+++ /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.
-
- 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 pcfjudg2judg ec (cj:PCFJudg ec) :=
- match cj with (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 (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 FCJudg Γ (Δ:CoercionEnv Γ) :=
- @prod (Tree ??(LeveledHaskType Γ ★)) (Tree ??(LeveledHaskType Γ ★)).
- Definition fcjudg2judg {Γ}{Δ}(fc:FCJudg Γ Δ) :=
- match fc with
- (x,y) => Γ > Δ > x |- y
- end.
- Coercion fcjudg2judg : FCJudg >-> Judg.
-
- Definition pcfjudg2fcjudg {Γ}{Δ} ec (fc:PCFJudg Γ ec) : FCJudg Γ Δ :=
- match fc with
- (x,y) => (x @@@ (ec::nil),y @@@ (ec::nil))
- end.
-
- (* 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 Γ Δ : Tree ??(FCJudg Γ Δ) -> Tree ??(FCJudg Γ Δ) -> Type :=
-
- | org_fc : forall (h c:Tree ??(FCJudg Γ Δ))
- (r:Rule (mapOptionTree fcjudg2judg h) (mapOptionTree fcjudg2judg c)),
- Rule_Flat r ->
- OrgR _ _ h c
-
- | org_pcf : forall ec h c,
- ND (PCFRule Γ Δ ec) h c ->
- OrgR Γ Δ (mapOptionTree (pcfjudg2fcjudg ec) h) (mapOptionTree (pcfjudg2fcjudg ec) c).
-
- 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.
-*)
- (* any proof in organized form can be "dis-organized" *)
- (*
- Definition unOrgR : forall Γ Δ h c, OrgR Γ Δ 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 Γ Δ h c : ND (OrgR Γ Δ) h c -> ND Rule h c := nd_map (unOrgR Γ Δ).
- *)
-
- 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
- }.
-
- Definition SystemFCa_cut Γ Δ : forall a b c, ND (OrgR Γ Δ) ([(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.
- eapply nd_rule.
- set (@org_fc) as ofc.
- set (RArrange Γ Δ _ _ _ (RuCanL [l0])) as rule.
- apply org_fc with (r:=RArrange _ _ _ _ _ (RuCanL [_])).
- 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.
- *)
- admit.
- 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_sequents Γ Δ : @SequentND _ (OrgR Γ Δ) _ _ :=
- { snd_cut := SystemFCa_cut Γ Δ }.
- apply Build_SequentND.
- 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:=RVoid _ _ ).
- auto.
- eapply nd_comp.
- eapply nd_comp; [ apply nd_llecnac | idtac ].
- apply (nd_prod IHa1 IHa2).
- apply nd_rule.
- apply org_fc with (r:=RJoin _ _ _ _ _ _).
- auto.
- admit.
- *)
- admit.
- admit.
- admit.
- admit.
- Defined.
-
- Definition SystemFCa_left Γ Δ a b c : ND (OrgR Γ Δ) [(b,c)] [((a,,b),(a,,c))].
- admit.
- (*
- eapply nd_comp; [ apply nd_llecnac | eapply nd_comp; [ idtac | idtac ] ].
- eapply nd_prod; [ apply snd_initial | apply nd_id ].
- apply nd_rule.
- apply org_fc with (r:=RJoin Γ Δ a b a c).
- auto.
- *)
- Defined.
-
- Definition SystemFCa_right Γ Δ a b c : ND (OrgR Γ Δ) [(b,c)] [((b,,a),(c,,a))].
- admit.
- (*
- eapply nd_comp; [ apply nd_rlecnac | eapply nd_comp; [ idtac | idtac ] ].
- eapply nd_prod; [ apply nd_id | apply snd_initial ].
- apply nd_rule.
- apply org_fc with (r:=RJoin Γ Δ b a c a).
- auto.
- *)
- Defined.
-
- Instance SystemFCa_sequent_join Γ Δ : @ContextND _ _ _ _ (SystemFCa_sequents Γ Δ) :=
- { cnd_expand_left := fun a b c => SystemFCa_left Γ Δ c a b
- ; cnd_expand_right := fun a b c => SystemFCa_right Γ Δ c a b }.
- (*
- intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ ((RArrange _ _ _ _ _ (RCossa _ _ _)))).
- auto.
-
- intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RAssoc _ _ _))); auto.
-
- intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RCanL _))); auto.
-
- intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RCanR _))); auto.
-
- intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RuCanL _))); auto.
-
- intros; apply nd_rule. simpl.
- apply (org_fc _ _ _ _ (RArrange _ _ _ _ _ (RuCanR _))); auto.
- *)
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- Defined.
-
- Instance OrgFC Γ Δ : @ND_Relation _ (OrgR Γ Δ).
- Admitted.
-
- Instance OrgFC_SequentND_Relation Γ Δ : SequentND_Relation (SystemFCa_sequent_join Γ Δ) (OrgFC Γ Δ).
- admit.
- Defined.
-
- Definition OrgFC_ContextND_Relation Γ Δ
- : @ContextND_Relation _ _ _ _ _ (SystemFCa_sequent_join Γ Δ) (OrgFC Γ Δ) (OrgFC_SequentND_Relation Γ Δ).
- admit.
- Defined.
-
- (* 5.1.2 *)
- Instance SystemFCa Γ Δ : @ProgrammingLanguage (LeveledHaskType Γ ★) _ :=
- { pl_eqv := OrgFC_ContextND_Relation Γ Δ
- ; pl_snd := SystemFCa_sequents Γ Δ
- }.
-
-End HaskProofStratified.
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