct H1.
+     left; auto.
+     right; apply H0; auto.
+     right; apply H0; auto.
+   Qed.
 
-      destruct case_branch.
-        apply IBranch.
-        apply rest1.
-        apply rest2.
+  Lemma splitter {T} (l1:list T) l2 vf :
+      (In vf (app l1 l2) → False)
+      -> (In vf l1 → False)  /\ (In vf l2 → False).
+    intros.
+    split; intros; apply H; rewrite quark.
+    auto.
+    auto.
+    Qed.
+
+  Lemma helper
+    : forall T Z {eqdt:EqDecidable T}(tl:Tree ??T)(vf:T) ξ (q:Z),
+      (In vf (leaves tl) -> False) ->
+      mapOptionTree (fun v' => if eqd_dec vf v' then q else ξ v') tl = 
+      mapOptionTree ξ tl.
+    intros.
+    induction tl;
+      try destruct a;
+        simpl in *.
+    set (eqd_dec vf t) as x in *.
+    destruct x.
+    subst.
+      assert False.
+      apply H.
+      left; auto.
+      inversion H0.
+    auto.
+    auto.
+    apply splitter in H.
+    destruct H.
+    rewrite (IHtl1 H).
+    rewrite (IHtl2 H0).
+    reflexivity.
+    Qed.
+    
+  Lemma fresh_lemma'' Γ 
+    : forall types ξ lev, 
+    FreshM { varstypes : _
+      |  mapOptionTree (update_ξ(Γ:=Γ)   ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
+      /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
+  admit.
+  Defined.
+
+  Lemma fresh_lemma' Γ 
+    : forall types vars Σ ξ lev, Σ = mapOptionTree ξ vars ->
+    FreshM { varstypes : _
+      |  mapOptionTree (update_ξ(Γ:=Γ) ξ lev (leaves varstypes)) vars = Σ
+      /\ mapOptionTree (update_ξ       ξ lev (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = (types @@@ lev)
+      /\ distinct (leaves (mapOptionTree (@fst _ _) varstypes)) }.
+    induction types.
+      intros; destruct a.
+        refine (bind vf = fresh (leaves vars) ; return _).
+          apply FreshMon.
+          destruct vf as [ vf vf_pf ].
+          exists [(vf,h)].
+          split; auto.
+          simpl.
+          set (helper VV _ vars vf ξ (h@@lev) vf_pf) as q.
+          rewrite q.
+          symmetry; auto.
+          simpl.
+          destruct (eqd_dec vf vf); [ idtac | set (n (refl_equal _)) as n'; inversion n' ]; auto.
+          split; auto.
+          apply distinct_cons.
+          intro.
+          inversion H0.
+          apply distinct_nil.
+        refine (return _).
+          exists []; auto.
+          split.
+          simpl.
+          symmetry; auto.
+          split.
+          simpl.
+          reflexivity.
+          simpl.
+          apply distinct_nil.
+        intros vars Σ ξ lev pf; refine (bind x2 = IHtypes2 vars Σ ξ lev pf; _).
+          apply FreshMon.
+          destruct x2 as [vt2 [pf21 [pf22 pfdist]]].
+          refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,(types2@@@lev)) (update_ξ ξ lev
+            (leaves vt2)) _ _; return _).
+          apply FreshMon.
+          simpl.
+          rewrite pf21.
+          rewrite pf22.
+          reflexivity.
+          clear IHtypes1 IHtypes2.
+          destruct x1 as [vt1 [pf11 pf12]].
+          exists (vt1,,vt2); split; auto.
+
+          set (update_branches Γ ξ lev (leaves vt1) (leaves vt2)) as q.
+          set (mapOptionTree_extensional _ _ q) as q'.
+          rewrite q'.
+          clear q' q.
+          inversion pf11.
+          reflexivity.
+
+          simpl.
+          set (update_branches Γ ξ lev (leaves vt1) (leaves vt2)) as q.
+          set (mapOptionTree_extensional _ _ q) as q'.
+          rewrite q'.
+          rewrite q'.
+          clear q' q.
+          rewrite <- mapOptionTree_compose.
