(**************************************************************************)
include "basic_2/rt_computation/fpbg_fpbs.ma".
-include "basic_2/rt_computation/fsb_ffdeq.ma".
+include "basic_2/rt_computation/fsb_fdeq.ma".
(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
(* Properties with parallel rst-computation for closures ********************)
-lemma fsb_fpbs_trans: ∀h,o,G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄.
-#h #o #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄.
+#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_ffdeq_trans/
+[ -IH /2 width=5 by fsb_fdeq_trans/
| -H1 * /2 width=5 by/
]
qed-.
(* Properties with proper parallel rst-computation for closures *************)
-lemma fsb_intro_fpbg: ∀h,o,G1,L1,T1. (
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄
- ) → ≥[h, o] 𝐒⦃G1, L1, T1⦄.
+lemma fsb_intro_fpbg: ∀h,G1,L1,T1. (
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄
+ ) → ≥[h] 𝐒⦃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,o. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma fsb_ind_fpbg_fpbs: ∀h. ∀Q:relation3 genv lenv term.
+ (∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
-#h #o #Q #IH1 #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+ ∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → 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
[ -IH /2 width=5 by fsb_fpbs_trans/
]
qed-.
-lemma fsb_ind_fpbg: ∀h,o. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+lemma fsb_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
+ (∀G1,L1,T1. ≥[h] 𝐒⦃G1, L1, T1⦄ →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ >[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
Q G1 L1 T1
) →
- ∀G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ → Q G1 L1 T1.
-#h #o #Q #IH #G1 #L1 #T1 #H @(fsb_ind_fpbg_fpbs … H) -H
+ ∀G1,L1,T1. ≥[h] 𝐒⦃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⦄ → ⊥.
+#h #G #L #T #H
+@(fsb_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #_ #IH #H
+/2 width=5 by/
+qed-.