(* Basic_2A1: was just: lleq_cpxs_trans *)
lemma reqg_cpxs_trans (S) (G):
reflexive … S → symmetric … S →
- â\88\80L0,T0,T1. â\9dªG,L0â\9d« â\8a¢ T0 â¬\88* T1 â\86\92 â\88\80L2. L2 â\89\9b[S,T0] L0 â\86\92 â\9dªG,L2â\9d« ⊢ T0 ⬈* T1.
+ â\88\80L0,T0,T1. â\9d¨G,L0â\9d© â\8a¢ T0 â¬\88* T1 â\86\92 â\88\80L2. L2 â\89\9b[S,T0] L0 â\86\92 â\9d¨G,L2â\9d© ⊢ T0 ⬈* T1.
#S #G #H1S #H2S #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 //
#T0 #T #H0T0 #_ #IH #L2 #HL2
lapply (reqg_cpx_trans … HL2 … H0T0) // #H2T0
(* Basic_2A1: was just: cpxs_lleq_conf *)
lemma cpxs_reqg_conf (S) (G):
reflexive … S → symmetric … S →
- â\88\80L0,T0,T1. â\9dªG,L0â\9d« â\8a¢ T0 â¬\88* T1 â\86\92 â\88\80L2. L0 â\89\9b[S,T0] L2 â\86\92 â\9dªG,L2â\9d« ⊢ T0 ⬈* T1.
+ â\88\80L0,T0,T1. â\9d¨G,L0â\9d© â\8a¢ T0 â¬\88* T1 â\86\92 â\88\80L2. L0 â\89\9b[S,T0] L2 â\86\92 â\9d¨G,L2â\9d© ⊢ T0 ⬈* T1.
/3 width=8 by reqg_cpxs_trans, reqg_sym/ qed-.
(* Basic_2A1: was just: lleq_conf_sn *)
lemma cpxs_reqg_conf_sn (S) (G):
- â\88\80L1,T1,T2. â\9dªG,L1â\9d« ⊢ T1 ⬈* T2 →
+ â\88\80L1,T1,T2. â\9d¨G,L1â\9d© ⊢ T1 ⬈* T2 →
∀L2. L1 ≛[S,T1] L2 → L1 ≛[S,T2] L2.
#S #G #L1 #T1 #T2 #H @(cpxs_ind … H) -T2
/3 width=6 by cpx_reqg_conf_sn/
(* Basic_2A1: was just: cpxs_lleq_conf_dx *)
lemma cpxs_reqg_conf_dx (S) (G):
reflexive … S → symmetric … S →
- â\88\80L2,T1,T2. â\9dªG,L2â\9d« ⊢ T1 ⬈* T2 →
+ â\88\80L2,T1,T2. â\9d¨G,L2â\9d© ⊢ T1 ⬈* T2 →
∀L1. L1 ≛[S,T1] L2 → L1 ≛[S,T2] L2.
/4 width=4 by cpxs_reqg_conf_sn, reqg_sym/ qed-.