(* Properties with parallel rst-computation for closures ********************)
-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⦄.
+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
(* 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⦄ → ≥[h] 𝐒⦃G2, L2, T2⦄
- ) → ≥[h] 𝐒⦃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. ∀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) →
+ (∀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] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≥[h] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+ ∀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
qed-.
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) →
+ (∀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] 𝐒⦃G1, L1, T1⦄ → Q G1 L1 T1.
+ ∀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] 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/