X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FHaskProofToStrong.v;h=ece5801d25faa2ebf3a172747b183801e5c1542a;hp=40268e28331f8b471f6852182f7164b12f6b0137;hb=4359342db4052c77b802ce256856df71387e7a62;hpb=2a887b5df62e3202c859c953d83918872bda31e4

diff --git a/src/HaskProofToStrong.v b/src/HaskProofToStrong.v
index 40268e2..ece5801 100644
--- a/src/HaskProofToStrong.v
+++ b/src/HaskProofToStrong.v
@@ -16,51 +16,243 @@ Require Import HaskProof.
 
 Section HaskProofToStrong.
 
-  Context {VV:Type} {eqdec_vv:EqDecidable VV}.
+  Context {VV:Type} {eqdec_vv:EqDecidable VV} {freshM:FreshMonad VV}.
 
-  Definition Exprs Γ Δ ξ τ :=
-    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) => forall (ξ:VV -> LeveledHaskType Γ ★ ) vars, Σ=mapOptionTree ξ vars -> 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.
 
-  Definition judges2exprType (j:Tree ??Judg) : Type :=
-    ITree _ judg2exprType j.
+  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 urule2expr Γ Δ : forall h j (r:@URule Γ Δ h j),
-    judges2exprType (mapOptionTree UJudg2judg h) -> judges2exprType (mapOptionTree UJudg2judg j).
+  Definition ileaf `(it:ITree X F [t]) : F t.
+    inversion it.
+    apply X0.
+    Defined.
 
-    intros h j r.
+  Lemma update_branches : forall Γ (ξ:VV -> LeveledHaskType Γ ★) l1 l2 q,
+    update_ξ ξ (app l1 l2) q = update_ξ (update_ξ ξ l2) l1 q.
+    intros.
+    induction l1.
+      reflexivity.
+      simpl.
+      destruct a; simpl.
+      rewrite IHl1.
+      reflexivity.
+      Qed.
+
+  Lemma mapOptionTree_extensional {A}{B}(f g:A->B) : (forall a, f a = g a) -> (forall t, mapOptionTree f t = mapOptionTree g t).
+    intros.
+    induction t.
+    destruct a; auto.
+    simpl; rewrite H; auto.
+    simpl; rewrite IHt1; rewrite IHt2; auto.
+    Qed.
+
+   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.
+
+     destruct IHl1.
+     simpl in *.
+     intro.
+     destruct H1.
+     left; left; auto.
+     set (H H1) as q.
+     destruct q.
+     left; right; auto.
+     right; auto.
+     simpl.
+
+     destruct IHl1.
+     simpl in *.
+     intro.
+     destruct H1.
+     destruct H1.
+     left; auto.
+     right; apply H0; auto.
+     right; apply H0; auto.
+   Qed.
+
+  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 vars Σ ξ, Σ = mapOptionTree ξ vars ->
+    FreshM { varstypes : _
+      |  mapOptionTree (update_ξ(Γ:=Γ) ξ (leaves varstypes)) vars = Σ
+      /\ mapOptionTree (update_ξ       ξ (leaves varstypes)) (mapOptionTree (@fst _ _) varstypes) = types }.
+    induction types.
+      intros; destruct a.
+        refine (bind vf = fresh (leaves vars) ; return _).
+          apply FreshMon.
+          destruct vf as [ vf vf_pf ].
+          exists [(vf,l)].
+          split; auto.
+          simpl.
+          set (helper VV _ vars vf ξ l vf_pf) as q.
+          rewrite q.
+          symmetry; auto.
+          simpl.
+          destruct (eqd_dec vf vf); [ idtac | set (n (refl_equal _)) as n'; inversion n' ]; auto.
+        refine (return _).
+          exists []; auto.
+        intros vars Σ ξ pf; refine (bind x2 = IHtypes2 vars Σ ξ pf; _).
+          apply FreshMon.
+          destruct x2 as [vt2 [pf21 pf22]].
+          refine (bind x1 = IHtypes1 (vars,,(mapOptionTree (@fst _ _) vt2)) (Σ,,types2) (update_ξ ξ (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 (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 (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 *.
+          rewrite pf12.
+          inversion pf11.
+          rewrite <- mapOptionTree_compose.
+          reflexivity.
+        Defined.
 
