X-Git-Url: http://git.megacz.com/?p=coq-hetmet.git;a=blobdiff_plain;f=src%2FHaskProofCategory.v;h=18da307b34816ce21eafa61ef2ca1a54aea9f9e1;hp=a608b33088810d341d343cf7d2750abb4d95e558;hb=fd337b235014f43000773eb0d95168d89e93a893;hpb=e15a8b6d72e0b28af765acfda7ddad21b50704ee

diff --git a/src/HaskProofCategory.v b/src/HaskProofCategory.v
index a608b33..18da307 100644
--- a/src/HaskProofCategory.v
+++ b/src/HaskProofCategory.v
@@ -40,77 +40,543 @@ Require Import HaskStrongToProof.
 Require Import HaskProofToStrong.
 Require Import ProgrammingLanguage.
 
+Open Scope nd_scope.
+
 Section HaskProofCategory.
 
-  Definition unitType {Γ} : RawHaskType Γ ★.
+  Context (ndr_systemfc:@ND_Relation _ Rule).
+
+  (* 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} : ∀ h c, Rule (mapOptionTree (@UJudg2judg Γ Δ) h) (mapOptionTree (@UJudg2judg Γ Δ) c) -> Type :=
+  | PCF_RArrange         : ∀ x y              ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RCanL t a              ))
+  | PCF_RCanR            : ∀ t a              ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RCanR t a              ))
+  | PCF_RuCanL           : ∀ t a              ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RuCanL t a             ))
+  | PCF_RuCanR           : ∀ t a              ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RuCanR t a             ))
+  | PCF_RAssoc           : ∀ t a b c          ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RAssoc t a b c         ))
+  | PCF_RCossa           : ∀ t a b c          ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RCossa t a b c         ))
+  | PCF_RLeft            : ∀ h c x            ,  Rule (mapOptionTree (ext_tree_left  x) h) (mapOptionTree (ext_tree_left  x) c)
+  | PCF_RRight           : ∀ h c x            ,  Rule (mapOptionTree (ext_tree_right x) h) (mapOptionTree (ext_tree_right x) c)
+  | PCF_RExch            : ∀ t a b            ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RExch t a b            ))
+  | PCF_RWeak            : ∀ t a              ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RWeak t a              ))
+  | PCF_RCont            : ∀ t a              ,  Rule_PCF  [_>>_>_|-_] [_>>_>_|-_] (RURule (RCont t a              ))
+
+  | PCF_RLit             : ∀ Σ τ              ,  Rule_PCF               [         ]  [_>>_>_|-_] (RLit   Γ Δ  Σ τ          )
+  | PCF_RNote            : ∀ Σ τ l n          ,  Rule_PCF               [_>>_>_|-_]  [_>>_>_|-_] (RNote  Γ Δ  Σ τ       l n)
+  | PCF_RVar             : ∀ σ               l,  Rule_PCF               [         ]  [_>>_>_|-_] (RVar   Γ Δ    σ         l)
+  | PCF_RLam             : ∀ Σ tx te    q     ,  Rule_PCF               [_>>_>_|-_]  [_>>_>_|-_] (RLam   Γ Δ  Σ tx te   q  )
+  | PCF_RApp             : ∀ Σ tx te   p     l,  Rule_PCF ([_>>_>_|-_],,[_>>_>_|-_]) [_>>_>_|-_] (RApp   Γ Δ  Σ tx te   p l)
+  | PCF_RLet             : ∀ Σ σ₁ σ₂   p     l,  Rule_PCF ([_>>_>_|-_],,[_>>_>_|-_]) [_>>_>_|-_] (RLet   Γ Δ  Σ σ₁ σ₂   p l)
+  | PCF_RBindingGroup    : ∀ b c d e          ,  Rule_PCF ([_>>_>_|-_],,[_>>_>_|-_]) [_>>_>_|-_] (RBindingGroup _ _ b c d e)
+  | PCF_REmptyGroup      :                       Rule_PCF  [                      ]  [_>>_>_|-_] (REmptyGroup   _ _        ).
+
+(*  | PCF_RLetRec          : ∀ Σ₁ τ₁ τ₂ lev     ,  Rule_PCF [_>>_>_|-_] [_>>_>_|-_] (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).*)
+  Implicit Arguments Rule_PCF [ ].
+
+(* need int/boolean case *)
+(*  | PCF_RCase            : ∀ T κlen κ θ l x   ,  Rule_PCF (RCase Γ Δ T κlen κ θ l x)   (* FIXME: only for boolean and int *)*)
+
+  Definition PCFR Γ Δ h c := { r:_ & Rule_PCF Γ Δ h c r }.
+
+  (* this wraps code-brackets, with the specified environment classifier, around a type *)
+  Definition brakifyType {Γ} (ec:HaskTyVar Γ ★)(lt:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
+    match lt with
+      t @@ l => HaskBrak ec t @@ l
+    end.
+ 
+  Definition brakifyu {Γ}{Δ}(v:HaskTyVar Γ ★)(j:UJudg Γ Δ) : UJudg Γ Δ :=
+    match j with
+      mkUJudg Σ τ =>
