[((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 *)
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.
{ 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)].
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