+          rewrite <- mapOptionTree_compose.
+          rewrite <- mapOptionTree_compose in *.
+          split.
+          destruct pf12.
+          rewrite H.
+          inversion pf11.
+          rewrite <- mapOptionTree_compose.
+          reflexivity.
+
+          admit.
         Defined.
 
-  Definition proof2expr Γ Δ τ Σ : ND Rule [] [Γ > Δ > Σ |- [τ]] -> { ξ:VV -> LeveledHaskType Γ ★ & Expr Γ Δ ξ τ }.
-    intro pf.
-    set (closedFromSCND _ _ (mkSCND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) cnd_weak) as cnd.
-    apply closed2expr in cnd.
-    inversion cnd; subst.
+  Lemma fresh_lemma Γ ξ vars Σ Σ' lev
+    : Σ = mapOptionTree ξ vars ->
+    FreshM { vars' : _
+      |  mapOptionTree (update_ξ(Γ:=Γ) ξ lev ((vars',Σ')::nil)) vars = Σ
+      /\ mapOptionTree (update_ξ ξ lev ((vars',Σ')::nil)) [vars'] = [Σ' @@ lev] }.
+    intros.
+    set (fresh_lemma' Γ [Σ'] vars Σ ξ lev H) as q.
+    refine (q >>>= fun q' => return _).
+    apply FreshMon.
+    clear q.
+    destruct q' as [varstypes [pf1 [pf2 pfdist]]].
+    destruct varstypes; try destruct o; try destruct p; simpl in *.
+      destruct (eqd_dec v v); [ idtac | set (n (refl_equal _)) as n'; inversion n' ].    
+      inversion pf2; subst.
+      exists v.
+      destruct (eqd_dec v v); [ idtac | set (n (refl_equal _)) as n'; inversion n' ].
+      split; auto.
+      inversion pf2.
+      inversion pf2.
+    Defined.
+
+  Definition urule2expr  : forall Γ Δ h j (r:@URule Γ Δ h j) (ξ:VV -> LeveledHaskType Γ ★),
+    ITree _ (ujudg2exprType ξ) h -> ITree _ (ujudg2exprType ξ) j.
+
+      refine (fix urule2expr Γ Δ h j (r:@URule Γ Δ h j) ξ {struct r} : 
+        ITree _ (ujudg2exprType ξ) h -> ITree _ (ujudg2exprType ξ) j :=
+        match r as R in URule H C return ITree _ (ujudg2exprType ξ) H -> ITree _ (ujudg2exprType ξ) C with
+          | RLeft   h c ctx r => let case_RLeft  := tt in (fun e => _) (urule2expr _ _ _ _ r)
+          | RRight  h c ctx r => let case_RRight := tt in (fun e => _) (urule2expr _ _ _ _ r)
+          | RCanL   t a       => let case_RCanL  := tt in _
+          | RCanR   t a       => let case_RCanR  := tt in _
+          | RuCanL  t a       => let case_RuCanL := tt in _
+          | RuCanR  t a       => let case_RuCanR := tt in _
+          | RAssoc  t a b c   => let case_RAssoc := tt in _
+          | RCossa  t a b c   => let case_RCossa := tt in _
+          | RExch   t a b     => let case_RExch  := tt in _
+          | RWeak   t a       => let case_RWeak  := tt in _
+          | RCont   t a       => let case_RCont  := tt in _
+          end); clear urule2expr; intros.
+
+      destruct case_RCanL.
+        apply ILeaf; simpl; intros.
+        inversion X.
+        simpl in X0.
+        apply (X0 ([],,vars)).
+        simpl; rewrite <- H; auto.
+
+      destruct case_RCanR.
+        apply ILeaf; simpl; intros.
+        inversion X.
+        simpl in X0.
+        apply (X0 (vars,,[])).
+        simpl; rewrite <- H; auto.
+
+      destruct case_RuCanL.
+        apply ILeaf; simpl; intros.
+        destruct vars; try destruct o; inversion H.
+        inversion X.
+        simpl in X0.
+        apply (X0 vars2); auto.
+
+      destruct case_RuCanR.