-      refine (match r as R in URule H C
-                  return judges2exprType (mapOptionTree UJudg2judg H) -> judges2exprType (mapOptionTree UJudg2judg C) 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   ); intros.
+  Lemma fresh_lemma Γ ξ vars Σ Σ'
+    : Σ = mapOptionTree ξ vars ->
+    FreshM { vars' : _
+      |  mapOptionTree (update_ξ(Γ:=Γ) ξ ((vars',Σ')::nil)) vars = Σ
+      /\ mapOptionTree (update_ξ ξ ((vars',Σ')::nil)) [vars'] = [Σ'] }.
+    intros.
+    set (fresh_lemma' Γ [Σ'] vars Σ ξ H) as q.
+    refine (q >>>= fun q' => return _).
+    apply FreshMon.
+    clear q.
+    destruct q' as [varstypes [pf1 pf2]].
+    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.
+
+  Lemma manyFresh : forall Γ Σ (ξ0:VV -> LeveledHaskType Γ ★),
+    FreshM { vars : _ & { ξ : VV -> LeveledHaskType Γ ★ & Σ = mapOptionTree ξ vars } }.
+    intros.
+    set (fresh_lemma' Γ Σ []  []  ξ0 (refl_equal _)) as q.
+    refine (q >>>= fun q' => return _).
+    apply FreshMon.
+    clear q.
+    destruct q' as [varstypes [pf1 pf2]].
+    exists (mapOptionTree (@fst _ _) varstypes).
+    exists (update_ξ ξ0 (leaves varstypes)).
+    symmetry; auto.
+    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 ξ ([],,vars)).
+        apply (X0 ([],,vars)).
         simpl; rewrite <- H; auto.
 
       destruct case_RCanR.
         apply ILeaf; simpl; intros.
         inversion X.
         simpl in X0.
-        apply (X0 ξ (vars,,[])).
+        apply (X0 (vars,,[])).
         simpl; rewrite <- H; auto.
 
       destruct case_RuCanL.
@@ -68,14 +260,14 @@ Section HaskProofToStrong.
         destruct vars; try destruct o; inversion H.
         inversion X.
         simpl in X0.
-        apply (X0 ξ vars2); auto.
+        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 ξ vars1); auto.
+        apply (X0 vars1); auto.
 
       destruct case_RAssoc.
         apply ILeaf; simpl; intros.
@@ -83,7 +275,7 @@ Section HaskProofToStrong.
         simpl in X0.
         destruct vars; try destruct o; inversion H.
         destruct vars1; try destruct o; inversion H.
-        apply (X0 ξ (vars1_1,,(vars1_2,,vars2))).
+        apply (X0 (vars1_1,,(vars1_2,,vars2))).
         subst; auto.
 
       destruct case_RCossa.
@@ -92,242 +284,469 @@ Section HaskProofToStrong.
         simpl in X0.
         destruct vars; try destruct o; inversion H.
         destruct vars2; try destruct o; inversion H.
-        apply (X0 ξ ((vars1,,vars2_1),,vars2_2)).
+        apply (X0 ((vars1,,vars2_1),,vars2_2)).
         subst; auto.
 
       destruct case_RLeft.
-        (* this will require recursion *)
-        admit.
+        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.
-        (* this will require recursion *)
-        admit.
+        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 ξ (vars2,,vars1)).
+        apply (X0 (vars2,,vars1)).
         inversion H; subst; auto.
         
       destruct case_RWeak.
         apply ILeaf; simpl; intros.
         inversion X.
         simpl in X0.
-        apply (X0 ξ []).
+        apply (X0 []).
         auto.
         
       destruct case_RCont.
         apply ILeaf; simpl; intros.
         inversion X.
         simpl in X0.
-        apply (X0 ξ (vars,,vars)).
+        apply (X0 (vars,,vars)).
         simpl.
         rewrite <- H.
         auto.
         Defined.
 
