(* Properties with extended context-sensitive parallel rt-computation *******)
lemma cpxs_fpbs:
- â\88\80G,L,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88* T2 â\86\92 â\9dªG,L,T1â\9d« â\89¥ â\9dªG,L,T2â\9d«.
+ â\88\80G,L,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\88* T2 â\86\92 â\9d¨G,L,T1â\9d© â\89¥ â\9d¨G,L,T2â\9d©.
#G #L #T1 #T2 #H @(cpxs_ind … H) -T2
/3 width=5 by cpx_fpb, fpbs_strap1/
qed.
lemma fpbs_cpxs_trans:
- â\88\80G1,G,L1,L,T1,T. â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG,L,Tâ\9d« →
- â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â¬\88* T2 â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG,L,T2â\9d«.
+ â\88\80G1,G,L1,L,T1,T. â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G,L,Tâ\9d© →
+ â\88\80T2. â\9d¨G,Lâ\9d© â\8a¢ T â¬\88* T2 â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G,L,T2â\9d©.
#G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2
/3 width=5 by fpbs_strap1, cpx_fpb/
qed-.
lemma cpxs_fpbs_trans:
- â\88\80G1,G2,L1,L2,T,T2. â\9dªG1,L1,Tâ\9d« â\89¥ â\9dªG2,L2,T2â\9d« →
- â\88\80T1. â\9dªG1,L1â\9d« â\8a¢ T1 â¬\88* T â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,G2,L1,L2,T,T2. â\9d¨G1,L1,Tâ\9d© â\89¥ â\9d¨G2,L2,T2â\9d© →
+ â\88\80T1. â\9d¨G1,L1â\9d© â\8a¢ T1 â¬\88* T â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
#G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1
/3 width=5 by fpbs_strap2, cpx_fpb/
qed-.
lemma cpxs_teqg_fpbs_trans (S):
reflexive … S → symmetric … S →
- â\88\80G1,L1,T1,T. â\9dªG1,L1â\9d« ⊢ T1 ⬈* T → ∀T0. T ≛[S] T0 →
- â\88\80G2,L2,T2. â\9dªG1,L1,T0â\9d« â\89¥ â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,L1,T1,T. â\9d¨G1,L1â\9d© ⊢ T1 ⬈* T → ∀T0. T ≛[S] T0 →
+ â\88\80G2,L2,T2. â\9d¨G1,L1,T0â\9d© â\89¥ â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=6 by cpxs_fpbs_trans, teqg_fpbs_trans/ qed-.
lemma cpxs_teqg_fpbs (S):
reflexive … S → symmetric … S →
- â\88\80G,L,T1,T. â\9dªG,Lâ\9d« ⊢ T1 ⬈* T →
- â\88\80T2. T â\89\9b[S] T2 â\86\92 â\9dªG,L,T1â\9d« â\89¥ â\9dªG,L,T2â\9d«.
+ â\88\80G,L,T1,T. â\9d¨G,Lâ\9d© ⊢ T1 ⬈* T →
+ â\88\80T2. T â\89\9b[S] T2 â\86\92 â\9d¨G,L,T1â\9d© â\89¥ â\9d¨G,L,T2â\9d©.
/4 width=5 by cpxs_fpbs_trans, feqg_fpbs, teqg_feqg/ qed.
(* Properties with plus-iterated structural successor for closures **********)
lemma cpxs_fqup_fpbs:
- â\88\80G1,L1,T1,T. â\9dªG1,L1â\9d« ⊢ T1 ⬈* T →
- â\88\80G2,L2,T2. â\9dªG1,L1,Tâ\9d« â¬\82+ â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,L1,T1,T. â\9d¨G1,L1â\9d© ⊢ T1 ⬈* T →
+ â\88\80G2,L2,T2. â\9d¨G1,L1,Tâ\9d© â¬\82+ â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
(* Properties with star-iterated structural successor for closures **********)
lemma cpxs_fqus_fpbs:
- â\88\80G1,L1,T1,T. â\9dªG1,L1â\9d« ⊢ T1 ⬈* T →
- â\88\80G2,L2,T2. â\9dªG1,L1,Tâ\9d« â¬\82* â\9dªG2,L2,T2â\9d« â\86\92 â\9dªG1,L1,T1â\9d« â\89¥ â\9dªG2,L2,T2â\9d«.
+ â\88\80G1,L1,T1,T. â\9d¨G1,L1â\9d© ⊢ T1 ⬈* T →
+ â\88\80G2,L2,T2. â\9d¨G1,L1,Tâ\9d© â¬\82* â\9d¨G2,L2,T2â\9d© â\86\92 â\9d¨G1,L1,T1â\9d© â\89¥ â\9d¨G2,L2,T2â\9d©.
/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.