X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FHaskProofStratified.v;h=177bd6d3911e9bacc798cf47077a93d71976fef1;hp=8f70b3113bd9da4cab63f7c572410a212b61b8be;hb=9e7ea73d3a6f4bbfba279164a806490cf95efec4;hpb=e3e2ce9cb83acdd8191049b4e9bd3d4fcf6a4db4

diff --git a/src/HaskProofStratified.v b/src/HaskProofStratified.v
index 8f70b31..177bd6d 100644
--- a/src/HaskProofStratified.v
+++ b/src/HaskProofStratified.v
@@ -55,18 +55,22 @@ Open Scope nd_scope.
  *)
 Section HaskProofStratified.
 
+  Section PCF.
+
   Context (ndr_systemfc:@ND_Relation _ Rule).
 
-  Inductive PCFJudg Γ (Δ:CoercionEnv Γ) (ec:HaskTyVar Γ ★) :=
-    pcfjudg : Tree ??(HaskType Γ ★) -> Tree ??(HaskType Γ ★) -> PCFJudg Γ Δ ec.
-    Implicit Arguments pcfjudg [ [Γ] [Δ] [ec] ].
+  Context Γ (Δ:CoercionEnv Γ).
+  Definition PCFJudg (ec:HaskTyVar Γ ★) :=
+    @prod (Tree ??(HaskType Γ ★)) (Tree ??(HaskType Γ ★)).
+  Definition pcfjudg (ec:HaskTyVar Γ ★) :=
+    @pair (Tree ??(HaskType Γ ★)) (Tree ??(HaskType Γ ★)).
 
   (* given an PCFJudg at depth (ec::depth) we can turn it into an PCFJudg
    * from depth (depth) by wrapping brackets around everything in the
    * succedent and repopulating *)
-  Definition brakify {Γ}{Δ}{ec} (j:PCFJudg Γ Δ ec) : Judg :=
+  Definition brakify {ec} (j:PCFJudg ec) : Judg :=
     match j with
-      pcfjudg Σ τ => Γ > Δ > (Σ@@@(ec::nil)) |- (mapOptionTree (fun t => HaskBrak ec t) τ @@@ nil)
+      (Σ,τ) => Γ > Δ > (Σ@@@(ec::nil)) |- (mapOptionTree (fun t => HaskBrak ec t) τ @@@ nil)
       end.
 
   Definition pcf_vars {Γ}(ec:HaskTyVar Γ ★)(t:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★)
@@ -77,13 +81,13 @@ Section HaskProofStratified.
                               end
       end) t.
 
-  Inductive MatchingJudgments {Γ}{Δ}{ec} : Tree ??(PCFJudg Γ Δ ec) -> Tree ??Judg -> Type :=
+  Inductive MatchingJudgments {ec} : Tree ??(PCFJudg ec) -> Tree ??Judg -> Type :=
     | match_nil    : MatchingJudgments [] []
     | match_branch : forall a b c d, MatchingJudgments a b -> MatchingJudgments c d -> MatchingJudgments (a,,c) (b,,d)
     | match_leaf   : 
       forall Σ τ lev,
         MatchingJudgments
-          [pcfjudg (pcf_vars ec Σ)                               τ         ]
+          [((pcf_vars ec Σ)         ,                              τ        )]
           [Γ > Δ >              Σ  |- (mapOptionTree (HaskBrak ec) τ @@@ lev)].
 
   Definition fc_vars {Γ}(ec:HaskTyVar Γ ★)(t:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★)
@@ -94,56 +98,69 @@ Section HaskProofStratified.
                               end
       end) t.
 
-  Definition pcfjudg2judg {Γ}{Δ:CoercionEnv Γ} ec (cj:PCFJudg Γ Δ ec) :=
-    match cj with pcfjudg Σ τ => Γ > Δ > (Σ @@@ (ec::nil)) |- (τ @@@ (ec::nil)) end.
+  Definition pcfjudg2judg ec (cj:PCFJudg ec) :=
+    match cj with (Σ,τ) => Γ > Δ > (Σ @@@ (ec::nil)) |- (τ @@@ (ec::nil)) end.
 
