(* *)
(**************************************************************************)
-include "basic_2/rt_computation/fpbg_fpbs.ma".
-include "basic_2/rt_computation/fsb_fdeq.ma".
+include "basic_2/rt_computation/fpbg_fqup.ma".
+include "basic_2/rt_computation/fpbg_feqg.ma".
+include "basic_2/rt_computation/fsb_feqg.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⦄ →
- ∀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
+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 … 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/
-| -H1 * /2 width=5 by/
+[ -IH /2 width=9 by fsb_feqg_trans/
+| -H1 #H elim (fpbg_inv_fpbc_fpbs … H)
+ /2 width=5 by/
]
qed-.
+(* Properties with parallel rst-transition for closures *********************)
+
+lemma fsb_fpb_trans:
+ ∀G1,L1,T1. ≥𝐒 ❨G1,L1,T1❩ →
+ ∀G2,L2,T2. ❨G1,L1,T1❩ ≽ ❨G2,L2,T2❩ → ≥𝐒 ❨G2,L2,T2❩.
+/3 width=5 by fsb_fpbs_trans, fpb_fpbs/ qed-.
+
(* 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⦄.
-/4 width=1 by fsb_intro, fpb_fpbg/ qed.
+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, fpbc_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) →
- 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.
-#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 … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
@IH1 -IH1
[ -IH /2 width=5 by fsb_fpbs_trans/
| -H1 #G0 #L0 #T0 #H10
- elim (fpbs_fpbg_trans … H12 … H10) -G2 -L2 -T2
+ lapply (fpbs_fpbg_trans … H12 … H10) -G2 -L2 -T2 #H
+ elim (fpbg_inv_fpbc_fpbs … H) -H #G #L #T #H1 #H0
/2 width=5 by/
]
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) →
- 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
+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):
- ≥[h] 𝐒⦃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-.