Require Import HaskProof.
Require Import HaskStrongToProof.
Require Import HaskProofToStrong.
-(*Require Import FreydCategories.*)
-(*Require Import ProgrammingLanguage.*)
+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 brakifyType {Γ} (lt:LeveledHaskType Γ ★) : LeveledHaskType (★ ::Γ) ★ :=
+ 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 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.
- Definition brakify (j:Judg) : Judg :=
- match j with
- Γ > Δ > Σ |- τ =>
- (★ ::Γ) > weakCE Δ > mapOptionTree brakifyType Σ |- mapOptionTree brakifyType τ
+ 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.
- (* Rules allowed in PCF; i.e. rules we know how to turn into GArrows *)
- Section RulePCF.
+ (*
+ * 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.
- Context {Γ:TypeEnv}{Δ:CoercionEnv Γ}.
+ 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.
- Inductive Rule_PCF : forall {h}{c}, Rule h c -> Prop :=
- | PCF_RURule : ∀ h c r , Rule_PCF (RURule Γ Δ h c r)
- | 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).
+ destruct case_RuCanL.
+ (* ga_uncancell *)
+ admit.
+
+ destruct case_RuCanR.
+ (* ga_uncancelr *)
+ admit.
+
+ destruct case_RAssoc.
+ (* ga_assoc *)
+ admit.
+
+ destruct case_RCossa.
+ (* ga_unassoc *)
+ admit.
- Inductive BoundedRule : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
+ 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.
- (* any proof using only PCF rules is an n-bounded proof for any n>0 *)
- | br_pcf : forall n h c (r:Rule h c), Rule_PCF r -> BoundedRule n h c
+ (* 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.
- (* any n-bounded proof may be used as an (n+1)-bounded proof by prepending Esc and appending Brak *)
- | br_nest : forall n h c, ND (BoundedRule n) h c -> BoundedRule (S n) (mapOptionTree brakify h) (mapOptionTree brakify c)
- .
+ destruct case_RLam.
+ (* ga_curry, but try to avoid this someday in the future if the argument type isn't a function *)
+ admit.
- Context (ndr:forall n, @ND_Relation _ (BoundedRule n)).
+ destruct case_RApp.
+ (* ga_apply *)
+ admit.
- (* for every n we have a category of n-bounded proofs *)
- Definition JudgmentsN n := @Judgments_Category_CartesianCat _ (BoundedRule n) (ndr n).
+ destruct case_RLet.
+ (* ga_comp! perhaps this means the ga_curry avoidance can be done by turning lambdas into lets? *)
+ admit.
- Open Scope nd_scope.
- Open Scope pf_scope.
+ destruct case_RBindingGroup.
+ (* ga_first+ga_second; technically this assumes a specific evaluation order, which is bad *)
+ admit.
- Definition TypesNmor (n:nat) (t1 t2:Tree ??(LeveledHaskType Γ ★)) : JudgmentsN n := [Γ > Δ > t1 |- t2].
- Definition TypesN_id n (t:Tree ??(LeveledHaskType Γ ★)) : ND (BoundedRule n) [] [ Γ > Δ > t |- t ].
+ 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 TypesN_comp n t1 t2 t3 : ND (BoundedRule n) ([Γ > nil > t1 |- t2],,[Γ > nil > t2 |- t3]) [ Γ > nil > t1 |- t3 ].
+
+ Definition exponent {Γ} : LeveledHaskType Γ ★ -> LeveledHaskType Γ ★ -> LeveledHaskType Γ ★.
admit.
Defined.
- Definition TypesN n : ECategory (JudgmentsN n) (Tree ??(LeveledHaskType Γ ★)) (TypesNmor n).
-(*
- apply {| eid := TypesN_id n ; ecomp := TypesN_comp n |}; intros; simpl.
- apply (@MonoidalCat_all_central _ _ (JudgmentsN n) _ _ _ (JudgmentsN n)).
- apply (@MonoidalCat_all_central _ _ (JudgmentsN n) _ _ _ (JudgmentsN n)).
- admit.
- admit.
-*)
+
+ 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 > Δ > [] |- [brakify Σ τ]
+ end.
+
+ Definition brakifyUJudg (j:Judg) : Judg :=
+ match j with
+ Γ > Δ > Σ |- τ =>
+ Γ > Δ > [] |- [brakify Σ τ]
+ end.
+ *)
+
+ End RuleSystemFC.
+
+ Context (ndr_pcf :forall n Γ Δ, @ND_Relation _ (@RulePCF Γ Δ n)).
+
+
+ Instance PCF n Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) (@RulePCF Γ Δ n) :=
+ { 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*)
+ ; pl_sequent_join := _ (*@SequentExpansion Judg Rule T sequent pl_eqv pl_sc pl_subst*)
+ }.
+ 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.
Defined.
- (* 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.
+ Definition SystemFCa_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
+ refine {| plsmme_pl := SystemFCa n Γ Δ |}.
admit.
Defined.
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.
+
+ (* 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 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 makeTree : Tree ??(LeveledHaskType Γ ★) -> HaskType Γ ★.
admit.
Defined.
Implicit Arguments Rule_PCF [ [h] [c] ].
Implicit Arguments BoundedRule [ ].
-
+*)
(*
Definition code2garrow0 {Γ}(ec t1 t2:RawHaskType Γ ★) : RawHaskType Γ ★.
admit.
:= 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),
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