-  Definition rule2expr : forall h j (r:Rule h j), judges2exprType h -> judges2exprType j.
+  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 ξ' :
+    ITree (LeveledHaskType Γ ★)
+         (fun t : LeveledHaskType Γ ★ => Expr Γ Δ ξ' t)
+         (mapOptionTree (ξ' ○ (@fst _ _)) varstypes)
+         -> ELetRecBindings Γ Δ ξ' l
+         (mapOptionTree (fun x : VV * LeveledHaskType Γ ★ => ⟨fst x, unlev (snd x) ⟩) varstypes).
+    intros.
+    induction varstypes.
+    destruct a; simpl in *.
+    destruct p.
+    destruct l0 as [τ l'].
+    simpl.
+    apply ileaf in X. simpl in X.
+    assert (unlev (ξ' v) = τ).
+      admit.
+      rewrite <- H.
+      apply ELR_leaf.
+      rewrite H.
+      destruct (ξ' v).
+      rewrite <- H.
+      simpl.
+      assert (h0=l). admit.
+        rewrite H0 in X.
+        apply X.
+
+    apply ELR_nil.
+
+    simpl; apply ELR_branch.
+      apply IHvarstypes1.
+      simpl in X.
+      inversion X; auto.
+      apply IHvarstypes2.
+      simpl in X.
+      inversion X; auto.
+
+    Defined.
+
+
+(*
+  Definition case_helper tc Γ Δ lev tbranches avars ξ (Σ:Tree ??VV) tys :
+    forall pcb : ProofCaseBranch tc Γ Δ lev tbranches avars,
+  judg2exprType (pcb_judg pcb) -> FreshM
+  {scb : StrongCaseBranchWithVVs VV eqdec_vv tc avars &
+    Expr (sac_Γ scb Γ) (sac_Δ scb Γ avars (weakCK'' Δ)) 
+    (scbwv_ξ scb ξ lev) (weakLT' (tbranches @@  lev))}.
+    intros.
+    simpl in X.
+    destruct pcb.
+    simpl in *.
+    refine (bind ξvars = fresh_lemma' Γ pcb_freevars Σ [] ξ _ ; _). apply FreshMon.
+    destruct ξvars as [vars [ξ'
+  Defined.
+*)
+
+  Lemma itree_mapOptionTree : forall T T' F (f:T->T') t,
+    ITree _ F (mapOptionTree f t) ->
+    ITree _ (F ○ f) t.
+    intros.
+    induction t; try destruct a; simpl in *.
+      apply ILeaf.
+      inversion X; auto.
+      apply INone.
+      apply IBranch.
+        apply IHt1; inversion X; auto.
+        apply IHt2; inversion X; 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 judges2exprType H -> judges2exprType 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 _
-      | 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); intros.
+      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           => let case_RLetRec := tt       in _
+      end); intro X_; try apply ileaf in X_; simpl in X_.
 
   destruct case_RURule.
-    eapply urule2expr.
-    apply e.
-    apply X.
+    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; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
     apply EBrak.
-    inversion X.
-    set (X0 ξ vars H) as X'.
-    inversion X'.
-    apply X1.
+    apply (ileaf X).
 
   destruct case_REsc.
-    apply ILeaf; simpl; intros; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
     apply EEsc.
-    inversion X.
-    set (X0 ξ vars H) as X'.
-    inversion X'.
-    apply X1.
+    apply (ileaf X).
 
   destruct case_RNote.
-    apply ILeaf; simpl; intros; apply ILeaf.
-    inversion X.
-    apply ENote.
-    apply n.
-    simpl in X0.
-    set (X0 ξ vars H) as x1.
-    inversion x1.
-    apply X1.
+    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; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (return ILeaf _ _).
     apply ELit.
 
   destruct case_RVar.
-    apply ILeaf; simpl; intros; apply ILeaf.
+    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; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (return ILeaf _ _).
     apply EGlobal.
     apply wev.
 
   destruct case_RLam.
-    apply ILeaf; simpl; intros; apply ILeaf.
-    (* need a fresh variable here *)
-    admit.
+    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 ((⟨vnew, tx @@  x ⟩) :: 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; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
     eapply ECast.
     apply x.
-    inversion X.
-    simpl in X0.
-    set (X0 ξ vars H) as q.
-    inversion q.
-    apply X1.
+    apply ileaf in X. simpl in X.
+    apply X.
 
   destruct case_RBindingGroup.
     apply ILeaf; simpl; intros.
-    inversion X.
-    inversion X0.
-    inversion X1.
+    inversion X_.
+    apply ileaf in X.
+    apply ileaf in X0.
+    simpl in *.
     destruct vars; inversion H.
-    destruct o; inversion H5.
-    set (X2 _ _ H5) as q1.
-    set (X3 _ _ H6) as q2.
+    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; simpl; intros; apply ILeaf.
+    apply ILeaf.
+    inversion X_.
     inversion X.
     inversion X0.
-    inversion X1.
-    destruct vars; try destruct o; inversion H.
-    set (X2 _ _ H5) as q1.
-    set (X3 _ _ H6) as q2.
-    eapply EApp.
-      inversion q1.
-      apply X4.
-      inversion q2.
-      apply X4.
+    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; intros; apply ILeaf.
-    (* FIXME: need a var here, and other work *)
-    admit.
+    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 ((⟨vnew, σ₂ @@  p ⟩) :: 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; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars H >>>= fun X => return ILeaf _ _). apply FreshMon.
     apply ETyApp.
-    inversion X.
-    set (X0 _ _ H) as q.
-    inversion q.
-    apply X1.
+    apply (ileaf X).
 