+        apply ILeaf; simpl; intros.
+        destruct vars; try destruct o; inversion H.
+        inversion X.
+        simpl in X0.
+        apply (X0 vars1); auto.
+
+      destruct case_RAssoc.
+        apply ILeaf; simpl; intros.
+        inversion X.
+        simpl in X0.
+        destruct vars; try destruct o; inversion H.
+        destruct vars1; try destruct o; inversion H.
+        apply (X0 (vars1_1,,(vars1_2,,vars2))).
+        subst; auto.
+
+      destruct case_RCossa.
+        apply ILeaf; simpl; intros.
+        inversion X.
+        simpl in X0.
+        destruct vars; try destruct o; inversion H.
+        destruct vars2; try destruct o; inversion H.
+        apply (X0 ((vars1,,vars2_1),,vars2_2)).
+        subst; auto.
+
+      destruct case_RLeft.
+        destruct c; [ idtac | apply no_urules_with_multiple_conclusions in r0; inversion r0; exists c1; exists c2; auto ].
+        destruct o; [ idtac | apply INone ].
+        destruct u; simpl in *.
+        apply ILeaf; simpl; intros.
+        destruct vars; try destruct o; inversion H.
+        set (fun q => ileaf (e ξ q)) as r'.
+        simpl in r'.
+        apply r' with (vars:=vars2).
+        clear r' e.
+        clear r0.
+        induction h0.
+          destruct a.
+          destruct u.
+          simpl in X.
+          apply ileaf in X. 
+          apply ILeaf.
+          simpl.
+          simpl in X.
+          intros.
+          apply X with (vars:=vars1,,vars).
+          simpl.
+          rewrite H0.
+          rewrite H1.
+          reflexivity.
+          apply INone.
+          apply IBranch.
+          apply IHh0_1. inversion X; auto.
+          apply IHh0_2. inversion X; auto.
+          auto.
+        
+      destruct case_RRight.
+        destruct c; [ idtac | apply no_urules_with_multiple_conclusions in r0; inversion r0; exists c1; exists c2; auto ].
+        destruct o; [ idtac | apply INone ].
+        destruct u; simpl in *.
+        apply ILeaf; simpl; intros.
+        destruct vars; try destruct o; inversion H.
+        set (fun q => ileaf (e ξ q)) as r'.
+        simpl in r'.
+        apply r' with (vars:=vars1).
+        clear r' e.
+        clear r0.
+        induction h0.
+          destruct a.
+          destruct u.
+          simpl in X.
+          apply ileaf in X. 
+          apply ILeaf.
+          simpl.
+          simpl in X.
+          intros.
+          apply X with (vars:=vars,,vars2).
+          simpl.
+          rewrite H0.
+          rewrite H2.
+          reflexivity.
+          apply INone.
+          apply IBranch.
+          apply IHh0_1. inversion X; auto.
+          apply IHh0_2. inversion X; auto.
+          auto.
+
+      destruct case_RExch.
+        apply ILeaf; simpl; intros.
+        inversion X.
+        simpl in X0.
+        destruct vars; try destruct o; inversion H.
+        apply (X0 (vars2,,vars1)).
+        inversion H; subst; auto.
+        
+      destruct case_RWeak.
+        apply ILeaf; simpl; intros.
+        inversion X.
+        simpl in X0.
+        apply (X0 []).
+        auto.
+        
+      destruct case_RCont.
+        apply ILeaf; simpl; intros.
+        inversion X.
+        simpl in X0.
+        apply (X0 (vars,,vars)).
+        simpl.
+        rewrite <- H.
+        auto.
+        Defined.
+
+  Definition bridge Γ Δ (c:Tree ??(UJudg Γ Δ)) ξ :
+    ITree Judg judg2exprType (mapOptionTree UJudg2judg c) -> ITree (UJudg Γ Δ) (ujudg2exprType ξ) c.
+    intro it.
+    induction c.
+    destruct a.
+      destruct u; simpl in *.
+      apply ileaf in it.
+      apply ILeaf.
+      simpl in *.
+      intros; apply it with (vars:=vars); auto.