   (* Rules allowed in PCF; i.e. rules we know how to turn into GArrows     *)
   (* Rule_PCF consists of the rules allowed in flat PCF: everything except *)
   (* AppT, AbsT, AppC, AbsC, Cast, Global, and some Case statements        *)
-  Inductive Rule_PCF {Γ}{Δ:CoercionEnv Γ} (ec:HaskTyVar Γ ★)
-    : forall (h c:Tree ??(PCFJudg Γ Δ ec)), Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c) -> Type :=
-  | PCF_RArrange    : ∀ x y t     a,  Rule_PCF ec [pcfjudg _  _ ] [ pcfjudg _  _  ] (RArrange Γ Δ (x@@@(ec::nil)) (y@@@(ec::nil)) (t@@@(ec::nil)) a)
-  | PCF_RLit        : ∀ lit        ,  Rule_PCF ec [           ] [ pcfjudg [] [_] ] (RLit   Γ Δ  lit (ec::nil))
-  | PCF_RNote       : ∀ Σ τ   n    ,  Rule_PCF ec [pcfjudg _ [_]] [ pcfjudg _ [_] ] (RNote  Γ Δ  (Σ@@@(ec::nil)) τ         (ec::nil) n)
-  | PCF_RVar        : ∀ σ          ,  Rule_PCF ec [           ] [ pcfjudg [_] [_] ] (RVar   Γ Δ    σ         (ec::nil)  )
-  | PCF_RLam        : ∀ Σ tx te    ,  Rule_PCF ec [pcfjudg (_,,[_]) [_] ] [ pcfjudg _ [_] ] (RLam   Γ Δ  (Σ@@@(ec::nil)) tx te  (ec::nil)  )
+  Inductive Rule_PCF (ec:HaskTyVar Γ ★)
+    : forall (h c:Tree ??(PCFJudg ec)), Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c) -> Type :=
+  | PCF_RArrange    : ∀ x y t     a,  Rule_PCF ec [(_, _)] [(_, _)] (RArrange Γ Δ (x@@@(ec::nil)) (y@@@(ec::nil)) (t@@@(ec::nil)) a)
+  | PCF_RLit        : ∀ lit        ,  Rule_PCF ec [           ] [ ([],[_]) ] (RLit   Γ Δ  lit (ec::nil))
+  | PCF_RNote       : ∀ Σ τ   n    ,  Rule_PCF ec [(_,[_])] [(_,[_])] (RNote  Γ Δ  (Σ@@@(ec::nil)) τ         (ec::nil) n)
+  | PCF_RVar        : ∀ σ          ,  Rule_PCF ec [           ] [([_],[_])] (RVar   Γ Δ    σ         (ec::nil)  )
+  | PCF_RLam        : ∀ Σ tx te    ,  Rule_PCF ec [((_,,[_]),[_])] [(_,[_])] (RLam   Γ Δ  (Σ@@@(ec::nil)) tx te  (ec::nil)  )
 
   | PCF_RApp             : ∀ Σ Σ' tx te ,
-    Rule_PCF ec ([pcfjudg _ [_]],,[pcfjudg _ [_]]) [pcfjudg (_,,_) [_]]
+    Rule_PCF ec ([(_,[_])],,[(_,[_])]) [((_,,_),[_])]
     (RApp Γ Δ (Σ@@@(ec::nil))(Σ'@@@(ec::nil)) tx te (ec::nil))
 
   | PCF_RLet             : ∀ Σ Σ' σ₂   p,
-    Rule_PCF ec ([pcfjudg _ [_]],,[pcfjudg (_,,[_]) [_]]) [pcfjudg (_,,_) [_]]
+    Rule_PCF ec ([(_,[_])],,[((_,,[_]),[_])]) [((_,,_),[_])]
     (RLet Γ Δ (Σ@@@(ec::nil)) (Σ'@@@(ec::nil)) σ₂ p (ec::nil))
 
-  | PCF_REmptyGroup      :                 Rule_PCF ec [           ] [ pcfjudg []  [] ] (REmptyGroup   Γ Δ  )
+  | PCF_RVoid      :                 Rule_PCF ec [           ] [([],[])] (RVoid   Γ Δ  )
 (*| PCF_RLetRec          : ∀ Σ₁ τ₁ τ₂   ,  Rule_PCF (ec::nil) _ _ (RLetRec Γ Δ Σ₁ τ₁ τ₂ (ec::nil) )*)
-  | PCF_RBindingGroup    : ∀ Σ₁ Σ₂ τ₁ τ₂,  Rule_PCF ec ([pcfjudg _ _],,[pcfjudg _ _]) [pcfjudg (_,,_) (_,,_)]
-    (RBindingGroup Γ Δ (Σ₁@@@(ec::nil)) (Σ₂@@@(ec::nil)) (τ₁@@@(ec::nil)) (τ₂@@@(ec::nil))).
+  | PCF_RJoin    : ∀ Σ₁ Σ₂ τ₁ τ₂,  Rule_PCF ec ([(_,_)],,[(_,_)]) [((_,,_),(_,,_))]
+    (RJoin Γ Δ (Σ₁@@@(ec::nil)) (Σ₂@@@(ec::nil)) (τ₁@@@(ec::nil)) (τ₂@@@(ec::nil))).
   (* need int/boolean case *)
   Implicit Arguments Rule_PCF [ ].
 