+        Γ >> Δ > mapOptionTree (brakifyType v) Σ |- mapOptionTree (brakifyType v) τ
+    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 : bool -> nat -> forall Γ Δ, Tree ??(UJudg Γ Δ) -> Tree ??(UJudg Γ Δ) -> Type :=
+
+  | org_pcf       : forall n Γ Δ h c b,
+    PCFR Γ Δ h c -> OrgR b n Γ Δ h c
+
+  | org_fc        : forall n Γ Δ h c,
+    ND Rule (mapOptionTree UJudg2judg h) (mapOptionTree UJudg2judg c) -> OrgR true n Γ Δ h c
+
+  | org_nest      : forall n Γ Δ h c b v,
+    OrgR b    n  Γ Δ h c ->
+    OrgR b (S n) _ _ (mapOptionTree (brakifyu v) h) (mapOptionTree (brakifyu v) c)
+  . 
+
+  Definition OrgND b n Γ Δ := ND (OrgR b n Γ Δ).
+
+  Definition mkEsc {Γ}{Δ}(j:Tree ??(UJudg Γ Δ)) v h
+    : ND Rule
+    (mapOptionTree (@UJudg2judg Γ Δ) h)
+    (mapOptionTree (fun j => @UJudg2judg Γ Δ (brakifyu v j)) h).
+    admit.
+    Defined.
+
+  Definition mkBrak {Γ}{Δ}(j:Tree ??(UJudg Γ Δ)) v h
+    : ND Rule
+    (mapOptionTree (fun j => @UJudg2judg Γ Δ (brakifyu v j)) h)
+    (mapOptionTree (@UJudg2judg Γ Δ           ) h).
+    admit.
+    Defined.
+
+  (* any proof in organized form can be "dis-organized" *)
+  Definition unOrgR b n Γ Δ : forall h c, OrgR b n Γ Δ h c ->
+    ND Rule (mapOptionTree (@UJudg2judg Γ Δ) h) (mapOptionTree (@UJudg2judg Γ Δ) c).
+
+    intros.
+
+    induction X.
+      apply nd_rule.
+      destruct p.
+      apply x.
+
+    apply n0.
+
+    rewrite <- mapOptionTree_compose.
+      rewrite <- mapOptionTree_compose.
+      eapply nd_comp.
+      apply (mkBrak h).
+      eapply nd_comp; [ idtac |  apply (mkEsc c) ].
+      apply IHX.
+      Defined.
+
+    Definition unOrgND b n Γ Δ h c : 
+      ND (OrgR b n Γ Δ) h c -> ND Rule (mapOptionTree (@UJudg2judg Γ Δ) h) (mapOptionTree (@UJudg2judg Γ Δ) c)
+      := nd_map' (@UJudg2judg Γ Δ) (unOrgR b n Γ Δ).
+    
+    Instance OrgNDR b n Γ Δ : @ND_Relation _ (OrgR b n Γ Δ) :=
+    { 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.
+
+    Definition SystemFC_SC n : @SequentCalculus _ (RuleSystemFCa n) _ (mkJudg Γ Δ).
+      apply Build_SequentCalculus.
+      intro a.
+      induction a.
+      destruct a.
+      apply nd_rule.
+      destruct l.
+      apply sfc_flat with (r:=RVar _ _ _ _).
+      auto.
+      apply nd_rule.
+      apply sfc_flat with (r:=REmptyGroup _ _).
+      auto.
+      eapply nd_comp; [ apply nd_llecnac | idtac ].
+      eapply nd_comp; [ eapply nd_prod | idtac ].
+      apply IHa1.
+      apply IHa2.
+      apply nd_rule.
+      apply sfc_flat with (r:=RBindingGroup _ _ _ _ _ _ ).
+      auto.
+      Defined.
+
+    Existing Instance SystemFC_SC.
+
+    Lemma systemfc_cut n : ∀a b c,
+           ND (RuleSystemFCa n) ([Γ > Δ > a |- b],, [Γ > Δ > b |- c]) [Γ > Δ > a |- c]. 
+      intros.
+      admit.
+      Defined.
+
+    Lemma systemfc_cutrule n
+      : @CutRule _ (RuleSystemFCa n) _ (mkJudg Γ Δ) (ndr_systemfc n) (SystemFC_SC n).
+      apply Build_CutRule with (nd_cut:=systemfc_cut n).
+      admit.
+      admit.
+      admit.
+      Defined.
+
+    Definition systemfc_left n a b c : ND (RuleSystemFCa n) [Γ > Δ > b |- c] [Γ > Δ > a,, b |- a,, c].
+      eapply nd_comp; [ apply nd_llecnac | idtac ].
+      eapply nd_comp; [ eapply nd_prod | idtac ].
+      Focus 3.
+      apply nd_rule.
+      apply sfc_flat with (r:=RBindingGroup _ _ _ _ _ _ ).
+      auto.
+      idtac.
+      apply nd_seq_reflexive.
+      apply nd_id.
+      Defined.
+
+    Definition systemfc_right n a b c : ND (RuleSystemFCa n) [Γ > Δ > b |- c] [Γ > Δ > b,,a |- c,,a].
+      eapply nd_comp; [ apply nd_rlecnac | idtac ].
+      eapply nd_comp; [ eapply nd_prod | idtac ].
+      apply nd_id.
+      apply (nd_seq_reflexive a).
+      apply nd_rule.
+      apply sfc_flat with (r:=RBindingGroup _ _ _ _ _ _ ).
+      auto.
+      Defined.
+*)
+(*
+    Definition systemfc_expansion n
+      : @SequentExpansion _ (RuleSystemFCa n) _ (mkJudg Γ Δ) (ndr_systemfca n) (SystemFC_SC n) (systemfc_cutrule n).
+    Check  (@Build_SequentExpansion).