+    apply INone.
+    apply IBranch; [ apply IHc1 | apply IHc2 ]; inversion it; auto.
+    Defined.
+
+  Definition letrec_helper Γ Δ l (varstypes:Tree ??(VV * HaskType Γ ★)) ξ' :
+    ITree (LeveledHaskType Γ ★)
+         (fun t : LeveledHaskType Γ ★ => Expr Γ Δ ξ' t)
+         (mapOptionTree (ξ' ○ (@fst _ _)) varstypes)
+         -> ELetRecBindings Γ Δ ξ' l varstypes.
+    intros.
+    induction varstypes.
+    destruct a; simpl in *.
+    destruct p.
+    simpl.
+    apply ileaf in X. simpl in X.
+      apply ELR_leaf.
+      rename h into τ.
+      destruct (eqd_dec (unlev (ξ' v)) τ).
+      rewrite <- e.
+      destruct (ξ' v).
+      simpl.
+      destruct (eqd_dec h0 l).
+        rewrite <- e0.
+        apply X.
+      apply (Prelude_error "level mismatch; should never happen").
+      apply (Prelude_error "letrec type mismatch; should never happen").
+
+    apply ELR_nil.
+    apply ELR_branch.
+      apply IHvarstypes1; inversion X; auto.
+      apply IHvarstypes2; inversion X; auto.
+    Defined.
+
+  Definition unindex_tree {V}{F} : forall {t:Tree ??V}, ITree V F t -> Tree ??{ v:V & F v }.
+    refine (fix rec t it := match it as IT return Tree ??{ v:V & F v } with
+      | INone => T_Leaf None
+      | ILeaf x y => T_Leaf (Some _)
+      | IBranch _ _ b1 b2 => (rec _ b1),,(rec _ b2)
+            end).
+    exists x; auto.
+    Defined.
+
+  Definition fix_indexing X (F:X->Type)(J:X->Type)(t:Tree ??{ x:X & F x })
+    :  ITree { x:X & F x } (fun x => J (projT1 x))                                t
+    -> ITree X             (fun x:X => J x)   (mapOptionTree (@projT1 _ _) t).
+    intro it.
+    induction it; simpl in *.
+    apply INone.
+    apply ILeaf.
+    apply f.
+    simpl; apply IBranch; auto.
+    Defined.
+
+  Definition fix2 {X}{F} : Tree ??{ x:X & FreshM (F x) } -> Tree ??(FreshM { x:X & F x }).
+    refine (fix rec t := match t with
+      | T_Leaf None => T_Leaf None
+      | T_Leaf (Some x) => T_Leaf (Some _)
+      | T_Branch b1 b2 => T_Branch (rec b1) (rec b2)
+            end).
+    destruct x as [x fx].
+    refine (bind fx' = fx ; return _).
+    apply FreshMon.
+    exists x.
+    apply fx'.
+    Defined.
+  
+  Definition case_helper tc Γ Δ lev tbranches avars ξ :
+    forall pcb:{sac : StrongAltCon & ProofCaseBranch tc Γ Δ lev tbranches avars sac},
+     prod (judg2exprType (pcb_judg (projT2 pcb))) {vars' : Tree ??VV & pcb_freevars (projT2 pcb) = mapOptionTree ξ vars'} ->
+     ((fun sac => FreshM
+       { scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars sac
+         & Expr (sac_Γ sac Γ) (sac_Δ sac Γ avars (weakCK'' Δ)) (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@  lev)) }) (projT1 pcb)).
+    intro pcb.
+    intro X.
     simpl in X.
-    clear cnd pf.
+    simpl.
+    destruct pcb as [sac pcb].
+    simpl in *.
+
     destruct X.
-    exists x.
-    inversion e.
+    destruct s as [vars vars_pf].
+
+    refine (bind localvars = fresh_lemma' _ (unleaves  (vec2list (sac_types sac _ avars))) vars 
+      (mapOptionTree weakLT' (pcb_freevars pcb)) (weakLT' ○ ξ) (weakL' lev) _  ; _).
+      apply FreshMon.
+      rewrite vars_pf.