   destruct case_RAbsT.
-    apply ILeaf; simpl; intros; apply ILeaf.
+    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.
-    inversion X.
-    simpl in *.
-    set (X0 (weakLT ○ ξ) vars) as q.
-    rewrite mapOptionTree_compose in q.
-    rewrite <- H in q.
-    set (q (refl_equal _)) as q'.
-    inversion q'.
-    apply X1.
+    apply X.
 
   destruct case_RAppCo.
-    apply ILeaf; simpl; intros; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
+    auto.
     eapply ECoApp.
     apply γ.
-    inversion X.
-    set (X0 _ _ H) as q.
-    inversion q.
-    apply X1.
+    apply (ileaf X).
 
   destruct case_RAbsCo.
-    apply ILeaf; simpl; intros; apply ILeaf.
+    apply ILeaf; simpl; intros; refine (X_ ξ vars _ >>>= fun X => return ILeaf _ _). apply FreshMon.
+    auto.
     eapply ECoLam.
-    inversion X.
-    set (X0 _ _ H) as q.
-    inversion q; auto.
+    apply (ileaf X).
 
   destruct case_RLetRec.
-    admit.
+    apply ILeaf; simpl; intros.
+    refine (bind ξvars = fresh_lemma' _ y _ _ _ H; _). apply FreshMon.
+    destruct ξvars as [ varstypes [ pf1 pf2 ]].
+    refine (X_ ((update_ξ ξ (leaves varstypes)))
+      (vars,,(mapOptionTree (@fst _ _) varstypes)) _ >>>= fun X => return _); clear X_.  apply FreshMon.
+    simpl.
+    rewrite pf2.
+    rewrite pf1.
+    auto.
+    apply ILeaf.
+    destruct x as [τ l].
+    inversion X; subst; clear X.
+
+    (* getting rid of this will require strengthening RLetRec *)
+    assert ((mapOptionTree (fun x : VV * LeveledHaskType Γ ★ => ⟨fst x, unlev (snd x) @@  l ⟩) varstypes) = varstypes) as HHH.
+      admit.
+
+    apply (@ELetRec _ _ _ _ _ _ _ (mapOptionTree (fun x => ((fst x),unlev (snd x))) varstypes));
+      rewrite mapleaves; rewrite <- map_compose; simpl;
+      [ idtac
+      | rewrite <- mapleaves; rewrite HHH; apply (ileaf X0) ].
+
+    clear X0.
+    rewrite <- mapOptionTree_compose in X1.
+    set (fun x : VV * LeveledHaskType Γ ★ => ⟨fst x, unlev (snd x) @@  l ⟩) as ξ' in *.
+    rewrite <- mapleaves.
+    rewrite HHH.
+
+    apply (letrec_helper _ _ _ _ _ X1).
 
   destruct case_RCase.
-    admit.
+  apply ILeaf.
+simpl.
+intros.
+apply (Prelude_error "FIXME").
 
-      Defined.
+
+(*
+    apply ILeaf; simpl; intros.
+    inversion X_.
+    clear X_.
+    subst.
+    apply ileaf in X0.
+    simpl in X0.
+    set (mapOptionTreeAndFlatten pcb_freevars alts) as Σalts in *.
+    refine (bind ξvars = fresh_lemma' _ (Σalts,,Σ) _ _ _ H ; _).
+      apply FreshMon.
+      destruct vars; try destruct o; inversion H; clear H.
+      rename vars1 into varsalts.
+      rename vars2 into varsΣ.
+    
+    refine (X0 ξ varsΣ _ >>>= fun X => return ILeaf _ _); auto. apply FreshMon.
+      clear X0.
+      eapply (ECase _ _ _ _ _ _ _ (ileaf X1)).
+      clear X1.
+
+      destruct ξvars as [varstypes [pf1 pf2]].
+      
+      apply itree_mapOptionTree in X.
+      refine (itree_to_tree (itmap _ X)).
+      apply case_helper.
+*)
+    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)
+      | 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.
-
-      destruct case_nil.
-        apply INone.
-
-      destruct case_rule.
-        eapply rule2expr.
-        apply r.
-        apply rest.
-
-      destruct case_branch.
-        apply IBranch.
-        apply rest1.
-        apply rest2.
         Defined.
 
-  Definition proof2expr Γ Δ τ Σ : ND Rule [] [Γ > Δ > Σ |- [τ]] -> { ξ:VV -> LeveledHaskType Γ ★ & Expr Γ Δ ξ τ }.
+  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.
-    inversion cnd; subst.
-    simpl in X.
-    clear cnd pf.
-    destruct X.
-    exists x.
-    inversion e.
-    subst.
-    apply X.
+    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.