--- /dev/null
+(*********************************************************************************************************************************)
+(* HaskProofFlattener: *)
+(* *)
+(* The Flattening Functor. *)
+(* *)
+(*********************************************************************************************************************************)
+
+Generalizable All Variables.
+Require Import Preamble.
+Require Import General.
+Require Import NaturalDeduction.
+Require Import Coq.Strings.String.
+Require Import Coq.Lists.List.
+
+Require Import HaskKinds.
+Require Import HaskCoreTypes.
+Require Import HaskLiteralsAndTyCons.
+Require Import HaskStrongTypes.
+Require Import HaskProof.
+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 ProgrammingLanguage.
+Require Import HaskProofStratified.
+
+Open Scope nd_scope.
+
+
+(*
+ * The flattening transformation. Currently only TWO-level languages are
+ * supported, and the level-1 sublanguage is rather limited.
+*
+ * This file abuses terminology pretty badly. For purposes of this file,
+ * "PCF" means "the level-1 sublanguage" and "FC" (aka System FC) means
+ * the whole language (level-0 language including bracketed level-1 terms)
+ *)
+Section HaskProofFlattener.
+
+
+(*
+ 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 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 ??(HaskType Γ ★)) : HaskType Γ ★.
+ admit.
+ Defined.
+
+ Definition mkGA {Γ} : HaskType Γ ★ -> HaskType Γ ★ -> HaskType Γ ★.
+ admit.
+ Defined.
+
+ Definition guestJudgmentAsGArrowType {Γ}{Δ}{ec}(lt:PCFJudg Γ Δ ec) : HaskType Γ ★ :=
+ match lt with
+ pcfjudg x y =>
+ (mkGA (tensorizeType x) (tensorizeType y))
+ end.
+
+ Definition obact {Γ}{Δ} ec (X:Tree ??(PCFJudg Γ Δ ec)) : Tree ??(LeveledHaskType Γ ★) :=
+ mapOptionTree guestJudgmentAsGArrowType X @@@ nil.
+
+ Hint Constructors Rule_Flat.
+ Context {ndr:@ND_Relation _ Rule}.
+
+ (*
+ * 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}{ec}
+ : forall h c,
+ (h~~{JudgmentsL _ _ (PCF _ Γ Δ ec)}~~>c) ->
+ ((obact ec h)~~{TypesL _ _ (SystemFCa _ Γ Δ)}~~>(obact ec c)).
+
+ set (@nil (HaskTyVar Γ ★)) as lev.
+
+ 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 _ _ (REmptyGroup _ _ )). auto.
+
+ (* the proof from hypothesis X of conclusion X (nd_id1) becomes RVar *)
+ apply nd_rule; apply (org_fc _ _ (RVar _ _ _ _)). auto.
+
+ (* the proof from hypothesis X of no conclusions (nd_weak) becomes RWeak;;REmptyGroup *)
+ eapply nd_comp;
+ [ idtac
+ | eapply nd_rule
+ ; eapply (org_fc _ _ (RArrange _ _ _ _ _ (RWeak _)))
+ ; auto ].
+ eapply nd_rule.
+ eapply (org_fc _ _ (REmptyGroup _ _)); auto.
+
+ (* 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 _ _ (RArrange _ _ _ _ _ (RCont _))) ].
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ set (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa _ Γ Δ))
+ (mapOptionTree guestJudgmentAsGArrowType h @@@ lev)) as q.
+ eapply nd_comp.
+ eapply nd_prod.
+ apply q.
+ apply q.
+ apply nd_rule.
+ eapply (org_fc _ _ (RBindingGroup _ _ _ _ _ _ )).
+ auto.
+ auto.
+
+ (* nd_prod becomes nd_llecnac;;nd_prod;;RBindingGroup *)
+ eapply nd_comp.
+ apply (nd_llecnac ;; nd_prod IHX1 IHX2).
+ apply nd_rule.
+ eapply (org_fc _ _ (RBindingGroup _ _ _ _ _ _ )).
+ auto.
+
+ (* 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 _ _ _ _ (SystemFCa _ Γ Δ))).
