(* *)
(**************************************************************************)
-include "basic_2/relocation/lifts_tdeq.ma".
-include "basic_2/static/lfdeq.ma".
-include "basic_2/rt_transition/lfpx.ma".
+include "basic_2/static/lfdeq_lfdeq.ma".
+include "basic_2/rt_transition/lfpx_fsle.ma".
-(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
-(* Properties with degree-based equivalence for terms ***********************)
+(* Properties with degree-based equivalence for local environments **********)
-(* Basic_2A1: was just: cpx_lleq_conf *)
-lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
-#h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
-[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
- elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
- /4 width=3 by tdeq_sort, deg_next, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(tdeq_inv_lref1 … H0) -H0
- elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
- elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
- elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
- elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(tdeq_inv_lref1 … H0) -H0
- elim (lfpx_inv_lref_pair_sn … H1) -H1 #K1 #X1 #HK01 #H destruct
- elim (lfdeq_inv_lref_pair_sn … H2) -H2 #K2 #X2 #HK02 #H destruct
- elim (IH … HK01 … HK02) [|*: //] -K0 -V0 #V #HV1 #HV2
- elim (tdeq_lifts … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
-| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -T0
- /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
-| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (lfdeq_pair_repl_dx … H2 … HV02) -H2 -HV02 #H2
- elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
- elim (tdeq_inv_lifts … HT1 … HUT1) -T1
- /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=3 by cpx_eps, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
- elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
- /3 width=3 by cpx_ee, ex2_intro/
-| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
- elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
-| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
- elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (lfdeq_pair_repl_dx … H2LT0 … HW02) -H2LT0 #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- elim (tdeq_lifts … HV1 … HVU1) -V1
- /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
-]
-qed-.
-(*
-lemma cpx_lleq_conf: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
- ∀L1. L2 ≡[T1, 0] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h, o] T2.
-/3 width=3 by lleq_cpx_trans, lleq_sym/ qed-.
+(* Basic_2A1: was just: cpx_lleq_conf_sn *)
+lemma cpx_lfdeq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h G) (lfdeq h o).
+/3 width=6 by cpx_lfxs_conf/ qed-.
-lemma cpx_lleq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h o G) (lleq 0).
-/3 width=6 by cpx_llpx_sn_conf, lift_mono, ex2_intro/ qed-.
-
-lemma cpx_lleq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h, o] T2 →
- ∀L1. L1 ≡[T1, 0] L2 → L1 ≡[T2, 0] L2.
-/4 width=6 by cpx_lleq_conf_sn, lleq_sym/ qed-.
-*)
\ No newline at end of file
+(* Basic_2A1: was just: cpx_lleq_conf_dx *)
+lemma cpx_lfdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
+/4 width=4 by cpx_lfdeq_conf_sn, lfdeq_sym/ qed-.
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