-  Definition PCFRule Γ Δ lev h c := { r:_ & @Rule_PCF Γ Δ lev h c r }.
+  Definition PCFRule lev h c := { r:_ & @Rule_PCF lev h c r }.
+  End PCF.
+
+  Definition FCJudg Γ (Δ:CoercionEnv Γ) :=
+    @prod (Tree ??(LeveledHaskType Γ ★)) (Tree ??(LeveledHaskType Γ ★)).
+  Definition fcjudg2judg {Γ}{Δ}(fc:FCJudg Γ Δ) :=
+    match fc with
+      (x,y) => Γ > Δ > x |- y
+        end.
+  Coercion fcjudg2judg : FCJudg >-> Judg.
+
+  Definition pcfjudg2fcjudg {Γ}{Δ} ec (fc:PCFJudg Γ ec) : FCJudg Γ Δ :=
+    match fc with
+      (x,y) => (x @@@ (ec::nil),y @@@ (ec::nil))
+        end.
 
   (* An organized deduction has been reorganized into contiguous blocks whose
    * hypotheses (if any) and conclusion have the same Γ and Δ and a fixed nesting depth.  The boolean
    * indicates if non-PCF rules have been used *)
-  Inductive OrgR : Tree ??Judg -> Tree ??Judg -> Type :=
+  Inductive OrgR Γ Δ : Tree ??(FCJudg Γ Δ) -> Tree ??(FCJudg Γ Δ) -> Type :=
 
-  | org_fc        : forall h c (r:Rule h c),
+  | org_fc        : forall (h c:Tree ??(FCJudg Γ Δ))
+    (r:Rule (mapOptionTree fcjudg2judg h) (mapOptionTree fcjudg2judg c)),
     Rule_Flat r ->
-    OrgR h c
+    OrgR _ _ h c
 
-  | org_pcf      : forall Γ Δ ec h h' c c',
-    MatchingJudgments    h  h' ->
-    MatchingJudgments    c  c' ->
-    ND (PCFRule Γ Δ ec)  h  c  ->
-    OrgR                 h' c'.
+  | org_pcf      : forall ec h c,
+    ND (PCFRule Γ Δ ec)  h c  ->
+    OrgR        Γ Δ     (mapOptionTree (pcfjudg2fcjudg ec) h)  (mapOptionTree (pcfjudg2fcjudg ec) c).
 
-  Definition mkEsc {Γ}{Δ}{ec}(h:Tree ??(PCFJudg Γ Δ ec))
+  Definition mkEsc Γ Δ ec (h:Tree ??(PCFJudg Γ ec))
     : ND Rule
-    (mapOptionTree brakify h)
-    (mapOptionTree (pcfjudg2judg ec) h).
+    (mapOptionTree (brakify Γ Δ) h)
+    (mapOptionTree (pcfjudg2judg Γ Δ ec) h).
     apply nd_replicate; intros.
     destruct o; simpl in *.
     induction t0.
@@ -154,10 +171,10 @@ Section HaskProofStratified.
     apply (Prelude_error "mkEsc got multi-leaf succedent").
     Defined.
 
-  Definition mkBrak {Γ}{Δ}{ec}(h:Tree ??(PCFJudg Γ Δ ec))
+  Definition mkBrak Γ Δ ec (h:Tree ??(PCFJudg Γ ec))
     : ND Rule
-    (mapOptionTree (pcfjudg2judg ec) h)
-    (mapOptionTree brakify h).
+    (mapOptionTree (pcfjudg2judg Γ Δ ec) h)
+    (mapOptionTree (brakify Γ Δ) h).
     apply nd_replicate; intros.
     destruct o; simpl in *.
     induction t0.
@@ -175,8 +192,8 @@ Section HaskProofStratified.
       pcf_vars ec Σ = snd vars }.
       *)
 
-  Definition pcfToND : forall Γ Δ ec h c,
-    ND (PCFRule Γ Δ ec) h c -> ND Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c).
+  Definition pcfToND Γ Δ : forall ec h c,
+    ND (PCFRule Γ Δ ec) h c -> ND Rule (mapOptionTree (pcfjudg2judg Γ Δ ec) h) (mapOptionTree (pcfjudg2judg Γ Δ ec) c).
     intros.
     eapply (fun q => nd_map' _ q X).
     intros.
@@ -186,21 +203,8 @@ Section HaskProofStratified.
     Defined.
     
   Instance OrgPCF Γ Δ lev : @ND_Relation _ (PCFRule Γ Δ lev) :=
-    { ndr_eqv := fun a b f g => (pcfToND _ _ _ _ _ f) === (pcfToND _ _ _ _ _ g) }.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      Defined.
+    { ndr_eqv := fun a b f g => (pcfToND  _ _ _ _ _ f) === (pcfToND _ _ _ _ _ g) }.
+    Admitted.
 