+      rewrite <- mapOptionTree_compose.
+      reflexivity.
+      destruct localvars as [localvars [localvars_pf1 [localvars_pf2 localvars_dist ]]].
+      set (mapOptionTree (@fst _ _) localvars) as localvars'.
+
+    set (list2vec (leaves localvars')) as localvars''.
+    cut (length (leaves localvars') = sac_numExprVars sac). intro H''.
+      rewrite H'' in localvars''.
+    cut (distinct (vec2list localvars'')). intro H'''.
+    set (@Build_StrongCaseBranchWithVVs _ _ _ _ avars sac localvars'' H''') as scb.
+
+    refine (bind q = (f (scbwv_ξ scb ξ lev) (vars,,(unleaves (vec2list (scbwv_exprvars scb)))) _) ; return _).
+      apply FreshMon.
+      simpl.
+      unfold scbwv_ξ.
+      rewrite vars_pf.
+      rewrite <- mapOptionTree_compose.
+      clear localvars_pf1.
+      simpl.
+      rewrite mapleaves'.
+
+    admit.
+
+    exists scb.
+    apply ileaf in q.
+    apply q.
+
+    admit.
+    admit.
+    Defined.
+
+  Fixpoint treeM {T}(t:Tree ??(FreshM T)) : FreshM (Tree ??T) :=
+    match t with
+      | T_Leaf None     => return []
+      | T_Leaf (Some x) => bind x' = x ; return [x']
+      | T_Branch b1 b2  => bind b1' = treeM b1 ; bind b2' = treeM b2 ; return (b1',,b2')
+    end.
+
+  Definition gather_branch_variables
+    Γ Δ (ξ:VV -> LeveledHaskType Γ ★) tc avars tbranches lev (alts:Tree ?? {sac : StrongAltCon &
+                ProofCaseBranch tc Γ Δ lev tbranches avars sac})
+    :
+    forall vars,
+    mapOptionTreeAndFlatten (fun x => pcb_freevars(Γ:=Γ) (projT2 x)) alts = mapOptionTree ξ vars
+    -> ITree Judg judg2exprType (mapOptionTree (fun x => pcb_judg (projT2 x)) alts)
+    -> ITree _ (fun q => prod (judg2exprType (pcb_judg (projT2 q))) 
+      { vars' : _ & pcb_freevars (projT2 q) = mapOptionTree ξ vars' })
+  alts.
+    induction alts;
+    intro vars;
+    intro pf;
+    intro source.
+    destruct a; [ idtac | apply INone ].
+    simpl in *.
+    apply ileaf in source.
+    apply ILeaf.
+    destruct s as [sac pcb].
+    simpl in *.
+    split.
+    intros.
+    eapply source.
+    apply H.
+    clear source.
+
+    exists vars.
+    auto.
+
+    simpl in pf.
+    destruct vars; try destruct o; simpl in pf; inversion pf.
+    simpl in source.
+    inversion source.
     subst.
+    apply IBranch.
+    apply (IHalts1 vars1 H0 X); auto.
+    apply (IHalts2 vars2 H1 X0); auto.
+
+    Defined.
+
+
+  Definition rule2expr : forall h j (r:Rule h j), ITree _ judg2exprType h -> ITree _ judg2exprType j.
+
+    intros h j r.