+apply (@Build_SequentExpansion _ _ _ _ (ndr_systemfca n)  _ _ (systemfc_left n) (systemfc_right n)).
+      refine {| se_expand_left:=systemfc_left n
+        ; se_expand_right:=systemfc_right n |}.
+
+*)
+
+    (* 5.1.2 *)
+    Instance SystemFCa n Γ Δ : @ProgrammingLanguage _ _ (@mkUJudg Γ Δ) (OrgR true n Γ Δ) :=
+    { pl_eqv                := OrgNDR true n Γ Δ
+    ; pl_tsr                := _ (*@TreeStructuralRules Judg Rule T sequent*)
+    ; pl_sc                 := _
+    ; pl_subst              := _
+    ; pl_sequent_join       := _
+    }.
+      apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
+      apply nd_rule; apply org_fc; apply nd_rule; simpl.  apply  (RURule _ _ _ _ (RCossa _ a b c)).
+      apply nd_rule; apply org_fc; apply nd_rule; simpl;  apply  (RURule _ _ _ _ (RAssoc _ a b c)).
+      apply nd_rule; apply org_fc; apply nd_rule; simpl;  apply  (RURule _ _ _ _ (RCanL  _ a    )).
+      apply nd_rule; apply org_fc; apply nd_rule; simpl;  apply  (RURule _ _ _ _ (RCanR  _ a    )).
+      apply nd_rule; apply org_fc; apply nd_rule; simpl;  apply  (RURule _ _ _ _ (RuCanL _ a    )).
+      apply nd_rule; apply org_fc; apply nd_rule; simpl;  apply  (RURule _ _ _ _ (RuCanR _ a    )).
+      Admitted.
+
+    (* "flat" SystemFC: no brackets allowed *)
+    Instance SystemFC Γ Δ : @ProgrammingLanguage _ _ (@mkUJudg Γ Δ) (OrgR true O Γ Δ) :=
+    { pl_eqv                := OrgNDR true O Γ Δ
+    ; pl_tsr                := _ (*@TreeStructuralRules Judg Rule T sequent*)
+    ; pl_sc                 := _
+    ; pl_subst              := _
+    ; pl_sequent_join       := _
+    }.
+      Admitted.
+
+    (* 5.1.3 *)
+    Instance PCF n Γ Δ : @ProgrammingLanguage _ _ (@mkUJudg Γ Δ) (OrgR false n Γ Δ) :=
+    { pl_eqv                := OrgNDR false n Γ Δ
+    ; pl_tsr                := _ (*@TreeStructuralRules Judg Rule T sequent*)
+    ; pl_sc                 := _
+    ; pl_subst              := _
+    ; pl_sequent_join       := _
+    }.
+      apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
+      apply nd_rule; apply org_pcf; simpl; exists (RCossa _ a b c); apply (PCF_RURule [_>>_>_|-_] [_>>_>_|-_]).
+      apply nd_rule; apply org_pcf; simpl; exists (RAssoc _ a b c); apply (PCF_RURule [_>>_>_|-_] [_>>_>_|-_]).
+      apply nd_rule; apply org_pcf; simpl; exists (RCanL  _ a    ); apply (PCF_RURule [_>>_>_|-_] [_>>_>_|-_]).
+      apply nd_rule; apply org_pcf; simpl; exists (RCanR  _ a    ); apply (PCF_RURule [_>>_>_|-_] [_>>_>_|-_]).
+      apply nd_rule; apply org_pcf; simpl; exists (RuCanL _ a    ); apply (PCF_RURule [_>>_>_|-_] [_>>_>_|-_]).
+      apply nd_rule; apply org_pcf; simpl; exists (RuCanR _ a    ); apply (PCF_RURule [_>>_>_|-_] [_>>_>_|-_]).
+    Admitted.
+
+(*
+  Definition code2garrow Γ (ec t:RawHaskType Γ ★) :=
+      match t with
+(*        | TApp ★ ★ (TApp _ ★ TArrow tx) t' => code2garrow0 ec tx       t'*)
+        |                               _  => code2garrow0 ec unitType t
+      end.
+  Opaque code2garrow.
+  Fixpoint typeMap {TV}{κ}(ty:@RawHaskType TV κ) : @RawHaskType TV κ :=
+      match ty as TY in RawHaskType _ K return RawHaskType TV K with
+        | TCode ec t        => code2garrow _ ec t
+        | TApp _ _ t1 t2    => TApp (typeMap t1) (typeMap t2)
+        | TAll _ f          => TAll _ (fun tv => typeMap (f tv))
+        | TCoerc _ t1 t2 t3 => TCoerc (typeMap t1) (typeMap t2) (typeMap t3)
+        | TVar   _ v        => TVar v
+        | TArrow            => TArrow
+        | TCon  tc          => TCon tc 
+        | TyFunApp  tf rhtl => (* FIXME *) TyFunApp tf rhtl
+      end.
+          
+  Definition typeMapL {Γ}(lht:LeveledHaskType Γ ★) : LeveledHaskType Γ ★  :=
+    match lht with
+(*      | t @@ nil       => (fun TV ite => typeMap (t TV ite)) @@ lev*)
+      | t @@ lev => (fun TV ite => typeMap (t TV ite)) @@ lev
+    end.
+*)
+
+  (* gathers a tree of guest-language types into a single host-language types via the tensor *)
+  Definition tensorizeType {Γ} (lt:Tree ??(LeveledHaskType Γ ★)) : HaskType Γ ★.
     admit.
     Defined.
 