+
+ (* nd_cancell becomes RVar;;RuCanL *)
+ eapply nd_comp;
+ [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanL _))) ].
+ apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa _ Γ Δ))).
+ auto.
+
+ (* nd_cancelr becomes RVar;;RuCanR *)
+ eapply nd_comp;
+ [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanR _))) ].
+ apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa _ Γ Δ))).
+ auto.
+
+ (* nd_llecnac becomes RVar;;RCanL *)
+ eapply nd_comp;
+ [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanL _))) ].
+ apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa _ Γ Δ))).
+ auto.
+
+ (* nd_rlecnac becomes RVar;;RCanR *)
+ eapply nd_comp;
+ [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanR _))) ].
+ apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa _ Γ Δ))).
+ auto.
+
+ (* nd_assoc becomes RVar;;RAssoc *)
+ eapply nd_comp;
+ [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RAssoc _ _ _))) ].
+ apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa _ Γ Δ))).
+ auto.
+
+ (* nd_cossa becomes RVar;;RCossa *)
+ eapply nd_comp;
+ [ idtac | eapply nd_rule; apply (org_fc _ _ (RArrange _ _ _ _ _ (RCossa _ _ _))) ].
+ apply (nd_seq_reflexive(SequentCalculus:=@pl_sc _ _ _ _ (SystemFCa _ Γ Δ))).
+ auto.
+
+ destruct r as [r rp].
+ refine (match rp as R in @Rule_PCF _ _ _ H C _ with
+ | PCF_RArrange h c r q => let case_RURule := tt in _
+ | PCF_RLit lit => let case_RLit := tt in _
+ | PCF_RNote Σ τ n => let case_RNote := tt in _
+ | PCF_RVar σ => let case_RVar := tt in _
+ | PCF_RLam Σ tx te => let case_RLam := tt in _
+ | PCF_RApp Σ tx te p => let case_RApp := tt in _
+ | PCF_RLet Σ σ₁ σ₂ p => 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 rp.
+ clear r h c.
+ rename r0 into r; rename h0 into h; rename c0 into c.
+
+ destruct case_RURule.
+ refine (match q with
+ | RLeft a b c r => let case_RLeft := tt in _
+ | RRight a b c r => let case_RRight := tt in _
+ | RCanL b => let case_RCanL := tt in _
+ | RCanR b => let case_RCanR := tt in _
+ | RuCanL b => let case_RuCanL := tt in _
+ | RuCanR b => let case_RuCanR := tt in _
+ | RAssoc b c d => let case_RAssoc := tt in _
+ | RCossa b c d => let case_RCossa := tt in _
+ | RExch b c => let case_RExch := tt in _
+ | RWeak b => let case_RWeak := tt in _
+ | RCont b => let case_RCont := tt in _
+ | RComp a b c f g => let case_RComp := 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_RExch.
+ (* ga_swap *)
+ admit.
+
+ destruct case_RWeak.
+ (* ga_drop *)
+ admit.
+
+ destruct case_RCont.
+ (* ga_copy *)
+ admit.
+
+ destruct case_RLeft.
+ (* ga_second *)
+ admit.
+
+ destruct case_RRight.
+ (* ga_first *)
+ admit.
+
+ destruct case_RComp.
+ (* ga_comp *)
+ admit.
+
+ destruct case_RLit.
+ (* ga_literal *)
+ admit.
+
+ (* hey cool, I figured out how to pass CoreNote's through... *)
+ destruct case_RNote.
+ eapply nd_comp.
+ eapply nd_rule.
+ eapply (org_fc _ _ (RVar _ _ _ _)) . auto.
+ apply nd_rule.
+ apply (org_fc _ _ (RNote _ _ _ _ _ n)). auto.
+
+ 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_REmptyGroup.
+ (* ga_id u *)
+ admit.
+
+ destruct case_RBindingGroup.
+ (* ga_first+ga_second; technically this assumes a specific evaluation order, which is bad *)
+ admit.
+
+ Defined.
+
+ Instance FlatteningFunctor {Γ}{Δ}{ec} : Functor (JudgmentsL _ _ (PCF _ Γ Δ ec)) (TypesL _ _ (SystemFCa _ Γ Δ)) (obact ec) :=
+ { fmor := FlatteningFunctor_fmor }.
