(* Main preservation properties *********************************************)
(* Basic_2A1: uses: snv_preserve *)
-lemma cnv_preserve (h) (a): â\88\80G,L,T. â¦\83G,Lâ¦\84 ⊢ T ![h,a] →
+lemma cnv_preserve (h) (a): â\88\80G,L,T. â\9d¨G,Lâ\9d© ⊢ T ![h,a] →
∧∧ IH_cnv_cpms_conf_lpr h a G L T
& IH_cnv_cpm_trans_lpr h a G L T.
#h #a #G #L #T #HT
(* Advanced preservation properties *****************************************)
lemma cnv_cpms_conf (h) (a) (G) (L):
- â\88\80T0. â¦\83G,Lâ¦\84 ⊢ T0 ![h,a] →
- â\88\80n1,T1. â¦\83G,Lâ¦\84 â\8a¢ T0 â\9e¡*[n1,h] T1 â\86\92 â\88\80n2,T2. â¦\83G,Lâ¦\84 â\8a¢ T0 â\9e¡*[n2,h] T2 →
- â\88\83â\88\83T. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n2-n1,h] T & â¦\83G,Lâ¦\84 â\8a¢ T2 â\9e¡*[n1-n2,h] T.
+ â\88\80T0. â\9d¨G,Lâ\9d© ⊢ T0 ![h,a] →
+ â\88\80n1,T1. â\9d¨G,Lâ\9d© â\8a¢ T0 â\9e¡*[h,n1] T1 â\86\92 â\88\80n2,T2. â\9d¨G,Lâ\9d© â\8a¢ T0 â\9e¡*[h,n2] T2 →
+ â\88\83â\88\83T. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,n2-n1] T & â\9d¨G,Lâ\9d© â\8a¢ T2 â\9e¡*[h,n1-n2] T.
/2 width=8 by cnv_cpms_conf_lpr/ qed-.
(* Basic_2A1: uses: snv_cprs_lpr *)
qed-.
lemma cnv_cpm_trans (h) (a) (G) (L):
- â\88\80T1. â¦\83G,Lâ¦\84 ⊢ T1 ![h,a] →
- â\88\80n,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡[n,h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T2 ![h,a].
+ â\88\80T1. â\9d¨G,Lâ\9d© ⊢ T1 ![h,a] →
+ â\88\80n,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡[h,n] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T2 ![h,a].
/2 width=6 by cnv_cpm_trans_lpr/ qed-.
(* Note: this is the preservation property *)
lemma cnv_cpms_trans (h) (a) (G) (L):
- â\88\80T1. â¦\83G,Lâ¦\84 ⊢ T1 ![h,a] →
- â\88\80n,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â\9e¡*[n,h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T2 ![h,a].
+ â\88\80T1. â\9d¨G,Lâ\9d© ⊢ T1 ![h,a] →
+ â\88\80n,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â\9e¡*[h,n] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T2 ![h,a].
/2 width=6 by cnv_cpms_trans_lpr/ qed-.
lemma cnv_lpr_trans (h) (a) (G):
- â\88\80L1,T. â¦\83G,L1â¦\84 â\8a¢ T ![h,a] â\86\92 â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â\9e¡[h] L2 â\86\92 â¦\83G,L2â¦\84 ⊢ T ![h,a].
+ â\88\80L1,T. â\9d¨G,L1â\9d© â\8a¢ T ![h,a] â\86\92 â\88\80L2. â\9d¨G,L1â\9d© â\8a¢ â\9e¡[h,0] L2 â\86\92 â\9d¨G,L2â\9d© ⊢ T ![h,a].
/2 width=6 by cnv_cpm_trans_lpr/ qed-.
lemma cnv_lprs_trans (h) (a) (G):
- â\88\80L1,T. â¦\83G,L1â¦\84 â\8a¢ T ![h,a] â\86\92 â\88\80L2. â¦\83G,L1â¦\84 â\8a¢ â\9e¡*[h] L2 â\86\92 â¦\83G,L2â¦\84 ⊢ T ![h,a].
+ â\88\80L1,T. â\9d¨G,L1â\9d© â\8a¢ T ![h,a] â\86\92 â\88\80L2. â\9d¨G,L1â\9d© â\8a¢ â\9e¡*[h,0] L2 â\86\92 â\9d¨G,L2â\9d© ⊢ T ![h,a].
#h #a #G #L1 #T #HT #L2 #H
@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
qed-.