(* Inversion lemmas with simple terms ***************************************)
-lemma cnx_inv_appl: â\88\80h,G,L,V,T. â¦\83G,Lâ¦\84 â\8a¢ â¬\88[h] ð\9d\90\8dâ¦\83â\93\90V.Tâ¦\84 →
- â\88§â\88§ â¦\83G,Lâ¦\84 â\8a¢ â¬\88[h] ð\9d\90\8dâ¦\83Vâ¦\84 & â¦\83G,Lâ¦\84 â\8a¢ â¬\88[h] ð\9d\90\8dâ¦\83Tâ¦\84 & ð\9d\90\92â¦\83Tâ¦\84.
+lemma cnx_inv_appl: â\88\80h,G,L,V,T. â\9dªG,Lâ\9d« â\8a¢ â¬\88[h] ð\9d\90\8dâ\9dªâ\93\90V.Tâ\9d« →
+ â\88§â\88§ â\9dªG,Lâ\9d« â\8a¢ â¬\88[h] ð\9d\90\8dâ\9dªVâ\9d« & â\9dªG,Lâ\9d« â\8a¢ â¬\88[h] ð\9d\90\8dâ\9dªTâ\9d« & ð\9d\90\92â\9dªTâ\9d«.
#h #G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2
- #H elim (tdeq_inv_pair … H) -H //
+ #H elim (teqx_inv_pair … H) -H //
| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2
- #H elim (tdeq_inv_pair … H) -H //
+ #H elim (teqx_inv_pair … H) -H //
| generalize in match HVT1; -HVT1 elim T1 -T1 * //
#p * #W1 #U1 #_ #_ #H
- [ elim (lifts_total V1 (ð\9d\90\94â\9d´1â\9dµ)) #V2 #HV12
- lapply (H (ⓓ{p}W1.ⓐV2.U1) ?) -H /2 width=3 by cpx_theta/ -HV12
- #H elim (tdeq_inv_pair … H) -H #H destruct
- | lapply (H (ⓓ{p}ⓝW1.V1.U1) ?) -H /2 width=1 by cpx_beta/
- #H elim (tdeq_inv_pair … H) -H #H destruct
+ [ elim (lifts_total V1 (ð\9d\90\94â\9d¨1â\9d©)) #V2 #HV12
+ lapply (H (ⓓ[p]W1.ⓐV2.U1) ?) -H /2 width=3 by cpx_theta/ -HV12
+ #H elim (teqx_inv_pair … H) -H #H destruct
+ | lapply (H (ⓓ[p]ⓝW1.V1.U1) ?) -H /2 width=1 by cpx_beta/
+ #H elim (teqx_inv_pair … H) -H #H destruct
]
]
qed-.
(* Properties with simple terms *********************************************)
-lemma cnx_appl_simple: â\88\80h,G,L,V,T. â¦\83G,Lâ¦\84 â\8a¢ â¬\88[h] ð\9d\90\8dâ¦\83Vâ¦\84 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â¬\88[h] ð\9d\90\8dâ¦\83Tâ¦\84 â\86\92 ð\9d\90\92â¦\83Tâ¦\84 →
- â¦\83G,Lâ¦\84 â\8a¢ â¬\88[h] ð\9d\90\8dâ¦\83â\93\90V.Tâ¦\84.
+lemma cnx_appl_simple: â\88\80h,G,L,V,T. â\9dªG,Lâ\9d« â\8a¢ â¬\88[h] ð\9d\90\8dâ\9dªVâ\9d« â\86\92 â\9dªG,Lâ\9d« â\8a¢ â¬\88[h] ð\9d\90\8dâ\9dªTâ\9d« â\86\92 ð\9d\90\92â\9dªTâ\9d« →
+ â\9dªG,Lâ\9d« â\8a¢ â¬\88[h] ð\9d\90\8dâ\9dªâ\93\90V.Tâ\9d«.
#h #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H //
#V0 #T0 #HV0 #HT0 #H destruct
-@tdeq_pair [ @HV | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
+@teqx_pair [ @HV | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
qed.