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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
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15 include "basic_2/syntax/tdeq_tdeq.ma".
16 include "basic_2/rt_transition/lfpx_fqup.ma".
17 include "basic_2/rt_transition/lfpx_lfdeq.ma".
18 include "basic_2/rt_computation/cpxs.ma".
20 (* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
22 (* Properties with degree-based equivalence for terms ***********************)
24 lemma tdeq_cpxs_trans: ∀h,o,U1,T1. U1 ≛[h, o] T1 → ∀G,L,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 →
25 ∃∃U2. ⦃G, L⦄ ⊢ U1 ⬈*[h] U2 & U2 ≛[h, o] T2.
26 #h #o #U1 #T1 #HUT1 #G #L #T2 #HT12 @(cpxs_ind … HT12) -T2 /2 width=3 by ex2_intro/
27 #T #T2 #_ #HT2 * #U #HU1 #HUT elim (tdeq_cpx_trans … HUT … HT2) -T -T1
28 /3 width=3 by ex2_intro, cpxs_strap1/
31 (* Note: this requires tdeq to be symmetric *)
32 lemma cpxs_tdneq_inv_step_sn: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬈*[h] T2 → (T1 ≛[h, o] T2 → ⊥) →
33 ∃∃T,T0. ⦃G, L⦄ ⊢ T1 ⬈[h] T & T1 ≛[h, o] T → ⊥ & ⦃G, L⦄ ⊢ T ⬈*[h] T0 & T0 ≛[h, o] T2.
34 #h #o #G #L #T1 #T2 #H @(cpxs_ind_dx … H) -T1
36 | #T1 #T #H1 #H2 #IH #Hn12 elim (tdeq_dec h o T1 T) #H destruct
37 [ -H1 -H2 elim IH -IH /3 width=3 by tdeq_trans/ -Hn12
38 #X #X2 #HTX #HnTX #HX2 #HXT2 elim (tdeq_cpx_trans … H … HTX) -HTX
39 #X1 #HTX1 #HX1 elim (tdeq_cpxs_trans … HX1 … HX2) -HX2
40 /5 width=8 by tdeq_canc_sn, tdeq_trans, ex4_2_intro/ (* Note: 2 tdeq_trans *)
41 | -IH -Hn12 /3 width=6 by ex4_2_intro/