(* *)
(**************************************************************************)
-include "basic_2/rt_transition/fpb_ffdeq.ma".
+include "basic_2/rt_transition/fpb_feqx.ma".
include "basic_2/rt_transition/fpbq.ma".
(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
(* Properties with proper parallel rst-transition for closures **************)
-lemma fpb_fpbq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lfpx, fqu_fquq/
+lemma fpb_fpbq:
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫ →
+ ❪G1,L1,T1❫ ≽ ❪G2,L2,T2❫.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=1 by fpbq_fquq, fpbq_cpx, fpbq_lpx, fqu_fquq/
qed.
(* Basic_2A1: fpb_fpbq_alt *)
-lemma fpb_fpbq_ffdneq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ∧∧ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ & (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥).
-/3 width=10 by fpb_fpbq, fpb_inv_ffdeq, conj/ qed-.
+lemma fpb_fpbq_fneqx:
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫ →
+ ∧∧ ❪G1,L1,T1❫ ≽ ❪G2,L2,T2❫ & (❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ⊥).
+/3 width=10 by fpb_fpbq, fpb_inv_feqx, conj/ qed-.
(* Inversrion lemmas with proper parallel rst-transition for closures *******)
(* Basic_2A1: uses: fpbq_ind_alt *)
-lemma fpbq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- ∨∨ ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+lemma fpbq_inv_fpb:
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≽ ❪G2,L2,T2❫ →
+ ∨∨ ❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫
+ | ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 * [2: * #H1 #H2 #H3 destruct ]
- /3 width=1 by fpb_fqu, ffdeq_intro_sn, or_intror, or_introl/
-| #T2 #H elim (tdeq_dec h o T1 T2)
- /4 width=1 by fpb_cpx, ffdeq_intro_sn, or_intror, or_introl/
-| #L2 elim (lfdeq_dec h o L1 L2 T1)
- /4 width=1 by fpb_lfpx, ffdeq_intro_sn, or_intror, or_introl/
+ /3 width=1 by fpb_fqu, feqx_intro_sn, or_intror, or_introl/
+| #T2 #H elim (teqx_dec T1 T2)
+ /4 width=1 by fpb_cpx, feqx_intro_sn, or_intror, or_introl/
+| #L2 elim (reqx_dec L1 L2 T1)
+ /4 width=1 by fpb_lpx, feqx_intro_sn, or_intror, or_introl/
| /2 width=1 by or_introl/
]
qed-.
(* Basic_2A1: fpbq_inv_fpb_alt *)
-lemma fpbq_ffdneq_inv_fpb: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
- (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥) → ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #H0
+lemma fpbq_fneqx_inv_fpb:
+ ∀G1,G2,L1,L2,T1,T2. ❪G1,L1,T1❫ ≽ ❪G2,L2,T2❫ →
+ (❪G1,L1,T1❫ ≛ ❪G2,L2,T2❫ → ⊥) → ❪G1,L1,T1❫ ≻ ❪G2,L2,T2❫.
+#G1 #G2 #L1 #L2 #T1 #T2 #H #H0
elim (fpbq_inv_fpb … H) -H // #H elim H0 -H0 //
qed-.