   (*
    * An intermediate representation necessitated by Coq's termination
@@ -220,33 +224,33 @@ Section HaskProofStratified.
       Alternating c
 
     | alt_pcf    : forall Γ Δ ec h c h' c',
-      MatchingJudgments    h  h' ->
-      MatchingJudgments    c  c' ->
+      MatchingJudgments Γ Δ  h  h' ->
+      MatchingJudgments Γ Δ  c  c' ->
       Alternating h' ->
       ND (PCFRule Γ Δ ec) h c ->
       Alternating c'.
 
   Require Import Coq.Logic.Eqdep.
-
+(*
   Lemma magic a b c d ec e :
-    ClosedND(Rule:=Rule) [a > b > c |- [d @@  (ec :: e)]] ->
-    ClosedND(Rule:=Rule) [a > b > pcf_vars ec c @@@ (ec :: nil) |- [d @@  (ec :: nil)]].
-  admit.
-  Defined.
+    ClosedSIND(Rule:=Rule) [a > b > c |- [d @@  (ec :: e)]] ->
+    ClosedSIND(Rule:=Rule) [a > b > pcf_vars ec c @@@ (ec :: nil) |- [d @@  (ec :: nil)]].
+    admit.
+    Defined.
 
-  Definition orgify : forall Γ Δ Σ τ (pf:ClosedND(Rule:=Rule) [Γ > Δ > Σ |- τ]), Alternating [Γ > Δ > Σ |- τ].
+  Definition orgify : forall Γ Δ Σ τ (pf:ClosedSIND(Rule:=Rule) [Γ > Δ > Σ |- τ]), Alternating [Γ > Δ > Σ |- τ].
 
     refine (
-      fix  orgify_fc' Γ Δ Σ τ (pf:ClosedND [Γ > Δ > Σ |- τ]) {struct pf} : Alternating [Γ > Δ > Σ |- τ] :=
+      fix  orgify_fc' Γ Δ Σ τ (pf:ClosedSIND [Γ > Δ > Σ |- τ]) {struct pf} : Alternating [Γ > Δ > Σ |- τ] :=
         let case_main := tt in _
-      with orgify_fc c (pf:ClosedND c) {struct pf} : Alternating c :=
+      with orgify_fc c (pf:ClosedSIND c) {struct pf} : Alternating c :=
       (match c as C return C=c -> Alternating C with
         | T_Leaf None                    => fun _ => alt_nil
         | T_Leaf (Some (Γ > Δ > Σ |- τ)) => let case_leaf := tt in fun eqpf => _
         | T_Branch b1 b2                 => let case_branch := tt in fun eqpf => _
       end (refl_equal _))
-      with orgify_pcf   Γ Δ ec pcfj j (m:MatchingJudgments pcfj j)
-        (pf:ClosedND (mapOptionTree (pcfjudg2judg ec) pcfj)) {struct pf} : Alternating j :=
+      with orgify_pcf   Γ Δ ec pcfj j (m:MatchingJudgments Γ Δ pcfj j)
+        (pf:ClosedSIND (mapOptionTree (pcfjudg2judg Γ Δ ec) pcfj)) {struct pf} : Alternating j :=
         let case_pcf := tt in _
       for orgify_fc').
 
@@ -297,13 +301,13 @@ Section HaskProofStratified.
     Admitted.
 
   Definition pcfify Γ Δ ec : forall Σ τ,
-    ClosedND(Rule:=Rule) [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]
-      -> ND (PCFRule Γ Δ ec) [] [pcfjudg Σ τ].
+    ClosedSIND(Rule:=Rule) [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]
+      -> ND (PCFRule Γ Δ ec) [] [(Σ,τ)].
 
