Require Import NaturalDeduction.
Require Import Coq.Strings.String.
Require Import Coq.Lists.List.
+
Require Import HaskKinds.
Require Import HaskCoreTypes.
Require Import HaskLiteralsAndTyCons.
Require Import NaturalDeduction.
Require Import NaturalDeductionCategory.
+Require Import Algebras_ch4.
+Require Import Categories_ch1_3.
+Require Import Functors_ch1_4.
+Require Import Isomorphisms_ch1_5.
+Require Import ProductCategories_ch1_6_1.
+Require Import OppositeCategories_ch1_6_2.
+Require Import Enrichment_ch2_8.
+Require Import Subcategories_ch7_1.
+Require Import NaturalTransformations_ch7_4.
+Require Import NaturalIsomorphisms_ch7_5.
+Require Import MonoidalCategories_ch7_8.
+Require Import Coherence_ch7_8.
+
+Require Import HaskStrongTypes.
+Require Import HaskStrong.
+Require Import HaskProof.
+Require Import HaskStrongToProof.
+Require Import HaskProofToStrong.
+(*Require Import FreydCategories.*)
+(*Require Import ProgrammingLanguage.*)
+
Section HaskProofCategory.
+
+ Definition unitType {Γ} : RawHaskType Γ ★.
+ admit.
+ Defined.
+
+ Definition brakifyType {Γ} (lt:LeveledHaskType Γ ★) : LeveledHaskType (★ ::Γ) ★ :=
+ match lt with
+ t @@ l => HaskBrak (FreshHaskTyVar ★) (weakT t) @@ weakL l
+ end.
+
+ Definition brakify (j:Judg) : Judg :=
+ match j with
+ Γ > Δ > Σ |- τ =>
+ (★ ::Γ) > weakCE Δ > mapOptionTree brakifyType Σ |- mapOptionTree brakifyType τ
+ end.
+
+ (* Rules allowed in PCF; i.e. rules we know how to turn into GArrows *)
+ Section RulePCF.
+
+ Context {Γ:TypeEnv}{Δ:CoercionEnv Γ}.
+
+ 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).
+
+ Inductive BoundedRule : 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
+
+ (* 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)
+ .
+
+ Context (ndr:forall n, @ND_Relation _ (BoundedRule n)).
+
+ (* for every n we have a category of n-bounded proofs *)
+ Definition JudgmentsN n := @Judgments_Category_CartesianCat _ (BoundedRule n) (ndr n).
+
+ Open Scope nd_scope.
+ Open Scope pf_scope.
+
+ 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)).
+ 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.
+ admit.
+ admit.
+ Defined.
+
+ Definition ReificationFunctorMonoidal n : MonoidalFunctor (JudgmentsN n) (JudgmentsN (S n)) (ReificationFunctor n).
+ admit.
+ Defined.
+
+ Definition makeTree : Tree ??(LeveledHaskType Γ ★) -> HaskType Γ ★.
+ admit.
+ Defined.
+
+ Definition flattenType n (j:JudgmentsN n) : TypesN n.
+ unfold eob_eob.
+ unfold ob in j.
+ refine (mapOptionTree _ j).
+ clear j; intro j.
+ destruct j as [ Γ' Δ' Σ τ ].
+ assert (Γ'=Γ). admit.
+ rewrite H in *.
+ clear H Γ'.
+ refine (_ @@ nil).
+ refine (HaskBrak _ ( (makeTree Σ) ---> (makeTree τ) )); intros.
+ admit.
+ Defined.
+
+ Definition FlattenFunctor_fmor n :
+ forall h c,
+ (h~~{JudgmentsN n}~~>c) ->
+ ((flattenType n h)~~{TypesN n}~~>(flattenType n c)).
+ intros.
+ unfold hom in *; simpl.
+ unfold mon_i.
+ unfold ehom.
+ unfold TypesNmor.
+
+ admit.
+ Defined.
+
+ Definition FlattenFunctor n : Functor (JudgmentsN n) (TypesN n) (flattenType n).
+ refine {| fmor := FlattenFunctor_fmor n |}; intros.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ End RulePCF.
+ Implicit Arguments Rule_PCF [ [h] [c] ].
+ Implicit Arguments BoundedRule [ ].
+
+
+(*
+ Definition code2garrow0 {Γ}(ec t1 t2:RawHaskType Γ ★) : RawHaskType Γ ★.
+ admit.
+ Defined.
+ 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.
+
+ Definition coMap {Γ}(ck:HaskCoercionKind Γ) :=
+ fun TV ite => match ck TV ite with
+ | mkRawCoercionKind _ t1 t2 => mkRawCoercionKind _ (typeMap t1) (typeMap t2)
+ end.
+
+ Definition flattenCoercion {Γ}{Δ}{ck}(hk:HaskCoercion Γ Δ ck) : HaskCoercion Γ (map coMap Δ) (coMap ck).
