(* Sub confluence propery with restricted rt-transition for terms ***********)
fact cnv_cpms_teqx_strip_lpr_aux (h) (a) (G0) (L0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G,L,Tâ¦\84 → IH_cnv_cpm_trans_lpr h a G L T) →
- (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G,L,Tâ¦\84 → IH_cnv_cpms_conf_lpr h a G L T) →
- â\88\80n1,T1. â¦\83G0,L0â¦\84 â\8a¢ T0 â\9e¡*[n1,h] T1 â\86\92 â¦\83G0,L0â¦\84 ⊢ T0 ![h,a] → T0 ≛ T1 →
- â\88\80n2,T2. â¦\83G0,L0â¦\84 ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
- â\88\80L1. â¦\83G0,L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G0,L0â¦\84 ⊢ ➡[h] L2 →
- â\88\83â\88\83T. â¦\83G0,L1â¦\84 â\8a¢ T1 â\9e¡[n2-n1,h] T & T1 â\89\9b T & â¦\83G0,L2â¦\84 ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
+ (â\88\80G,L,T. â\9dªG0,L0,T0â\9d« >[h] â\9dªG,L,Tâ\9d« → IH_cnv_cpm_trans_lpr h a G L T) →
+ (â\88\80G,L,T. â\9dªG0,L0,T0â\9d« >[h] â\9dªG,L,Tâ\9d« → IH_cnv_cpms_conf_lpr h a G L T) →
+ â\88\80n1,T1. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡*[n1,h] T1 â\86\92 â\9dªG0,L0â\9d« ⊢ T0 ![h,a] → T0 ≛ T1 →
+ â\88\80n2,T2. â\9dªG0,L0â\9d« ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« ⊢ ➡[h] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ T1 â\9e¡[n2-n1,h] T & T1 â\89\9b T & â\9dªG0,L2â\9d« ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
#h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
@(cpms_teqx_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0
[ #H0T1 #n2 #T2 #H1T12 #H2T12 #L1 #HL01 #L2 #HL02
qed-.
fact cnv_cpms_teqx_conf_lpr_aux (h) (a) (G0) (L0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G,L,Tâ¦\84 → IH_cnv_cpm_trans_lpr h a G L T) →
- (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 >[h] â¦\83G,L,Tâ¦\84 → IH_cnv_cpms_conf_lpr h a G L T) →
- â\88\80n1,T1. â¦\83G0,L0â¦\84 â\8a¢ T0 â\9e¡*[n1,h] T1 â\86\92 â¦\83G0,L0â¦\84 ⊢ T0 ![h,a] → T0 ≛ T1 →
- â\88\80n2,T2. â¦\83G0,L0â¦\84 ⊢ T0 ➡*[n2,h] T2 → T0 ≛ T2 →
- â\88\80L1. â¦\83G0,L0â¦\84 â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â¦\83G0,L0â¦\84 ⊢ ➡[h] L2 →
- â\88\83â\88\83T. â¦\83G0,L1â¦\84 â\8a¢ T1 â\9e¡*[n2-n1,h] T & T1 â\89\9b T & â¦\83G0,L2â¦\84 ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
+ (â\88\80G,L,T. â\9dªG0,L0,T0â\9d« >[h] â\9dªG,L,Tâ\9d« → IH_cnv_cpm_trans_lpr h a G L T) →
+ (â\88\80G,L,T. â\9dªG0,L0,T0â\9d« >[h] â\9dªG,L,Tâ\9d« → IH_cnv_cpms_conf_lpr h a G L T) →
+ â\88\80n1,T1. â\9dªG0,L0â\9d« â\8a¢ T0 â\9e¡*[n1,h] T1 â\86\92 â\9dªG0,L0â\9d« ⊢ T0 ![h,a] → T0 ≛ T1 →
+ â\88\80n2,T2. â\9dªG0,L0â\9d« ⊢ T0 ➡*[n2,h] T2 → T0 ≛ T2 →
+ â\88\80L1. â\9dªG0,L0â\9d« â\8a¢ â\9e¡[h] L1 â\86\92 â\88\80L2. â\9dªG0,L0â\9d« ⊢ ➡[h] L2 →
+ â\88\83â\88\83T. â\9dªG0,L1â\9d« â\8a¢ T1 â\9e¡*[n2-n1,h] T & T1 â\89\9b T & â\9dªG0,L2â\9d« ⊢ T2 ➡*[n1-n2,h] T & T2 ≛ T.
#h #a #G #L0 #T0 #IH2 #IH1 #n1 #T1 #H1T01 #H0T0 #H2T01
generalize in match IH1; generalize in match IH2;
@(cpms_teqx_ind_sn … H1T01 H0T0 H2T01 IH1 IH2) -n1 -T0