admit.
Defined.
- Definition brakifyType {Γ} (lt:LeveledHaskType Γ ★) : LeveledHaskType (★ ::Γ) ★ :=
+ Definition brakifyType {Γ} (v:HaskTyVar Γ ★)(lt:LeveledHaskType Γ ★) : LeveledHaskType Γ ★ :=
match lt with
- t @@ l => HaskBrak (FreshHaskTyVar ★) (weakT t) @@ weakL l
+ t @@ l => HaskBrak v t @@ l
end.
-
- Definition brakify (j:Judg) : Judg :=
+
+ Definition brakifyu {Γ}{Δ}(v:HaskTyVar Γ ★)(j:UJudg Γ Δ) : UJudg Γ Δ :=
match j with
- Γ > Δ > Σ |- τ =>
- (★ ::Γ) > weakCE Δ > mapOptionTree brakifyType Σ |- mapOptionTree brakifyType τ
+ mkUJudg Σ τ =>
+ Γ >> Δ > mapOptionTree (brakifyType v) Σ |- mapOptionTree (brakifyType v) τ
end.
- (* Rules allowed in PCF; i.e. rules we know how to turn into GArrows *)
- Section RulePCF.
+ 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.
+
+ Context (ndr_systemfc:@ND_Relation _ Rule).
- Context {Γ:TypeEnv}{Δ:CoercionEnv Γ}.
+ Open Scope nd_scope.
- (* Rule_PCF consists of the rules allowed in flat PCF: everything except
+ (* 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 : 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.
+ Inductive Rule_PCF {Γ:TypeEnv}{Δ:CoercionEnv Γ} : forall {h}{c}, Rule h c -> Type :=
+ | PCF_RURule : ∀ h c r , Rule_PCF (RURule Γ Δ h c r)
+ | 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 : ∀ 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 *)
+ | PCF_REmptyGroup : ∀ q a , Rule_PCF (REmptyGroup q a)
+ | PCF_RLetRec : ∀ Σ₁ τ₁ τ₂ lev , Rule_PCF (RLetRec Γ Δ Σ₁ τ₁ τ₂ lev).
+ Implicit Arguments Rule_PCF [ ].
+
+ Definition PCFR Γ Δ h c := { r:_ & Rule_PCF Γ Δ h c r }.
+
+ (* 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 (mapOptionTree UJudg2judg h) (mapOptionTree UJudg2judg 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).
+
+ (* 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).
- Section RuleSystemFC.
+ intros.
- Context {Γ:TypeEnv}{Δ:CoercionEnv Γ}.
+ induction X.
+ apply nd_rule.
+ destruct p.
+ apply x.
- (* "RuleSystemFCa n" is the type of rules permitted in SystemFC^\alpha with n-level-deep nesting
- * in a fixed Γ,Δ context. This is a subcategory of the "complete" SystemFCa, but it's enough to
- * do the flattening transformation *)
- 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, ND (@RulePCF Γ Δ n) h c -> RuleSystemFCa (S n) h c
- .
+ apply n0.
- Context (ndr_systemfca:forall n, @ND_Relation _ (RuleSystemFCa n)).
+ 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).
Defined.
Lemma systemfc_cutrule n
- : @CutRule _ (RuleSystemFCa n) _ (mkJudg Γ Δ) (ndr_systemfca n) (SystemFC_SC n).
+ : @CutRule _ (RuleSystemFCa n) _ (mkJudg Γ Δ) (ndr_systemfc n) (SystemFC_SC n).
apply Build_CutRule with (nd_cut:=systemfc_cut n).
admit.
admit.
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).
refine {| se_expand_left:=systemfc_left n
; se_expand_right:=systemfc_right n |}.
+*)
(* 5.1.2 *)
- Instance SystemFCa n : @ProgrammingLanguage _ Judg (mkJudg Γ Δ) (RuleSystemFCa n) :=
- { pl_eqv := ndr_systemfca n
+ 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 := SystemFC_SC n
- ; pl_subst := systemfc_cutrule n
- ; pl_sequent_join := systemfc_expansion n
+ ; pl_sc := _
+ ; pl_subst := _
+ ; pl_sequent_join := _
}.
apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
- apply sfc_flat with (r:=RURule _ _ _ _ (RCossa _ a b c)); auto. apply Flat_RURule.
- apply sfc_flat with (r:=RURule _ _ _ _ (RAssoc _ a b c)); auto. apply Flat_RURule.
- apply sfc_flat with (r:=RURule _ _ _ _ (RCanL _ a )); auto. apply Flat_RURule.
- apply sfc_flat with (r:=RURule _ _ _ _ (RCanR _ a )); auto. apply Flat_RURule.
- apply sfc_flat with (r:=RURule _ _ _ _ (RuCanL _ a )); auto. apply Flat_RURule.
- apply sfc_flat with (r:=RURule _ _ _ _ (RuCanR _ a )); auto. apply Flat_RURule.
-Show Existentials.
+ 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.
- 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.
+
+ (* 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
+ 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 *)
+
+
+ Definition magic {Γ}{Δ} h c n x :
+Rule_PCF Γ Δ (mapOptionTree (@UJudg2judg Γ Δ) h) (mapOptionTree (@UJudg2judg Γ Δ) c) x
+-> ND (OrgR true n Γ Δ) []
+ [Γ >> Δ > mapOptionTree guestJudgmentAsGArrowType h
+ |- mapOptionTree guestJudgmentAsGArrowType c].
+ admit.
+ Defined.
+
+ (*
+ * 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 G D 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 q a b c d e => let case_RBindingGroup := tt in _
+ | PCF_RCase T κlen κ θ l x => let case_RCase := tt in _
+ | PCF_REmptyGroup q a => let case_REmptyGroup := tt in _
+ | PCF_RLetRec Σ₁ τ₁ τ₂ lev => let case_RLetRec := tt in _
+ end).
+ rename Γ
+
+ apply (magic _ _ _ _ r0).
+ 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 > 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)).
- (* 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_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.
-
- 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.
+ 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 |}.
:= 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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