+ admit.
+ admit.
+ admit.
+ 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 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.
+
+ 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.
+*)
+ (* 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 *)
+
+End HaskProofFlattener.
+
--- /dev/null
+(*********************************************************************************************************************************)
+(* HaskProofStratified: *)
+(* *)
+(* An alternate representation for HaskProof which ensures that deductions on a given level are grouped into contiguous *)
+(* blocks. This representation lacks the attractive compositionality properties of HaskProof, but makes it easier to *)
+(* perform the flattening process. *)
+(* *)
+(*********************************************************************************************************************************)
+
+Generalizable All Variables.
+Require Import Preamble.
+Require Import General.
+Require Import NaturalDeduction.
+Require Import Coq.Strings.String.
+Require Import Coq.Lists.List.
+
+Require Import HaskKinds.
+Require Import HaskCoreTypes.
+Require Import HaskLiteralsAndTyCons.
+Require Import HaskStrongTypes.
+Require Import HaskProof.
+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 ProgrammingLanguage.
+
+Open Scope nd_scope.
+
+
+(*
+ * The flattening transformation. Currently only TWO-level languages are
+ * supported, and the level-1 sublanguage is rather limited.
+*
+ * This file abuses terminology pretty badly. For purposes of this file,
+ * "PCF" means "the level-1 sublanguage" and "FC" (aka System FC) means
+ * the whole language (level-0 language including bracketed level-1 terms)
+ *)
+Section HaskProofStratified.
+
+ Context (ndr_systemfc:@ND_Relation _ Rule).
+
+ Inductive PCFJudg Γ (Δ:CoercionEnv Γ) (ec:HaskTyVar Γ ★) :=
+ pcfjudg : Tree ??(HaskType Γ ★) -> Tree ??(HaskType Γ ★) -> PCFJudg Γ Δ ec.
+ Implicit Arguments pcfjudg [ [Γ] [Δ] [ec] ].
+
+ (* given an PCFJudg at depth (ec::depth) we can turn it into an PCFJudg
+ * from depth (depth) by wrapping brackets around everything in the
+ * succedent and repopulating *)
+ Definition brakify {Γ}{Δ}{ec} (j:PCFJudg Γ Δ ec) : Judg :=
+ match j with
+ pcfjudg Σ τ => Γ > Δ > (Σ@@@(ec::nil)) |- (mapOptionTree (fun t => HaskBrak ec t) τ @@@ nil)
+ end.
+
+ Definition pcf_vars {Γ}(ec:HaskTyVar Γ ★)(t:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★)
+ := mapOptionTreeAndFlatten (fun lt =>
+ match lt with t @@ l => match l with
+ | ec'::nil => if eqd_dec ec ec' then [t] else []
+ | _ => []
+ end
+ end) t.
+
+ Inductive MatchingJudgments {Γ}{Δ}{ec} : Tree ??(PCFJudg Γ Δ ec) -> Tree ??Judg -> Type :=
+ | match_nil : MatchingJudgments [] []
+ | match_branch : forall a b c d, MatchingJudgments a b -> MatchingJudgments c d -> MatchingJudgments (a,,c) (b,,d)
+ | match_leaf :
+ forall Σ τ lev,
+ MatchingJudgments
+ [pcfjudg (pcf_vars ec Σ) τ ]
+ [Γ > Δ > Σ |- (mapOptionTree (HaskBrak ec) τ @@@ lev)].
+
+ Definition fc_vars {Γ}(ec:HaskTyVar Γ ★)(t:Tree ??(LeveledHaskType Γ ★)) : Tree ??(HaskType Γ ★)
+ := mapOptionTreeAndFlatten (fun lt =>
+ match lt with t @@ l => match l with
+ | ec'::nil => if eqd_dec ec ec' then [] else [t]
+ | _ => []
+ end
+ end) t.