-  Definition brakifyType {Γ} (lt:LeveledHaskType Γ ★) : LeveledHaskType (★ ::Γ) ★ :=
+  Definition mkGA {Γ} : HaskType Γ ★ -> HaskType Γ ★ -> HaskType Γ ★. 
+    admit.
+    Defined.
+
+  Definition guestJudgmentAsGArrowType {Γ}{Δ} (lt:UJudg Γ Δ) : LeveledHaskType Γ ★ :=
     match lt with
-      t @@ l => HaskBrak (FreshHaskTyVar ★) (weakT t) @@ weakL l
+      mkUJudg x y =>
+      (mkGA (tensorizeType x) (tensorizeType y)) @@ nil
+    end.
+
+  Fixpoint obact n {Γ}{Δ}(X:Tree ??(UJudg Γ Δ)) : Tree ??(LeveledHaskType Γ ★) :=
+    match n with
+      | 0    => mapOptionTree guestJudgmentAsGArrowType X
+      | S n' => let t := obact n' X
+                in  [guestJudgmentAsGArrowType (Γ >> Δ > [] |- t )]
     end.
 
-  Definition brakify (j:Judg) : Judg :=
+  (*
+   * Here it is, what you've all been waiting for!  When reading this,
+   * it might help to have the definition for "Inductive ND" (see
+   * NaturalDeduction.v) handy as a cross-reference.
+   *)
+  Definition FlatteningFunctor_fmor {Γ}{Δ}
+    : forall h c,
+      (h~~{JudgmentsL _ _ (PCF 0 Γ Δ)}~~>c) ->
+      ((obact 0 h)~~{TypesL _ _ (SystemFC Γ Δ)}~~>(obact 0 c)).
+    unfold hom; unfold ob; unfold ehom; simpl; unfold mon_i; unfold obact; intros.
+
+    induction X; simpl.
+
+    (* the proof from no hypotheses of no conclusions (nd_id0) becomes REmptyGroup *)
+    apply nd_rule; apply org_fc; simpl. apply nd_rule. apply REmptyGroup.
+
+    (* the proof from hypothesis X of conclusion X (nd_id1) becomes RVar *)
+    apply nd_rule; apply org_fc; simpl. apply nd_rule. destruct (guestJudgmentAsGArrowType h). apply RVar.
+
+    (* the proof from hypothesis X of no conclusions (nd_weak) becomes RWeak;;REmptyGroup *)
+    apply nd_rule; apply org_fc; simpl.
+      eapply nd_comp; [ idtac | apply (nd_rule (RURule _ _ _ _ (RWeak _ _))) ].
+      apply nd_rule. apply REmptyGroup.      
+
+    (* the proof from hypothesis X of two identical conclusions X,,X (nd_copy) becomes RVar;;RBindingGroup;;RCont *)
+    eapply nd_comp; [ idtac | eapply nd_rule; eapply org_fc; apply (nd_rule (RURule _ _ _ _ (RCont _ _))) ].
+      eapply nd_comp; [ apply nd_llecnac | idtac ].
+      set (nd_seq_reflexive(SequentCalculus:=@pl_sc  _ _ _ _ (SystemFC Γ Δ)) (mapOptionTree guestJudgmentAsGArrowType h)) as q.
+      eapply nd_comp.
+      eapply nd_prod.
+      apply q.
+      apply q.
+      apply nd_rule; eapply org_fc.
+      simpl.
+      apply nd_rule.
+      apply RBindingGroup.
+
+    (* nd_prod becomes nd_llecnac;;nd_prod;;RBindingGroup *)
+    eapply nd_comp.
+      apply (nd_llecnac ;; nd_prod IHX1 IHX2).
+      apply nd_rule; apply org_fc; simpl.
+      eapply nd_rule. apply RBindingGroup.
+
+    (* nd_comp becomes pl_subst (aka nd_cut) *)
+    eapply nd_comp.
+      apply (nd_llecnac ;; nd_prod IHX1 IHX2).
+      clear IHX1 IHX2 X1 X2.
+      apply (@nd_cut _ _ _ _ _ _ (@pl_subst _ _ _ _ (SystemFC Γ Δ))).
+
+    (* nd_cancell becomes RVar;;RuCanL *)
+    eapply nd_comp;
+      [ idtac | eapply nd_rule; apply org_fc; simpl; apply nd_rule; apply (RURule _ _ _ _ (RuCanL _ _)) ].
+      apply (nd_seq_reflexive(SequentCalculus:=@pl_sc  _ _ _ _ (SystemFC Γ Δ))).
+
+    (* nd_cancelr becomes RVar;;RuCanR *)
+    eapply nd_comp;
+      [ idtac | eapply nd_rule; apply org_fc; simpl; apply nd_rule; apply (RURule _ _ _ _ (RuCanR _ _)) ].
+      apply (nd_seq_reflexive(SequentCalculus:=@pl_sc  _ _ _ _ (SystemFC Γ Δ))).
+
+    (* nd_llecnac becomes RVar;;RCanL *)
+    eapply nd_comp;
+      [ idtac | eapply nd_rule; apply org_fc; simpl; apply nd_rule; apply (RURule _ _ _ _ (RCanL _ _)) ].
+      apply (nd_seq_reflexive(SequentCalculus:=@pl_sc  _ _ _ _ (SystemFC Γ Δ))).
+
+    (* nd_rlecnac becomes RVar;;RCanR *)
+    eapply nd_comp;
+      [ idtac | eapply nd_rule; apply org_fc; simpl; apply nd_rule; apply (RURule _ _ _ _ (RCanR _ _)) ].
+      apply (nd_seq_reflexive(SequentCalculus:=@pl_sc  _ _ _ _ (SystemFC Γ Δ))).
+
+    (* nd_assoc becomes RVar;;RAssoc *)
+    eapply nd_comp;
+      [ idtac | eapply nd_rule; apply org_fc; simpl; apply nd_rule; apply (RURule _ _ _ _ (RAssoc _ _ _ _)) ].
+      apply (nd_seq_reflexive(SequentCalculus:=@pl_sc  _ _ _ _ (SystemFC Γ Δ))).
+
+    (* nd_coss becomes RVar;;RCossa *)
+    eapply nd_comp;
+      [ idtac | eapply nd_rule; apply org_fc; simpl; apply nd_rule; apply (RURule _ _ _ _ (RCossa _ _ _ _)) ].
+      apply (nd_seq_reflexive(SequentCalculus:=@pl_sc  _ _ _ _ (SystemFC Γ Δ))).
+
+    (* now, the interesting stuff: the inference rules of Judgments(PCF) become GArrow-parameterized terms *)
+      refine (match r as R in OrgR B N G D H C return
+                match N with
+                  | S _ => True
+                  | O   => if B then True
+                           else ND (OrgR true 0 G D)
+                                  []
+                                  [G >> D > mapOptionTree guestJudgmentAsGArrowType H |- mapOptionTree guestJudgmentAsGArrowType C]
