*)
Section HaskProofStratified.
+ Section PCF.
+
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] ].
+ 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 {Γ}{Δ}{ec} (j:PCFJudg Γ Δ ec) : Judg :=
+ Definition brakify {ec} (j:PCFJudg ec) : Judg :=
match j with
- pcfjudg Σ τ => Γ > Δ > (Σ@@@(ec::nil)) |- (mapOptionTree (fun t => HaskBrak ec t) τ @@@ nil)
+ (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 ★)
end
end) t.
- Inductive MatchingJudgments {Γ}{Δ}{ec} : Tree ??(PCFJudg Γ Δ ec) -> Tree ??Judg -> Type :=
+ 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
- [pcfjudg (pcf_vars ec Σ) τ ]
+ [((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 ★)
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.
+ 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 {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) )
+ 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 ([pcfjudg _ [_]],,[pcfjudg _ [_]]) [pcfjudg (_,,_) [_]]
+ 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 ([pcfjudg _ [_]],,[pcfjudg (_,,[_]) [_]]) [pcfjudg (_,,_) [_]]
+ Rule_PCF ec ([(_,[_])],,[((_,,[_]),[_])]) [((_,,_),[_])]
(RLet Γ Δ (Σ@@@(ec::nil)) (Σ'@@@(ec::nil)) σ₂ p (ec::nil))
- | PCF_RVoid : Rule_PCF ec [ ] [ pcfjudg [] [] ] (RVoid Γ Δ )
+ | 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 ([pcfjudg _ _],,[pcfjudg _ _]) [pcfjudg (_,,_) (_,,_)]
+ | 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 Γ Δ lev h c := { r:_ & @Rule_PCF Γ Δ lev h c r }.
+ 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 : Tree ??Judg -> Tree ??Judg -> Type :=
+ Inductive OrgR Γ Δ : Tree ??(FCJudg Γ Δ) -> Tree ??(FCJudg Γ Δ) -> Type :=
- | org_fc : forall h c (r:Rule h c),
+ | org_fc : forall (h c:Tree ??(FCJudg Γ Δ))
+ (r:Rule (mapOptionTree fcjudg2judg h) (mapOptionTree fcjudg2judg c)),
Rule_Flat r ->
- OrgR h c
+ 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'.
+ | 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 Δ ec))
+ Definition mkEsc Γ Δ ec (h:Tree ??(PCFJudg Γ ec))
: ND Rule
- (mapOptionTree brakify h)
- (mapOptionTree (pcfjudg2judg ec) h).
+ (mapOptionTree (brakify Γ Δ) h)
+ (mapOptionTree (pcfjudg2judg Γ Δ ec) h).
apply nd_replicate; intros.
destruct o; simpl in *.
induction t0.
apply (Prelude_error "mkEsc got multi-leaf succedent").
Defined.
- Definition mkBrak {Γ}{Δ}{ec}(h:Tree ??(PCFJudg Γ Δ ec))
+ Definition mkBrak Γ Δ ec (h:Tree ??(PCFJudg Γ ec))
: ND Rule
- (mapOptionTree (pcfjudg2judg ec) h)
- (mapOptionTree brakify h).
+ (mapOptionTree (pcfjudg2judg Γ Δ ec) h)
+ (mapOptionTree (brakify Γ Δ) h).
apply nd_replicate; intros.
destruct o; simpl in *.
induction t0.
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).
+ 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.
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.
- Defined.
+ { ndr_eqv := fun a b f g => (pcfToND _ _ _ _ _ f) === (pcfToND _ _ _ _ _ g) }.
+ Admitted.
(*
* An intermediate representation necessitated by Coq's termination
Alternating c
| alt_pcf : forall Γ Δ ec h c h' c',
- MatchingJudgments h h' ->
- MatchingJudgments c 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.
+ admit.
+ Defined.
Definition orgify : forall Γ Δ Σ τ (pf:ClosedSIND(Rule:=Rule) [Γ > Δ > Σ |- τ]), Alternating [Γ > Δ > Σ |- τ].
| 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:ClosedSIND (mapOptionTree (pcfjudg2judg ec) pcfj)) {struct pf} : Alternating j :=
+ 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').
Definition pcfify Γ Δ ec : forall Σ τ,
ClosedSIND(Rule:=Rule) [ Γ > Δ > Σ@@@(ec::nil) |- τ @@@ (ec::nil)]
- -> ND (PCFRule Γ Δ ec) [] [pcfjudg Σ τ].
+ -> 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) [] [pcfjudg Σ τ] :=
+ : 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 _
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.
+ (*
+ 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).
apply n.
*)
Admitted.
-
- Definition unOrgND h c : ND OrgR h c -> ND Rule h c := nd_map unOrgR.
+ Definition unOrgND Γ Δ h c : ND (OrgR Γ Δ) h c -> ND Rule h c := nd_map (unOrgR Γ Δ).
+ *)
- 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.
- Defined.
-
Hint Constructors Rule_Flat.
- Definition PCF_Arrange {Γ}{Δ}{lev} : forall x y z, Arrange x y -> ND (PCFRule Γ Δ lev) [pcfjudg x z] [pcfjudg y z].
+ 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) ([ pcfjudg a b ],,[ pcfjudg b c ]) [ pcfjudg a c ].
