-
-(* Properties concerning sn parallel reduction on local environments ********)
-
-(* Basic_1: was just: pr3_pr2_pr2_t *)
-(* Basic_1: includes: pr3_pr0_pr2_t *)
-lemma lpr_cpr_trans: ∀G. b_c_transitive … (cpr G) (λ_. lpr G).
-#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3 by/
-| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (lpr_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
- /4 width=6 by cprs_strap2, cprs_delta/
-|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/
-|4,6: /3 width=1 by cprs_flat, cprs_eps/
-|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/
-]
-qed-.
-
-lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=5 by lpr_cpr_trans, cprs_bind_dx, lpr_pair/ qed.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma lpr_cprs_trans: ∀G. b_rs_transitive … (cpr G) (λ_. lpr G).
-#G @b_c_trans_CTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
-qed-.
-
-(* Basic_1: was: pr3_strip *)
-(* Basic_1: includes: pr1_strip *)
-lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
-normalize /4 width=3 by cpr_conf, TC_strip1/ qed-.
-
-lemma cprs_lpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 → ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #H @(cprs_ind … H) -T1 /2 width=3 by ex2_intro/
-#T #T1 #_ #HT1 #IHT0 #L1 #HL01 elim (IHT0 … HL01)
-#T2 #HT2 #HT02 elim (lpr_cpr_conf_dx … HT1 … HL01) -L0
-#T3 #HT3 #HT13 elim (cprs_strip … HT2 … HT3) -T
-/4 width=5 by cprs_strap2, cprs_strap1, ex2_intro/
-qed-.
-
-lemma cprs_lpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡ L1 →
- ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
-#G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01
-/3 width=3 by lpr_cprs_trans, ex2_intro/
-qed-.
-
-lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=5 by lpr_cprs_trans, cprs_bind_dx, lpr_pair/ qed.