+
+ Definition pcfjudg2judg {Γ}{Δ:CoercionEnv Γ} ec (cj:PCFJudg Γ Δ ec) :=
+ match cj with pcfjudg Σ τ => Γ > Δ > (Σ @@@ (ec::nil)) |- (τ @@@ (ec::nil)) end.
+
+ (* 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:CoercionEnv Γ} (ec:HaskTyVar Γ ★)
+ : forall (h c:Tree ??(PCFJudg Γ Δ ec)), Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c) -> Type :=
+ | PCF_RArrange : ∀ x y t a, Rule_PCF ec [pcfjudg _ _ ] [ pcfjudg _ _ ] (RArrange Γ Δ (x@@@(ec::nil)) (y@@@(ec::nil)) (t@@@(ec::nil)) a)
+ | PCF_RLit : ∀ lit , Rule_PCF ec [ ] [ pcfjudg [] [_] ] (RLit Γ Δ lit (ec::nil))
+ | PCF_RNote : ∀ Σ τ n , Rule_PCF ec [pcfjudg _ [_]] [ pcfjudg _ [_] ] (RNote Γ Δ (Σ@@@(ec::nil)) τ (ec::nil) n)
+ | PCF_RVar : ∀ σ , Rule_PCF ec [ ] [ pcfjudg [_] [_] ] (RVar Γ Δ σ (ec::nil) )
+ | PCF_RLam : ∀ Σ tx te , Rule_PCF ec [pcfjudg (_,,[_]) [_] ] [ pcfjudg _ [_] ] (RLam Γ Δ (Σ@@@(ec::nil)) tx te (ec::nil) )
+
+ | PCF_RApp : ∀ Σ Σ' tx te ,
+ Rule_PCF ec ([pcfjudg _ [_]],,[pcfjudg _ [_]]) [pcfjudg (_,,_) [_]]
+ (RApp Γ Δ (Σ@@@(ec::nil))(Σ'@@@(ec::nil)) tx te (ec::nil))
+
+ | PCF_RLet : ∀ Σ Σ' σ₂ p,
+ Rule_PCF ec ([pcfjudg _ [_]],,[pcfjudg (_,,[_]) [_]]) [pcfjudg (_,,_) [_]]
+ (RLet Γ Δ (Σ@@@(ec::nil)) (Σ'@@@(ec::nil)) σ₂ p (ec::nil))
+
+ | PCF_REmptyGroup : Rule_PCF ec [ ] [ pcfjudg [] [] ] (REmptyGroup Γ Δ )
+(*| PCF_RLetRec : ∀ Σ₁ τ₁ τ₂ , Rule_PCF (ec::nil) _ _ (RLetRec Γ Δ Σ₁ τ₁ τ₂ (ec::nil) )*)
+ | PCF_RBindingGroup : ∀ Σ₁ Σ₂ τ₁ τ₂, Rule_PCF ec ([pcfjudg _ _],,[pcfjudg _ _]) [pcfjudg (_,,_) (_,,_)]
+ (RBindingGroup Γ Δ (Σ₁@@@(ec::nil)) (Σ₂@@@(ec::nil)) (τ₁@@@(ec::nil)) (τ₂@@@(ec::nil))).
+ (* need int/boolean case *)
+ Implicit Arguments Rule_PCF [ ].
+
+ Definition PCFRule Γ Δ lev h c := { r:_ & @Rule_PCF Γ Δ lev 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 : Tree ??Judg -> Tree ??Judg -> Type :=
+
+ | org_fc : forall h c (r:Rule h c),
+ Rule_Flat r ->
+ OrgR h c
+
+ | org_pcf : forall Γ Δ ec h h' c c',
+ MatchingJudgments h h' ->
+ MatchingJudgments c c' ->
+ ND (PCFRule Γ Δ ec) h c ->
+ OrgR h' c'.
+
+ Definition mkEsc {Γ}{Δ}{ec}(h:Tree ??(PCFJudg Γ Δ ec))
+ : ND Rule
+ (mapOptionTree brakify h)
+ (mapOptionTree (pcfjudg2judg ec) h).
+ apply nd_replicate; intros.
+ destruct o; simpl in *.
+ induction t0.
+ destruct a; simpl.
+ apply nd_rule.
+ apply REsc.