+                end with
+                | org_pcf  n Γ Δ h c b r => _
+                | org_fc   n Γ Δ h c r => _
+                | org_nest n Γ Δ h c b v q => _
+              end); destruct n; try destruct b; try apply I.
+      destruct r0.
+
+      clear r h c Γ Δ.
+      rename r0 into r; rename h0 into h; rename c0 into c; rename Γ0 into Γ; rename Δ0 into Δ.
+
+      refine (match r as R in Rule_PCF _ _ H C _ with
+                | PCF_RURule           h c r            => let case_RURule := tt in _
+                | PCF_RLit             Σ τ              => let case_RLit := tt in _
+                | PCF_RNote            Σ τ l n          => let case_RNote := tt in _
+                | PCF_RVar             σ               l=> let case_RVar := tt in _
+                | PCF_RLam             Σ tx te    q     => let case_RLam := tt in _
+                | PCF_RApp             Σ tx te   p     l=> let case_RApp := tt in _
+                | PCF_RLet             Σ σ₁ σ₂   p     l=> let case_RLet := tt in _
+                | PCF_RBindingGroup    b c d e          => let case_RBindingGroup := tt in _
+                | PCF_REmptyGroup                       => let case_REmptyGroup := tt in _
+(*                | PCF_RCase            T κlen κ θ l x   => let case_RCase := tt in _*)
+(*                | PCF_RLetRec          Σ₁ τ₁ τ₂ lev     => let case_RLetRec := tt in _*)
+              end ); simpl in *.
+      clear x r h c.
+      rename r0 into r; rename h0 into h; rename c0 into c.
+
+      destruct case_RURule.
+        refine (match r with
+          | RLeft   a b c r => let case_RLeft  := tt in _
+          | RRight  a b c r => let case_RRight := tt in _
+          | RCanL   a b     => let case_RCanL  := tt in _
+          | RCanR   a b     => let case_RCanR  := tt in _
+          | RuCanL  a b     => let case_RuCanL := tt in _
+          | RuCanR  a b     => let case_RuCanR := tt in _
+          | RAssoc  a b c d => let case_RAssoc := tt in _
+          | RCossa  a b c d => let case_RCossa := tt in _
+          | RExch   a b c   => let case_RExch  := tt in _
+          | RWeak   a b     => let case_RWeak  := tt in _
+          | RCont   a b     => let case_RCont  := tt in _
+        end).
+
+      destruct case_RCanL.
+        (* ga_cancell *)
+        admit.
+        
+      destruct case_RCanR.
+        (* ga_cancelr *)
+        admit.
+
+      destruct case_RuCanL.
+        (* ga_uncancell *)
+        admit.
+        
+      destruct case_RuCanR.
+        (* ga_uncancelr *)
+        admit.
+        
+      destruct case_RAssoc.
+        (* ga_assoc *)
+        admit.
+        
+      destruct case_RCossa.
+        (* ga_unassoc *)
+        admit.
+
+      destruct case_RLeft.
+        (* ga_second *)
+        admit.
+        
+      destruct case_RRight.
+        (* ga_first *)
+        admit.
+        
+      destruct case_RExch.
+        (* ga_swap *)
+        admit.
+        
+      destruct case_RWeak.
+        (* ga_drop *)
+        admit.
+        
+      destruct case_RCont.
+        (* ga_copy *)
+        admit.
+        
+      destruct case_RLit.
+        (* ga_literal *)
+        admit.
+
+      (* hey cool, I figured out how to pass CoreNote's through... *)
+      destruct case_RNote.
+        apply nd_rule.
+        apply org_fc.
+        eapply nd_comp.
+        eapply nd_rule.
+        apply RVar.
+        apply nd_rule.
+        apply RNote.
+        apply n.
+        
+      destruct case_RVar.
+        (* ga_id *)
+        admit.
+
+      destruct case_RLam.
+        (* ga_curry, but try to avoid this someday in the future if the argument type isn't a function *)
+        admit.
+
+      destruct case_RApp.
+        (* ga_apply *)
+        admit.
+
+      destruct case_RLet.
+        (* ga_comp! perhaps this means the ga_curry avoidance can be done by turning lambdas into lets? *)
+        admit.
+
+      destruct case_RBindingGroup.
+        (* ga_first+ga_second; technically this assumes a specific evaluation order, which is bad *)
+        admit.
+
+      destruct case_REmptyGroup.
+        (* ga_id u *)
+        admit.
+      Defined.
+
+  Instance FlatteningFunctor {n}{Γ}{Δ} : Functor (JudgmentsL _ _ (PCF n Γ Δ)) (TypesL _ _ (SystemFCa n Γ Δ)) obact :=
+    { fmor := FlatteningFunctor_fmor }.
+    unfold hom; unfold ob; unfold ehom; intros; simpl.
+
+
+  Definition productifyType {Γ} (lt:Tree ??(LeveledHaskType Γ ★)) : LeveledHaskType Γ ★.
+    admit.
+    Defined.
+
+  Definition exponent {Γ} : LeveledHaskType Γ ★ -> LeveledHaskType Γ ★ -> LeveledHaskType Γ ★. 
+    admit.
+    Defined.
+
+  Definition brakify {Γ}(Σ τ:Tree ??(LeveledHaskType Γ ★))  := exponent (productifyType Σ) (productifyType τ).
+
+  Definition brakifyJudg (j:Judg) : Judg :=
     match j with
       Γ > Δ > Σ |- τ =>
-        (★ ::Γ) > weakCE Δ > mapOptionTree brakifyType Σ |- mapOptionTree brakifyType τ
+        Γ > Δ > [] |- [brakify Σ τ]
     end.
 
