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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/rt_transition/cpm_drops.ma".
16 include "basic_2/rt_transition/cpt_drops.ma".
18 (* T-BOUND CONTEXT-SENSITIVE PARALLEL T-TRANSITION FOR TERMS ****************)
20 (* Properties with t-bound rt-transition for terms **************************)
22 lemma cpm_cpt_cpr (h) (n) (G) (L):
23 ∀T1,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 →
24 ∃∃T0. ⦃G,L⦄ ⊢ T1 ⬆[h,n] T0 & ⦃G,L⦄ ⊢ T0 ➡[h] T2.
25 #h #n #G #L #T1 #T2 #H
26 @(cpm_ind … H) -n -G -L -T1 -T2
27 [ #I #G #L /2 width=3 by ex2_intro/
28 | #G #L #s /3 width=3 by cpm_sort, ex2_intro/
29 | #n #G #K #V1 #V2 #W2 #_ * #V0 #HV10 #HV02 #HVW2
30 elim (lifts_total V0 (𝐔❴1❵)) #W0 #HVW0
31 lapply (cpm_lifts_bi … HV02 (Ⓣ) … (K.ⓓV1) … HVW0 … HVW2) -HVW2
32 [ /3 width=1 by drops_refl, drops_drop/ ] -HV02 #HW02
33 /3 width=3 by cpt_delta, ex2_intro/
34 | #n #G #K #V1 #V2 #W2 #_ * #V0 #HV10 #HV02 #HVW2
35 elim (lifts_total V0 (𝐔❴1❵)) #W0 #HVW0
36 lapply (cpm_lifts_bi … HV02 (Ⓣ) … (K.ⓛV1) … HVW0 … HVW2) -HVW2
37 [ /3 width=1 by drops_refl, drops_drop/ ] -HV02 #HW02
38 /3 width=3 by cpt_ell, ex2_intro/
39 | #n #I #G #K #T2 #U2 #i #_ * #T0 #HT0 #HT02 #HTU2
40 elim (lifts_total T0 (𝐔❴1❵)) #U0 #HTU0
41 lapply (cpm_lifts_bi … HT02 (Ⓣ) … (K.ⓘ{I}) … HTU0 … HTU2) -HTU2
42 [ /3 width=1 by drops_refl, drops_drop/ ] -HT02 #HU02
43 /3 width=3 by cpt_lref, ex2_intro/
44 | #n #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #_ #_ * #T0 #HT10 #HT02
45 /3 width=5 by cpt_bind, cpm_bind, ex2_intro/
46 | #n #G #L #V1 #V2 #T1 #T2 #HV12 #_ #_ * #T0 #HT10 #HT02
47 /3 width=5 by cpt_appl, cpm_appl, ex2_intro/
48 | #n #G #L #V1 #V2 #T1 #T2 #_ #_ * #V0 #HV10 #HV02 * #T0 #HT10 #HT02
49 /3 width=5 by cpt_cast, cpm_cast, ex2_intro/
50 | #n #G #L #V #U1 #T1 #T2 #HTU1 #_ * #T0 #HT10 #HT02
51 elim (cpt_lifts_sn … HT10 (Ⓣ) … (L.ⓓV) … HTU1) -T1
52 [| /3 width=1 by drops_refl, drops_drop/ ] #U0 #HTU0 #HU10
53 /3 width=6 by cpt_bind, cpm_zeta, ex2_intro/
54 | #n #G #L #U #T1 #T2 #_ * #T0 #HT10 #HT02
55 | #n #G #L #U1 #U2 #T #_ * #U0 #HU10 #HU02
56 /3 width=3 by cpt_ee, ex2_intro/
57 | #n #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #_ #_ #_ * #T0 #HT10 #HT02
58 /4 width=7 by cpt_appl, cpt_bind, cpm_beta, ex2_intro/
59 | #n #p #G #L #V1 #V2 #V0 #W1 #W2 #T1 #T2 #HV12 #HW12 #_ #_ #_ * #T0 #HT10 #HT02 #HV20
60 /4 width=9 by cpt_appl, cpt_bind, cpm_theta, ex2_intro/
63 (* Forward lemmas with t-bound rt-transition for terms **********************)
65 lemma cpt_fwd_cpm (h) (n) (G) (L):
66 ∀T1,T2. ⦃G,L⦄ ⊢ T1 ⬆[h,n] T2 → ⦃G,L⦄ ⊢ T1 ➡[n,h] T2.