(**************************************************************************)
include "basic_2/rt_computation/cpxs.ma".
-include "basic_2/rt_computation/fpbs_fqup.ma".
+include "basic_2/rt_computation/fpbs_feqg.ma".
+include "basic_2/rt_computation/fpbs_fqus.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
-(* Properties with unbound context-sensitive parallel rt-computation ********)
+(* Properties with extended context-sensitive parallel rt-computation *******)
-lemma cpxs_fpbs: ∀h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈*[h] T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
-#h #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbq_cpx, fpbs_strap1/
+lemma cpxs_fpbs:
+ ∀G,L,T1,T2. ❨G,L❩ ⊢ T1 ⬈* T2 → ❨G,L,T1❩ ≥ ❨G,L,T2❩.
+#G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=5 by cpx_fpb, fpbs_strap1/
qed.
-lemma fpbs_cpxs_trans: ∀h,G1,G,L1,L,T1,T. ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T⦄ →
- ∀T2. ⦃G,L⦄ ⊢ T ⬈*[h] T2 → ⦃G1,L1,T1⦄ ≥[h] ⦃G,L,T2⦄.
-#h #G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2
-/3 width=5 by fpbs_strap1, fpbq_cpx/
+lemma fpbs_cpxs_trans:
+ ∀G1,G,L1,L,T1,T. ❨G1,L1,T1❩ ≥ ❨G,L,T❩ →
+ ∀T2. ❨G,L❩ ⊢ T ⬈* T2 → ❨G1,L1,T1❩ ≥ ❨G,L,T2❩.
+#G1 #G #L1 #L #T1 #T #H1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=5 by fpbs_strap1, cpx_fpb/
qed-.
-lemma cpxs_fpbs_trans: ∀h,G1,G2,L1,L2,T,T2. ⦃G1,L1,T⦄ ≥[h] ⦃G2,L2,T2⦄ →
- ∀T1. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
-#h #G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1
-/3 width=5 by fpbs_strap2, fpbq_cpx/
+lemma cpxs_fpbs_trans:
+ ∀G1,G2,L1,L2,T,T2. ❨G1,L1,T❩ ≥ ❨G2,L2,T2❩ →
+ ∀T1. ❨G1,L1❩ ⊢ T1 ⬈* T → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
+#G1 #G2 #L1 #L2 #T #T2 #H1 #T1 #H @(cpxs_ind_dx … H) -T1
+/3 width=5 by fpbs_strap2, cpx_fpb/
qed-.
-lemma cpxs_teqx_fpbs_trans: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
- ∀T0. T ≛ T0 →
- ∀G2,L2,T2. ⦃G1,L1,T0⦄ ≥[h] ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
-/3 width=3 by cpxs_fpbs_trans, teqx_fpbs_trans/ qed-.
+lemma cpxs_teqg_fpbs_trans (S):
+ reflexive … S → symmetric … S →
+ ∀G1,L1,T1,T. ❨G1,L1❩ ⊢ T1 ⬈* T → ∀T0. T ≛[S] T0 →
+ ∀G2,L2,T2. ❨G1,L1,T0❩ ≥ ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
+/3 width=6 by cpxs_fpbs_trans, teqg_fpbs_trans/ qed-.
-lemma cpxs_teqx_fpbs: ∀h,G,L,T1,T. ⦃G,L⦄ ⊢ T1 ⬈*[h] T →
- ∀T2. T ≛ T2 → ⦃G,L,T1⦄ ≥[h] ⦃G,L,T2⦄.
-/4 width=3 by cpxs_fpbs_trans, feqx_fpbs, teqx_feqx/ qed.
-
-(* Properties with star-iterated structural successor for closures **********)
-
-lemma cpxs_fqus_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1,L1,T⦄ ⬂* ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
-/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.
+lemma cpxs_teqg_fpbs (S):
+ reflexive … S → symmetric … S →
+ ∀G,L,T1,T. ❨G,L❩ ⊢ T1 ⬈* T →
+ ∀T2. T ≛[S] T2 → ❨G,L,T1❩ ≥ ❨G,L,T2❩.
+/4 width=5 by cpxs_fpbs_trans, feqg_fpbs, teqg_feqg/ qed.
(* Properties with plus-iterated structural successor for closures **********)
-lemma cpxs_fqup_fpbs: ∀h,G1,L1,T1,T. ⦃G1,L1⦄ ⊢ T1 ⬈*[h] T →
- ∀G2,L2,T2. ⦃G1,L1,T⦄ ⬂+ ⦃G2,L2,T2⦄ → ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄.
+lemma cpxs_fqup_fpbs:
+ ∀G1,L1,T1,T. ❨G1,L1❩ ⊢ T1 ⬈* T →
+ ∀G2,L2,T2. ❨G1,L1,T❩ ⬂+ ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
/3 width=5 by fpbs_fqup_trans, cpxs_fpbs/ qed.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma cpxs_fqus_fpbs:
+ ∀G1,L1,T1,T. ❨G1,L1❩ ⊢ T1 ⬈* T →
+ ∀G2,L2,T2. ❨G1,L1,T❩ ⬂* ❨G2,L2,T2❩ → ❨G1,L1,T1❩ ≥ ❨G2,L2,T2❩.
+/3 width=5 by fpbs_fqus_trans, cpxs_fpbs/ qed.