-  (* Rules allowed in PCF; i.e. rules we know how to turn into GArrows *)
-  Section RulePCF.
-
-    Context {Γ:TypeEnv}{Δ:CoercionEnv Γ}.
-
-    (* 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 : forall {h}{c}, Rule h c -> Prop :=
-    | PCF_RURule           : ∀ h c r            ,  Rule_PCF (RURule Γ Δ  h c r)
-    | PCF_RLit             : ∀ Γ Δ Σ τ          ,  Rule_PCF (RLit Γ Δ Σ τ  )
-    | PCF_RNote            : ∀ Σ τ l n          ,  Rule_PCF (RNote Γ Δ Σ τ l n)
-    | PCF_RVar             : ∀ σ               l,  Rule_PCF (RVar Γ Δ  σ         l)
-    | PCF_RLam             : ∀ Σ tx te    q     ,  Rule_PCF (RLam Γ Δ  Σ tx te      q )
-    | PCF_RApp             : ∀ Σ tx te   p     l,  Rule_PCF (RApp Γ Δ  Σ tx te   p l)
-    | PCF_RLet             : ∀ Σ σ₁ σ₂   p     l,  Rule_PCF (RLet Γ Δ  Σ σ₁ σ₂   p l)
-    | PCF_RBindingGroup    : ∀ q a b c d e      ,  Rule_PCF (RBindingGroup q a b c d e)
-    | PCF_RCase            : ∀ T κlen κ θ l x   ,  Rule_PCF (RCase Γ Δ T κlen κ θ l x)   (* FIXME: only for boolean and int *)
-    | Flat_REmptyGroup     : ∀ q a              ,  Rule_PCF (REmptyGroup q a)
-    | Flat_RLetRec         : ∀ Γ Δ Σ₁ τ₁ τ₂ lev ,  Rule_PCF (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
-
-    (* "RulePCF n" is the type of rules permitted in PCF with n-level deep nesting (so, "RulePCF 0" is flat PCF) *)
-    Inductive RulePCF : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
-
-    (* any proof using only PCF rules is an n-bounded proof for any n>0 *)
-    | pcf_flat : forall n h c (r:Rule h c), Rule_PCF  r  -> RulePCF n h c
-
-    (* any n-bounded proof may be used as an (n+1)-bounded proof by prepending Esc and appending Brak *)
-    | pfc_nest : forall n h c, ND (RulePCF n) h c    -> RulePCF (S n) (mapOptionTree brakify h) (mapOptionTree brakify c)
-    .
-  End RulePCF.
+  Definition brakifyUJudg (j:Judg) : Judg :=
+    match j with
+      Γ > Δ > Σ |- τ =>
+        Γ > Δ > [] |- [brakify Σ τ]
+    end.
+  *)
+
+  End RuleSystemFC.
 