+ apply nd_id.
+ apply (Prelude_error "mkEsc got multi-leaf succedent").
+ Defined.
+
+ Definition mkBrak {Γ}{Δ}{ec}(h:Tree ??(PCFJudg Γ Δ ec))
+ : ND Rule
+ (mapOptionTree (pcfjudg2judg ec) h)
+ (mapOptionTree brakify h).
+ apply nd_replicate; intros.
+ destruct o; simpl in *.
+ induction t0.
+ destruct a; simpl.
+ apply nd_rule.
+ apply RBrak.
+ apply nd_id.
+ apply (Prelude_error "mkBrak got multi-leaf succedent").
+ Defined.
+
+ (*
+ Definition Partition {Γ} ec (Σ:Tree ??(LeveledHaskType Γ ★)) :=
+ { vars:(_ * _) |
+ fc_vars ec Σ = fst vars /\
+ pcf_vars ec Σ = snd vars }.
+ *)
+
+ Definition pcfToND : forall Γ Δ ec h c,
+ ND (PCFRule Γ Δ ec) h c -> ND Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c).
+ intros.
+ eapply (fun q => nd_map' _ q X).
+ intros.
+ destruct X0.
+ apply nd_rule.
+ apply x.
+ Defined.
+
+ Instance OrgPCF Γ Δ lev : @ND_Relation _ (PCFRule Γ Δ lev) :=
+ { ndr_eqv := fun a b f g => (pcfToND _ _ _ _ _ f) === (pcfToND _ _ _ _ _ g) }.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ (*
+ * An intermediate representation necessitated by Coq's termination
+ * conditions. This is basically a tree where each node is a
+ * subproof which is either entirely level-1 or entirely level-0
+ *)
+ Inductive Alternating : Tree ??Judg -> Type :=
+
+ | alt_nil : Alternating []
+
+ | alt_branch : forall a b,
+ Alternating a -> Alternating b -> Alternating (a,,b)
+
+ | alt_fc : forall h c,
+ Alternating h ->
+ ND Rule h c ->
+ Alternating c
+
+ | alt_pcf : forall Γ Δ ec h c h' c',
+ MatchingJudgments h h' ->
+ MatchingJudgments c c' ->
+ Alternating h' ->
+ ND (PCFRule Γ Δ ec) h c ->
+ Alternating c'.
+
+ Require Import Coq.Logic.Eqdep.
+
+ Lemma magic a b c d ec e :
+ ClosedND(Rule:=Rule) [a > b > c |- [d @@ (ec :: e)]] ->
+ ClosedND(Rule:=Rule) [a > b > pcf_vars ec c @@@ (ec :: nil) |- [d @@ (ec :: nil)]].
+ admit.
+ Defined.
+
+ Definition orgify : forall Γ Δ Σ τ (pf:ClosedND(Rule:=Rule) [Γ > Δ > Σ |- τ]), Alternating [Γ > Δ > Σ |- τ].
+
+ refine (
+ fix orgify_fc' Γ Δ Σ τ (pf:ClosedND [Γ > Δ > Σ |- τ]) {struct pf} : Alternating [Γ > Δ > Σ |- τ] :=
+ let case_main := tt in _
+ with orgify_fc c (pf:ClosedND c) {struct pf} : Alternating c :=
+ (match c as C return C=c -> Alternating C with
+ | T_Leaf None => fun _ => alt_nil
+ | T_Leaf (Some (Γ > Δ > Σ |- τ)) => let case_leaf := tt in fun eqpf => _
+ | T_Branch b1 b2 => let case_branch := tt in fun eqpf => _
+ end (refl_equal _))
+ with orgify_pcf Γ Δ ec pcfj j (m:MatchingJudgments pcfj j)
+ (pf:ClosedND (mapOptionTree (pcfjudg2judg ec) pcfj)) {struct pf} : Alternating j :=
+ let case_pcf := tt in _
+ for orgify_fc').
+
+ destruct case_main.
+ inversion pf; subst.
+ set (alt_fc _ _ (orgify_fc _ X) (nd_rule X0)) as backup.
