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❫ ⊢ ⬈𝐍[h] ⓐV.T →
- ∧∧ ❪G,L❫ ⊢ ⬈𝐍[h] V & ❪G,L❫ ⊢ ⬈𝐍[h] 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
(* Properties with simple terms *********************************************)
-lemma cnx_appl_simple: ∀h,G,L,V,T. ❪G,L❫ ⊢ ⬈𝐍[h] V → ❪G,L❫ ⊢ ⬈𝐍[h] T → 𝐒❪T❫ →
- ❪G,L❫ ⊢ ⬈𝐍[h] ⓐ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.