+(*********************************************************************************************************************************)
+(* PCF: *)
+(* *)
+(* 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 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 HaskKinds.
+Require Import HaskCoreTypes.
+Require Import HaskLiteralsAndTyCons.
+Require Import HaskStrongTypes.
+Require Import HaskProof.
+Require Import NaturalDeduction.
+Require Import NaturalDeductionCategory.
+
+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 PCF.
+
+ Section PCF.
+
+ Context {ndr_systemfc:@ND_Relation _ Rule}.
+ Context Γ (Δ:CoercionEnv Γ).
+
+ Definition PCFJudg (ec:HaskTyVar Γ ★) :=
+ @prod (Tree ??(HaskType Γ ★)) (Tree ??(HaskType Γ ★)).
+ Definition pcfjudg (ec:HaskTyVar Γ ★) :=
+ @pair (Tree ??(HaskType Γ ★)) (Tree ??(HaskType Γ ★)).
+
+ (* 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 {ec} (j:PCFJudg ec) : Judg :=
+ match j with
+ (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 {ec} : Tree ??(PCFJudg 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
+ [((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 FCJudg :=
+ @prod (Tree ??(LeveledHaskType Γ ★)) (Tree ??(LeveledHaskType Γ ★)).
+ Definition fcjudg2judg (fc:FCJudg) :=
+ match fc with
+ (x,y) => Γ > Δ > x |- y
+ end.
+ Coercion fcjudg2judg : FCJudg >-> Judg.
+
+ Definition pcfjudg2judg ec (cj:PCFJudg ec) :=
+ match cj with (Σ,τ) => Γ > Δ > (Σ @@@ (ec::nil)) |- (τ @@@ (ec::nil)) end.
+
+ Definition pcfjudg2fcjudg ec (fc:PCFJudg ec) : FCJudg :=
+ match fc with
+ (x,y) => (x @@@ (ec::nil),y @@@ (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 (ec:HaskTyVar Γ ★)
+ : forall (h c:Tree ??(PCFJudg ec)), Rule (mapOptionTree (pcfjudg2judg ec) h) (mapOptionTree (pcfjudg2judg ec) c) -> Type :=
+ | PCF_RArrange : ∀ x y t a, Rule_PCF ec [(_, _)] [(_, _)] (RArrange Γ Δ (x@@@(ec::nil)) (y@@@(ec::nil)) (t@@@(ec::nil)) a)
+ | PCF_RLit : ∀ lit , Rule_PCF ec [ ] [ ([],[_]) ] (RLit Γ Δ lit (ec::nil))
+ | PCF_RNote : ∀ Σ τ n , Rule_PCF ec [(_,[_])] [(_,[_])] (RNote Γ Δ (Σ@@@(ec::nil)) τ (ec::nil) n)
+ | PCF_RVar : ∀ σ , Rule_PCF ec [ ] [([_],[_])] (RVar Γ Δ σ (ec::nil) )
+ | PCF_RLam : ∀ Σ tx te , Rule_PCF ec [((_,,[_]),[_])] [(_,[_])] (RLam Γ Δ (Σ@@@(ec::nil)) tx te (ec::nil) )
+
+ | PCF_RApp : ∀ Σ Σ' tx te ,
+ Rule_PCF ec ([(_,[_])],,[(_,[_])]) [((_,,_),[_])]
+ (RApp Γ Δ (Σ@@@(ec::nil))(Σ'@@@(ec::nil)) tx te (ec::nil))
+
+ | PCF_RLet : ∀ Σ Σ' σ₂ p,
+ Rule_PCF ec ([(_,[_])],,[((_,,[_]),[_])]) [((_,,_),[_])]
+ (RLet Γ Δ (Σ@@@(ec::nil)) (Σ'@@@(ec::nil)) σ₂ p (ec::nil))
+
+ | PCF_RVoid : Rule_PCF ec [ ] [([],[])] (RVoid Γ Δ )
+(*| PCF_RLetRec : ∀ Σ₁ τ₁ τ₂ , Rule_PCF (ec::nil) _ _ (RLetRec Γ Δ Σ₁ τ₁ τ₂ (ec::nil) )*)
+ | PCF_RJoin : ∀ Σ₁ Σ₂ τ₁ τ₂, Rule_PCF ec ([(_,_)],,[(_,_)]) [((_,,_),(_,,_))]
+ (RJoin Γ Δ (Σ₁@@@(ec::nil)) (Σ₂@@@(ec::nil)) (τ₁@@@(ec::nil)) (τ₂@@@(ec::nil))).
+ (* need int/boolean case *)
+ Implicit Arguments Rule_PCF [ ].
+
+ Definition PCFRule lev h c := { r:_ & @Rule_PCF lev h c r }.
+ End PCF.
+
+ Definition mkEsc Γ Δ ec (h:Tree ??(PCFJudg Γ ec))
+ : ND Rule
+ (mapOptionTree (brakify Γ Δ) h)
+ (mapOptionTree (pcfjudg2judg Γ Δ 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 ec))
+ : ND Rule
+ (mapOptionTree (pcfjudg2judg Γ Δ ec) h)
+ (mapOptionTree (brakify Γ Δ) 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 Γ Δ : forall ec h c,
+ ND (PCFRule Γ Δ ec) h c -> ND Rule (mapOptionTree (pcfjudg2judg Γ Δ ec) h) (mapOptionTree (pcfjudg2judg Γ Δ 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) }.
+ Admitted.
