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15 include "basic_2/rt_transition/cpm_simple.ma".
16 include "basic_2/rt_transition/cnr.ma".
18 (* NORMAL TERMS FOR CONTEXT-SENSITIVE R-TRANSITION **************************)
20 (* Inversion lemmas with simple terms ***************************************)
22 lemma cnr_inv_appl (h) (G) (L):
23 ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄ → ∧∧ ⦃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 cpr_pair_sn/ -HV2 #H destruct //
26 | #T2 #HT2 lapply (HVT1 (ⓐV1.T2) ?) -HVT1 /2 width=1 by cpr_flat/ -HT2 #H destruct //
27 | generalize in match HVT1; -HVT1 elim T1 -T1 * // #a * #W1 #U1 #_ #_ #H
28 [ elim (lifts_total V1 𝐔❴1❵) #V2 #HV12
29 lapply (H (ⓓ{a}W1.ⓐV2.U1) ?) -H /2 width=3 by cpm_theta/ -HV12 #H destruct
30 | lapply (H (ⓓ{a}ⓝW1.V1.U1) ?) -H /2 width=1 by cpm_beta/ #H destruct
35 (* Properties with simple terms *********************************************)
37 (* Basic_1: was only: nf2_appl_lref *)
38 lemma cnr_appl_simple (h) (G) (L):
39 ∀V,T. ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃V⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃T⦄ → 𝐒⦃T⦄ → ⦃G,L⦄ ⊢ ➡[h] 𝐍⦃ⓐV.T⦄.
40 #h #G #L #V #T #HV #HT #HS #X #H
41 elim (cpm_inv_appl1_simple … H) -H // #V0 #T0 #HV0 #HT0 #H destruct
42 <(HV … HV0) -V0 <(HT … HT0) -T0 //