(* Properties with parallel rst-computation for closures ********************)
-lemma fsb_fpbs_trans: ∀h,G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ →
- ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → ≥𝐒[h] ❪G2,L2,T2❫.
-#h #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+lemma fsb_fpbs_trans:
+ ∀G1,L1,T1. ≥𝐒 ❪G1,L1,T1❫ →
+ ∀G2,L2,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ → ≥𝐒 ❪G2,L2,T2❫.
+#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_feqx_trans/
(* Properties with proper parallel rst-computation for closures *************)
-lemma fsb_intro_fpbg: ∀h,G1,L1,T1. (
- ∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → ≥𝐒[h] ❪G2,L2,T2❫
- ) → ≥𝐒[h] ❪G1,L1,T1❫.
+lemma fsb_intro_fpbg:
+ ∀G1,L1,T1.
+ (∀G2,L2,T2. ❪G1,L1,T1❫ > ❪G2,L2,T2❫ → ≥𝐒 ❪G2,L2,T2❫) →
+ ≥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. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ →
- (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) →
- ∀G1,L1,T1. ≥𝐒[h] ❪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
+lemma fsb_ind_fpbg_fpbs (Q:relation3 …):
+ (∀G1,L1,T1. ≥𝐒 ❪G1,L1,T1❫ →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ > ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G1,L1,T1. ≥𝐒 ❪G1,L1,T1❫ →
+ ∀G2,L2,T2. ❪G1,L1,T1❫ ≥ ❪G2,L2,T2❫ → Q G2 L2 T2.
+#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. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ≥𝐒[h] ❪G1,L1,T1❫ →
- (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
- 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
+lemma fsb_ind_fpbg (Q:relation3 …):
+ (∀G1,L1,T1. ≥𝐒 ❪G1,L1,T1❫ →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ > ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G1,L1,T1. ≥𝐒 ❪G1,L1,T1❫ → Q G1 L1 T1.
+#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):
- ≥𝐒[h] ❪G,L,T❫ → ❪G,L,T❫ >[h] ❪G,L,T❫ → ⊥.
-#h #G #L #T #H
+lemma fsb_fpbg_refl_false (G) (L) (T):
+ ≥𝐒 ❪G,L,T❫ → ❪G,L,T❫ > ❪G,L,T❫ → ⊥.
+#G #L #T #H
@(fsb_ind_fpbg … H) -G -L -T #G1 #L1 #T1 #_ #IH #H
/2 width=5 by/
qed-.