+ admit.
+ Defined.
+
+ Lemma update_typeMap Γ (lev:HaskLevel Γ) ξ v t
+ : (typeMap ○ (update_ξ ξ lev ((⟨v, t ⟩) :: nil)))
+ = ( update_ξ (typeMap ○ ξ) lev ((⟨v, typeMap_ t ⟩) :: nil)).
+ admit.
+ Qed.
+
+ Lemma foo κ Γ σ τ : typeMap_ (substT σ τ) = substT(Γ:=Γ)(κ₁:=κ) (fun TV ite => typeMap ○ σ TV ite) τ.
+ admit.
+ Qed.
+
+ Lemma lit_lemma lit Γ : typeMap_ (literalType lit) = literalType(Γ:=Γ) lit.
+ admit.
+ Qed.
+*)
+(*
+ Definition flatten : forall h c, Rule h c -> @ND Judg Rule (mapOptionTree flattenJudgment h) (mapOptionTree flattenJudgment c).
+ intros h c r.
+ refine (match r as R in Rule H C return ND Rule (mapOptionTree flattenJudgment H) (mapOptionTree flattenJudgment C) with
+ | RURule a b c d e => let case_RURule := tt in _
+ | RNote Γ Δ Σ τ l n => let case_RNote := tt in _
+ | RLit Γ Δ l _ => let case_RLit := tt in _
+ | RVar Γ Δ σ p => let case_RVar := tt in _
+ | RGlobal Γ Δ σ l wev => let case_RGlobal := tt in _
+ | RLam Γ Δ Σ tx te x => let case_RLam := tt in _
+ | RCast Γ Δ Σ σ τ γ x => let case_RCast := tt in _
+ | RAbsT Γ Δ Σ κ σ a => let case_RAbsT := tt in _
+ | RAppT Γ Δ Σ κ σ τ y => let case_RAppT := tt in _
+ | RAppCo Γ Δ Σ κ σ₁ σ₂ γ σ l => let case_RAppCo := tt in _
+ | RAbsCo Γ Δ Σ κ σ σ₁ σ₂ y => let case_RAbsCo := tt in _
+ | RApp Γ Δ Σ₁ Σ₂ tx te p => let case_RApp := tt in _
+ | RLet Γ Δ Σ₁ Σ₂ σ₁ σ₂ p => let case_RLet := tt in _
+ | RBindingGroup Γ p lri m x q => let case_RBindingGroup := tt in _
+ | REmptyGroup _ _ => let case_REmptyGroup := tt in _
+ | RBrak Σ a b c n m => let case_RBrak := tt in _
+ | REsc Σ a b c n m => let case_REsc := tt in _
+ | RCase Γ Δ lev tc Σ avars tbranches alts => let case_RCase := tt in _
+ | RLetRec Γ Δ lri x y t => let case_RLetRec := tt in _
+ end).
+
+ destruct case_RURule.
+ admit.
+
+ destruct case_RBrak.
+ simpl.
+ admit.
+
+ destruct case_REsc.
+ simpl.
+ admit.
+
+ destruct case_RNote.
+ eapply nd_rule. simpl. apply RNote; auto.
+
+ destruct case_RLit.
+ simpl.
+
+ set (@RNote Γ Δ Σ τ l) as q.
+ Defined.
+
+ Definition flatten' {h}{c} (pf:ND Rule h c) := nd_map' flattenJudgment flatten pf.
+
+
+ @ND Judgment1 Rule1 (mapOptionTree f h) (mapOptionTree f c).
+
+ refine (fix flatten : forall Γ Δ Σ τ
+ (pf:SCND Rule [] [Γ > Δ > Σ |- τ ]) :
+ SCND Rule [] [Γ > Δ > mapOptionTree typeMap Σ |- mapOptionTree typeMap τ ] :=
+ match pf as SCND _ _
+ | scnd_comp : forall ht ct c , SCND ht ct -> Rule ct [c] -> SCND ht [c]
+ | scnd_weak : forall c , SCND c []
+ | scnd_leaf : forall ht c , SCND ht [c] -> SCND ht [c]
+ | scnd_branch : forall ht c1 c2, SCND ht c1 -> SCND ht c2 -> SCND ht (c1,,c2)
+ Expr Γ Δ ξ τ -> Expr Γ (map coMap Δ) (typeMap ○ ξ) (typeMap τ).
+*)
+
(*
+ Lemma all_lemma Γ κ σ l :
+(@typeMap (κ :: Γ)
+ (@HaskTApp (κ :: Γ) κ (@weakF Γ κ ★ σ) (@FreshHaskTyVar Γ κ) @@
+ @weakL Γ κ l)) = (@typeMap Γ (@HaskTAll Γ κ σ @@ l)).