+ 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.
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]|=[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.
+ *)
+ Admitted.
- Instance PCF_sequents Γ Δ lev : @SequentND _ (PCFRule Γ Δ lev) _ pcfjudg :=
+ 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.
admit.
Defined.
- Definition PCF_left Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (a,,b) (a,,c)].
+ 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'.
- Defined.
+ Admitted.
- Definition PCF_right Γ Δ lev a b c : ND (PCFRule Γ Δ lev) [pcfjudg b c] [pcfjudg (b,,a) (c,,a)].
+ 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'.
- Defined.
+ Admitted.
- Instance PCF_sequent_join Γ Δ lev : @ContextND _ (PCFRule Γ Δ lev) _ pcfjudg _ :=
+ 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 }.
- admit.
- admit.
- admit.
- admit.
- admit.
- admit.
- Defined.
- Instance OrgPCF_SequentND_Relation Γ Δ lev : SequentND_Relation (PCF_sequent_join Γ Δ lev) OrgND.
- admit.
- Defined.
-
- Instance OrgPCF_ContextND_Relation Γ Δ lev : ContextND_Relation (PCF_sequent_join Γ Δ lev).
- admit.
- Defined.
-
- (* 5.1.3 *)
- Instance PCF Γ Δ lev : @ProgrammingLanguage _ _ pcfjudg (PCFRule Γ Δ lev) :=
- { pl_eqv := OrgPCF_ContextND_Relation Γ Δ lev
- ; pl_snd := PCF_sequents Γ Δ lev
- }.
- (*
- apply Build_TreeStructuralRules; intros; unfold eqv; unfold hom; simpl.
-
- apply nd_rule. unfold PCFRule. simpl.
+ intros; apply nd_rule. unfold PCFRule. simpl.
exists (RArrange _ _ _ _ _ (RCossa _ _ _)).
- apply (PCF_RArrange lev ((a,,b),,c) (a,,(b,,c)) x).
+ apply (PCF_RArrange _ _ lev ((a,,b),,c) (a,,(b,,c)) x).
- apply nd_rule. unfold PCFRule. simpl.
+ intros; apply nd_rule. unfold PCFRule. simpl.
exists (RArrange _ _ _ _ _ (RAssoc _ _ _)).
- apply (PCF_RArrange lev (a,,(b,,c)) ((a,,b),,c) x).
+ apply (PCF_RArrange _ _ lev (a,,(b,,c)) ((a,,b),,c) x).
- apply nd_rule. unfold PCFRule. simpl.
+ intros; apply nd_rule. unfold PCFRule. simpl.
exists (RArrange _ _ _ _ _ (RCanL _)).
- apply (PCF_RArrange lev ([],,a) _ _).
+ apply (PCF_RArrange _ _ lev ([],,a) _ _).
- apply nd_rule. unfold PCFRule. simpl.
+ intros; apply nd_rule. unfold PCFRule. simpl.
exists (RArrange _ _ _ _ _ (RCanR _)).
- apply (PCF_RArrange lev (a,,[]) _ _).
+ apply (PCF_RArrange _ _ lev (a,,[]) _ _).
- apply nd_rule. unfold PCFRule. simpl.
+ intros; apply nd_rule. unfold PCFRule. simpl.
exists (RArrange _ _ _ _ _ (RuCanL _)).
- apply (PCF_RArrange lev _ ([],,a) _).
+ apply (PCF_RArrange _ _ lev _ ([],,a) _).
- apply nd_rule. unfold PCFRule. simpl.
+ intros; apply nd_rule. unfold PCFRule. simpl.
exists (RArrange _ _ _ _ _ (RuCanR _)).
- apply (PCF_RArrange lev _ (a,,[]) _).
+ apply (PCF_RArrange _ _ lev _ (a,,[]) _).
Defined.
-*)
- Definition SystemFCa_cut Γ Δ : forall a b c, ND OrgR ([ Γ > Δ > a |- b ],,[ Γ > Δ > b |- c ]) [ Γ > Δ > a |- c ].
+ 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 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])).
+ 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.
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 _ (mkJudg Γ Δ) :=
+ 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 _ _ _ _).
apply org_fc with (r:=RJoin _ _ _ _ _ _).
auto.
admit.
- Defined.
+ *)
+ admit.
+ admit.
+ admit.
+ admit.
+ Defined.
- Definition SystemFCa_left Γ Δ a b c : ND OrgR [Γ > Δ > b |- c] [Γ > Δ > (a,,b) |- (a,,c)].
+ 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] [Γ > Δ > (b,,a) |- (c,,a)].
+ 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 : @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.
+ 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 _ _ (mkJudg Γ Δ) OrgR.
-(*
- { pl_eqv := OrgNDR
- ; pl_sn := SystemFCa_sequents Γ Δ
- ; pl_subst := SystemFCa_cutrule Γ Δ
- ; pl_sequent_join := SystemFCa_sequent_join Γ Δ
+ Instance SystemFCa Γ Δ : @ProgrammingLanguage (LeveledHaskType Γ ★) _ :=
+ { pl_eqv := OrgFC_ContextND_Relation Γ Δ
+ ; pl_snd := SystemFCa_sequents 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.
-*)
-admit.
- Defined.
End HaskProofStratified.
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