include "basic_2/rt_transition/cpx_simple.ma".
include "basic_2/rt_transition/cnx.ma".
-(* NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ********)
+(* NORMAL TERMS FOR EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION *******)
(* Inversion lemmas with simple terms ***************************************)
-lemma cnx_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⦄.
-#h #G #L #V1 #T1 #HVT1 @and3_intro
+lemma cnx_inv_appl (G) (L):
+ ∀V,T. ❪G,L❫ ⊢ ⬈𝐍 ⓐV.T →
+ ∧∧ ❪G,L❫ ⊢ ⬈𝐍 V & ❪G,L❫ ⊢ ⬈𝐍 T & 𝐒❪T❫.
+#G #L #V1 #T1 #HVT1 @and3_intro
[ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2
#H elim (teqx_inv_pair … H) -H //
| #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2
#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
+ [ 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/
+ | lapply (H (ⓓ[p]ⓝW1.V1.U1) ?) -H /2 width=1 by cpx_beta/
#H elim (teqx_inv_pair … H) -H #H destruct
]
]
(* Properties with simple terms *********************************************)
-lemma cnx_appl_simple: ∀h,G,L,V,T. ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ →
- ⦃G,L⦄ ⊢ ⬈[h] 𝐍⦃ⓐV.T⦄.
-#h #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H //
+lemma cnx_appl_simple (G) (L):
+ ∀V,T. ❪G,L❫ ⊢ ⬈𝐍 V → ❪G,L❫ ⊢ ⬈𝐍 T → 𝐒❪T❫ → ❪G,L❫ ⊢ ⬈𝐍 ⓐV.T.
+#G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H //
#V0 #T0 #HV0 #HT0 #H destruct
@teqx_pair [ @HV | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)
qed.