Require Import HaskProof.
Require Import HaskStrongToProof.
Require Import HaskProofToStrong.
-(*Require Import FreydCategories.*)
-(*Require Import ProgrammingLanguage.*)
+Require Import ProgrammingLanguage.
Section HaskProofCategory.
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).
+ | 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).
- Inductive BoundedRule : nat -> Tree ??Judg -> Tree ??Judg -> Type :=
+ (* "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 *)
- | br_pcf : forall n h c (r:Rule h c), Rule_PCF r -> BoundedRule n h c
+ | 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 *)
- | br_nest : forall n h c, ND (BoundedRule n) h c -> BoundedRule (S n) (mapOptionTree brakify h) (mapOptionTree brakify c)
+ | pfc_nest : forall n h c, ND (RulePCF n) h c -> RulePCF (S n) (mapOptionTree brakify h) (mapOptionTree brakify c)
.
+ End RulePCF.
- Context (ndr:forall n, @ND_Relation _ (BoundedRule n)).
+ Section RuleSystemFC.
- (* for every n we have a category of n-bounded proofs *)
- Definition JudgmentsN n := @Judgments_Category_CartesianCat _ (BoundedRule n) (ndr n).
+ Context {Γ:TypeEnv}{Δ:CoercionEnv Γ}.
- Open Scope nd_scope.
- Open Scope pf_scope.
+ (* "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
+ .
- 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 ].
- admit.
- Defined.
- Definition TypesN_comp n t1 t2 t3 : ND (BoundedRule n) ([Γ > nil > t1 |- t2],,[Γ > nil > t2 |- t3]) [ Γ > nil > t1 |- t3 ].
- 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)).
+ Context (ndr_systemfca:forall n, @ND_Relation _ (RuleSystemFCa n)).
+
+ 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_systemfca 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 : @ProgrammingLanguage _ Judg (mkJudg Γ Δ) (RuleSystemFCa n) :=
+ { pl_eqv := ndr_systemfca n
+ ; pl_tsr := _ (*@TreeStructuralRules Judg Rule T sequent*)
+ ; pl_sc := SystemFC_SC n
+ ; pl_subst := systemfc_cutrule n
+ ; pl_sequent_join := systemfc_expansion n
+ }.
+ 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.
+
+ Admitted.
+
+ 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_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.
- (* 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.
+ Definition PCF_SMME (n:nat)(Γ:TypeEnv)(Δ:CoercionEnv Γ) : ProgrammingLanguageSMME.
+ refine {| plsmme_pl := PCF n Γ Δ |}.
admit.
+ Defined.
+
+ 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.
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