+
+      refine (match r as R in Rule H C return ITree _ judg2exprType H -> ITree _ judg2exprType C with
+      | RURule a b c d e              => let case_RURule := tt        in _
+      | RNote   Γ Δ Σ τ l n           => let case_RNote := tt         in _
+      | RLit    Γ Δ l     _           => let case_RLit := tt          in _
+      | RVar    Γ Δ σ         p       => let case_RVar := tt          in _
+      | RGlobal Γ Δ σ l wev           => let case_RGlobal := tt       in _
+      | RLam    Γ Δ Σ tx te     x     => let case_RLam := tt          in _
+      | RCast   Γ Δ Σ σ τ γ     x     => let case_RCast := tt         in _
+      | RAbsT   Γ Δ Σ κ σ a           => let case_RAbsT := tt         in _
+      | RAppT   Γ Δ Σ κ σ τ     y     => let case_RAppT := tt         in _
+      | RAppCo  Γ Δ Σ κ σ₁ σ₂ γ σ l   => let case_RAppCo := tt        in _
+      | RAbsCo  Γ Δ Σ κ σ  σ₁ σ₂  y   => let case_RAbsCo := tt        in _
+      | RApp    Γ Δ Σ₁ Σ₂ tx te p     => let case_RApp := tt          in _
+      | RLet    Γ Δ Σ₁ Σ₂ σ₁ σ₂ p     => let case_RLet := tt          in _
+      | RBindingGroup Γ p lri m x q   => let case_RBindingGroup := tt in _
+      | REmptyGroup _ _               => let case_REmptyGroup := tt   in _
+      | RBrak   Σ a b c n m           => let case_RBrak := tt         in _
+      | REsc    Σ a b c n m           => let case_REsc := tt          in _
+      | RCase   Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt         in _
+      | RLetRec Γ Δ lri x y t         => let case_RLetRec := tt       in _
+      end); intro X_; try apply ileaf in X_; simpl in X_.
+
+  destruct case_RURule.
+    destruct d; try destruct o.
+      apply ILeaf; destruct u; simpl; intros.
+      set (@urule2expr a b _ _ e ξ) as q.
+      set (fun z => ileaf (q z)) as q'.
+      simpl in q'.
+      apply q' with (vars:=vars).
+      clear q' q.
+      apply bridge.
+      apply X_.
+      auto.
+      apply no_urules_with_empty_conclusion in e; inversion e; auto.
+      apply no_urules_with_multiple_conclusions in e; inversion e; auto; exists d1; exists d2; auto.
+
+  destruct case_RBrak.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
+    apply EBrak.
+    apply (ileaf X).
+
+  destruct case_REsc.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
+    apply EEsc.
+    apply (ileaf X).
+
+  destruct case_RNote.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
+    apply ENote; auto.
+    apply (ileaf X).
+
+  destruct case_RLit.
+    apply ILeaf; simpl; intros; refine (return ILeaf _ _).
+    apply ELit.
+
+  destruct case_RVar.
+    apply ILeaf; simpl; intros; refine (return ILeaf _ _).
+    destruct vars; simpl in H; inversion H; destruct o. inversion H1. rewrite H2.
+    apply EVar.
+    inversion H.
+
+  destruct case_RGlobal.
+    apply ILeaf; simpl; intros; refine (return ILeaf _ _).
+    apply EGlobal.
+    apply wev.
+
+  destruct case_RLam.
+    apply ILeaf.
+    simpl in *; intros.
+    refine (fresh_lemma _ ξ vars _ tx x H >>>= (fun pf => _)).
+    apply FreshMon.
+    destruct pf as [ vnew [ pf1 pf2 ]].
+    set (update_ξ ξ x ((⟨vnew, tx  ⟩) :: nil)) as ξ' in *.
+    refine (X_ ξ' (vars,,[vnew]) _ >>>= _).
+    apply FreshMon.
+    simpl.
+    rewrite pf1.
+    rewrite <- pf2.
+    simpl.
+    reflexivity.
+    intro hyp.
+    refine (return _).
+    apply ILeaf.
+    apply ELam with (ev:=vnew).
+    apply ileaf in hyp.
+    simpl in hyp.
+    unfold ξ' in hyp.
+    apply hyp.
+
+  destruct case_RCast.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
+    eapply ECast.
+    apply x.
+    apply ileaf in X. simpl in X.
     apply X.
+
+  destruct case_RBindingGroup.
+    apply ILeaf; simpl; intros.
+    inversion X_.
+    apply ileaf in X.
+    apply ileaf in X0.
+    simpl in *.
+    destruct vars; inversion H.
+    destruct o; inversion H3.
+    refine (X ξ vars1 H3 >>>= fun X' => X0 ξ vars2 H4 >>>= fun X0' => return _).
+    apply FreshMon.
+    apply FreshMon.
+    apply IBranch; auto.
+
+  destruct case_RApp.    
+    apply ILeaf.
