(* Basic_2A1: uses: fpb *)
definition fpbc (G1) (L1) (T1) (G2) (L2) (T2): Prop ≝
- â\88§â\88§ â\9dªG1,L1,T1â\9d« â\89½ â\9dªG2,L2,T2â\9d«
- & (â\9dªG1,L1,T1â\9d« â\89\85 â\9dªG2,L2,T2â\9d« → ⊥).
+ â\88§â\88§ â\9d¨G1,L1,T1â\9d© â\89½ â\9d¨G2,L2,T2â\9d©
+ & (â\9d¨G1,L1,T1â\9d© â\89\85 â\9d¨G2,L2,T2â\9d© → ⊥).
interpretation
"proper parallel rst-transition (closure)"
(* Basic_2A1: fpbq_inv_fpb_alt *)
lemma fpbc_intro (G1) (L1) (T1) (G2) (L2) (T2):
- â\9dªG1,L1,T1â\9d« â\89½ â\9dªG2,L2,T2â\9d« â\86\92 (â\9dªG1,L1,T1â\9d« â\89\85 â\9dªG2,L2,T2â\9d« → ⊥) →
- â\9dªG1,L1,T1â\9d« â\89» â\9dªG2,L2,T2â\9d«.
+ â\9d¨G1,L1,T1â\9d© â\89½ â\9d¨G2,L2,T2â\9d© â\86\92 (â\9d¨G1,L1,T1â\9d© â\89\85 â\9d¨G2,L2,T2â\9d© → ⊥) →
+ â\9d¨G1,L1,T1â\9d© â\89» â\9d¨G2,L2,T2â\9d©.
/3 width=1 by conj/ qed.
lemma rpx_fpbc (G) (T):
- â\88\80L1,L2. â\9dªG,L1â\9d« â\8a¢ â¬\88[T] L2 â\86\92 (L1 â\89\85[T] L2 â\86\92 â\8a¥) â\86\92 â\9dªG,L1,Tâ\9d« â\89» â\9dªG,L2,Tâ\9d«.
+ â\88\80L1,L2. â\9d¨G,L1â\9d© â\8a¢ â¬\88[T] L2 â\86\92 (L1 â\89\85[T] L2 â\86\92 â\8a¥) â\86\92 â\9d¨G,L1,Tâ\9d© â\89» â\9d¨G,L2,Tâ\9d©.
/4 width=4 by fpbc_intro, rpx_fpb, feqg_fwd_reqg_sn/ qed.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: fpb_fpbq_alt *)
lemma fpbc_inv_gen (S):
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â\89» â\9dªG2,L2,T2â\9d« →
- â\88§â\88§ â\9dªG1,L1,T1â\9d« â\89½ â\9dªG2,L2,T2â\9d« & (â\9dªG1,L1,T1â\9d« â\89\9b[S] â\9dªG2,L2,T2â\9d« → ⊥).
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â\89» â\9d¨G2,L2,T2â\9d© →
+ â\88§â\88§ â\9d¨G1,L1,T1â\9d© â\89½ â\9d¨G2,L2,T2â\9d© & (â\9d¨G1,L1,T1â\9d© â\89\9b[S] â\9d¨G2,L2,T2â\9d© → ⊥).
#S #G1 #G2 #L1 #L2 #T1 #T2 *
/4 width=2 by feqg_feqx, conj/
qed-.
(* Basic_2A1: uses: fpb_fpbq *)
lemma fpbc_fwd_fpb:
- â\88\80G1,G2,L1,L2,T1,T2. â\9dªG1,L1,T1â\9d« â\89» â\9dªG2,L2,T2â\9d« →
- â\9dªG1,L1,T1â\9d« â\89½ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,G2,L1,L2,T1,T2. â\9d¨G1,L1,T1â\9d© â\89» â\9d¨G2,L2,T2â\9d© →
+ â\9d¨G1,L1,T1â\9d© â\89½ â\9d¨G2,L2,T2â\9d©.
#G1 #G2 #L1 #L2 #T1 #T2 * //
qed-.