(* Properties with parallel rst-computation for closures ********************)
-lemma fsb_fpbs_trans: â\88\80h,G1,L1,T1. â\89¥[h] ð\9d\90\92â¦\83G1,L1,T1â¦\84 →
- â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 â\89¥[h] â¦\83G2,L2,T2â¦\84 â\86\92 â\89¥[h] ð\9d\90\92â¦\83G2,L2,T2â¦\84.
+lemma fsb_fpbs_trans: â\88\80h,G1,L1,T1. â\89¥[h] ð\9d\90\92â\9dªG1,L1,T1â\9d« →
+ â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« â\89¥[h] â\9dªG2,L2,T2â\9d« â\86\92 â\89¥[h] ð\9d\90\92â\9dªG2,L2,T2â\9d«.
#h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
elim (fpbs_inv_fpbg … H12) -H12
(* Properties with proper parallel rst-computation for closures *************)
lemma fsb_intro_fpbg: ∀h,G1,L1,T1. (
- â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 >[h] â¦\83G2,L2,T2â¦\84 â\86\92 â\89¥[h] ð\9d\90\92â¦\83G2,L2,T2â¦\84
- ) â\86\92 â\89¥[h] ð\9d\90\92â¦\83G1,L1,T1â¦\84.
+ â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d« â\86\92 â\89¥[h] ð\9d\90\92â\9dªG2,L2,T2â\9d«
+ ) â\86\92 â\89¥[h] ð\9d\90\92â\9dªG1,L1,T1â\9d«.
/4 width=1 by fsb_intro, fpb_fpbg/ qed.
(* Eliminators with proper parallel rst-computation for closures ************)
lemma fsb_ind_fpbg_fpbs: ∀h. ∀Q:relation3 genv lenv term.
- (â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â¦\83G1,L1,T1â¦\84 →
- (â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 >[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2) →
+ (â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â\9dªG1,L1,T1â\9d« →
+ (â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d« → Q G2 L2 T2) →
Q G1 L1 T1
) →
- â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â¦\83G1,L1,T1â¦\84 →
- â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 â\89¥[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2.
+ â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â\9dªG1,L1,T1â\9d« →
+ â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« â\89¥[h] â\9dªG2,L2,T2â\9d« → Q G2 L2 T2.
#h #Q #IH1 #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
@IH1 -IH1
qed-.
lemma fsb_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
- (â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â¦\83G1,L1,T1â¦\84 →
- (â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 >[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2) →
+ (â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â\9dªG1,L1,T1â\9d« →
+ (â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d« → Q G2 L2 T2) →
Q G1 L1 T1
) →
- â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â¦\83G1,L1,T1â¦\84 → Q G1 L1 T1.
+ â\88\80G1,L1,T1. â\89¥[h] ð\9d\90\92â\9dªG1,L1,T1â\9d« → Q G1 L1 T1.
#h #Q #IH #G1 #L1 #T1 #H @(fsb_ind_fpbg_fpbs … H) -H
/3 width=1 by/
qed-.
(* Inversion lemmas with proper parallel rst-computation for closures *******)
lemma fsb_fpbg_refl_false (h) (G) (L) (T):
- â\89¥[h] ð\9d\90\92â¦\83G,L,Tâ¦\84 â\86\92 â¦\83G,L,Tâ¦\84 >[h] â¦\83G,L,Tâ¦\84 → ⊥.
+ â\89¥[h] ð\9d\90\92â\9dªG,L,Tâ\9d« â\86\92 â\9dªG,L,Tâ\9d« >[h] â\9dªG,L,Tâ\9d« → ⊥.
#h #G #L #T #H
@(fsb_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #_ #IH #H
/2 width=5 by/