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15 include "basic_2/rt_transition/cpx_simple.ma".
16 include "basic_2/rt_transition/cnx.ma".
18 (* NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ********)
20 (* Inversion lemmas with simple terms ***************************************)
22 lemma cnx_inv_appl: ∀h,G,L,V,T. ❪G,L❫ ⊢ ⬈[h] 𝐍❪ⓐV.T❫ →
23 ∧∧ ❪G,L❫ ⊢ ⬈[h] 𝐍❪V❫ & ❪G,L❫ ⊢ ⬈[h] 𝐍❪T❫ & 𝐒❪T❫.
24 #h #G #L #V1 #T1 #HVT1 @and3_intro
25 [ #V2 #HV2 lapply (HVT1 (ⓐV2.T1) ?) -HVT1 /2 width=1 by cpx_pair_sn/ -HV2
26 #H elim (teqx_inv_pair … H) -H //
27 | #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpx_flat/ -HT2
28 #H elim (teqx_inv_pair … H) -H //
29 | generalize in match HVT1; -HVT1 elim T1 -T1 * //
31 [ elim (lifts_total V1 (𝐔❨1❩)) #V2 #HV12
32 lapply (H (ⓓ[p]W1.ⓐV2.U1) ?) -H /2 width=3 by cpx_theta/ -HV12
33 #H elim (teqx_inv_pair … H) -H #H destruct
34 | lapply (H (ⓓ[p]ⓝW1.V1.U1) ?) -H /2 width=1 by cpx_beta/
35 #H elim (teqx_inv_pair … H) -H #H destruct
40 (* Properties with simple terms *********************************************)
42 lemma cnx_appl_simple: ∀h,G,L,V,T. ❪G,L❫ ⊢ ⬈[h] 𝐍❪V❫ → ❪G,L❫ ⊢ ⬈[h] 𝐍❪T❫ → 𝐒❪T❫ →
44 #h #G #L #V #T #HV #HT #HS #X #H elim (cpx_inv_appl1_simple … H) -H //
45 #V0 #T0 #HV0 #HT0 #H destruct
46 @teqx_pair [ @HV | @HT ] // (**) (* auto fails because δ-expansion gets in the way *)