     refine ((
-      fix pcfify Σ τ (pn:@ClosedND _ Rule [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]) {struct pn}
-      : ND (PCFRule Γ Δ ec) [] [pcfjudg Σ τ] :=
-     (match pn in @ClosedND _ _ J return J=[Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)] -> _ with
+      fix pcfify Σ τ (pn:@ClosedSIND _ Rule [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]) {struct pn}
+      : ND (PCFRule Γ Δ ec) [] [(Σ,τ)] :=
+     (match pn in @ClosedSIND _ _ J return J=[Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)] -> _ with
       | cnd_weak             => let case_nil    := tt in _
       | cnd_rule h c cnd' r  => let case_rule   := tt in _
       | cnd_branch _ _ c1 c2 => let case_branch := tt in _
@@ -314,15 +318,14 @@ Section HaskProofStratified.
       destruct c; try destruct o; inversion H.
       destruct j.
       Admitted.
-
+*)
   (* any proof in organized form can be "dis-organized" *)
-  Definition unOrgR : forall h c, OrgR h c -> ND Rule h c.
+  (*
+  Definition unOrgR : forall Γ Δ h c, OrgR Γ Δ h c -> ND Rule h c.
     intros.
-
     induction X.
       apply nd_rule.
       apply r.
-
     eapply nd_comp.
       (*
       apply (mkEsc h).
@@ -331,52 +334,16 @@ Section HaskProofStratified.
       apply n.
       *)
       Admitted.
-
-  Definition unOrgND h c :  ND OrgR h c -> ND Rule h c := nd_map unOrgR.
+  Definition unOrgND Γ Δ h c :  ND (OrgR Γ Δ) h c -> ND Rule h c := nd_map (unOrgR Γ Δ).
+  *)
     
-  Instance OrgNDR : @ND_Relation _ OrgR :=
-    { ndr_eqv := fun a b f g => (unOrgND _ _ f) === (unOrgND _ _ g) }.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      admit.
-      Defined.
-
   Hint Constructors Rule_Flat.
 
-  Instance PCF_sequents Γ Δ lev : @SequentCalculus _ (PCFRule Γ Δ lev) _ pcfjudg.
-    apply Build_SequentCalculus.
-    intros.
-    induction a.
-    destruct a; simpl.
-    apply nd_rule.
-      exists (RVar _ _ _ _).
-      apply PCF_RVar.
-    apply nd_rule.
-      exists (REmptyGroup _ _ ).
-      apply PCF_REmptyGroup.
-    eapply nd_comp.
-      eapply nd_comp; [ apply nd_llecnac | idtac ].
-      apply (nd_prod IHa1 IHa2).
-      apply nd_rule.
-        exists (RBindingGroup _ _ _ _ _ _). 
-        apply PCF_RBindingGroup.
-        Defined.
-
-  Definition PCF_Arrange {Γ}{Δ}{lev} : forall x y z, Arrange x y -> ND (PCFRule Γ Δ lev) [pcfjudg x z] [pcfjudg y z].
+  Definition PCF_Arrange {Γ}{Δ}{lev} : forall x y z, Arrange x y -> ND (PCFRule Γ Δ lev) [(x,z)] [(y,z)].
     admit.
     Defined.
 
-  Definition PCF_cut Γ Δ lev : forall a b c, ND (PCFRule Γ Δ lev) ([ pcfjudg a b ],,[ pcfjudg b c ]) [ pcfjudg a c ].
+  Definition PCF_cut Γ Δ lev : forall a b c, ND (PCFRule Γ Δ lev) ([(a,b)],,[(b,c)]) [(a,c)].
     intros.
     destruct b.
     destruct o.
@@ -391,7 +358,7 @@ Section HaskProofStratified.
       apply RuCanL.
       eapply nd_comp; [ idtac | apply (PCF_Arrange ([],,a) a [h0]); apply RCanL ].
       apply nd_rule.
-(*
+      (*
       set (@RLet Γ Δ [] (a@@@(ec::nil)) h0 h (ec::nil)) as q.
       exists q.
       apply (PCF_RLet _ [] a h0 h).
@@ -399,103 +366,93 @@ Section HaskProofStratified.
     apply (Prelude_error "cut rule invoked with [a|=[b]] [[b]|=[x,,y]]").
     apply (Prelude_error "cut rule invoked with [a|=[]]  [[]|=c]").
     apply (Prelude_error "cut rule invoked with [a|=[b,,c]] [[b,,c]|=z]").
-*)
-    admit.
-    admit.
-    admit.
-    admit.
-    admit.
-    Defined.
+    *)
+    Admitted.
 
-  Instance PCF_cutrule Γ Δ lev : CutRule (PCF_sequents Γ Δ lev) :=
-    { nd_cut := PCF_cut Γ Δ lev }.
-    admit.
-    admit.
-    admit.
-    Defined.
+  Instance PCF_sequents Γ Δ lev ec : @SequentND _ (PCFRule Γ Δ lev) _ (pcfjudg Γ ec) :=
+    { snd_cut := PCF_cut Γ Δ lev }.
+    apply Build_SequentND.
+    intros.
+    induction a.
+    destruct a; simpl.
+    apply nd_rule.
+      exists (RVar _ _ _ _).
+      apply PCF_RVar.
+    apply nd_rule.
+      exists (RVoid _ _ ).
+      apply PCF_RVoid.
+    eapply nd_comp.
+      eapply nd_comp; [ apply nd_llecnac | idtac ].
+      apply (nd_prod IHa1 IHa2).
+      apply nd_rule.
+        exists (RJoin _ _ _ _ _ _). 
+        apply PCF_RJoin.
+      admit.
+        Defined.
 