+ refine (match X0 as R in Rule H C return
+ match C with
+ | T_Leaf (Some (Γ > Δ > Σ |- τ)) =>
+ h=H -> Alternating [Γ > Δ > Σ |- τ] -> Alternating [Γ > Δ > Σ |- τ]
+ | _ => True
+ end
+ with
+ | RBrak Σ a b c n m => let case_RBrak := tt in fun pf' backup => _
+ | REsc Σ a b c n m => let case_REsc := tt in fun pf' backup => _
+ | _ => fun pf' x => x
+ end (refl_equal _) backup).
+ clear backup0 backup.
+
+ destruct case_RBrak.
+ rename c into ec.
+ set (@match_leaf Σ0 a ec n [b] m) as q.
+ set (orgify_pcf Σ0 a ec _ _ q) as q'.
+ apply q'.
+ simpl.
+ rewrite pf' in X.
+ apply magic in X.
+ apply X.
+
+ destruct case_REsc.
+ apply (Prelude_error "encountered Esc in wrong side of mkalt").
+
+ destruct case_leaf.
+ apply orgify_fc'.
+ rewrite eqpf.
+ apply pf.
+
+ destruct case_branch.
+ rewrite <- eqpf in pf.
+ inversion pf; subst.
+ apply no_rules_with_multiple_conclusions in X0.
+ inversion X0.
+ exists b1. exists b2.
+ auto.
+ apply (alt_branch _ _ (orgify_fc _ X) (orgify_fc _ X0)).
+
+ destruct case_pcf.
+ Admitted.
+
+ Definition pcfify Γ Δ ec : forall Σ τ,
+ ClosedND(Rule:=Rule) [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]
+ -> ND (PCFRule Γ Δ ec) [] [pcfjudg Σ τ].
+
+ refine ((
+ fix pcfify Σ τ (pn:@ClosedND _ Rule [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]) {struct pn}
+ : ND (PCFRule Γ Δ ec) [] [pcfjudg Σ τ] :=
+ (match pn in @ClosedND _ _ J return J=[Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)] -> _ with
+ | cnd_weak => let case_nil := tt in _
+ | cnd_rule h c cnd' r => let case_rule := tt in _
+ | cnd_branch _ _ c1 c2 => let case_branch := tt in _
+ end (refl_equal _)))).
+ intros.
+ inversion H.
+ intros.
+ destruct c; try destruct o; inversion H.
+ destruct j.
+ Admitted.
+
+ (* any proof in organized form can be "dis-organized" *)
+ Definition unOrgR : forall h c, OrgR h c -> ND Rule h c.
+ intros.
+
+ induction X.
+ apply nd_rule.
+ apply r.
+
+ eapply nd_comp.
+ (*
+ apply (mkEsc h).
+ eapply nd_comp; [ idtac | apply (mkBrak c) ].
+ apply pcfToND.
+ apply n.
+ *)
+ Admitted.
+
+ Definition unOrgND h c : ND OrgR h c -> ND Rule h c := nd_map unOrgR.
+
+ Instance OrgNDR : @ND_Relation _ OrgR :=
+ { 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.
+
+ Instance PCF_sequents Γ Δ lev : @SequentCalculus _ (PCFRule Γ Δ lev) _ pcfjudg.
+ apply Build_SequentCalculus.
+ intros.
+ induction a.
+ destruct a; simpl.
+ apply nd_rule.
+ exists (RVar _ _ _ _).
+ apply PCF_RVar.
+ apply nd_rule.
+ exists (REmptyGroup _ _ ).
+ apply PCF_REmptyGroup.
+ eapply nd_comp.
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply (nd_prod IHa1 IHa2).
+ apply nd_rule.
+ exists (RBindingGroup _ _ _ _ _ _).
+ apply PCF_RBindingGroup.
+ Defined.
+
+ Definition PCF_Arrange {Γ}{Δ}{lev} : forall x y z, Arrange x y -> ND (PCFRule Γ Δ lev) [pcfjudg x z] [pcfjudg y z].
+ admit.
+ Defined.
+
+ Definition PCF_cut Γ Δ lev : forall a b c, ND (PCFRule Γ Δ lev) ([ pcfjudg a b ],,[ pcfjudg b c ]) [ pcfjudg a c ].
