(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_tc.ma".
-include "basic_2/rt_computation/cprs_lpr.ma".
-include "basic_2/rt_computation/lprs.ma".
+include "basic_2/rt_computation/lprs_tc.ma".
(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
(* Eliminators with r-transition for full local environments ****************)
(* Basic_2A1: was: lprs_ind_dx *)
-lemma lprs_ind_sn (h) (G) (L2): ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L → ⦃G, L⦄ ⊢ ➡*[h] L2 → R L → R L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → R L1.
-/3 width=7 by lpr_cprs_trans, cpr_refl, lex_CTC_ind_sn/ qed-.
+lemma lprs_ind_sn (h) (G) (L2):
+ ∀Q:predicate lenv. Q L2 →
+ (∀L1,L. ❪G,L1❫ ⊢ ➡[h,0] L → ❪G,L❫ ⊢ ➡*[h,0] L2 → Q L → Q L1) →
+ ∀L1. ❪G,L1❫ ⊢ ➡*[h,0] L2 → Q L1.
+/4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_sn/ qed-.
(* Basic_2A1: was: lprs_ind *)
-lemma lprs_ind_dx (h) (G) (L1): ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h] L → ⦃G, L⦄ ⊢ ➡[h] L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → R L2.
-/3 width=7 by lpr_cprs_trans, cpr_refl, lex_CTC_ind_dx/ qed-.
+lemma lprs_ind_dx (h) (G) (L1):
+ ∀Q:predicate lenv. Q L1 →
+ (∀L,L2. ❪G,L1❫ ⊢ ➡*[h,0] L → ❪G,L❫ ⊢ ➡[h,0] L2 → Q L → Q L2) →
+ ∀L2. ❪G,L1❫ ⊢ ➡*[h,0] L2 → Q L2.
+/4 width=8 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, cpr_refl, lex_CTC_ind_dx/ qed-.
-(* Properties with unbound rt-transition for full local environments ********)
+(* Properties with extended rt-transition for full local environments *******)
-lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
-/3 width=3 by lpr_cprs_trans, lex_CTC_inj/ qed.
+lemma lpr_lprs (h) (G):
+ ∀L1,L2. ❪G,L1❫ ⊢ ➡[h,0] L2 → ❪G,L1❫ ⊢ ➡*[h,0] L2.
+/4 width=3 by lprs_CTC, lpr_cprs_trans, lex_CTC_inj/ qed.
(* Basic_2A1: was: lprs_strap2 *)
-lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L →
- ∀L2.⦃G, L⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
-/3 width=3 by lpr_cprs_trans, lex_CTC_step_sn/ qed-.
+lemma lprs_step_sn (h) (G):
+ ∀L1,L. ❪G,L1❫ ⊢ ➡[h,0] L →
+ ∀L2.❪G,L❫ ⊢ ➡*[h,0] L2 → ❪G,L1❫ ⊢ ➡*[h,0] L2.
+/4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_sn/ qed-.
(* Basic_2A1: was: lpxs_strap1 *)
-lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡*[h] L →
- ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
-/3 width=3 by lpr_cprs_trans, lex_CTC_step_dx/ qed-.
+lemma lprs_step_dx (h) (G):
+ ∀L1,L. ❪G,L1❫ ⊢ ➡*[h,0] L →
+ ∀L2. ❪G,L❫ ⊢ ➡[h,0] L2 → ❪G,L1❫ ⊢ ➡*[h,0] L2.
+/4 width=3 by lprs_inv_CTC, lprs_CTC, lpr_cprs_trans, lex_CTC_step_dx/ qed-.
+
+lemma lprs_strip (h) (G):
+ confluent2 … (lprs h 0 G) (lpr h 0 G).
+#h #G #L0 #L1 #HL01 #L2 #HL02
+elim (TC_strip1 … L1 ?? HL02) -HL02
+[ /3 width=3 by lprs_TC, ex2_intro/ | skip
+| /2 width=1 by lprs_inv_TC/
+| /2 width=3 by lpr_conf/
+]
+qed-.