+*)
+
+(*
+ Definition flatten : forall Γ Δ ξ τ, Expr Γ Δ ξ τ -> Expr Γ (map coMap Δ) (typeMap ○ ξ) (typeMap τ).
+ refine (fix flatten Γ' Δ' ξ' τ' (exp:Expr Γ' Δ' ξ' τ') : Expr Γ' (map coMap Δ') (typeMap ○ ξ') (typeMap τ') :=
+ match exp as E in Expr G D X T return Expr G (map coMap D) (typeMap ○ X) (typeMap T) with
+ | EGlobal Γ Δ ξ t wev => EGlobal _ _ _ _ wev
+ | EVar Γ Δ ξ ev => EVar _ _ _ ev
+ | ELit Γ Δ ξ lit lev => let case_ELit := tt in _
+ | EApp Γ Δ ξ t1 t2 lev e1 e2 => EApp _ _ _ _ _ _ (flatten _ _ _ _ e1) (flatten _ _ _ _ e2)
+ | ELam Γ Δ ξ t1 t2 lev v e => let case_ELam := tt in _
+ | ELet Γ Δ ξ tv t l ev elet ebody => let case_ELet := tt in _
+ | ELetRec Γ Δ ξ lev t tree branches ebody => let case_ELetRec := tt in _
+ | ECast Γ Δ ξ t1 t2 γ lev e => let case_ECast := tt in _
+ | ENote Γ Δ ξ t n e => ENote _ _ _ _ n (flatten _ _ _ _ e)
+ | ETyLam Γ Δ ξ κ σ l e => let case_ETyLam := tt in _
+ | ECoLam Γ Δ κ σ σ₁ σ₂ ξ l e => let case_ECoLam := tt in _
+ | ECoApp Γ Δ κ σ₁ σ₂ γ σ ξ l e => let case_ECoApp := tt in _
+ | ETyApp Γ Δ κ σ τ ξ l e => let case_ETyApp := tt in _
+ | ECase Γ Δ ξ l tc tbranches atypes e alts' => let case_ECase := tt in _
+
+ | EEsc Γ Δ ξ ec t lev e => let case_EEsc := tt in _
+ | EBrak Γ Δ ξ ec t lev e => let case_EBrak := tt in _
+ end); clear exp ξ' τ' Γ' Δ'.
+
+ destruct case_ELit.
+ simpl.
+ rewrite lit_lemma.
+ apply ELit.
+
+ destruct case_ELam.
+ set (flatten _ _ _ _ e) as q.
+ rewrite update_typeMap in q.
+ apply (@ELam _ _ _ _ _ _ _ _ v q).
+
+ destruct case_ELet.
+ set (flatten _ _ _ _ ebody) as ebody'.
+ set (flatten _ _ _ _ elet) as elet'.
+ rewrite update_typeMap in ebody'.
+ apply (@ELet _ _ _ _ _ _ _ _ _ elet' ebody').
- Context (flat_dynamic_semantics : @ND_Relation _ Rule).
- Context (ml_dynamic_semantics : @ND_Relation _ Rule).
+ destruct case_EEsc.
+ admit.
+ destruct case_EBrak.
+ admit.
+
+ destruct case_ECast.
+ apply flatten in e.
+ eapply ECast in e.
+ apply e.
+ apply flattenCoercion in γ.
+ apply γ.
+
+ destruct case_ETyApp.
+ apply flatten in e.
+ simpl in e.
+ unfold HaskTAll in e.
+ unfold typeMap_ in e.
+ simpl in e.
+ eapply ETyApp in e.
+ rewrite <- foo in e.
+ apply e.
+
+ destruct case_ECoLam.
+ apply flatten in e.
+ simpl in e.
+ set (@ECoLam _ _ _ _ _ _ _ _ _ _ e) as x.
+ simpl in x.
+ simpl.
+ unfold typeMap_.
+ simpl.
+ apply x.
+
+ destruct case_ECoApp.
+ simpl.
+ apply flatten in e.
+ eapply ECoApp.
+ unfold mkHaskCoercionKind in *.
+ simpl in γ.
+ apply flattenCoercion in γ.
+ unfold coMap in γ at 2.
+ apply γ.
+ apply e.
+
+ destruct case_ETyLam.
+ apply flatten in e.
+ set (@ETyLam Unique _ Γ (map coMap Δ) (typeMap ○ ξ) κ (fun ite x => typeMap (σ x ite))) as e'.
+ unfold HaskTAll in *.
+ unfold typeMap_ in *.
+ rewrite <- foo in e'.
+ unfold typeMap in e'.
+ simpl in e'.
+ apply ETyLam.
+
+Set Printing Implicit.
+idtac.
+idtac.
+
+
+ admit.
+
+ destruct case_ECase.
+ admit.
+
+ destruct case_ELetRec.
+ admit.
+ Defined.