-  Definition PCF_left Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (a,,b) (a,,c)].
+  Definition PCF_left Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [(b,c)] [((a,,b),(a,,c))].
     eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
-    eapply nd_prod; [ apply nd_seq_reflexive | apply nd_id ].
+    eapply nd_prod; [ apply snd_initial | apply nd_id ].
     apply nd_rule.
-    set (@PCF_RBindingGroup Γ Δ lev a b a c) as q'.
+    set (@PCF_RJoin Γ Δ lev a b a c) as q'.
     refine (existT _ _ _).
     apply q'.
-    Defined.
+    Admitted.
 
-  Definition PCF_right Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (b,,a) (c,,a)].
+  Definition PCF_right Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [(b,c)] [((b,,a),(c,,a))].
     eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
-    eapply nd_prod; [ apply nd_id | apply nd_seq_reflexive ].
+    eapply nd_prod; [ apply nd_id | apply snd_initial ].
     apply nd_rule.
-    set (@PCF_RBindingGroup Γ Δ lev b a c a) as q'.
+    set (@PCF_RJoin Γ Δ lev b a c a) as q'.
     refine (existT _ _ _).
     apply q'.
-    Defined.
-
-  Instance PCF_sequent_join Γ Δ lev : @SequentExpansion _ _ _ _ _ (PCF_sequents Γ Δ lev) (PCF_cutrule Γ Δ lev) :=
-  { se_expand_left  := PCF_left  Γ Δ lev
-  ; se_expand_right := PCF_right Γ Δ lev }.
-    admit.
-    admit.
-    admit.
-    admit.
-    Defined.
+    Admitted.
 
-  (* 5.1.3 *)
-  Instance PCF Γ Δ lev : @ProgrammingLanguage _ _ pcfjudg (PCFRule Γ Δ lev) :=
-  { pl_eqv                := OrgPCF Γ Δ lev
-  ; pl_sc                 := PCF_sequents Γ Δ lev
-  ; pl_subst              := PCF_cutrule Γ Δ lev
-  ; pl_sequent_join       := PCF_sequent_join Γ Δ lev
-  }.
-    apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
+  Instance PCF_sequent_join Γ Δ lev : @ContextND _ (PCFRule Γ Δ lev) _ (pcfjudg Γ lev) _ :=
+  { cnd_expand_left  := fun a b c => PCF_left  Γ Δ lev c a b
+  ; cnd_expand_right := fun a b c => PCF_right Γ Δ lev c a b }.
 
-    apply nd_rule. unfold PCFRule. simpl.
+    intros; apply nd_rule. unfold PCFRule. simpl.
       exists (RArrange _ _ _ _ _ (RCossa _ _ _)).
-      apply (PCF_RArrange lev ((a,,b),,c) (a,,(b,,c)) x).
+      apply (PCF_RArrange _ _ lev ((a,,b),,c) (a,,(b,,c)) x).
 
-    apply nd_rule. unfold PCFRule. simpl.
+    intros; apply nd_rule. unfold PCFRule. simpl.
       exists (RArrange _ _ _ _ _ (RAssoc _ _ _)).
-      apply (PCF_RArrange lev (a,,(b,,c)) ((a,,b),,c) x).
+      apply (PCF_RArrange _ _ lev (a,,(b,,c)) ((a,,b),,c) x).
 
-    apply nd_rule. unfold PCFRule. simpl.
+    intros; apply nd_rule. unfold PCFRule. simpl.
       exists (RArrange _ _ _ _ _ (RCanL _)).
-      apply (PCF_RArrange lev ([],,a) _ _).
+      apply (PCF_RArrange _ _ lev ([],,a) _ _).
 
-    apply nd_rule. unfold PCFRule. simpl.
+    intros; apply nd_rule. unfold PCFRule. simpl.
       exists (RArrange _ _ _ _ _ (RCanR _)).
-      apply (PCF_RArrange lev (a,,[]) _ _).
+      apply (PCF_RArrange _ _ lev (a,,[]) _ _).
 
-    apply nd_rule. unfold PCFRule. simpl.
+    intros; apply nd_rule. unfold PCFRule. simpl.
       exists (RArrange _ _ _ _ _ (RuCanL _)).
-      apply (PCF_RArrange lev _ ([],,a) _).
+      apply (PCF_RArrange _ _ lev _ ([],,a) _).
 
-    apply nd_rule. unfold PCFRule. simpl.
+    intros; apply nd_rule. unfold PCFRule. simpl.
       exists (RArrange _ _ _ _ _ (RuCanR _)).
-      apply (PCF_RArrange lev _ (a,,[]) _).
+      apply (PCF_RArrange _ _ lev _ (a,,[]) _).
       Defined.
 