+
+ (*
+ * 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 Γ Δ h h' ->
+ MatchingJudgments Γ Δ 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 :
+ ClosedSIND(Rule:=Rule) [a > b > c |- [d @@ (ec :: e)]] ->
+ ClosedSIND(Rule:=Rule) [a > b > pcf_vars ec c @@@ (ec :: nil) |- [d @@ (ec :: nil)]].
+ admit.
+ Defined.
+
+ Definition orgify : forall Γ Δ Σ τ (pf:ClosedSIND(Rule:=Rule) [Γ > Δ > Σ |- τ]), Alternating [Γ > Δ > Σ |- τ].
+
+ refine (
+ fix orgify_fc' Γ Δ Σ τ (pf:ClosedSIND [Γ > Δ > Σ |- τ]) {struct pf} : Alternating [Γ > Δ > Σ |- τ] :=
+ let case_main := tt in _
+ with orgify_fc c (pf:ClosedSIND 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 Γ Δ pcfj j)
+ (pf:ClosedSIND (mapOptionTree (pcfjudg2judg Γ Δ 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,
+ ClosedSIND(Rule:=Rule) [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]
+ -> ND (PCFRule Γ Δ ec) [] [(Σ,τ)].
+
+ refine ((
+ fix pcfify Σ τ (pn:@ClosedSIND _ Rule [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]) {struct pn}
+ : ND (PCFRule Γ Δ ec) [] [(Σ,τ)] :=
+ (match pn in @ClosedSIND _ _ 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.
+*)
+
+ Hint Constructors Rule_Flat.
+
+ Definition PCF_Arrange {Γ}{Δ}{lev} : forall x y z, Arrange x y -> ND (PCFRule Γ Δ lev) [(x,z)] [(y,z)].
+ admit.
+ Defined.
+
+ Definition PCF_cut Γ Δ lev : forall a b c, ND (PCFRule Γ Δ lev) ([(a,b)],,[(b,c)]) [(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]").
+ *)
+ Admitted.
+
+ Instance PCF_sequents Γ Δ lev ec : @SequentND _ (PCFRule Γ Δ lev) _ (pcfjudg Γ ec) :=
+ { snd_cut := PCF_cut Γ Δ lev }.
+ apply Build_SequentND.
+ intros.
+ induction a.
+ destruct a; simpl.
+ apply nd_rule.
+ exists (RVar _ _ _ _).
+ apply PCF_RVar.
+ apply nd_rule.
+ exists (RVoid _ _ ).
+ apply PCF_RVoid.
+ eapply nd_comp.
+ eapply nd_comp; [ apply nd_llecnac | idtac ].
+ apply (nd_prod IHa1 IHa2).
+ apply nd_rule.
+ exists (RJoin _ _ _ _ _ _).
+ apply PCF_RJoin.
+ admit.
+ Defined.
+
+ Definition PCF_left Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [(b,c)] [((a,,b),(a,,c))].
+ eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
+ eapply nd_prod; [ apply snd_initial | apply nd_id ].
+ apply nd_rule.
+ set (@PCF_RJoin Γ Δ lev a b a c) as q'.
+ refine (existT _ _ _).
+ apply q'.
+ Admitted.
+
+ Definition PCF_right Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [(b,c)] [((b,,a),(c,,a))].
+ eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
+ eapply nd_prod; [ apply nd_id | apply snd_initial ].
+ apply nd_rule.
+ set (@PCF_RJoin Γ Δ lev b a c a) as q'.
+ refine (existT _ _ _).
+ apply q'.
+ Admitted.
+
+ Instance PCF_sequent_join Γ Δ lev : @ContextND _ (PCFRule Γ Δ lev) _ (pcfjudg Γ lev) _ :=
+ { cnd_expand_left := fun a b c => PCF_left Γ Δ lev c a b
+ ; cnd_expand_right := fun a b c => PCF_right Γ Δ lev c a b }.
+
+ intros; apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RCossa _ _ _)).
+ apply (PCF_RArrange _ _ lev ((a,,b),,c) (a,,(b,,c)) x).
+
+ intros; apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RAssoc _ _ _)).
+ apply (PCF_RArrange _ _ lev (a,,(b,,c)) ((a,,b),,c) x).
+
+ intros; apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RCanL _)).
+ apply (PCF_RArrange _ _ lev ([],,a) _ _).
+
+ intros; apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RCanR _)).
+ apply (PCF_RArrange _ _ lev (a,,[]) _ _).
+
+ intros; apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RuCanL _)).
+ apply (PCF_RArrange _ _ lev _ ([],,a) _).
+
+ intros; apply nd_rule. unfold PCFRule. simpl.
+ exists (RArrange _ _ _ _ _ (RuCanR _)).
+ apply (PCF_RArrange _ _ lev _ (a,,[]) _).
+ Defined.
+
+ Instance OrgPCF_SequentND_Relation Γ Δ lev : SequentND_Relation (PCF_sequent_join Γ Δ lev) (OrgPCF Γ Δ lev).
+ admit.
+ Defined.
+
+ Definition OrgPCF_ContextND_Relation Γ Δ lev
+ : @ContextND_Relation _ _ _ _ _ (PCF_sequent_join Γ Δ lev) (OrgPCF Γ Δ lev) (OrgPCF_SequentND_Relation Γ Δ lev).
+ admit.
+ Defined.
+
+ (* 5.1.3 *)
+ Instance PCF Γ Δ lev : ProgrammingLanguage :=
+ { pl_cnd := PCF_sequent_join Γ Δ lev
+ ; pl_eqv := OrgPCF_ContextND_Relation Γ Δ lev
+ }.
+
+End PCF.
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