(* Main preservation properties *********************************************)
(* Basic_2A1: uses: snv_preserve *)
-lemma cnv_preserve (a) (h): ∀G,L,T. ⦃G,L⦄ ⊢ T ![a,h] →
- ∧∧ IH_cnv_cpms_conf_lpr a h G L T
- & IH_cnv_cpm_trans_lpr a h G L T.
-#a #h #G #L #T #HT
-letin o ≝ (sd_O h)
-lapply (cnv_fwd_fsb … o … HT) -HT #H
+lemma cnv_preserve (h) (a): ∀G,L,T. ❨G,L❩ ⊢ 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
+lapply (cnv_fwd_fsb … HT) -HT #H
@(fsb_ind_fpbg … H) -G -L -T #G #L #T #_ #IH
@conj [ letin aux ≝ cnv_cpms_conf_lpr_aux | letin aux ≝ cnv_cpm_trans_lpr_aux ]
-@(aux … o … G L T) // #G0 #L0 #T0 #H
+@(aux … G L T) // #G0 #L0 #T0 #H
elim (IH … H) -IH -H //
qed-.
-theorem cnv_cpms_conf_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_conf_lpr a h G L T.
-#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=1 by/
+theorem cnv_cpms_conf_lpr (h) (a) (G) (L) (T): IH_cnv_cpms_conf_lpr h a G L T.
+#h #a #G #L #T #HT elim (cnv_preserve … HT) /2 width=1 by/
qed-.
(* Basic_2A1: uses: snv_cpr_lpr *)
-theorem cnv_cpm_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpm_trans_lpr a h G L T.
-#a #h #G #L #T #HT elim (cnv_preserve … HT) /2 width=2 by/
+theorem cnv_cpm_trans_lpr (h) (a) (G) (L) (T): IH_cnv_cpm_trans_lpr h a G L T.
+#h #a #G #L #T #HT elim (cnv_preserve … HT) /2 width=2 by/
qed-.
(* Advanced preservation properties *****************************************)
+lemma cnv_cpms_conf (h) (a) (G) (L):
+ ∀T0. ❨G,L❩ ⊢ T0 ![h,a] →
+ ∀n1,T1. ❨G,L❩ ⊢ T0 ➡*[h,n1] T1 → ∀n2,T2. ❨G,L❩ ⊢ T0 ➡*[h,n2] T2 →
+ ∃∃T. ❨G,L❩ ⊢ T1 ➡*[h,n2-n1] T & ❨G,L❩ ⊢ T2 ➡*[h,n1-n2] T.
+/2 width=8 by cnv_cpms_conf_lpr/ qed-.
+
(* Basic_2A1: uses: snv_cprs_lpr *)
-lemma cnv_cpms_trans_lpr (a) (h) (G) (L) (T): IH_cnv_cpms_trans_lpr a h G L T.
-#a #h #G #L1 #T1 #HT1 #n #T2 #H
+lemma cnv_cpms_trans_lpr (h) (a) (G) (L) (T): IH_cnv_cpms_trans_lpr h a G L T.
+#h #a #G #L1 #T1 #HT1 #n #T2 #H
@(cpms_ind_dx … H) -n -T2 /3 width=6 by cnv_cpm_trans_lpr/
qed-.
+
+lemma cnv_cpm_trans (h) (a) (G) (L):
+ ∀T1. ❨G,L❩ ⊢ T1 ![h,a] →
+ ∀n,T2. ❨G,L❩ ⊢ T1 ➡[h,n] T2 → ❨G,L❩ ⊢ 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):
+ ∀T1. ❨G,L❩ ⊢ T1 ![h,a] →
+ ∀n,T2. ❨G,L❩ ⊢ T1 ➡*[h,n] T2 → ❨G,L❩ ⊢ T2 ![h,a].
+/2 width=6 by cnv_cpms_trans_lpr/ qed-.
+
+lemma cnv_lpr_trans (h) (a) (G):
+ ∀L1,T. ❨G,L1❩ ⊢ T ![h,a] → ∀L2. ❨G,L1❩ ⊢ ➡[h,0] L2 → ❨G,L2❩ ⊢ T ![h,a].
+/2 width=6 by cnv_cpm_trans_lpr/ qed-.
+
+lemma cnv_lprs_trans (h) (a) (G):
+ ∀L1,T. ❨G,L1❩ ⊢ T ![h,a] → ∀L2. ❨G,L1❩ ⊢ ➡*[h,0] L2 → ❨G,L2❩ ⊢ T ![h,a].
+#h #a #G #L1 #T #HT #L2 #H
+@(lprs_ind_dx … H) -L2 /2 width=3 by cnv_lpr_trans/
+qed-.