(**************************************************************************)
include "basic_2/rt_computation/fpbg_fpbs.ma".
-include "basic_2/rt_computation/fsb_fdeq.ma".
+include "basic_2/rt_computation/fsb_feqx.ma".
(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
(* Properties with parallel rst-computation for closures ********************)
-lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
- 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: ∀h,G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ →
+ â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« â\89¥[h] â\9dªG2,L2,T2â\9d« â\86\92 â\89¥ð\9d\90\92[h] â\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
-[ -IH /2 width=5 by fsb_fdeq_trans/
+[ -IH /2 width=5 by fsb_feqx_trans/
| -H1 * /2 width=5 by/
]
qed-.
(* 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
- ) → ≥[h] 𝐒⦃G1,L1,T1⦄.
+ â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d« â\86\92 â\89¥ð\9d\90\92[h] â\9dªG2,L2,T2â\9d«
+ ) → ≥𝐒[h] ❪G1,L1,T1❫.
/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.
- (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
- (â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 >[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2) →
+ (∀G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ →
+ (â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d« → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
- â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 â\89¥[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2.
+ ∀G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ →
+ 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.
- (∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ →
- (â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 >[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2) →
+ (∀G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ →
+ (â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d« → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h] 𝐒⦃G1,L1,T1⦄ → Q G1 L1 T1.
+ ∀G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ → 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):
- ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L,T⦄ >[h] ⦃G,L,T⦄ → ⊥.
+ ≥𝐒[h] ❪G,L,T❫ → ❪G,L,T❫ >[h] ❪G,L,T❫ → ⊥.
#h #G #L #T #H
@(fsb_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #_ #IH #H
/2 width=5 by/