+    inversion X_.
+    inversion X.
+    inversion X0.
+    simpl in *.
+    intros.
+    destruct vars. try destruct o; inversion H.
+    simpl in H.
+    inversion H.
+    set (X1 ξ vars1 H5) as q1.
+    set (X2 ξ vars2 H6) as q2.
+    refine (q1 >>>= fun q1' => q2 >>>= fun q2' => return _).
+    apply FreshMon.
+    apply FreshMon.
+    apply ILeaf.
+    apply ileaf in q1'.
+    apply ileaf in q2'.
+    simpl in *.
+    apply (EApp _ _ _ _ _ _ q1' q2').
+
+  destruct case_RLet.
+    apply ILeaf.
+    simpl in *; intros.
+    destruct vars; try destruct o; inversion H.
+    refine (fresh_lemma _ ξ vars1 _ σ₂ p H1 >>>= (fun pf => _)).
+    apply FreshMon.
+    destruct pf as [ vnew [ pf1 pf2 ]].
+    set (update_ξ ξ p ((⟨vnew, σ₂  ⟩) :: nil)) as ξ' in *.
+    inversion X_.
+    apply ileaf in X.
+    apply ileaf in X0.
+    simpl in *.
+    refine (X0 ξ  vars2 _ >>>= fun X0' => _).
+    apply FreshMon.
+    auto.
+    refine (X  ξ' (vars1,,[vnew]) _ >>>= fun X1' => _).
+    apply FreshMon.
+    rewrite H1.
+    simpl.
+    rewrite pf2.
+    rewrite pf1.
+    rewrite H1.
+    reflexivity.
+    refine (return _).
+    apply ILeaf.
+    apply ileaf in X0'.
+    apply ileaf in X1'.
+    simpl in *.
+    apply ELet with (ev:=vnew)(tv:=σ₂).
+    apply X0'.
+    apply X1'.
+
+  destruct case_REmptyGroup.
+    apply ILeaf; simpl; intros.
+    refine (return _).
+    apply INone.
+
+  destruct case_RAppT.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
+    apply ETyApp.
+    apply (ileaf X).
+
+  destruct case_RAbsT.
+    apply ILeaf; simpl; intros; refine (X_ (weakLT ○ ξ) vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
+    rewrite mapOptionTree_compose.
+    rewrite <- H.
+    reflexivity.
+    apply ileaf in X. simpl in *.
+    apply ETyLam.
+    apply X.
+
+  destruct case_RAppCo.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
+    auto.
+    eapply ECoApp.
+    apply γ.
+    apply (ileaf X).
+
+  destruct case_RAbsCo.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
+    auto.
+    eapply ECoLam.
+    apply (ileaf X).
+
+  destruct case_RLetRec.
+    apply ILeaf; simpl; intros.
+    refine (bind ξvars = fresh_lemma' _ y _ _ _ t H; _). apply FreshMon.
+    destruct ξvars as [ varstypes [ pf1[ pf2 pfdist]]].
+    refine (X_ ((update_ξ ξ t (leaves varstypes)))
+      (vars,,(mapOptionTree (@fst _ _) varstypes)) _ >>>= fun X => return _); clear X_.  apply FreshMon.
+    simpl.
+    rewrite pf2.
+    rewrite pf1.
+    auto.
+    apply ILeaf.
+    inversion X; subst; clear X.
+
+    apply (@ELetRec _ _ _ _ _ _ _ varstypes).
+    apply (@letrec_helper Γ Δ t varstypes).
+    rewrite <- pf2 in X1.
+    rewrite mapOptionTree_compose.
+    apply X1.
+    apply ileaf in X0.
+    apply X0.
+
+  destruct case_RCase.
+    apply ILeaf; simpl; intros.
+    inversion X_.
+    clear X_.
+    subst.
+    apply ileaf in X0.
+    simpl in X0.
+
+    (* body_freevars and alts_freevars are the types of variables in the body and alternatives (respectively) which are free
+     * from the viewpoint just outside the case block -- i.e. not bound by any of the branches *)
+    rename Σ into body_freevars_types.
+    rename vars into all_freevars.