-  (* "RuleSystemFCa n" is the type of rules permitted in SystemFC^\alpha with n-level-deep nesting
-   * (so, "RuleSystemFCa 0" is damn close to System FC) *)
-  Inductive RuleSystemFCa : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
-  | sfc_flat : forall n h c     (r:Rule h c), Rule_Flat r -> RuleSystemFCa    n  h c
-  | sfc_nest : forall n h c Γ Δ, ND (@RulePCF Γ Δ n) h c  -> RuleSystemFCa (S n) h c
-  .
-  
   Context (ndr_pcf      :forall n Γ Δ, @ND_Relation _ (@RulePCF Γ Δ n)).
-  Context (ndr_systemfca:forall n, @ND_Relation _ (RuleSystemFCa n)).
 
-  (* 5.1.3 *)
+
   Instance PCF n Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) (@RulePCF Γ Δ n) :=
-  { pl_eqv                := _ (*@ND_Relation Judg Rule where "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2)*)
-  ; pl_tsr                := _ (*@TreeStructuralRules Judg Rule T sequent*)
-  ; pl_sc                 := _ (*@SequentCalculus Judg Rule _ sequent*)
-  ; pl_subst              := _ (*@CutRule Judg Rule _ sequent pl_eqv pl_sc*)
-  ; pl_sequent_join       := _ (*@SequentExpansion Judg Rule T sequent pl_eqv pl_sc pl_subst*)
-  }.
-  Admitted.
-  
-  (* 5.1.2 *)
-  Instance SystemFCa n Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) (RuleSystemFCa n) :=
-  { pl_eqv                := _ (*@ND_Relation Judg Rule where "pf1 === pf2" := (@ndr_eqv _ _ pl_eqv _ _ pf1 pf2)*)
+  { pl_eqv                := _
   ; pl_tsr                := _ (*@TreeStructuralRules Judg Rule T sequent*)
   ; pl_sc                 := _ (*@SequentCalculus Judg Rule _ sequent*)
   ; pl_subst              := _ (*@CutRule Judg Rule _ sequent pl_eqv pl_sc*)
@@ -118,6 +584,49 @@ Section HaskProofCategory.
   }.
   Admitted.
 