+ intros.
+ destruct b.
+ destruct o.
+ destruct c.
+ destruct o.
+
+ (* when the cut is a single leaf and the RHS is a single leaf: *)
+ eapply nd_comp.
+ eapply nd_prod.
+ apply nd_id.
+ apply (PCF_Arrange [h] ([],,[h]) [h0]).
+ apply RuCanL.
+ eapply nd_comp; [ idtac | apply (PCF_Arrange ([],,a) a [h0]); apply RCanL ].
+ apply nd_rule.
+(*
+ set (@RLet Γ Δ [] (a@@@(ec::nil)) h0 h (ec::nil)) as q.
+ exists q.
+ apply (PCF_RLet _ [] a h0 h).
+ apply (Prelude_error "cut rule invoked with [a|=[b]] [[b]|=[]]").
+ apply (Prelude_error "cut rule invoked with [a|=[b]] [[b]|=[x,,y]]").
+ apply (Prelude_error "cut rule invoked with [a|=[]] [[]|=c]").
+ apply (Prelude_error "cut rule invoked with [a|=[b,,c]] [[b,,c]|=z]").
+*)
+ admit.
+ admit.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ Instance PCF_cutrule Γ Δ lev : CutRule (PCF_sequents Γ Δ lev) :=
+ { nd_cut := PCF_cut Γ Δ lev }.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ Definition PCF_left Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (a,,b) (a,,c)].
+ eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
+ eapply nd_prod; [ apply nd_seq_reflexive | apply nd_id ].
+ apply nd_rule.
+ set (@PCF_RBindingGroup Γ Δ lev a b a c) as q'.
+ refine (existT _ _ _).
+ apply q'.
+ Defined.
+
+ Definition PCF_right Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (b,,a) (c,,a)].
+ eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
+ eapply nd_prod; [ apply nd_id | apply nd_seq_reflexive ].
+ apply nd_rule.
+ set (@PCF_RBindingGroup Γ Δ lev b a c a) as q'.
+ refine (existT _ _ _).
+ apply q'.
+ Defined.
+
+ Instance PCF_sequent_join Γ Δ lev : @SequentExpansion _ _ _ _ _ (PCF_sequents Γ Δ lev) (PCF_cutrule Γ Δ lev) :=
+ { se_expand_left := PCF_left Γ Δ lev
+ ; se_expand_right := PCF_right Γ Δ lev }.
+ admit.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ (* 5.1.3 *)
+ Instance PCF Γ Δ lev : @ProgrammingLanguage _ _ pcfjudg (PCFRule Γ Δ lev) :=
+ { pl_eqv := OrgPCF Γ Δ lev
+ ; pl_sc := PCF_sequents Γ Δ lev
+ ; pl_subst := PCF_cutrule Γ Δ lev
+ ; pl_sequent_join := PCF_sequent_join Γ Δ lev
+ }.
+ apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
+
+ apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RCossa _ _ _)).
+ apply (PCF_RArrange lev ((a,,b),,c) (a,,(b,,c)) x).
+
+ apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RAssoc _ _ _)).
+ apply (PCF_RArrange lev (a,,(b,,c)) ((a,,b),,c) x).
+
+ apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RCanL _)).
+ apply (PCF_RArrange lev ([],,a) _ _).
+
+ apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RCanR _)).
+ apply (PCF_RArrange lev (a,,[]) _ _).
+
+ apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RuCanL _)).
+ apply (PCF_RArrange lev _ ([],,a) _).
+
+ apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RuCanR _)).
+ apply (PCF_RArrange lev _ (a,,[]) _).
+ Defined.
+
+ Instance SystemFCa_sequents Γ Δ : @SequentCalculus _ OrgR _ (mkJudg Γ Δ).
+ apply Build_SequentCalculus.
+ intros.
+ induction a.
+ destruct a; simpl.
+ apply nd_rule.
+ destruct l.
+ apply org_fc with (r:=RVar _ _ _ _).
+ auto.
+ apply nd_rule.
