(* Inversion lemmas with simple terms ***************************************)
lemma cnr_inv_appl (h) (G) (L):
- ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ & ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ & 𝐒⦃T⦄.
+ ∀V,T. ❪G,L❫ ⊢ ➡𝐍[h,0] ⓐV.T →
+ ∧∧ ❪G,L❫ ⊢ ➡𝐍[h,0] V & ❪G,L❫ ⊢ ➡𝐍[h,0] T & 𝐒❪T❫.
#h #G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpr_pair_sn/ -HV2 #H destruct //
| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct //
| generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
- [ elim (lifts_total V1 ð\9d\90\94â\9d´1â\9dµ) #V2 #HV12
- lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /2 width=3 by cpm_theta/ -HV12 #H destruct
- | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /2 width=1 by cpm_beta/ #H destruct
+ [ elim (lifts_total V1 ð\9d\90\94â\9d¨1â\9d©) #V2 #HV12
+ lapply (H (ⓓ[a]W1.ⓐV2.U1) ?) -H /2 width=3 by cpm_theta/ -HV12 #H destruct
+ | lapply (H (ⓓ[a]ⓝW1.V1.U1) ?) -H /2 width=1 by cpm_beta/ #H destruct
]
]
qed-.
(* Basic_1: was only: nf2_appl_lref *)
lemma cnr_appl_simple (h) (G) (L):
- â\88\80V,T. â¦\83G,Lâ¦\84 â\8a¢ â\9e¡[h] ð\9d\90\8dâ¦\83Vâ¦\84 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â\9e¡[h] ð\9d\90\8dâ¦\83Tâ¦\84 â\86\92 ð\9d\90\92â¦\83Tâ¦\84 â\86\92 â¦\83G,Lâ¦\84 â\8a¢ â\9e¡[h] ð\9d\90\8dâ¦\83â\93\90V.Tâ¦\84.
+ â\88\80V,T. â\9dªG,Lâ\9d« â\8a¢ â\9e¡ð\9d\90\8d[h,0] V â\86\92 â\9dªG,Lâ\9d« â\8a¢ â\9e¡ð\9d\90\8d[h,0] T â\86\92 ð\9d\90\92â\9dªTâ\9d« â\86\92 â\9dªG,Lâ\9d« â\8a¢ â\9e¡ð\9d\90\8d[h,0] â\93\90V.T.
#h #G #L #V #T #HV #HT #HS #X #H
elim (cpm_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
<(HV … HV0) -V0 <(HT … HT0) -T0 //