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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/rt_computation/lprs_tc.ma".
17 (* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
19 (* Eliminators with r-transition for full local environments ****************)
21 (* Basic_2A1: was: lprs_ind_dx *)
22 lemma lprs_ind_sn (h) (G) (L2): ∀Q:predicate lenv. Q L2 →
23 (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L → ⦃G, L⦄ ⊢ ➡*[h] L2 → Q L → Q L1) →
24 ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L1.
25 /4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_sn/ qed-.
27 (* Basic_2A1: was: lprs_ind *)
28 lemma lprs_ind_dx (h) (G) (L1): ∀Q:predicate lenv. Q L1 →
29 (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h] L → ⦃G, L⦄ ⊢ ➡[h] L2 → Q L → Q L2) →
30 ∀L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → Q L2.
31 /4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_dx/ qed-.
33 (* Properties with unbound rt-transition for full local environments ********)
35 lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
36 /4 width=3 by lprs_CTC, lpr_cprs_trans, lex_CTC_inj/ qed.
38 (* Basic_2A1: was: lprs_strap2 *)
39 lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L →
40 ∀L2.⦃G, L⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
41 /4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_sn/ qed-.
43 (* Basic_2A1: was: lpxs_strap1 *)
44 lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡*[h] L →
45 ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
46 /4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_dx/ qed-.
48 lemma lprs_strip (h) (G): confluent2 … (lprs h G) (lpr h G).
49 #h #G #L0 #L1 #HL01 #L2 #HL02
50 elim (TC_strip1 … L1 ?? HL02) -HL02
51 [ /3 width=3 by lprs_TC, ex2_intro/ | skip
52 | /2 width=1 by lprs_inv_TC/
53 | /2 width=3 by lpr_conf/