+ apply org_fc with (r:=REmptyGroup _ _ ).
+ auto.
+ eapply nd_comp.
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply (nd_prod IHa1 IHa2).
+ apply nd_rule.
+ apply org_fc with (r:=RBindingGroup _ _ _ _ _ _).
+ auto.
+ Defined.
+
+ Definition SystemFCa_cut Γ Δ : forall a b c, ND OrgR ([ Γ > Δ > a |- b ],,[ Γ > Δ > b |- c ]) [ Γ > Δ > a |- c ].
+ intros.
+ destruct b.
+ destruct o.
+ destruct c.
+ destruct o.
+
+ (* when the cut is a single leaf and the RHS is a single leaf: *)
+ eapply nd_comp.
+ eapply nd_prod.
+ apply nd_id.
+ eapply nd_rule.
+ apply org_fc with (r:=RArrange _ _ _ _ _ (RuCanL [l])).
+ auto.
+ eapply nd_comp; [ idtac | eapply nd_rule; apply org_fc with (r:=RArrange _ _ _ _ _ (RCanL _)) ].
+ apply nd_rule.
+ destruct l.
+ destruct l0.
+ assert (h0=h2). admit.
+ subst.
+ apply org_fc with (r:=@RLet Γ Δ [] a h1 h h2).
+ auto.
+ auto.
+ apply (Prelude_error "systemfc cut rule invoked with [a|=[b]] [[b]|=[]]").
+ apply (Prelude_error "systemfc cut rule invoked with [a|=[b]] [[b]|=[x,,y]]").
+ apply (Prelude_error "systemfc rule invoked with [a|=[]] [[]|=c]").
+ apply (Prelude_error "systemfc rule invoked with [a|=[b,,c]] [[b,,c]|=z]").
+ Defined.
+
+ Instance SystemFCa_cutrule Γ Δ : CutRule (SystemFCa_sequents Γ Δ) :=
+ { nd_cut := SystemFCa_cut Γ Δ }.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ Definition SystemFCa_left Γ Δ a b c : ND OrgR [Γ > Δ > b |- c] [Γ > Δ > (a,,b) |- (a,,c)].
+ eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
+ eapply nd_prod; [ apply nd_seq_reflexive | apply nd_id ].
+ apply nd_rule.
+ apply org_fc with (r:=RBindingGroup Γ Δ a b a c).
+ auto.
+ Defined.
+
+ Definition SystemFCa_right Γ Δ a b c : ND OrgR [Γ > Δ > b |- c] [Γ > Δ > (b,,a) |- (c,,a)].
+ eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
+ eapply nd_prod; [ apply nd_id | apply nd_seq_reflexive ].
+ apply nd_rule.
+ apply org_fc with (r:=RBindingGroup Γ Δ b a c a).
+ auto.
+ Defined.
+
+ Instance SystemFCa_sequent_join Γ Δ : @SequentExpansion _ _ _ _ _ (SystemFCa_sequents Γ Δ) (SystemFCa_cutrule Γ Δ) :=
+ { se_expand_left := SystemFCa_left Γ Δ
+ ; se_expand_right := SystemFCa_right Γ Δ }.
+ admit.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ (* 5.1.2 *)
+ Instance SystemFCa Γ Δ : @ProgrammingLanguage _ _ (mkJudg Γ Δ) OrgR :=
+ { pl_eqv := OrgNDR
+ ; pl_sc := SystemFCa_sequents Γ Δ
+ ; pl_subst := SystemFCa_cutrule Γ Δ
+ ; pl_sequent_join := SystemFCa_sequent_join Γ Δ
+ }.
+ apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
+ apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCossa a b c))). apply Flat_RArrange.
+ apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RAssoc a b c))). apply Flat_RArrange.
+ apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanL a ))). apply Flat_RArrange.
+ apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RCanR a ))). apply Flat_RArrange.
+ apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanL a ))). apply Flat_RArrange.
+ apply nd_rule. apply (org_fc _ _ (RArrange _ _ _ _ _ (RuCanR a ))). apply Flat_RArrange.
+ Defined.
+
+End HaskProofStratified.
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