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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/reducibility/cpr_ltpr.ma".
16 include "basic_2/equivalence/cpcs_cpcs.ma".
18 (* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
20 (* Properties about context-free parallel reduction on local environments ***)
22 (* Basic_1: was only: pc3_pr0_pr2_t *)
23 (* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
24 lemma ltpr_cpr_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2.
25 #L1 #L2 #HL12 #T1 #T2 #HT12
26 elim (cpr_ltpr_conf_eq … HT12 … HL12) -L1 #T #HT1 #HT2
27 @(cprs_div … T) /2 width=1/ /3 width=1/ (**) (* /4 width=3/ is too long *)
30 (* Basic_1: was: pc3_wcpr0_t *)
31 (* Basic_1: note: pc3_wcpr0_t should be renamed *)
32 lemma ltpr_cprs_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
33 #L1 #L2 #HL12 #T1 #T2 #H @(cprs_ind … H) -T2 //
35 @(cpcs_trans … IHT1) -T1 /2 width=3/
38 (* Basic_1: was: pc3_wcpr0 *)
39 lemma ltpr_cpcs_conf: ∀L1,L2. L1 ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
40 #L1 #L2 #HL12 #T1 #T2 #H
41 elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
42 @(cpcs_canc_dx … T) /2 width=3/