+  Inductive RuleX : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
+  | x_flat : forall n     h c   (r:Rule h c), Rule_Flat r -> RuleX    n  h c
+  | x_nest : forall n Γ Δ h c,  ND (@RulePCF Γ Δ n) h c   ->
+    RuleX (S n) (mapOptionTree brakifyJudg h) (mapOptionTree brakifyJudg c).
+
+  Section X.
+
+    Context (n:nat).
+    Context (ndr:@ND_Relation _ (RuleX (S n))).
+
+    Definition SystemFCa' := Judgments_Category ndr.
+
+    Definition ReificationFunctor_fmor Γ Δ
+      : forall h c,
+        (h~~{JudgmentsL _ _ (PCF n Γ Δ)}~~>c) ->
+        ((mapOptionTree brakifyJudg h)~~{SystemFCa'}~~>(mapOptionTree brakifyJudg c)).
+      unfold hom; unfold ob; simpl.
+      intros.
+      apply nd_rule.
+      eapply x_nest.
+      apply X.
+      Defined.
+
+    Definition ReificationFunctor Γ Δ : Functor (JudgmentsL _ _ (PCF n Γ Δ)) SystemFCa' (mapOptionTree brakifyJudg).
+      refine {| fmor := ReificationFunctor_fmor Γ Δ |}; unfold hom; unfold ob; simpl ; intros.
+      unfold ReificationFunctor_fmor; simpl.
+      admit.
+      unfold ReificationFunctor_fmor; simpl.
+      admit.
+      unfold ReificationFunctor_fmor; simpl.
+      admit.
+      Defined.
+
+    Definition FlatteningFunctor Γ Δ : Functor (JudgmentsL _ _ (PCF n Γ Δ)) SystemFCa' (mapOptionTree brakify).
+      refine {| fmor := ReificationFunctor_fmor Γ Δ |}; unfold hom; unfold ob; simpl ; intros.
+      unfold ReificationFunctor_fmor; simpl.
+      admit.
+      unfold ReificationFunctor_fmor; simpl.
+      admit.
+      unfold ReificationFunctor_fmor; simpl.
+      admit.
+      Defined.
+
   Definition PCF_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
     refine {| plsmme_pl := PCF n Γ Δ |}.
     admit.
@@ -128,30 +637,26 @@ Section HaskProofCategory.
     admit.
     Defined.
 
+  Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n).
+    admit.
+    Defined.
+
   (* 5.1.4 *)
   Definition PCF_SystemFCa_two_level n Γ Δ : TwoLevelLanguage (PCF_SMME n Γ Δ) (SystemFCa_SMME (S n) Γ Δ).
     admit.
+    (*  ... and the retraction exists *)
     Defined.
-  (*  ... and the retraction exists *)
 
   (* Any particular proof in HaskProof is only finitely large, so it uses only finitely many levels of nesting, so
    * it falls within (SystemFCa n) for some n.  This function calculates that "n" and performs the translation *)
-  (*Definition HaskProof_to_SystemFCa :
-    forall h c (pf:ND Judg h c),
-      { n:nat & h ~~{JudgmentsL SystemFCa_SMME n *)
+  (*
+  Definition HaskProof_to_SystemFCa :
+    forall h c (pf:ND Rule h c),
+      { n:nat & h ~~{JudgmentsL (SystemFCa_SMME n)}~~> c }.
+      *)
 
   (* for every n we have a functor from the category of (n+1)-bounded proofs to the category of n-bounded proofs *)
-    (*
-  Definition ReificationFunctor n : Functor (JudgmentsN n) (JudgmentsN (S n)) (mapOptionTree brakify).
-    refine {| fmor := fun h c (f:h~~{JudgmentsN n}~~>c) => nd_rule (br_nest _ _ _ f) |}; intros; simpl.
-    admit.
-    admit.
-    admit.
-    Defined.
 
-  Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n).
-    admit.
-    Defined.
 
   Definition makeTree : Tree ??(LeveledHaskType Γ ★) -> HaskType Γ ★.
     admit.
@@ -484,35 +989,6 @@ idtac.
         := mapOptionTree (fun q:Tree ??(@CoreType V) * Tree ??(@CoreType V) =>
           let (a,s):=q in (Σ₁,,(``a)^^^n,[`<[ n |- encodeTypeTree_flat s ]>])).
    
-      Definition append_brak
-      : forall {c}, ND_ML
-          (mapOptionTree (ob2judgment_flat ((⟨Σ₁,n⟩) :: past))                        c )
-          (mapOptionTree (ob2judgment                   past )  (EscBrak_Functor_Fobj c)).
-        intros.
-        unfold ND_ML.
-        unfold EscBrak_Functor_Fobj.
-        rewrite mapOptionTree_comp.
-        simpl in *.
-        apply nd_replicate.
-        intro o; destruct o.
-        apply nd_rule.
-        apply MLRBrak.
-        Defined.
-    
-      Definition prepend_esc
-      : forall {h}, ND_ML
-          (mapOptionTree (ob2judgment                  past )  (EscBrak_Functor_Fobj h))
-          (mapOptionTree (ob2judgment_flat ((⟨Σ₁,n⟩) :: past))                        h ).
-        intros.
-        unfold ND_ML.
-        unfold EscBrak_Functor_Fobj.
-        rewrite mapOptionTree_comp.
-        simpl in *.
-        apply nd_replicate.
-        intro o; destruct o.
-        apply nd_rule.
-        apply MLREsc.
-        Defined.
     
       Definition EscBrak_Functor_Fmor
         : forall a b (f:a~~{SystemFC_Cat_Flat ((Σ₁,n)::past)}~~>b),