+
+ (* This proof will work for any dynamic semantics you like, so
+ * long as those semantics are an ND_Relation (associativity,
+ * neutrality, etc) *)
+ Context (dynamic_semantics : @ND_Relation _ Rule).
Section SystemFC_Category.
- Context (encodeTypeTree_reduce : @LeveledHaskType V -> @LeveledHaskType V -> @LeveledHaskType V).
- Context (encodeTypeTree_empty : @LeveledHaskType V).
- Context (encodeTypeTree_flat_empty : @CoreType V).
- Context (encodeTypeTree_flat_reduce : @CoreType V -> @CoreType V -> @CoreType V).
-
- Definition encodeTypeTree :=
- @treeReduce _ _ (fun x => match x with None => encodeTypeTree_empty | Some q => q end) encodeTypeTree_reduce.
- Definition encodeTypeTree_flat :=
- @treeReduce _ _ (fun x => match x with None => encodeTypeTree_flat_empty | Some q => q end) encodeTypeTree_flat_reduce.
- (* the full category of judgments *)
- Definition ob2judgment past :=
- fun q:Tree ??(@LeveledHaskType V) * Tree ??(@LeveledHaskType V) =>
- let (a,s):=q in (Γ > past : a |- (encodeTypeTree s) ).
- Definition SystemFC_Cat past :=
- @Judgments_Category_monoidal _ Rule
- (@ml_dynamic_semantics V)
- (Tree ??(@LeveledHaskType V) * Tree ??(@LeveledHaskType V))
- (ob2judgment past).
-
- (* the category of judgments with no variables or succedents in the "future" –- still may have code types *)
- (* technically this should be a subcategory of SystemFC_Cat *)
- Definition ob2judgment_flat past :=
- fun q:Tree ??(@CoreType V) * Tree ??(@CoreType V) =>
- let (a,s):=q in (Γ > past : ``a |- `(encodeTypeTree_flat s) ).
- Definition SystemFC_Cat_Flat past :=
- @Judgments_Category_monoidal _ Rule
- (@flat_dynamic_semantics V)
- (Tree ??(@CoreType V) * Tree ??(@CoreType V))
- (ob2judgment_flat past).
+
+ Context {Γ:TypeEnv}
+ {Δ:CoercionEnv Γ}.
+
+ Definition Context := Tree ??(LeveledHaskType Γ ★).
+
+ Notation "a |= b" := (Γ >> Δ > a |- b).
+
+ (*
+ SystemFCa
+ PCF
+ SystemFCa_two_level
+ SystemFCa_initial_GArrow
+ *)
+
+ Context (nd_eqv:@ND_Relation _ (@URule Γ Δ)).
+ Check (@ProgrammingLanguage).
+ Context (PL:@ProgrammingLanguage (LeveledHaskType Γ ★)
+ (fun x y => match x with x1|=x2 => match y with y1|=y2 => @URule Γ Δ)).
+ Definition JudgmentsFC := @Judgments_Category_CartesianCat _ (@URule Γ Δ) nd_eqv.
+ Definition TypesFC := @TypesL _ (@URule Γ Δ) nd_eqv.
+
+ (* The full subcategory of SystemFC(Γ,Δ) consisting only of judgments involving types at a fixed level. Note that
+ * code types are still permitted! *)
+ Section SingleLevel.
+ Context (lev:HaskLevel Γ).
+
+ Inductive ContextAtLevel : Context -> Prop :=
+ | contextAtLevel_nil : ContextAtLevel []
+ | contextAtLevel_leaf : forall τ, ContextAtLevel [τ @@ lev]
+ | contextAtLevel_branch : forall b1 b2, ContextAtLevel b1 -> ContextAtLevel b2 -> ContextAtLevel (b1,,b2).
+
+ Inductive JudgmentsAtLevel : JudgmentsFC -> Prop :=
+ | judgmentsAtLevel_nil : JudgmentsAtLevel []
+ | judgmentsAtLevel_leaf : forall c1 c2, ContextAtLevel c1 -> ContextAtLevel c2 -> JudgmentsAtLevel [c1 |= c2]
+ | judgmentsAtLevel_branch : forall j1 j2, JudgmentsAtLevel j1 -> JudgmentsAtLevel j2 -> JudgmentsAtLevel (j1,,j2).
+ Definition JudgmentsFCAtLevel := FullSubcategory JudgmentsFC JudgmentsAtLevel.
+ Definition TypesFCAtLevel := FullSubcategory TypesFC ContextAtLevel.
+ End SingleLevel.
+
+ End SystemFC_Category.
+
+ Implicit Arguments TypesFC [ ].
+
+(*
Section EscBrak_Functor.
Context
(past:@Past V)
admit.
Defined.
*)
-End HaskProofCategory.
\ No newline at end of file
+*)
+End HaskProofCategory.
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