-  Instance SystemFCa_sequents Γ Δ : @SequentCalculus _ OrgR _ (mkJudg Γ Δ).
-    apply Build_SequentCalculus.
-    intros.
-    induction a.
-    destruct a; simpl.
-    apply nd_rule.
-      destruct l.
-      apply org_fc with (r:=RVar _ _ _ _).
-      auto.
-    apply nd_rule.
-      apply org_fc with (r:=REmptyGroup _ _ ).
-      auto.
-    eapply nd_comp.
-      eapply nd_comp; [ apply nd_llecnac | idtac ].
-      apply (nd_prod IHa1 IHa2).
-      apply nd_rule.
-        apply org_fc with (r:=RBindingGroup _ _ _ _ _ _). 
-        auto.
-        Defined.
+  Instance OrgPCF_SequentND_Relation Γ Δ lev : SequentND_Relation (PCF_sequent_join Γ Δ lev) (OrgPCF Γ Δ lev).
+    admit.
+    Defined.
+
+  Definition OrgPCF_ContextND_Relation Γ Δ lev
+    : @ContextND_Relation _ _ _ _ _ (PCF_sequent_join Γ Δ lev) (OrgPCF Γ Δ lev) (OrgPCF_SequentND_Relation Γ Δ lev).
+    admit.
+    Defined.
 
-  Definition SystemFCa_cut Γ Δ : forall a b c, ND OrgR ([ Γ > Δ > a |- b ],,[ Γ > Δ > b |- c ]) [ Γ > Δ > a |- c ].
+  (* 5.1.3 *)
+  Instance PCF Γ Δ lev : ProgrammingLanguage :=
+  { pl_cnd     := PCF_sequent_join Γ Δ lev
+  ; pl_eqv     := OrgPCF_ContextND_Relation Γ Δ lev
+  }.
+
+  Definition SystemFCa_cut Γ Δ : forall a b c, ND (OrgR Γ Δ) ([(a,b)],,[(b,c)]) [(a,c)].
     intros.
     destruct b.
     destruct o.
@@ -503,11 +460,14 @@ Section HaskProofStratified.
     destruct o.
 
     (* when the cut is a single leaf and the RHS is a single leaf: *)
+    (*
     eapply nd_comp.
       eapply nd_prod.
       apply nd_id.
       eapply nd_rule.
-      apply org_fc with (r:=RArrange _ _ _ _ _ (RuCanL [l])).
+      set (@org_fc) as ofc.
+      set (RArrange Γ Δ _ _ _ (RuCanL [l0])) as rule.
+      apply org_fc with (r:=RArrange _ _ _ _ _ (RuCanL [_])).
       auto.
       eapply nd_comp; [ idtac | eapply nd_rule; apply org_fc with (r:=RArrange _ _ _ _ _ (RCanL _)) ].
       apply nd_rule.
@@ -518,58 +478,111 @@ Section HaskProofStratified.
       apply org_fc with (r:=@RLet Γ Δ [] a h1 h h2). 
       auto.
       auto.
+      *)
+    admit.
     apply (Prelude_error "systemfc cut rule invoked with [a|=[b]] [[b]|=[]]").
     apply (Prelude_error "systemfc cut rule invoked with [a|=[b]] [[b]|=[x,,y]]").
     apply (Prelude_error "systemfc rule invoked with [a|=[]]  [[]|=c]").
     apply (Prelude_error "systemfc rule invoked with [a|=[b,,c]] [[b,,c]|=z]").
     Defined.
 
-  Instance SystemFCa_cutrule Γ Δ : CutRule (SystemFCa_sequents Γ Δ) :=
-    { nd_cut := SystemFCa_cut Γ Δ }.
-    admit.
-    admit.
-    admit.
-    Defined.
+  Instance SystemFCa_sequents Γ Δ : @SequentND _ (OrgR Γ Δ) _ _ :=
+  { snd_cut := SystemFCa_cut Γ Δ }.
+    apply Build_SequentND.
+    intros.
+    induction a.
+    destruct a; simpl.
+    (*
+    apply nd_rule.
+      destruct l.
+      apply org_fc with (r:=RVar _ _ _ _).
+      auto.
+    apply nd_rule.
+      apply org_fc with (r:=RVoid _ _ ).
+      auto.
+    eapply nd_comp.
+      eapply nd_comp; [ apply nd_llecnac | idtac ].
+      apply (nd_prod IHa1 IHa2).
+      apply nd_rule.
+        apply org_fc with (r:=RJoin _ _ _ _ _ _). 
+        auto.
+      admit.
+      *)
+      admit.
+      admit.
+      admit.
+      admit.
+      Defined.
 