+    rename X0 into body_expr.
+    rename X  into alts_exprs.
+
+    destruct all_freevars; try destruct o; inversion H.
+    rename all_freevars2 into body_freevars.
+    rename all_freevars1 into alts_freevars.
+
+    set (gather_branch_variables _ _ _ _ _ _ _ _ _ H1 alts_exprs) as q.
+    set (itmap (fun pcb alt_expr => case_helper tc Γ Δ lev tbranches avars ξ pcb alt_expr) q) as alts_exprs'.
+    apply fix_indexing in alts_exprs'.
+    simpl in alts_exprs'.
+    apply unindex_tree in alts_exprs'.
+    simpl in alts_exprs'.
+    apply fix2 in alts_exprs'.
+    apply treeM in alts_exprs'.
+
+    refine ( alts_exprs' >>>= fun Y =>
+      body_expr ξ _ _
+      >>>= fun X => return ILeaf _ (@ECase _ _ _ _ _ _ _ _ _ (ileaf X) Y)); auto.
+      apply FreshMon.
+      apply FreshMon.
+      apply H2.
+    Defined.
+
+  Definition closed2expr : forall c (pn:@ClosedND _ Rule c), ITree _ judg2exprType c.
+    refine ((
+      fix closed2expr' j (pn:@ClosedND _ Rule j) {struct pn} : ITree _ judg2exprType j :=
+      match pn in @ClosedND _ _ J return ITree _ judg2exprType J with
+      | cnd_weak             => let case_nil    := tt in INone _ _
+      | cnd_rule h c cnd' r  => let case_rule   := tt in rule2expr _ _ r (closed2expr' _ cnd')
+      | cnd_branch _ _ c1 c2 => let case_branch := tt in IBranch _ _ (closed2expr' _ c1) (closed2expr' _ c2)
+      end)); clear closed2expr'; intros; subst.
+        Defined.
+
+  Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
+    FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
+    intros Γ Σ.
+    induction Σ; intro ξ.
+    destruct a.
+    destruct l as [τ l].
+    set (fresh_lemma' Γ [τ] [] [] ξ l (refl_equal _)) as q.
+    refine (q >>>= fun q' => return _).
+    apply FreshMon.
+    clear q.
+    destruct q' as [varstypes [pf1 [pf2 distpf]]].
+    exists (mapOptionTree (@fst _ _) varstypes).
+    exists (update_ξ ξ l (leaves varstypes)).
+    symmetry; auto.
+    refine (return _).
+    exists [].
+    exists ξ; auto.
+    refine (bind f1 = IHΣ1 ξ ; _).
+    apply FreshMon.
+    destruct f1 as [vars1 [ξ1 pf1]].
+    refine (bind f2 = IHΣ2 ξ1 ; _).
+    apply FreshMon.
+    destruct f2 as [vars2 [ξ2 pf22]].
+    refine (return _).
+    exists (vars1,,vars2).
+    exists ξ2.
+    simpl.
+    rewrite pf22.
+    rewrite pf1.
+    admit.
+    Defined.
+
+  Definition proof2expr Γ Δ τ Σ (ξ0: VV -> LeveledHaskType Γ ★)
+    {zz:ToString VV} : ND Rule [] [Γ > Δ > Σ |- [τ]] ->
+    FreshM (???{ ξ : _ & Expr Γ Δ ξ τ}).
+    intro pf.
+    set (closedFromSCND _ _ (mkSCND systemfc_all_rules_one_conclusion _ _ _ pf (scnd_weak [])) cnd_weak) as cnd.
+    apply closed2expr in cnd.
+    apply ileaf in cnd.
+    simpl in *.
+    clear pf.
+    refine (bind ξvars = manyFresh _ Σ ξ0; _).
+    apply FreshMon.
+    destruct ξvars as [vars ξpf].
+    destruct ξpf as [ξ pf].
+    refine (cnd ξ vars _ >>>= fun it => _).
+    apply FreshMon.
+    auto.
+    refine (return OK _).
+    exists ξ.
+    apply (ileaf it).
     Defined.
 
 End HaskProofToStrong.