(* Basic_2A1: uses: fpbq *)
definition fpb (G1) (L1) (T1) (G2) (L2) (T2): Prop ≝
- â\88\83â\88\83L,T. â\9dªG1,L1,T1â\9d« â¬\82⸮ â\9dªG2,L,Tâ\9d« & â\9dªG2,Lâ\9d« â\8a¢ T â¬\88 T2 & â\9dªG2,Lâ\9d« ⊢ ⬈[T] L2.
+ â\88\83â\88\83L,T. â\9d¨G1,L1,T1â\9d© â¬\82⸮ â\9d¨G2,L,Tâ\9d© & â\9d¨G2,Lâ\9d© â\8a¢ T â¬\88 T2 & â\9d¨G2,Lâ\9d© ⊢ ⬈[T] L2.
interpretation
"parallel rst-transition (closure)"
(* Basic properties *********************************************************)
lemma fpb_intro (G1) (L1) (T1) (G2) (L2) (T2):
- â\88\80L,T. â\9dªG1,L1,T1â\9d« â¬\82⸮ â\9dªG2,L,Tâ\9d« â\86\92 â\9dªG2,Lâ\9d« ⊢ T ⬈ T2 →
- â\9dªG2,Lâ\9d« â\8a¢ â¬\88[T] L2 â\86\92 â\9dªG1,L1,T1â\9d« â\89½ â\9dªG2,L2,T2â\9d«.
+ â\88\80L,T. â\9d¨G1,L1,T1â\9d© â¬\82⸮ â\9d¨G2,L,Tâ\9d© â\86\92 â\9d¨G2,Lâ\9d© ⊢ T ⬈ T2 →
+ â\9d¨G2,Lâ\9d© â\8a¢ â¬\88[T] L2 â\86\92 â\9d¨G1,L1,T1â\9d© â\89½ â\9d¨G2,L2,T2â\9d©.
/2 width=5 by ex3_2_intro/ qed.
lemma rpx_fpb (G) (T):
- â\88\80L1,L2. â\9dªG,L1â\9d« â\8a¢ â¬\88[T] L2 â\86\92 â\9dªG,L1,Tâ\9d« â\89½ â\9dªG,L2,Tâ\9d«.
+ â\88\80L1,L2. â\9d¨G,L1â\9d© â\8a¢ â¬\88[T] L2 â\86\92 â\9d¨G,L1,Tâ\9d© â\89½ â\9d¨G,L2,Tâ\9d©.
/2 width=5 by fpb_intro/ qed.
(* Basic inversion lemmas ***************************************************)
lemma fpb_inv_gen (G1) (L1) (T1) (G2) (L2) (T2):
- â\9dªG1,L1,T1â\9d« â\89½ â\9dªG2,L2,T2â\9d« →
- â\88\83â\88\83L,T. â\9dªG1,L1,T1â\9d« â¬\82⸮ â\9dªG2,L,Tâ\9d« & â\9dªG2,Lâ\9d« â\8a¢ T â¬\88 T2 & â\9dªG2,Lâ\9d« ⊢ ⬈[T] L2.
+ â\9d¨G1,L1,T1â\9d© â\89½ â\9d¨G2,L2,T2â\9d© →
+ â\88\83â\88\83L,T. â\9d¨G1,L1,T1â\9d© â¬\82⸮ â\9d¨G2,L,Tâ\9d© & â\9d¨G2,Lâ\9d© â\8a¢ T â¬\88 T2 & â\9d¨G2,Lâ\9d© ⊢ ⬈[T] L2.
// qed-.
(* Basic_2A1: removed theorems 2: