(* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****)
-lemma fsb_inv_csx: â\88\80h,G,L,T. â\89¥[h] ð\9d\90\92â¦\83G,L,Tâ¦\84 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84.
+lemma fsb_inv_csx: â\88\80h,G,L,T. â\89¥[h] ð\9d\90\92â\9dªG,L,Tâ\9d« â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªTâ\9d«.
#h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
qed-.
(* Propreties with context-sensitive stringly rt-normalizing terms **********)
-lemma csx_fsb_fpbs: â\88\80h,G1,L1,T1. â¦\83G1,L1â¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83T1â¦\84 →
- â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 â\89¥[h] â¦\83G2,L2,T2â¦\84 â\86\92 â\89¥[h] ð\9d\90\92â¦\83G2,L2,T2â¦\84.
+lemma csx_fsb_fpbs: â\88\80h,G1,L1,T1. â\9dªG1,L1â\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªT1â\9d« →
+ â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« â\89¥[h] â\9dªG2,L2,T2â\9d« â\86\92 â\89¥[h] ð\9d\90\92â\9dªG2,L2,T2â\9d«.
#h #G1 #L1 #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
-#G0 #L0 #T0 #IHu #H10
+#G0 #L0 #T0 #IHu #H10
lapply (fpbs_csx_conf … H10) // -HT1 #HT0
generalize in match IHu; -IHu generalize in match H10; -H10
@(rsx_ind … (csx_rsx … HT0)) -L0
#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
[ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
| #T2 #HT02 #HnT02
- elim (fpbs_cpx_tdneq_trans … H10 … HT02 HnT02) -T0
+ elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0
/3 width=4 by/
| #L2 #HL02 #HnL02 @(IHd … HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
[ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
| #G3 #L3 #T3 #H03 #_
elim (lpx_fqup_trans … H03 … HL02) -L2 #L4 #T4 #HT04 #H43 #HL43
- elim (tdeq_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
- [ elim (tdeq_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
- /4 width=7 by fsb_fpbs_trans, tdeq_rdeq_lpx_fpbs, fpbs_fqup_trans/
- | elim (cpxs_tdneq_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
- elim (fpbs_cpx_tdneq_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
- /3 width=14 by fpbs_cpxs_tdeq_fqup_lpx_trans/
+ elim (teqx_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
+ [ elim (teqx_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
+ /4 width=7 by fsb_fpbs_trans, teqx_reqx_lpx_fpbs, fpbs_fqup_trans/
+ | elim (cpxs_tneqx_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
+ elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
+ /3 width=14 by fpbs_cpxs_teqx_fqup_lpx_trans/
]
]
]
qed.
-lemma csx_fsb: â\88\80h,G,L,T. â¦\83G,Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84 â\86\92 â\89¥[h] ð\9d\90\92â¦\83G,L,Tâ¦\84.
+lemma csx_fsb: â\88\80h,G,L,T. â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªTâ\9d« â\86\92 â\89¥[h] ð\9d\90\92â\9dªG,L,Tâ\9d«.
/2 width=5 by csx_fsb_fpbs/ qed.
(* Advanced eliminators *****************************************************)
lemma csx_ind_fpb: ∀h. ∀Q:relation3 genv lenv term.
- (â\88\80G1,L1,T1. â¦\83G1,L1â¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83T1â¦\84 →
- (â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 â\89»[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2) →
+ (â\88\80G1,L1,T1. â\9dªG1,L1â\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªT1â\9d« →
+ (â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« â\89»[h] â\9dªG2,L2,T2â\9d« → Q G2 L2 T2) →
Q G1 L1 T1
) →
- â\88\80G,L,T. â¦\83G,Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84 → Q G L T.
+ â\88\80G,L,T. â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªTâ\9d« → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
lemma csx_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
- (â\88\80G1,L1,T1. â¦\83G1,L1â¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83T1â¦\84 →
- (â\88\80G2,L2,T2. â¦\83G1,L1,T1â¦\84 >[h] â¦\83G2,L2,T2â¦\84 → Q G2 L2 T2) →
+ (â\88\80G1,L1,T1. â\9dªG1,L1â\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªT1â\9d« →
+ (â\88\80G2,L2,T2. â\9dªG1,L1,T1â\9d« >[h] â\9dªG2,L2,T2â\9d« → Q G2 L2 T2) →
Q G1 L1 T1
) →
- â\88\80G,L,T. â¦\83G,Lâ¦\84 â\8a¢ â¬\88*[h] ð\9d\90\92â¦\83Tâ¦\84 → Q G L T.
+ â\88\80G,L,T. â\9dªG,Lâ\9d« â\8a¢ â¬\88*[h] ð\9d\90\92â\9dªTâ\9d« → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.