-  Definition SystemFCa_left Γ Δ a b c : ND OrgR [Γ > Δ > b |- c] [Γ > Δ > (a,,b) |- (a,,c)].
+  Definition SystemFCa_left Γ Δ a b c : ND (OrgR Γ Δ) [(b,c)] [((a,,b),(a,,c))].
+    admit.
+    (*
     eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
-    eapply nd_prod; [ apply nd_seq_reflexive | apply nd_id ].
+    eapply nd_prod; [ apply snd_initial | apply nd_id ].
     apply nd_rule.
-    apply org_fc with (r:=RBindingGroup Γ Δ a b a c).
+    apply org_fc with (r:=RJoin Γ Δ a b a c).
     auto.
+    *)
     Defined.
 
-  Definition SystemFCa_right Γ Δ a b c : ND OrgR [Γ > Δ > b |- c] [Γ > Δ > (b,,a) |- (c,,a)].
+  Definition SystemFCa_right Γ Δ a b c : ND (OrgR Γ Δ) [(b,c)] [((b,,a),(c,,a))].
+    admit.
+    (*
     eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
-    eapply nd_prod; [ apply nd_id | apply nd_seq_reflexive ].
+    eapply nd_prod; [ apply nd_id | apply snd_initial ].
     apply nd_rule.
-    apply org_fc with (r:=RBindingGroup Γ Δ b a c a).
+    apply org_fc with (r:=RJoin Γ Δ b a c a).
     auto.
+    *)
     Defined.
 
-  Instance SystemFCa_sequent_join Γ Δ : @SequentExpansion _ _ _ _ _ (SystemFCa_sequents Γ Δ) (SystemFCa_cutrule Γ Δ) :=
-  { se_expand_left  := SystemFCa_left  Γ Δ 
-  ; se_expand_right := SystemFCa_right Γ Δ }.
-    admit.
-    admit.
+  Instance SystemFCa_sequent_join Γ Δ : @ContextND _ _ _ _ (SystemFCa_sequents Γ Δ) :=
+  { cnd_expand_left  := fun a b c => SystemFCa_left Γ Δ c a b
+  ; cnd_expand_right := fun a b c => SystemFCa_right Γ Δ c a b }.
+    (*
+    intros; apply nd_rule. simpl.
+      apply (org_fc _ _ _ _ ((RArrange _ _ _ _ _ (RCossa _ _ _)))).
+      auto.
+
+    intros; apply nd_rule. simpl.
+      apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RAssoc _ _ _))); auto.
+
+    intros; apply nd_rule. simpl.
+      apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RCanL _))); auto.
+
+    intros; apply nd_rule. simpl.
+      apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RCanR _))); auto.
+
+    intros; apply nd_rule. simpl.
+      apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RuCanL _))); auto.
+
+    intros; apply nd_rule. simpl.
+      apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RuCanR _))); auto.
+      *)
+      admit.
+      admit.
+      admit.
+      admit.
+      admit.
+      admit.
+      Defined.
+
+  Instance OrgFC Γ Δ : @ND_Relation _ (OrgR Γ Δ).
+    Admitted.
+
+  Instance OrgFC_SequentND_Relation Γ Δ : SequentND_Relation (SystemFCa_sequent_join Γ Δ) (OrgFC Γ Δ).
     admit.
+    Defined.
+
+  Definition OrgFC_ContextND_Relation Γ Δ
+    : @ContextND_Relation _ _ _ _ _ (SystemFCa_sequent_join Γ Δ) (OrgFC Γ Δ) (OrgFC_SequentND_Relation Γ Δ).
     admit.
     Defined.
 
   (* 5.1.2 *)
-  Instance SystemFCa Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) OrgR :=
-  { pl_eqv                := OrgNDR
-  ; pl_sc                 := SystemFCa_sequents     Γ Δ
-  ; pl_subst              := SystemFCa_cutrule      Γ Δ
-  ; pl_sequent_join       := SystemFCa_sequent_join Γ Δ
+  Instance SystemFCa Γ Δ : @ProgrammingLanguage (LeveledHaskType Γ ★) _ :=
+  { pl_eqv                := OrgFC_ContextND_Relation Γ Δ
+  ; pl_snd                := SystemFCa_sequents Γ Δ
   }.
-    apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
-    apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCossa a b c))). apply Flat_RArrange.
-    apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RAssoc a b c))). apply Flat_RArrange.
-    apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanL  a    ))). apply Flat_RArrange.
-    apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanR  a    ))). apply Flat_RArrange.
-    apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanL a    ))). apply Flat_RArrange.
-    apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanR a    ))). apply Flat_RArrange.
-    Defined.
 
 End HaskProofStratified.